# The families analytic index for $1|1$-dimensional Euclidean field theories

Daniel Berwick-Evans

###### Abstract.

We construct a KO-valued families index for a class of $1|1$-dimensional
Euclidean field theories. This realizes a conjectured cocycle map in the
Stolz–Teichner program. We further show that a bundle of spin manifolds leads
to a family of partially-defined $1|1$-Euclidean field theories, yielding a
cocycle refinement of the families analytic index. The methods are chosen with
the goal of generalizing to $2|1$-dimensional field theories, where analogous
structures are expected to provide an analytic index valued in topological
modular forms.

##  1\. Introduction and statement of results

Stolz and Teichner have outlined constructions of categories of $1|1$\- and
$2|1$-dimensional Euclidean field theories over a smooth manifold $M$ indexed
by a degree $n\in{\mathbb{Z}}$ [ST04, HST10, ST11]. They conjecture that these
categories determine geometric cocycle models for real ${\rm K}$-theory and
topological modular forms [ST11, §1.5-1.6],

(6)

It is further expected that these cocycle models inherit an analytic
pushforward from quantization of field theories. The ultimate goal is to use
these pushforwards to prove a TMF-generalization of the Atiyah–Singer index
theorem.

The feeling has always been that a proper understanding of the relationship
between $1|1$-Euclidean field theories and the index theorem in K-theory will
provide significant insight into the $2|1$-dimensional case [Seg88, §6],
[ST04, §3]. With this in mind, the goal of this paper is to formulate and
prove a version of (6) when $d=1$ that communicates with the families analytic
index. We use a variation on the notion of degree $n$ field theories over $M$,
denoted $1|1\hbox{-{\sf eft}}^{n}(M)$, that incorporates reflection positivity
and reality data as well as a condition demanding the existence of smooth
cutoff theories; see §1.2 below.

######  Theorem 1.1.

An object of $1|1\hbox{-{\sf eft}}^{n}(M)$ determines a self-adjoint, Clifford
linear superconnection ${\mathbb{A}}$ on a real vector bundle $V\to M$ (not
necessarily finite-rank). The families index construction for self-adjoint
superconnections determines a cocycle map

(7) $\displaystyle 1|1\hbox{-{\sf eft}}^{n}(M)\xrightarrow{{\rm
Index}}\widehat{\rm KO}{}^{n}(M)$

valued in the differential ${\rm KO}$-theory of $M$. If the renormalization
group flow sends an object $E\in 1|1\hbox{-{\sf eft}}^{n}(M)$ to the zero
theory, then its underlying class in ${\rm KO}^{n}(M)$ is zero.

The categories $1|1\hbox{-{\sf eft}}^{n}(M)$ also accommodate a refinement of
the KO-valued families analytic index for a bundle of spin manifolds.

######  Theorem 1.2.

The Bismut superconnection of a family of Riemannian spin manifolds $X\to M$
determines an object $\alpha(X)\in 1|1\hbox{-{\sf eft}}^{-n}(M)$ whose image
under (7) is the families differential analytic index of $X\to M$.

With the knowledge that the existing definitions of supersymmetric Euclidean
field theory are unlikely to be the final word (see Remark B.1 and B.11), the
cocycle map (7) is designed to be as robust as possible. Indeed, the
categories $1|1\hbox{-{\sf eft}}^{n}(M)$ and Theorem 1.1 come from evaluating
a field theory on specific families of $1|1$-Euclidean bordisms, and the
structures relevant to index theory are inherited from the geometry of these
bordisms. The upshot is that any reasonable category of $1|1$-Euclidean field
theories over $M$ has a subcategory with a cocycle map (7); see Remark 1.13.
Perhaps more importantly, the categories $1|1\hbox{-{\sf eft}}^{n}(M)$ have a
generalization to the $d=2$ version of the conjecture (6), and the techniques
behind Theorems 1.1 and 1.2 lay the groundwork for this $d=2$ sequel [BE23c].

###  1.1. The main idea

In the physics literature, a $1|1$-dimensional field theory is more commonly
called _$\mathcal{N}=1$ supersymmetric quantum mechanics._ The basic datum is
a ${\mathbb{Z}}/2$-graded inner product space $\mathcal{H}=\mathcal{H}^{\rm
ev}\oplus\mathcal{H}^{\rm odd}$ and an odd operator $Q\colon\mathcal{H}^{{\rm
ev}/{\rm odd}}\to\mathcal{H}^{{\rm odd}/{\rm ev}}$ [Wit82, §1]. This
determines a representation of the super semigroup

(8) $\displaystyle{\mathbb{R}}^{1|1}_{\geq 0}\to{\rm
End}(\mathcal{H}),\qquad(t,\theta)\mapsto e^{-tQ^{2}+\theta Q},$

that enhances the usual time-evolution semigroup in Wick-rotated quantum
mechanics. Depending on the type of theory desired, one demands additional
structure and property on (8). For example, _compact_ theories require that
(8) takes values in trace-class operators, and _reflection-positive_ theories
require $Q$ to be self-adjoint. The main example motivating (6) takes
$Q=\not{D}$ to be the Dirac operator for a compact spin manifold $X$ [ST04,
Example 3.2.9]. This theory is both compact and reflection positive.

The super semigroup ${\mathbb{R}}^{1|1}_{\geq 0}$ has a geometric
interpretation as a moduli space of super intervals [ST04, §3] for which
multiplication in the super semigroup corresponds to gluing of super
intervals. Super intervals comprise a subcategory of the $1|1$-Euclidean
bordism category [HST10, §6],

(15) $\displaystyle\left\\{\\!\\!\begin{array}[]{c}{\rm super}\\\ {\rm
intervals}\end{array}\\!\\!\right\\}\xhookrightarrow{\iota}\left\\{\\!\\!\begin{array}[]{c}1|1\hbox{-}{\rm
Euclidean}\\\ {\rm
bordisms}\end{array}\\!\\!\right\\}\implies\left\\{\\!\\!\begin{array}[]{c}1|1\hbox{-}{\rm
Euclidean}\\\ {\rm field\
theories}\end{array}\\!\\!\right\\}\xrightarrow{\iota^{*}}{\rm
Rep}({\mathbb{R}}^{1|1}_{\geq 0})$

and $1|1$-Euclidean field theories are defined as representations of this
bordism category. In particular, a functor out of the $1|1$-Euclidean bordism
category restricts along $\iota$ to a representation of
${\mathbb{R}}^{1|1}_{\geq 0}$. The representations (8) that come from this
restriction have additional properties, e.g., compactness is automatic [ST12,
§2]. The initial definition of $1|1$-Euclidean field theory from [ST04, §3]
simply equips representations of ${\mathbb{R}}^{1|1}_{\geq 0}$ with these
expected additional properties. The more recent definition from [ST11] instead
gives a precise notion of the $1|1$-Euclidean bordism category, and defines a
$1|1$-Euclidean field theory as a representation of this bordism category. One
upshot of the newer definition is a natural generalization to field theories
over a smooth manifold, gotten by equipping $1|1$-Euclidean bordisms with a
map to a (background) manifold $M$, see Definition 1.6 below.

Super intervals with a map to $M$ determine the category of _superpaths_ in
$M$. This leads to a families-generalization of (15)

(24)

where a functor out of the $1|1$-Euclidean bordism category over $M$
determines a representation of the category of superpaths in $M$ by
restriction. Examples of representations of the category of superpaths arise
from _super parallel transport_ along a superconnection [Dum08]; a folklore
theorem extends these representations to $1|1$-Euclidean field theories over
$M$ [ST11, §1.3].

The basic mechanism behind Theorem 1.1 is an inverse to this super parallel
transport construction: a $1|1$-Euclidean field theory over $M$ determines a
superconnection via restriction to _nearly constant superpaths_ in $M$,
denoted ${\sf sPath}_{0}(M)$. These are superpaths whose maps to $M$ factor
through the projection ${\mathbb{R}}^{1|1}\to{\mathbb{R}}^{0|1}$.

######  Proposition 1.3 (Proposition 6.20).

There is a functor

${\rm Rep}({\sf sPath}_{0}(M))\to{\sf Vect}^{{\mathbb{A}}}(M)$

from (not necessarily finite-rank) representations of ${\sf sPath}_{0}(M)$ to
the groupoid of (not necessarily finite-rank) super vector bundles with
superconnection on $M$.

This leads to a natural candidate for the cocycle map (6)

(27) $\displaystyle\left\\{\\!\\!\\!\begin{array}[]{c}1|1\hbox{-}{\rm
Euclidean\ field}\\\ {\rm theories\ over}\
M\end{array}\\!\\!\\!\right\\}\xrightarrow{\iota^{*}}{\rm Rep}({\sf
sPath}_{0}(M))\xrightarrow{{\rm Prop~{}\ref{lem:mainA}}}{\sf
Vect}^{{\mathbb{A}}}(M)\stackrel{{\scriptstyle{\rm
Index}}}{{\dashrightarrow}}{\rm KO}^{0}(M)$

where the last arrow is the families index construction applied to a
superconnection, e.g., see [BGV92, Chapter 9].

For an arbitrary superconnection, the index construction may not be valid—one
needs certain additional properties. Some of these properties are automatic
when a representation of ${\sf sPath}_{0}(M)$ comes from the restriction of a
$1|1$-Euclidean field theory. For example, the operators
$e^{-t{\mathbb{A}}^{2}}$ are self-adjoint and trace class for $t>0$. However,
these conditions alone do not guarantee the existence of an index bundle. The
precise conditions on the superconnection can be complicated to state and
depend sensitively on the analytical framework, e.g., compare Lott and
Gorokhovsky’s superconnections on Hilbert bundles [GL15].

From the field theory point of view, the additional property required to
construct (27) is quite straightforward: one needs the existence of _smooth
energy cutoffs_. Roughly, this means that a $1|1$-Euclidean field theory over
$M$ determines a compatible ${\mathbb{R}}_{>0}$-family of theories that
discard states of energy greater than $\lambda\in{\mathbb{R}}_{>0}$. The
states with energy $<\lambda$ are used to construct the families index. The
existence of smooth energy cutoffs (and hence the existence of a families
index) is a _property_ of a field theory, not additional data. Furthermore,
this property makes sense for Euclidean field theories in any super dimension,
in particular dimension $2|1$. The existence of smooth cutoffs is analogous to
the ellipticity property for differential operators, see Definition 1.11 and
Remark 1.14. The categories $1|1\hbox{-{\sf eft}}^{n}(M)$ are roughly defined
as the value of a field a $1|1$-Euclidean field theory on nearly constant
superpaths (see §1.2) where in additional we require a smooth cutoff property;
the following provides an index-theoretic characterization.

######  Proposition 1.4 (Proposition 7.3).

There is an equivalence of categories

(28) $\displaystyle{\rm sPar}\colon{\sf Vect}^{{\mathbb{A}}}_{\rm
ind}(M)\xrightarrow{\sim}1|1\hbox{-{\sf eft}}^{0}(M)$

from the category of real, self-adjoint superconnections admitting smooth
index bundles to the category $1|1\hbox{-{\sf eft}}^{0}(M)$.

In short, ${\sf Vect}^{{\mathbb{A}}}_{\rm ind}(M)\subset{\sf
Vect}^{{\mathbb{A}}}(M)$ is the subcategory of superconnections on which the
(differential) index map is defined. The functor ${\rm sPar}$ is totally
explicit, e.g.,

(29) $\displaystyle
e^{-t{\mathbb{A}}^{2}+\theta{\mathbb{A}}}\colon{\mathbb{R}}^{1|1}_{\geq
0}\to{\rm End}(\Omega^{\bullet}(M;V_{\mathbb{C}}))$

determines the underlying representation of ${\sf sPath}_{0}(M)$ under (28).
The formula (29) also coincides with super parallel transport along a
superconnection ${\mathbb{A}}$ on a finite rank vector bundle; e.g., see
[Dum12, Theorem 1], [Sto19, Proposition 6.7] or [BEH16, §2]. This motivates
the notation ${\rm sPar}$ for the functor (28), as it provides a partial
construction of super parallel transport for more general superconnections;
see Remark 1.5 below. Proposition 1.4 allows us to construct the cocycle map
in Theorem 1.1 as the composition (27).

######  Remark 1.5.

Dumitrescu’s super parallel transport [Dum08] extends (28) to a functor

${\sf Vect}^{\mathbb{A}}_{{\rm fin}}(M)\to 1|1\hbox{-{\sf EFT}}^{0}(M),$

from superconnection on finite-rank super vector bundles; compare Corollary
7.4. An infinite-rank generalization of super parallel transport would provide
a complete dictionary between $1|1$-Euclidean field theories and
superconnections. However, in our present analytical setting (following
[ST11])) super parallel transport for infinite rank bundles need not be
defined owing to the failure of existence and uniqueness results for
differential equations in topological vector spaces, e.g., see [Ham82, Example
5.6.1] for counterexamples in Fréchet spaces. From the physics point of view,
one expects field theories to be determined by infinitesimal data. We note
that some of the prior definitions of $1|1$-Euclidean field theory leverage
existence and uniqueness results in Hilbert spaces [HST10, §6]. In a different
direction, there are existence and uniqueness results for tame Fréchet spaces
and tame maps. Adopting a more Wilsonian approach, one could define
$1|1$-Euclidean field theories as an ${\rm Ind}$-completion of finite-rank
objects with compatibilities via the energy filtration; this also forces
existence and uniqueness. When attempting to weigh these options, the main
concern is that the correct analytical context for the $d=2$ case in
conjecture (6) is not yet clear. As such, we leave these choices of analysis
flexible to avoid over-constraining the possible definitions in this $d=2$
generalization.

###  1.2. Technical overview

The categories $1|1\hbox{-{\sf eft}}^{n}(M)$ in Theorems 1.1 and 1.2 come from
distilling the information of a degree $n$ Euclidean field theory down to the
essential pieces required to formulate the index map (7). We work in the
framework for field theories developed by Stolz and Teichner in [ST11], and
begin with a brief recollection of their main definition; see §B for a more
detailed overview.

######  Definition 1.6 ([ST11, §4-5]).

For $n\in{\mathbb{Z}}$ the _degree $n$ twist_ is a $\otimes$-invertible
functor

(30) $\displaystyle\mathscr{T}^{n}\colon 1|1\hbox{-}{\sf EBord}({\rm
pt})\to{\sf{Alg}}$

where ${\sf{Alg}}$ is a Morita category of (topological) algebras, bimodules,
and bimodule maps. Pulling back along the canonical functor $1|1\hbox{-}{\sf
EBord}(M)\to 1|1\hbox{-}{\sf EBord}({\rm pt})$ we obtain a twist for the
$1|1$-Euclidean bordism category over $M$ that we will also denote by
$\mathscr{T}^{n}$. A _degree $n$ field theory over $M$_ is a natural
transformation

(31)

######  Remark 1.7.

Some details of the above definition have not appeared in the literature; we
refer to §5 for further discussion that culminates in Hypothesis 5.9.

######  Definition 1.8 (Definitions B.30, B.32, and 5.16, assuming Hypothesis
5.9).

Define a category $1|1\hbox{-{\sf EFT}}^{n}(M)$ whose objects are real,
reflection positive degree $n$ field theories (31). Morphisms in
$1|1\hbox{-{\sf EFT}}^{n}(M)$ are isomorphisms between internal natural
transformations $E\simeq E^{\prime}$ that are compatible with the real and
reflection positive structures.

######  Remark 1.9.

Reflection positivity is perhaps unnecessary in the $d=1$ version of
conjecture (6): unoriented, real field theories (with a different positivity
condition) recover KO-theory [HST10], see Remark 5.18. However, unoriented and
real structures do not generalize to the $2|1$-dimensional Euclidean bordism
category, and the Miller character ${\rm TMF}\to{\rm KO}(\\!(q)\\!)$
necessitates a map from $2|1$-Euclidean field theories to real K-theory if
conjecture (6) is to hold. Reflection positivity is one way to obtain real
structures on Hilbert spaces of states in dimension $2|1$; see [GPPV21, 3.2.2]
and [BE23c].

In §3-§4, we construct a category ${\sf sPath}_{0}(M)$ of nearly constant
super Euclidean paths in $M$ with an internal functor $\iota$

(32)

When applied to degree zero theories, restriction along $\iota$ makes (24)
precise. Together with Proposition 1.3, we obtain the composition (27) that
extract a superconnection from a $1|1$-Euclidean field theory.

######  Proposition 1.10.

The restriction of a degree $n$ Euclidean field theory (31) along the functor
(32) determines:

  1. (1)

vector bundles $V_{+},V_{-}\to M$ with fiberwise ${\mathbb{C}}{\rm l}_{+n}$\-
and ${\mathbb{C}}{\rm l}_{-n}$-actions and Clifford linear superconnections
${\mathbb{A}}_{+}$ and ${\mathbb{A}}_{-}$, respectively (Proposition 6.20);

  2. (2)

a pairing between section-valued differential forms (Proposition 6.26)

(33) $\displaystyle\langle-,-\rangle\colon{\mathbb{C}}{\rm
l}_{n}\otimes_{{\mathbb{C}}{\rm l}_{-n}\otimes{\mathbb{C}}{\rm
l}_{n}}\Big{(}\Omega^{\bullet}(M;V_{-})\otimes\Omega^{\bullet}(M;V_{+})\Big{)}\to\Omega^{\bullet}(M)$

for which the superconnections satisfy
$\langle{\mathbb{A}}_{-}x,y\rangle=(-1)^{|x|}\langle
x,{\mathbb{A}}_{+}y\rangle$;

  3. (3)

differential forms
$Z,Z_{\ell}\in\Omega^{\bullet}(M;C^{\infty}({\mathbb{R}}_{>0}))$ satisfying

$dZ=0,\ \ \partial_{\ell}Z=dZ_{\ell},\qquad{\rm deg}(Z)=n\mod 2,\ \ {\rm
deg}(Z_{\ell})=n-1\mod 2$

where $\ell$ is the standard coordinate on ${\mathbb{R}}_{>0}$ (Proposition
6.44).

The renormalization group flow on degree $n$ field theories induces the
Getzler rescaling of superconnections (Proposition 6.22): renormalization flow
by $\mu\in{\mathbb{R}}_{>0}$ restricts to

(34)
$\displaystyle{\mathbb{A}}_{\pm}\mapsto\mu({\mathbb{A}}_{\pm})_{0}+({\mathbb{A}}_{\pm})_{1}+\mu^{-1}({\mathbb{A}}_{\pm})_{2}+\mu^{-2}({\mathbb{A}}_{\pm})_{3}+\dots$

Reflection and real structures restrict along (32) as follows.

  1. (1)

A reflection structure determines isomorphisms
$V_{\pm}\simeq\overline{V}_{\mp}$ of bundles of Clifford modules relative to
the $*$-superalgebra structure ${\mathbb{C}}{\rm l}_{\pm
n}\to\overline{{\mathbb{C}}{\rm l}}_{\mp n}$ defined in (202). Using this
data, the pairing (33) determines graded hermitian pairings on $V_{+}$ and
$V_{-}$ for which the superconnections ${\mathbb{A}}_{+}$ and
${\mathbb{A}}_{-}$ are graded self-adjoint. The positivity condition requires
the hermitian pairing to be positive (Proposition 6.32).

  2. (2)

A real structure determines isomorphisms $V_{\pm}\simeq\overline{V}_{\pm}$ of
bundles of modules relative to the standard real structure on
${\mathbb{C}}{\rm l}_{n}$. The superconnections are compatible with this real
structure on $V_{\pm}$. For a reflection positive theory with real structure,
the hermitian inner product is also real (Proposition 6.37).

As described before Proposition 1.4, the upshot of Proposition 1.10 is that a
degree $n$ Euclidean field theory (31) with real and reflection positive
structures determines a superconnection for which a ${\rm KO}^{n}$-valued
families index construction _might_ apply. The existence of this index bundle
can be a rather subtle analytical question. The main condition one requires is
that a certain direct sum of eigenspaces for eigenvalues less than a fixed
$\lambda\in{\mathbb{R}}_{>0}$ form a smooth vector bundle over (an open
submanifold) of $M$; see Definition C.16. Physically, this subbundle of
eigenspaces is the subspace of states with energy less than $\lambda$.
Rephrasing in physical language, the existence of the index bundle is the same
as demanding the existence of a smooth cutoff theory in the sense of Wilsonian
effective field theory, e.g., see [Cos11a, §1.3] for a mathematical exposition
of the key ideas. The following definition captures the precise pieces of a
$1|1$-Euclidean field theory required to extract a KO-valued families index.

######  Definition 1.11.

The groupoid $1|1\hbox{-{\sf eft}}^{n}(M)$ has objects degree $n$
representations of ${\sf sPath}_{0}(M)$ (Definition 6.18) that are reflection
positive (Definition 6.30), trace class (Definition 6.41), real (Definition
6.35), and admit cutoffs (Definition 7.1). Morphisms are isomorphisms between
degree $n$ representations of ${\sf sPath}_{0}(M)$ that are compatible with
the reflection structure, real structure, and trace.

######  Corollary 1.12.

Theorem 1.1 constructs a cocycle map for a subcategory of Stolz and Teichner’s
degree $n$ theories,

$\displaystyle\\{E\in 1|1\hbox{-{\sf EFT}}^{n}(M)\mid E\ {\rm admits\ energy\
cutoffs}\\}\xrightarrow{{\rm cocycle}}\widehat{\rm KO}{}^{n}(M).$

###### Proof.

The definition of $2|1\hbox{-{\sf eft}}^{n}(M)$ is set up so that an object
$E\in 1|1\hbox{-{\sf EFT}}^{n}(M)$ determines the _data_ of an object in
$1|1\hbox{-{\sf eft}}^{n}(M)$ by restriction to specific families of super
Euclidean bordisms; it remains to check that these data satisfy the required
properties. The positivity property for an object in $1|1\hbox{-{\sf
eft}}^{n}(M)$ follows from reflection positivity for $1|1\hbox{-{\sf
EFT}}^{n}(M)$. Field theories satisfying the cutoff property provide a
subcategory of field theories, and we have a span

(35) $\displaystyle 1|1\hbox{-{\sf EFT}}^{n}(M)\hookleftarrow\\{E\in
1|1\hbox{-{\sf EFT}}^{n}(M)\mid E\ {\rm admits\ energy\
cutoffs}\\}\xrightarrow{{\rm restrict}}1|1\hbox{-{\sf eft}}^{n}(M).$

Postcomposing the map on the right with (7) provides the claimed map to
$\widehat{\rm KO}{}^{n}(M)$. ∎

Theorem 1.2 follows from turning Proposition 1.10 around: the Bismut
superconnection determines a representation of ${\sf sPath}_{0}(M)$ from the
formula (29), and this representation naturally comes with the additional data
and properties required to determine an object in $1|1\hbox{-{\sf
eft}}^{n}(M)$. In this example, the existence of cutoffs relies on Bismut’s
families index theorem [Bis86]. Despite only being part of the data of a field
theory (see Remark 1.15), $\alpha(X)\in 1|1\hbox{-{\sf eft}}^{-n}(M)$ contains
all the relevant data for the families differential index.

We conclude with several remarks about the framework developed in this paper.

######  Remark 1.13.

One can formulate the span (35) for any definition of degree $n$,
$1|1$-Euclidean field theories defined using a bordism category that contains
the families of small bordisms (see Definition 4.23) where the degree twist
satisfies Hypothesis 5.9. Hence, such field theories admit a subcategory with
a cocycle map valued in $\widehat{\rm KO}{}^{n}(M)$ by composing with (7). For
finite-rank examples, the cutoff condition is automatically satisfied, see
Corollary 7.4 which follows from [BGV92, §9.1]. Hence, the cutoff condition in
(35) is only interesting for infinite-rank examples.

######  Remark 1.14.

Cutoffs make sense for a broad class of field theories. Indeed, in Witten’s
study of the $\mathcal{N}=(0,1)$ supersymmetric sigma model [Wit88, §1], he
suggested that cutoffs be used to analyze the field theories in the $d=2$
version of the conjectures (6). For $1|1$-dimensional field theories, the
property of having smooth energy cutoffs is analogous to the property of a
differential operator being elliptic; in particular, cutoffs require
eigenfunctions of the degree zero part of ${\mathbb{A}}^{2}$ to determine
smooth sections over $M$.

######  Remark 1.15.

Assuming a generators and relations description of the category
$1|1\hbox{-}{\sf EBord}(M)$ (following [HST10, 6.7] and [ST11, 3.2]), the data
of an object of $1|1\hbox{-{\sf eft}}^{n}(M)$ is the data of a degree $n$
field theory defined on a subcategory of _small bordisms in $M$_

$1|1\hbox{-{\sf ebord}}(M)\hookrightarrow 1|1\hbox{-}{\sf EBord}(M)$

generated by the $S$-families of bordisms in Definition 4.23.

######  Remark 1.16.

By results of Hopkins–Hovey [HH92], Theorems 1.1 and 1.2 yield a surjection

(36) $\displaystyle 1|1\hbox{-{\sf eft}}^{n}(M)\twoheadrightarrow{\rm
KO}^{n}(M),$

verifying a version of the $d=1$ case of (6). A stronger form of the
conjectures (6) asserts that the map (36) induces an isomorphism on
concordance classes, so that the corresponding cohomology theories are a
complete invariant of field theories [ST11, §1.5]. It is easy to see that (36)
factors through concordance classes, but we do not expect an isomorphism in
our framework used below for reasons we explain presently. In view of the
factorization (27) of the cocycle map, the stronger form of conjecture (6) in
particular requires a category of superconnections whose concordance classes
are KO-theory. But in existing definitions the isomorphism class of the
underlying vector bundle is a concordance invariant. In particular, the rank
of a vector bundle is a concordance invariant, but is not an invariant of the
underlying ${\rm KO}$-class. We see two possible ways to avoid this problem.
The first option is to define field theories on (infinite rank) Hilbert
bundles, using that such bundles are always trivializable [Kui65]; this
resonates with previous definitions from [ST04] and [HST10], and likely makes
contact with Lott and Gorokhovsky’s model for differential K-theory [GL15].
The second option follows the suggestion in [ST11, Remark 3.16], and requires
a framework for superconnections defined on (not necessarily locally free)
sheaves of topological vector spaces, e.g., see [Ulr21] for finite-rank
results in this direction. It is not clear if either of these approaches
generalize to the desired statement in dimension $2|1$.

### Outline

The motivation from physics is explained in §2. In §3 we define categories of
superpaths in a smooth manifold. In §4 we show that superpaths give examples
of bordisms in Stolz and Teichner’s $1|1$-Euclidean bordism category from
[ST11]. In §5 we sketch a construction of the degree $n$ twist and the values
of the resulting degree $n$ twisted field theories. The main technical results
are proved in §6, where we restrict these values of field theories to the
nearly constant superpaths to prove Proposition 1.10. In §7, the relationship
between cutoff theories and the families index construction leads us to the
definition of the groupoids $1|1\hbox{-{\sf eft}}^{n}(M)$. Theorems 1.1 and
1.2 then follow from standard constructions in index theory. In §A we review
some background super algebra and super geometry, §B summarizes the main
definitions and constructions from [ST11] used in this paper, and §C reviews
the families index in differential ${\rm KO}$-theory from [BE23b].

Readers without detailed knowledge of the Stolz–Teichner program may prefer to
skim sections §4 and §5; these sections explain how our results fit with
[ST11], but are logically independent from the main theorems.

This paper is part of a series [BE23a, BE23b, BE23c] that solves a
1-categorical version of the conjectures (6). The zero categorical version is
the main result of [BE23a]. In the 1-categorical enhancement, the target of
(6) requires a new model for (differential) ${\rm KO}$-theory; this is not
directly related to field theories and is handled in [BE23b], reviewed in §C
below. The present paper connects $1|1$-Euclidean field theories with these
differential KO-cocycles via an index map. In [BE23c] we generalize this index
to $2|1$-dimensional field theories.

### Acknowledgements

It is a pleasure to thank Kevin Costello, Theo Johnson-Freyd, Matthias
Ludewig, Stephan Stolz, Peter Teichner, and Arnav Tripathy for stimulating
conversations on this material. This work was supported by the National
Science Foundation under grant number DMS-2205835.

###### Contents

  1. 1 Introduction and statement of results
  2. 2 Motivation from supersymmetric quantum mechanics
  3. 3 Super Euclidean paths in a manifold
  4. 4 Super Euclidean paths as bordisms
  5. 5 $1|1$-Euclidean field theories of degree $n$
  6. 6 Extracting geometric data from a representations of superpaths
  7. 7 Energy cutoffs and the families index
  8. A Super hermitian structures, $*$-superalgebras, and supermanifolds
  9. B Review of Stolz and Teichner’s Euclidean field theories
  10. C The families index in KO-theory

##  2\. Motivation from supersymmetric quantum mechanics

In this section we overview the arguments in supersymmetric quantum mechanics
that lead to Theorems 1.1 and 1.2 in the case that $M={\rm pt}$. The
discussion is intentionally informal with precise definitions delayed until
later sections.

###  2.1. Quantum mechanics on $X$

To build intuition, we start by explaining ordinary quantum mechanics without
supersymmetry.

Let $X$ be a closed, oriented Riemannian manifold. Quantum mechanics on $X$
has as its space of states $\mathcal{H}=C^{\infty}(X;{\mathbb{C}})$ endowed
with the hermitian inner product inherited from integration with respect to
the Riemannian volume form on $X$. The Laplacian $H=\Delta$ on $X$ is the
Hamiltonian of the theory, generating time-evolution. In the Wick-rotated
theory, this gives a representation of the semigroup ${\mathbb{R}}_{\geq 0}$,

(37) $\displaystyle{\mathbb{R}}_{\geq 0}\to{\rm End}(\mathcal{H}),\qquad
t\mapsto e^{-tH},$

where $t\in{\mathbb{R}}_{\geq 0}$ is identified with a time parameter. This
representation has important additional properties: $e^{-tH}$ is self-adjoint,
and for $t>0$ the trace is finite

(38) $\displaystyle Z:={\sf Tr}(e^{-tH})\in C^{\infty}({\mathbb{R}}_{>0})$

since $X$ was assumed to be compact. The function of $t$ given by (38) is
called the _partition function_. A general quantum mechanical system is given
by a semigroup representation (37) satisfying a self-adjoint and trace-class
property.

######  Remark 2.1.

Often one takes the Hilbert completion of the Fréchet space
$\mathcal{H}=C^{\infty}(X;{\mathbb{C}})$, resulting in a Hilbert space of
states and an unbounded Hamiltonian $H=\Delta$. For the definition of field
theory from [ST11] the domain of the operator $\Delta$ (as a Fréchet space)
turns out to be more fundamental. By contrast, the Hilbert completion is the
basic object of study for definitions of field theories in [ST04] and [HST10].
It is unclear which point of view generalizes more readily to the $d=2$
conjecture in (6). In this paper we will largely follow [ST11], though we will
ignore these functional analytic distinctions for the remainder of this
section.

In the formalism of [ST04, HST10, ST11], a quantum mechanical system
determines a _1-dimensional Euclidean field theory_ , defined as a
representation of a 1-dimensional oriented Riemannian bordism category
$1\hbox{-}{\sf EBord}$. The objects of $1\hbox{-}{\sf EBord}$ are finite
disjoint unions of $\bullet={\rm pt}^{+}$ and $\circ={\rm pt}^{-}$, the point
with its two choices of orientation. The morphisms are disjoint unions of
1-dimensional Riemannian bordisms:

(39)

The 1-manifold ${\sf S}_{t}$ is the circle of circumference $t$, viewed as a
bordism from the empty set to itself. The 1-manifolds underlying the bordisms
${\sf I}_{t}^{\pm}$, ${\sf R}_{t}^{\pm}$ and ${\sf L}_{t}^{\pm}$ are all
intervals of length $t$; the distinctions between them come from their source
and target data. The notation in (39) follows [HST10, ST11], where ${\sf
L}_{t}^{\pm}$ and ${\sf R}_{t}^{\pm}$ are the “left-” and “right-elbows” of
length $t$, respectively, where the $+$ or $-$ decoration refers to the
orientation, indicated pictorially in (39) by an arrow. We also follow the
convention from [HST10, ST11] to read these pictures of bordisms from right to
left. For example, ${\sf R}_{t}^{+}$ is a bordism from the empty set to ${\rm
pt}^{-}\coprod{\rm pt}^{+}$, while ${\sf L}_{t}^{+}$ is a bordism from ${\rm
pt}^{+}\coprod{\rm pt}^{-}$ to the empty set. Bordisms compose by gluing along
matching source and target data. The disjoint union of 0- and 1-manifolds
endows the category $1\hbox{-}{\sf EBord}$ with a symmetric monoidal
structure.

One can compute relations associated with compositions of bordisms. For
example, there are isometries of Riemannian 1-manifolds:

(40) $\displaystyle{\sf I}_{s}^{\pm}\circ{\sf I}_{t}^{\pm}\simeq{\sf
I}_{s+t}^{\pm},\qquad{\sf L}_{0}^{\pm}\circ\sigma^{\pm}\simeq{\sf
L}_{0}^{\mp},\qquad{\sf L}_{r}^{\pm}\circ({\sf I}_{t}^{\pm}\coprod{\sf
I}_{s}^{\mp})\simeq{\sf L}_{r+s+t}^{\pm},$

where in the middle identity $\sigma^{\pm}\colon{\rm pt}^{\mp}\coprod{\rm
pt}^{\pm}\stackrel{{\scriptstyle\sim}}{{\to}}{\rm pt}^{\pm}\coprod{\rm
pt}^{\mp}$ is the symmetry for the monoidal structure. The first identity is a
geometric counterpart to the semigroup ${\mathbb{R}}_{\geq 0}$ from (37): the
lengths of intervals add. Define the category ${\sf
Fun}^{\otimes}(1\hbox{-}{\sf EBord},{\sf Vect})$ whose objects are symmetric
monoidal functors

(41) $\displaystyle E\colon 1\hbox{-}{\sf EBord}\to{\sf Vect}$

and morphisms are natural isomorphisms of functors. Above, ${\sf Vect}$
denotes the category of super (i.e., ${\mathbb{Z}}/2$-graded) nuclear Fréchet
spaces over ${\mathbb{C}}$ endowed with the ${\mathbb{Z}}/2$-graded projective
tensor product. Physically, the ${\mathbb{Z}}/2$-grading
$\mathcal{H}=\mathcal{H}^{\rm ev}\oplus\mathcal{H}^{\rm odd}$ keeps track of
the even subspace $\mathcal{H}^{\rm ev}$ as _bosonic_ states and the odd
subspace $\mathcal{H}^{\rm odd}$ as _fermionic_ states. This grading will
become more important when we add supersymmetry.

For the data of a functor (41) to match the standard data of a quantum
mechanical system (37), one must demand an additional structure on the
functors (41) called _reflection positivity_ , e.g., see [FH21, §3] or [KS21,
page 18]. This begins with the ${\mathbb{Z}}/2$-action on $1\hbox{-}{\sf
EBord}$ generated by orientation reversal of 1-manifolds; in terms of (39),
this is

(42) $\displaystyle{\rm pt}^{\pm}\mapsto{\rm pt}^{\mp},\qquad{\sf
I}_{t}^{\pm}\mapsto{\sf I}_{t}^{\mp},\qquad{\sf R}_{t}^{\pm}\mapsto{\sf
R}_{t}^{\mp},\qquad{\sf L}_{t}^{\pm}\mapsto{\sf L}_{t}^{\mp}.$

Physically, this action reverses the arrow of time. There is also a
${\mathbb{Z}}/2$-action on vector spaces generated by complex conjugation. A
_reflection structure_ for the functor (41) is ${\mathbb{Z}}/2$-equivariant
data with respect to the actions on the source and target (e.g., see [FH21,
§B] or Definition B.13 below), which includes the data of an isomorphism

$\mathcal{H}:=E({\rm
pt}^{+})\stackrel{{\scriptstyle\sim}}{{\to}}\overline{E({\rm pt}^{-})}.$

A representation (41) with a reflection structure is _reflection positive_ if
the hermitian pairing

$\langle-,-\rangle\colon\overline{\mathcal{H}}\otimes\mathcal{H}\simeq E({\rm
pt}^{-})\otimes E({\rm pt}^{+})\xrightarrow{E({\sf L}_{0}^{-})}{\mathbb{C}}$

is positive. A _1-dimensional Euclidean field theory_ is a reflection positive
representation (41). A property demanded by reflection positivity is that
$E({\sf I}_{t}^{\pm})$ be a self-adjoint representation of the semigroup
${\mathbb{R}}_{\geq 0}$ on $\mathcal{H}$.

From the discussion above, a 1-dimensional Euclidean field theory is
determined by its restriction along (42) as the data of a self-adjoint
semigroup representation $(t\mapsto E({\sf I}_{t}^{+}))$ on the vector space
$\mathcal{H}=E({\rm pt}^{+})$ endowed with the hermitian inner product
$\langle-,-\rangle=E(L_{0}^{-})$. Furthermore, when restricted to
${\mathbb{R}}_{>0}\subset{\mathbb{R}}_{\geq 0}$, this semigroup representation
is trace class and the partition function

$Z={\sf Tr}(E({\sf I}_{t}^{+}))=E({\sf S}_{t}),\qquad t\in{\mathbb{R}}_{>0}$

can be viewed as a function on the moduli of metrized circles with
circumference $t>0$. Quantum mechanics on a closed, oriented Riemannian
manifold $X$ provides an example of a field theory with values

$E({\rm pt}^{+})=C^{\infty}(X;{\mathbb{C}}),\quad E({\sf
I}_{t}^{+})=e^{-t\Delta},\quad E({\sf
L}_{0}^{-})(s_{1},s_{2})=\int_{X}\overline{s_{1}}s_{2}\ d{\rm vol}.$

######  Remark 2.2.

We comment briefly on an important technical detail ignored above. Various
quantities in a 1-dimensional Euclidean field theory ought to depend smoothly
on geometric parameters, e.g., the semigroup $t\mapsto E({\sf I}_{t}^{+})$ and
the partition function $Z=E({\sf S}_{t})$ should depend smoothly on $t$. In
Stolz and Teichner’s formalism [HST10, ST11] this is addressed by defining
field theories as fibered functors, assigning smoothly varying quantities to
smoothly varying families of bordisms; see §B below.

###  2.2. Supersymmetric quantum mechanics

Let $X$ be a closed, even-dimensional Riemannian spin manifold, and
$\not{S}\to X$ be the ${\mathbb{Z}}/2$-graded spinor bundle with Dirac
operator $\not{D}$. Supersymmetric quantum mechanics on $X$ studies the space
of states $\mathcal{H}=\Gamma(X;\not{S})$ as a ${\mathbb{Z}}/2$-graded vector
space with hermitian inner product determined by pairing and integrating
spinors. The Dirac Laplacian $H=\not{D}{}^{2}$ is the Hamiltonian for this
theory, leading to an odd square root $Q=\not{D}$ of the Hamiltonian

(43) $\displaystyle\frac{1}{2}[Q,Q]=\frac{1}{2}(QQ+QQ)=Q^{2}=\not{D}^{2}=H,$

where $[-,-]$ is the graded commutator. The operator $Q$ defines a
_supersymmetry_ , e.g., see [Wit82, §1]. A first consequence of supersymmetry
is that the partition function

(44) $\displaystyle Z={\sf sTr}(e^{-tH})={\sf
sTr}(e^{-tQ^{2}})=\dim(\ker(Q))_{+}-\dim(\ker(Q))_{-}={\rm Ind}(Q)$

is independent of $t$ and equal to the super dimension of the kernel of $Q$,
i.e., the index of the Dirac operator $Q=\not{D}$. This follows from the
McKean–Singer formula [MS67]; see also [BGV92, Theorem 3.50]. It is crucial in
this argument that the partition function (44) uses the _supertrace_ , defined
as

${\sf sTr}(e^{-tH}):={\sf Tr}((-1)^{\sf F}\circ e^{-tH})$

where ${\sf Tr}$ is the ordinary trace, and $(-1)^{\sf F}$ acts by $+1$ on the
even subspace of $\mathcal{H}$ and $-1$ on the odd subspace.

The supersymmetry $Q$ allows one to extend the usual time-evolution (37) to a
representation of a _super semigroup_ ,

(45) $\displaystyle{\mathbb{R}}^{1|1}_{\geq 0}\to{\rm
End}(\mathcal{H})\qquad(t,\theta)\mapsto e^{-tH+\theta Q},$

where $(t,\theta)$ are coordinates on the supermanifold with boundary
${\mathbb{R}}^{1|1}_{\geq 0}$, defined by restricting the structure sheaf of
${\mathbb{R}}^{1|1}$ along the inclusion ${\mathbb{R}}_{\geq
0}\subset{\mathbb{R}}$ (see Example A.12). The semigroup multiplication on
${\mathbb{R}}^{1|1}_{\geq 0}$ is the restriction of multiplication in the
super Lie group defined as follows.

######  Definition 2.3.

Define the super Lie group ${\mathbb{R}}^{1|1}$ with multiplication

(46)
$\displaystyle(t,\theta)\cdot(s,\eta)=(t+s+\theta\eta,\theta+\eta)\qquad(t,\theta),(s,\eta)\in{\mathbb{R}}^{1|1}(S),$

where the above formula is understood in terms of the functor of points of
${\mathbb{R}}^{1|1}$.

######  Remark 2.4.

A closely related (and perhaps better known) example of supersymmetric quantum
mechanics takes as its space of states $\mathcal{H}=\Omega^{\bullet}(X)$,
differential forms on an oriented Riemannian manifold [Wit82, §2]. The
Hamiltonian is then defined to be the Hodge Laplacian $H=dd^{*}+d^{*}d$, which
has a pair of commuting odd square roots,

(47) $\displaystyle Q_{1}=d+d^{*},\quad
Q_{2}=i(d-d^{*}),\quad[Q_{1},Q_{2}]=0,\quad Q_{1}^{2}=Q_{2}^{2}=H.$

This is the data of $\mathcal{N}=2$ supersymmetry, rather than the
$\mathcal{N}=1$ example above. With our intended connections to index theory,
$\mathcal{N}=1$ supersymmetry is more fundamental. Of course, forgetting one
of the $Q_{i}$ in (47) extracts an $\mathcal{N}=1$ theory.

A general supersymmetric quantum mechanical system is given by a
supersemigroup representation (45) satisfying a self-adjoint and trace class
condition. In the formalism of [ST04, HST10, ST11], such data determine a
_$1|1$ -dimensional Euclidean field theory_, as we review below.

One can repackage supersymmetric quantum mechanics on $X$ as a representation
of a $1|1$-Euclidean bordism category $1|1\hbox{-}{\sf EBord}$, arising as a
generalization of $1\hbox{-}{\sf EBord}$. The objects of $1|1\hbox{-}{\sf
EBord}$ are finite disjoint unions of $\bullet={\rm spt}^{+}$ and $\circ={\rm
spt}^{-}$, the _super point_ with its two choices of orientation. The
morphisms can be visualized as

(48)

much the same as (39), but where the parameter $t$ measuring the Riemannian
length is replaced by a pair $(t,\theta)\in{\mathbb{R}}^{1|1}$ measuring a
“super” length. This super length has the usual even parameter $t$, but also
an odd parameter $\theta$; these parameters determine (super) families of
bordisms as we explain in §3.3. It turns out to be convenient111The
isomorphism ${\mathbb{R}}^{1|1}_{\geq 0}\simeq({\mathbb{R}}^{1|1}_{\leq
0})^{\sf{op}}$ equivalently allows one to view the bordisms ${\sf
I}^{-}_{t,\theta},{\sf R}^{-}_{t,\theta},{\sf L}^{-}_{t,\theta}$ and ${\sf
S}^{-}_{t,\theta}$ with $(t,\theta)$ in ${\mathbb{R}}^{1|1}_{\geq 0}$, but
then compositions involving these bordisms use the opposite super semigroup
structure. This is a non-issue for ordinary Euclidean bordisms, since
${\mathbb{R}}_{\geq 0}$ and ${\mathbb{R}}_{\leq 0}$ are isomorphic semigroups.
for ${\sf I}^{+}_{t,\theta},{\sf R}^{+}_{t,\theta},{\sf L}^{+}_{t,\theta}$ and
${\sf S}^{+}_{t,\theta}$ to have super length parameters $(t,\theta)$ in
${\mathbb{R}}^{1|1}_{\geq 0}$, whereas ${\sf I}^{-}_{t,\theta},{\sf
R}^{-}_{t,\theta},{\sf L}^{-}_{t,\theta}$ and ${\sf S}^{-}_{t,\theta}$ have
$(t,\theta)$ in ${\mathbb{R}}^{1|1}_{\leq 0}$. The supermanifolds underlying
the bordisms ${\sf I}_{t,\theta}^{\pm}$, ${\sf R}_{t,\theta}^{\pm}$ and ${\sf
L}_{t,\theta}^{\pm}$ are again all the same, being super intervals. We
similarly find relations associated with compositions of bordisms, e.g.,

(49) $\displaystyle{\sf I}_{s,\eta}^{\pm}\circ{\sf
I}_{t,\theta}^{\pm}\simeq{\sf I}_{s+t+\eta\theta,\eta+\theta}^{\pm},\quad{\sf
L}_{0}^{\pm}\circ\sigma^{\pm}\simeq{\sf L}_{0}^{\mp},\quad{\sf
L}_{0}^{\pm}\circ({\sf I}_{t,\theta}^{\pm}\coprod{\sf
I}_{s,\eta}^{\mp})\simeq{\sf L}_{(s,\eta)^{-1}(t,\theta)}^{\pm}$

where in the middle identity, $\sigma^{\pm}\colon{\rm spt}^{\mp}\coprod{\rm
spt}^{\pm}\stackrel{{\scriptstyle\sim}}{{\to}}{\rm spt}^{\pm}\coprod{\rm
spt}^{\mp}$ is again the symmetry for the monoidal structure. The first
identity is the geometric counterpart to multiplication in the super
semigroups ${\mathbb{R}}_{\geq 0}^{1|1}$ from (45). Note that the gluing of
super intervals encodes a non-commutative super group; this is expected as
(43) is not an abelian super Lie algebra.

With this in place, we may consider the category ${\sf
Fun}^{\otimes}(1|1\hbox{-}{\sf EBord},{\sf Vect})$ whose objects are symmetric
monoidal functors

(50) $\displaystyle E\colon 1|1\hbox{-}{\sf EBord}\to{\sf Vect}.$

For the data of a functor (50) to properly match with the physicists notion of
a supersymmetric quantum mechanical theory, we again must impose a version of
reflection positivity; this is defined analogously to the case without
supersymmetry. We define a _$1|1$ -dimensional Euclidean field theory_ to be a
reflection-positive functor (50). One consequence is that a $1|1$-dimensional
Euclidean field theory determines a self-adjoint representation of the super
semigroup ${\mathbb{R}}_{\geq 0}^{1|1}$ on a ${\mathbb{Z}}/2$-graded
topological vector space $\mathcal{H}$ with hermitian inner product
$\langle-,-\rangle$. This data satisfies a trace property on
${\mathbb{R}}_{>0}^{1|1}\subset{\mathbb{R}}^{1|1}_{\geq 0}$, where the trace
is the value of the functor (50) on a super circle (which turns out to be a
constant function of $(t,\theta)$). Supersymmetric quantum mechanics
determines a reflection positive functor (50) with values

(52) $\displaystyle\begin{array}[]{c}E({\rm spt}^{+})=\Gamma(X;\not{S}),\quad
E({\sf I}_{t,\theta}^{+})=e^{-t\not{D}+\theta\not{D}},\quad E({\sf
L}_{0}^{-})(s_{1},s_{2})=\int_{X}(s_{1},s_{2})d{\rm vol}.\end{array}$

######  Remark 2.5.

Restricting (45) along the homomorphism ${\mathbb{R}}_{\geq
0}\subset{\mathbb{R}}^{1|1}_{\geq 0}$ of semigroups forgets the supersymmetry,
extracting an ordinary quantum mechanical theory. Similarly, any
$1|1$-dimensional Euclidean field theory determines a 1-dimensional Euclidean
field theory; see [ST11, Equation 4.14].

######  Remark 2.6.

Supersymmetric quantum mechanics can be understood entirely from the vantage
of super Lie algebra representations (43) or super semigroup representations
(45). Our primary motivation for pursuing a description in terms of the
$1|1$-Euclidean bordism category is that there is an evident $2|1$-Euclidean
generalization, illuminating a candidate definition for $2|1$-Euclidean field
theory in the conjecture (6).

###  2.3. The renormalization group flow, low-energy cutoffs, and K-theory

Given $\mu\in{\mathbb{R}}_{>0}$, there is a functor $1\hbox{-}{\sf EBord}\to
1\hbox{-}{\sf EBord}$ that dilates the metric on Euclidean bordisms by the
factor $\mu$. In terms of the elementary bordisms (39), this has the effect

$\displaystyle{\rm pt}^{\pm}\mapsto{\rm pt}^{\pm},\qquad{\sf
I}_{t}^{\pm}\mapsto{\sf I}_{\mu t}^{\pm},\qquad{\sf R}_{t}^{\pm}\mapsto{\sf
R}_{\mu t}^{\pm},\qquad{\sf L}_{t}^{\pm}\mapsto{\sf L}_{\mu t}^{\pm},\quad{\sf
S}_{t}\mapsto{\sf S}_{\mu t}.$

Precomposing a functor (41) with this action defines the _renormalization
group flow_ acting on 1-dimensional Euclidean field theories. A field theory
is invariant under the renormalization group (RG) flow if and only if $E({\sf
I}_{t}^{+})={{\rm id}}_{\mathcal{H}}$ for all $t$. The trace class condition
on $E({\sf I}_{t}^{+})$ implies that RG-invariant field theories have a
finite-dimensional space of states $\mathcal{H}$, and hence determine a
1-dimensional topological field theory.

Similarly, there are functors $1|1\hbox{-}{\sf EBord}\to 1|1\hbox{-}{\sf
EBord}$ that dilate the super lengths as
$(t,\theta)\mapsto(\mu^{2}t,\mu\theta)$. On the elementary bordisms (48) this
gives

$\displaystyle{\rm pt}^{\pm}\mapsto{\rm pt}^{\pm},\qquad{\sf
I}_{t,\theta}^{\pm}\mapsto{\sf I}_{\mu^{2}t,\mu\theta}^{\pm},\qquad{\sf
R}_{t,\theta}^{\pm}\mapsto{\sf R}_{\mu^{2}t,\mu\theta}^{\pm},\qquad{\sf
L}_{t,\theta}^{\pm}\mapsto{\sf L}_{\mu^{2}t,\mu\theta}^{\pm},\quad{\sf
S}_{t,\theta}^{\pm}\mapsto{\sf S}_{\mu^{2}t,\mu\theta}^{\pm}.$

The nontrivial action on the odd part of the super length $\theta$ is
necessary to make the dilation action compatible with composition (and hence a
functor). Precomposing (50) with the above functor defines the renormalization
group flow on $1|1$-dimensional super Euclidean field theories. In the
presence of supersymmetry, the partition function is invariant under the
renormalization group flow: it is a constant function of the geometric
parameters $(t,\theta)$.

Given a quantum mechanical system of the form (37) or (45), let
$\mathcal{H}_{\lambda}\subset\mathcal{H}$ denote the $\lambda$-eigenspace of
the self-adjoint Hamiltonian $H$ (so $\lambda\in{\mathbb{R}}$). Then
$\mathcal{H}_{\lambda}$ is the space of _states with energy $\lambda$_. Define
the vector space of states with energy $<\lambda$,

(53)
$\displaystyle\mathcal{H}^{<\lambda}:=\bigoplus_{\nu<\lambda}\mathcal{H}_{\nu}$

with hermitian inner product given by the restriction of the inner product on
$\mathcal{H}$ to the subspaces $\mathcal{H}^{<\lambda}\subset\mathcal{H}$. The
trace class condition on $e^{-tH}$ implies that eigenspaces of $H$ are finite-
dimensional, the spectrum of $H$ is discrete, and
$\mathcal{H}_{\lambda}=\\{0\\}$ for $\lambda\ll 0$. Hence,
$\mathcal{H}^{<\lambda}$ is finite-dimensional. Define _$\lambda$ -cutoff
theory_ as the quantum mechanical system given by the finite-dimensional
vector space $\mathcal{H}^{<\lambda}$ equipped with the identity time-
evolution operator. The $\lambda$-cutoff theory is therefore invariant under
the renormalization group flow, i.e., a 1-dimensional topological field
theory.

######  Remark 2.7.

One could equally equip $\mathcal{H}^{<\lambda}$ with the restricted
Hamiltonian $H|_{\mathcal{H}^{<\lambda}}$ when defining the cutoff theory. For
our purposes, it turns out to be more convenient for the cutoff theory to be
RG-invariant.

Since $\mathcal{H}$ is ${\mathbb{Z}}/2$-graded and $H$ is an even map,
$\mathcal{H}_{\lambda}$ is ${\mathbb{Z}}/2$-graded for each
$\lambda\in{\mathbb{R}}$. Given a fixed choice of $\lambda$ and 1- or
$1|1$-dimensional Euclidean field theory $E$, consider the assignment

(54) $\displaystyle
E\mapsto[\mathcal{H}^{<\lambda}]=[(\mathcal{H}^{<\lambda})^{\rm
ev}]-[(\mathcal{H}^{<\lambda})^{\rm odd}]\in{\rm K}^{0}({\rm pt}).$

In the absence of supersymmetry, the map (54) generally depends on $\lambda$.
However, when $H$ has an odd square root $Q$, then the assignment is
independent of $\lambda$: $Q$ gives isomorphisms

$Q\colon\mathcal{H}_{\lambda}^{{\rm ev}/{\rm
odd}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{H}_{\lambda}^{{\rm odd}/{\rm
ev}},\qquad\lambda\neq 0$

and so

$[\mathcal{H}^{<\lambda}]=[\mathcal{H}_{0}]\in{\rm K}^{0}({\rm
pt})\simeq{\mathbb{Z}}$

for all $\lambda$. This integer-valued invariant in supersymmetric quantum
mechanics is the _Witten index_ , originally studied in [Wit82]. The Witten
index is a deformation invariant of a $1|1$-dimensional Euclidean field
theory.

######  Example 2.8.

For the theory with $Q=\not{D}$ the Dirac operator on an even dimensional spin
manifold, the Witten index agrees with the analytic index ${\rm
Ind}(\not{D})=[\mathcal{H}_{0}]\in{\rm K}^{0}({\rm pt})$ of the Dirac
operator. In the example discussed in Remark 2.4, the Witten index is the
Euler characteristic of $X$.

By contrast, 1-dimensional Euclidean field theories without supersymmetry form
a contractible space [HST10, Theorem 6.35] and so have no interesting
deformation invariants.

######  Remark 2.9.

We note that the McKean–Singer formula (44) equates the Witten index and the
partition function as elements of ${\mathbb{Z}}$. In families, this
identification is modified: the partition function is a differential form
which agrees with the Chern character of the Witten index.

###  2.4. Symmetries and real structures

Given a quantum mechanical system in the form (37) or (45), a super algebra
$A$ acting on $\mathcal{H}$ is an algebra of _symmetries_ if it commutes with
the action of $H$ or $Q$. Reflection positivity for a theory with $A$-symmetry
again endows $\mathcal{H}$ with a hermitian inner product, and additionally
requires the data of a $*$-structure on $A$ and the property that the
$A$-action on $\mathcal{H}$ be a $*$-action.

It is often convenient to organize field theories according to their
symmetries: for a given super algebra $A$ one can consider the collection of
theories with $A$-symmetry. Stolz and Teichner’s degree $n$ field theories are
supersymmetric quantum mechanical systems (45) with symmetry specified by the
the $n$th (complex) Clifford algebra

(57) $\displaystyle{\mathbb{C}}{\rm l}_{n}:=\left\\{\begin{array}[]{ll}\langle
f_{1},\dots f_{n}\mid[f_{j},f_{k}]=-2\delta_{jk}\rangle&n\geq 0\\\ \langle
e_{1},\dots e_{n}\mid[e_{j},e_{k}]=2\delta_{jk}\rangle&n<0.\end{array}\right.$

where $\\{e_{1},\dots,e_{n}\\}$ or $\\{f_{1},\dots,f_{n}\\}$ denotes the
standard basis of ${\mathbb{R}}^{n}$. These Clifford algebras carry a standard
$*$-structure; see §A.2.

One can also ask for a _real structure_ on a quantum mechanical system,
meaning a grading-preserving ${\mathbb{C}}$-antilinear involution of
$\mathcal{H}$ commuting with the Hamiltonian. When considering theories with
$A$-symmetry, one should demand a compatible real structure on $A$. In the
case of the Clifford algebras, this is equivalent to declaring the generators
in (57) to be real so that the real structure is determined by complex
conjugation. Using the Atiyah–Bott–Shapiro description of ${\rm KO}^{n}({\rm
pt})$ in terms of Clifford modules [ABS64], such real, degree $n$ theories
determine classes in ${\rm KO}^{n}({\rm pt})$ using the construction (54).

Examples of real, degree $n$ theories arise naturally in spin geometry. Let
$X$ be an $n$-dimensional spin manifold with principal spin bundle $P$, i.e.,
$P\to X$ is a ${\rm Spin}(n)$-principal bundle that double covers of the
oriented frame bundle of $TX$. Define the spinor bundle as

(58) $\displaystyle{\mathbb{S}}:=P\times_{{\rm Spin}(n)}{\rm
Cl}_{n}=P\times_{{\rm Spin}(n)}({\rm Cl}_{n}^{+}\oplus{\rm Cl}_{n}^{-})$

with ${\mathbb{Z}}/2$-grading inherited from the ${\mathbb{Z}}/2$-grading on
the real Clifford algebra ${\rm Cl}({\mathbb{R}}^{n})={\rm Cl}_{n}$. This
version of the spinor bundle carries a right ${\rm Cl}_{n}$-action, which we
will identify with a left ${\rm Cl}_{-n}$-action (see Example A.8). The odd,
self-adjoint, ${\rm Cl}_{-n}$-linear Dirac operator $\not{D}$ acts on global
sections $\Gamma(X;{\mathbb{S}})$. The same constructions as before give a
quantum mechanical system with additional structures: the space of states
$\mathcal{H}=\Gamma(X;\not{S})$ is a real, self-adjoint ${\rm Cl}_{-n}$-module
and the operator $Q=\not{D}$ commutes with the Clifford action. The energy
cutoff (53) gives a finite-dimensional Clifford module. Via the
Atiyah–Bott–Shapiro construction, the assignment (54) is the _Clifford index_
of the Dirac operator, taking values in ${\rm KO}^{-n}({\rm pt})$. This
fundamental geometric example motivates Stolz and Teichner’s definitions of
degree $n$ field theories in dimension $1|1$, see [ST04, Example 2.3.3].

######  Remark 2.10.

For an inner product space $W$, the Clifford algebra ${\rm Cl}(W)$ can be
understood as the quantization of a free (fermionic) particle moving in the
odd vector space $\Pi W$, e.g., see [Fre99, page 19]. Free fermions in higher
dimensions illuminate a path towards a general framework for degree $n$ field
theories. In particular, this suggests a $2|1$-dimensional generalization for
the second conjecture in (6).

##  3\. Super Euclidean paths in a manifold

###  3.1. Warm-up: The category of (non-super) Euclidean paths in a manifold

Let ${\mathbb{A}}^{1}$ be the affine real line, regarded as an oriented
Riemannian manifold. The isometry group of ${\mathbb{A}}^{1}$ is the Lie group
${\mathbb{R}}$, acting on ${\mathbb{A}}^{1}$ by translation. We will describe
Euclidean paths modeled on ${\mathbb{A}}^{1}$ with an eye towards the
generalization to $1|1$-Euclidean paths.

The Riemannian manifold ${\mathbb{A}}^{1}$, isometry group ${\mathbb{R}}$, and
a choice of connected codimension 1 submanifold
${\mathbb{A}}^{0}\subset{\mathbb{A}}^{1}$ determines the 1-dimensional
Euclidean geometry $({\mathbb{R}},{\mathbb{A}}^{1},{\mathbb{A}}^{0})$ in the
sense of Stolz and Teichner [ST11, §2.5]; see §B.1 for a review. Below, for
$s\in{\mathbb{R}}$ let $T_{s}\colon{\mathbb{A}}^{1}\to{\mathbb{A}}^{1}$ denote
translation by $s$.

######  Definition 3.1.

Let $M$ be a smooth manifold. A _positively oriented Euclidean path_ in $M$ is
a triple $(\phi,t_{\rm in},t_{\rm out})$ for a smooth map
$\phi\colon{\mathbb{A}}^{1}\to M$ and points $t_{\rm in},t_{\rm
out}\in{\mathbb{A}}^{1}$ with $t_{\rm out}-t_{\rm in}\in{\mathbb{R}}_{\geq
0}$. Similarly, a _negatively oriented Euclidean path_ in $M$ is a triple
$(\phi,t_{\rm in},t_{\rm out})$ with $t_{\rm out}-t_{\rm
in}\in{\mathbb{R}}_{\leq 0}$. A _Euclidean path_ is a positively or negatively
oriented Euclidean path. The point $\phi(t_{\rm in})\in M$ is the _incoming_
boundary and $\phi(t_{\rm out})\in M$ is the _outgoing_ boundary of a
Euclidean path. An _isometry_ $(\phi,t_{\rm in},t_{\rm
out})\stackrel{{\scriptstyle\sim}}{{\to}}(\phi^{\prime},t_{\rm
in}^{\prime},t_{\rm out}^{\prime})$ between a pair of Euclidean paths is the
data of $s\in{\mathbb{R}}$ such that $\phi^{\prime}=\phi\circ T_{s}^{-1}$,
$T_{s}(t_{\rm in})=t_{\rm in}^{\prime}$, and $T_{s}(t_{\rm out})=t_{\rm
out}^{\prime}$.

Euclidean paths and isometries form a groupoid that is equivalent to a
discrete groupoid (i.e., a groupoid with only identity morphisms) whose
objects are

(59) $\displaystyle{\mathbb{R}}_{\geq 0}\times{\sf
Map}({\mathbb{R}},M)\coprod{\mathbb{R}}_{\leq 0}\times{\sf
Map}({\mathbb{R}},M),$

where we regard ${\sf Map}({\mathbb{R}},M)$ as a presheaf on the site of
manifolds,

${\sf Map}({\mathbb{R}},M)\colon{\sf Mfld}^{\sf{op}}\to{\sf Set},\qquad
S\mapsto C^{\infty}(S\times{\mathbb{R}},M).$

Indeed, for a given Euclidean path $(\phi,t_{\rm in},t_{\rm out})$, the
basepoint $t_{\rm in}\in{\mathbb{A}}^{1}$ gives an identification
${\mathbb{A}}^{1}\simeq{\mathbb{R}}^{1}$, so that $\phi\colon{\mathbb{R}}\to
M$ and $t_{\rm out}$ is determined by a point in ${\mathbb{R}}_{\geq
0}\coprod{\mathbb{R}}_{\leq 0}$.

The 1-dimensional Euclidean geometry has some additional structures that lead
to additional structures on Euclidean paths. First, the basepoint
${\mathbb{A}}^{0}\subset{\mathbb{A}}^{1}$ provides an isomorphism
${\mathbb{A}}^{1}\simeq{\mathbb{R}}^{1}$ (that may be different than the
identification using $t_{\rm in}$). This gives a preferred orientation-
reversing map of ${\mathbb{A}}^{1}$,

(60) $\displaystyle{\mathbb{A}}^{1}\simeq{\mathbb{R}}^{1}\xrightarrow{{\sf
or}}{\mathbb{R}}^{1}\simeq{\mathbb{A}}^{1},\qquad{\sf or}(t)=-t,\
t\in{\mathbb{R}}.$

Next, for any $\mu\in{\mathbb{R}}_{>0}$ there is a diffeomorphism

(61) $\displaystyle{\mathbb{A}}^{1}\simeq{\mathbb{R}}^{1}\xrightarrow{{\sf
rg}_{\mu}}{\mathbb{R}}^{1}\simeq{\mathbb{A}}^{1},\qquad{\sf
rg}_{\mu}(t)=\mu\cdot t$

that dilates the metric on ${\mathbb{A}}^{1}$. The notation ${\sf rg}_{\mu}$
is in reference to the connection with the renormalization group. We emphasize
that (60) and (61) are not isometries in the 1-dimensional Euclidean geometry
$({\mathbb{R}},{\mathbb{A}}^{1},{\mathbb{A}}^{0})$.

The structures (60) and (61) provide the following additional structures on
Euclidean paths. The _orientation reversal_ of a Euclidean path $(\phi,t_{\rm
in},t_{\rm out})$ applies the orientation-reversing isometry (60), giving an
assignment

$(\phi,t_{\rm in},t_{\rm out})\mapsto(\phi\circ{\sf or}^{-1},{\sf or}(t_{\rm
in}),{\sf or}(t_{\rm out})).$

For $\mu\in{\mathbb{R}}_{>0}$ the _$\mu$ -dilation_ of a Euclidean path
$(\phi,t_{\rm in},t_{\rm out})$ applies the diffeomorphism (61) to a Euclidean
path so that

$(\phi,t_{\rm in},t_{\rm out})\mapsto(\phi\circ{\sf rg}_{\mu}^{-1},{\sf
rg}_{\mu}(t_{\rm in}),{\sf rg}_{\mu}(t_{\rm out})).$

In terms of the description (59) of Euclidean paths, orientation reversal and
$\mu$-dilation are

$\displaystyle{\mathbb{R}}_{\geq 0}\times{\sf
Map}({\mathbb{R}},M)\coprod{\mathbb{R}}_{\leq 0}\times{\sf
Map}({\mathbb{R}},M)$ $\displaystyle\to$ $\displaystyle{\mathbb{R}}_{\geq
0}\times{\sf Map}({\mathbb{R}},M)\coprod{\mathbb{R}}_{\leq 0}\times{\sf
Map}({\mathbb{R}},M)$ (62) $\displaystyle(t,\phi)$ $\displaystyle\mapsto$
$\displaystyle(-t,\phi\circ{\sf or}^{-1})$ $\displaystyle(t,\phi)$
$\displaystyle\mapsto$ $\displaystyle(\mu\cdot t,\phi\circ{\sf
rg}_{\mu}^{-1}).$

In particular, orientation reversal exchanges positively and negatively
oriented Euclidean paths. The incoming and outgoing boundary $\phi(t_{\rm
in}),\phi(t_{\rm out})\in M$ are preserved by orientation reversal and
$\mu$-dilation.

Next, we consider the extent to which Euclidean paths can be viewed as
morphisms in a category. There are maps

${\rm in},{\rm out}\colon{\mathbb{R}}_{\geq 0}\times{\sf
Map}({\mathbb{R}},M)\coprod{\mathbb{R}}_{\leq 0}\times{\sf
Map}({\mathbb{R}},M)\rightrightarrows M\coprod M$

sending a Euclidean path to its incoming or outgoing boundary. We would like
the above to determine the source and target maps in a category in which
composition comes from concatenation of paths. However, this runs into a
problem: concatenation is only piecewise smooth. There are different ways to
resolve this problem. One option is to consider collared paths in $M$, which
is the route followed by Stolz and Teichner [ST11]; see §B.3. Another option
is to encode cutting laws rather than gluing laws as in [DG20, LS21, BEP22].
The resulting bordism categories are not equivalent, and it isn’t clear which
one is “correct” when generalizing the quantum mechanical systems discussed in
§2.1. However, there is a subspace of (59) where the difficulties of
concatenation disappear: the _constant_ Euclidean paths in $M$.

######  Definition 3.2.

Define the (Lie) _category of constant paths in $M$_, denoted ${\sf
Path}_{0}(M)$, as having objects and morphisms the smooth manifolds

${\rm Ob}({\sf Path}_{0}(M))=M\coprod M,\qquad{\rm Mor}({\sf
Path}_{0}(M))={\mathbb{R}}_{\geq 0}\times M\coprod{\mathbb{R}}_{\leq 0}\times
M.$

The source and target maps in ${\sf Path}_{0}(M)$ are given by the
projections. The unit map is determined by inclusion at $0\in{\mathbb{R}}$,
and composition is determined by addition in ${\mathbb{R}}$.

Under the inclusion of the constant maps

${\rm Mor}({\sf Path}_{0}(M))={\mathbb{R}}_{\geq 0}\times
M\coprod{\mathbb{R}}_{\leq 0}\times M\subset{\mathbb{R}}_{\geq 0}\times{\sf
Map}({\mathbb{R}},M)\coprod{\mathbb{R}}_{\leq 0}\times{\sf
Map}({\mathbb{R}},M)$

concatenation of paths corresponds to composition in the category ${\sf
Path}_{0}(M)$. The assignments (62) furthermore extend to functors

(63) $\displaystyle{\sf Or}\colon{\sf Path}_{0}(M)\to{\sf
Path}_{0}(M),\quad{\sf RG}_{\mu}\colon{\sf Path}_{0}(M)\to{\sf Path}_{0}(M).$

Next we explain the relationship between Euclidean paths and Euclidean loops.
There is a ${\mathbb{Z}}$-action on (59) given by
$(t,\phi)\mapsto(t,n_{t}\cdot\phi)$ where $n_{t}\cdot\phi\colon{\mathbb{R}}\to
M$ is the map

$(n_{t}\cdot\phi)(s)=\phi(s+nt),\qquad n\in{\mathbb{Z}}.$

For paths of strictly positively length, the ${\mathbb{Z}}$-fixed subspace (as
a presheaf) is the _Euclidean loop space_ of $M$

$\mathcal{L}(M):=({\mathbb{R}}_{>0}\times{\sf
Map}({\mathbb{R}},M)\coprod{\mathbb{R}}_{<0}\times{\sf
Map}({\mathbb{R}},M))^{\mathbb{Z}}\subset{\mathbb{R}}_{\geq 0}\times{\sf
Map}({\mathbb{R}},M)\coprod{\mathbb{R}}_{\leq 0}\times{\sf
Map}({\mathbb{R}},M)$

whose points $(t,\phi)$ are loops in $M$ of circumference $t>0$. Furthermore,
$\mathcal{L}(M)$ has an $S^{1}$-action by rotation of loops. Any constant path
of positive length determines a loop, giving a span

(64) $\displaystyle{\rm Mor}({\sf
Path}_{0}(M))\hookleftarrow{\mathbb{R}}_{>0}\times
M\coprod{\mathbb{R}}_{<0}\times M\hookrightarrow\mathcal{L}(M),$

and the $S^{1}$-action on $\mathcal{L}(M)$ restricts to the trivial action on
${\rm Mor}({\sf Path}_{0}(M))$.

In summary, although constant Euclidean paths lack the geometric richness of
the full path space, they provide two technical simplifications: (1) constant
paths form a finite-dimensional smooth manifold, and (2) concatenation of
constant paths always results in a (smooth) constant path. When $M={\rm pt}$,
the category ${\sf Path}_{0}(M)$ recovers the semigroup of intervals (37), and
hence the categories ${\sf Path}_{0}(M)$ (and their representations) offer one
possible families-generalization of the quantum mechanical systems studied
previously. With this in mind, we purse a generalization of the above; the
goal of the remainder of the section is to generalize Definition 3.2 to a
category of (nearly) constant super Euclidean paths in $M$.

###  3.2. $1|1$-dimensional Euclidean geometry

Recall from (46) the super Lie group ${\mathbb{R}}^{1|1}$, and let
${\mathbb{A}}^{1|1}$ denote the (affine) supermanifold underlying the super
Lie group ${\mathbb{R}}^{1|1}$. This notation emphasizes that
${\mathbb{R}}^{1|1}$ has a distinguished basepoint (the identity
$0\in{\mathbb{R}}^{1|1}$) whereas ${\mathbb{A}}^{1|1}$ does not. Similarly,
let ${\mathbb{A}}^{0|1}$ denote the supermanifold underlying the super Lie
group ${\mathbb{R}}^{0|1}$. Let ${\mathbb{A}}^{0|1}\subset{\mathbb{A}}^{1|1}$
be the embedding modeled on the standard222We caution that
${\mathbb{R}}^{0|1}\subset{\mathbb{R}}^{1|1}$ is not a homomorphism for the
group structure (46) on ${\mathbb{R}}^{1|1}$. inclusion
${\mathbb{R}}^{0|1}\subset{\mathbb{R}}^{1|1}$. We again refer to §B.1 for a
review of rigid super geometries.

######  Definition 3.3 ([HST10, Example 6.16] and [ST11, §4.2]).

Form the semidirect product ${\mathbb{R}}^{1|1}\rtimes{\mathbb{Z}}/2$ where
${\mathbb{Z}}/2=\\{\pm 1\\}$ acts on ${\mathbb{R}}^{1|1}$ by
$(t,\theta)\mapsto(t,\pm\theta)$. The _$1|1$ -Euclidean geometry_ is specified
by the triple

$(G,{\mathbb{M}},{\mathbb{M}}^{c})=({\mathbb{R}}^{1|1}\rtimes{\mathbb{Z}}/2,{\mathbb{A}}^{1|1},{\mathbb{A}}^{0|1})$

for the evident action of ${\mathbb{R}}^{1|1}\rtimes{\mathbb{Z}}/2$ on
${\mathbb{A}}^{1|1}$ and the inclusion
${\mathbb{A}}^{0|1}\subset{\mathbb{A}}^{1|1}$ described above.

In the notation of Definition B.2, the stack
$({\mathbb{R}}^{1|1}\rtimes{\mathbb{Z}}/2,{\mathbb{A}}^{1|1})$-${\sf SMfld}$
of $1|1$-Euclidean manifolds has as objects submersions $Y\to S$ whose fibers
are $1|1$-dimensional manifolds endowed with a super Euclidean structure. The
stack of $1|1$-Euclidean pairs has as objects fiber bundles $Y\to S$ of
$1|1$-manifolds endowed with super Euclidean structure together with a family
of embedded $0|1$-dimensional supermanifolds $Y^{c}\subset Y$ over $S$.

######  Example 3.4.

Our main example of an $S$-family of $1|1$-Euclidean pairs depends on
$(t,\theta)\in{\mathbb{R}}^{1|1}(S)$ and is given by

(65) $\displaystyle
Y^{c}=S\times{\mathbb{R}}^{0|1}\xhookrightarrow{i_{t,\theta}}S\times{\mathbb{R}}^{1|1}=Y$

where

(66) $\displaystyle i_{t,\theta}\colon S\times{\mathbb{R}}^{0|1}\subset
S\times{\mathbb{R}}^{1|1}\xrightarrow{T_{t,\theta}}S\times{\mathbb{R}}^{1|1}$

using the standard inclusion and the super translation map $T_{t,\theta}$
defined as the composition

(67) $\displaystyle S\times{\mathbb{R}}^{1|1}\xrightarrow{{{\rm
id}}_{S}\times(t,\theta)\times{{\rm
id}}_{{\mathbb{R}}^{1|1}}}S\times{\mathbb{R}}^{1|1}\times{\mathbb{R}}^{1|1}\xrightarrow{{{\rm
id}}_{S}\times{\rm act}}S\times{\mathbb{R}}^{1|1},$

where the second arrow is the action of ${\mathbb{R}}^{1|1}$ on itself. For
any $(t,\theta)$ and $(s,\eta)$ the associated $1|1$-Euclidean pairs are
isomorphic via the isometry

$S\times{\mathbb{R}}^{1|1}\xrightarrow{T_{(s,\eta)\cdot(t,\theta)^{-1}}}S\times{\mathbb{R}}^{1|1}.$

In particular, all pairs (65) are isomorphic to the standard one
$S\times{\mathbb{R}}^{0|1}\subset S\times{\mathbb{R}}^{1|1}$.

The stack of $1|1$-dimensional super Euclidean manifolds has some extra
structures generalizing orientation reversal (60) and dilation (61) of
ordinary Euclidean 1-manifolds.

######  Definition 3.5.

_Orientation reversal_ is the automorphism of the $1|1$-Euclidean geometry
determined by the homomorphism

(68) $\displaystyle{\sf or}\colon{\mathbb{R}}^{1|1}\to{\mathbb{R}}^{1|1},$
$\displaystyle\qquad(t,\theta)\mapsto(-t,-i\theta),\qquad(t,\theta)\in{\mathbb{R}}^{1|1}(S).$

We observe that when restricted to ${\mathbb{R}}\subset{\mathbb{R}}^{1|1}$
(68) is the standard orientation reversing map (60) on ${\mathbb{R}}$.

######  Remark 3.6.

The map (68) is called the _pin generator_ in [HST10, Example 6.16].

######  Definition 3.7.

For $\mu\in{\mathbb{R}}_{>0}$, the _renormalization group action_ is the
automorphism of the $1|1$-Euclidean geometry determined by the homomorphism

(69) $\displaystyle{\sf
rg}_{\mu}\colon{\mathbb{R}}^{1|1}\to{\mathbb{R}}^{1|1},$
$\displaystyle\qquad(t,\theta)\mapsto(\mu^{2}t,\mu\theta),\qquad(t,\theta)\in{\mathbb{R}}^{1|1}(S).$

######  Definition 3.8 ([ST11, Example 2.39 and §4.1]).

The _spin flip_ is the automorphism of the $1|1$-Euclidean geometry determined
by the action of $-1\in\\{\pm
1\\}\simeq{\mathbb{Z}}/2<{\mathbb{R}}^{1|1}\rtimes{\mathbb{Z}}/2$

(70) $\displaystyle{\sf
fl}\colon{\mathbb{R}}^{1|1}\to{\mathbb{R}}^{1|1},\qquad(t,\theta)\mapsto(t,-\theta),\qquad(t,\theta)\in{\mathbb{R}}^{1|1}(S).$

Since ${\mathbb{Z}}/2<{\mathbb{R}}^{1|1}\rtimes{\mathbb{Z}}/2$ is in the
center, the spin flip commutes with the action of the isometry group.

The $1|1$-Euclidean geometry also has a real structure in the sense of
Definition B.5.

######  Lemma 3.9 ([HST10, Example 6.20]).

The super Lie group ${\mathbb{R}}^{1|1}$ has a real structure

(71) $\displaystyle{\sf
r}\colon{\mathbb{R}}^{1|1}\xrightarrow{\sim}\overline{{\mathbb{R}}}^{1|1},\qquad\overline{{\sf
r}}\circ{\sf r}={{\rm id}}_{{\mathbb{R}}^{1|1}}.$

###### Proof.

Following Example A.19, first choose an identification
${\mathbb{R}}^{1|1}\simeq\Pi(\underline{{\mathbb{R}}}_{\mathbb{C}})$ where
$\underline{{\mathbb{R}}}$ is the trivial real line bundle on ${\mathbb{R}}$
and $\underline{{\mathbb{R}}}_{\mathbb{C}}$ is its complexification. This
endows the supermanifold underlying ${\mathbb{R}}^{1|1}$ with a real
structure. To promote this to a real structure on the super Lie group, let
$\theta\in C^{\infty}({\mathbb{R}}^{1|1})^{\rm
odd}\simeq\Gamma({\mathbb{R}},\Lambda^{1}(\underline{{\mathbb{R}}}_{\mathbb{C}}))$
denote the function associated with a (real) trivialization of
$\underline{{\mathbb{R}}}$. With this choice, the group multiplication (46) is
real. ∎

######  Definition 3.10.

Define a real structure on the $1|1$-dimensional super Euclidean model
geometry from the real structure on the super Lie group
${\mathbb{R}}^{1|1}\rtimes{\mathbb{Z}}/2$ from Lemma 3.9, using that
${\mathbb{Z}}/2$ preserves this real structure on ${\mathbb{R}}^{1|1}$.

######  Example 3.11.

By Lemma B.4, the maps (68), (69) and (70) determine endofunctors of the stack
of $1|1$-Euclidean manifolds,

${\sf Or},{\sf RG}_{\mu},{\sf
Fl}\colon({\mathbb{R}}^{1|1}\rtimes{\mathbb{Z}}/2,{\mathbb{A}}^{1|1})\hbox{-}{\sf
SMfld}\to({\mathbb{R}}^{1|1}\rtimes{\mathbb{Z}}/2,{\mathbb{A}}^{1|1})\hbox{-}{\sf
SMfld},$

that send an $S$-family of $1|1$-Euclidean manifolds to another $S$-family.
Applied to the super Euclidean pair (65), we find

$\displaystyle{\sf
Or}(S\times{\mathbb{R}}^{0|1}\xhookrightarrow{i_{t,\theta}}S\times{\mathbb{R}}^{1|1})$
$\displaystyle=$
$\displaystyle(S\times{\mathbb{R}}^{0|1}\xhookrightarrow{i_{-t,-i\theta}}S\times{\mathbb{R}}^{1|1})$
$\displaystyle{\sf
RG}_{\mu}(S\times{\mathbb{R}}^{0|1}\xhookrightarrow{i_{t,\theta}}S\times{\mathbb{R}}^{1|1})$
$\displaystyle=$
$\displaystyle(S\times{\mathbb{R}}^{0|1}\xhookrightarrow{i_{\mu^{2}t,\mu\theta}}S\times{\mathbb{R}}^{1|1})$
$\displaystyle{\sf
Fl}(S\times{\mathbb{R}}^{0|1}\xhookrightarrow{i_{t,\theta}}S\times{\mathbb{R}}^{1|1})$
$\displaystyle=$
$\displaystyle(S\times{\mathbb{R}}^{0|1}\xhookrightarrow{i_{t,-\theta}}S\times{\mathbb{R}}^{1|1})$

Similarly, the map (71) determines a map of stacks

${\sf
R}\colon({\mathbb{R}}^{1|1}\rtimes{\mathbb{Z}}/2,{\mathbb{A}}^{1|1})\hbox{-}{\sf
SMfld}\to({\mathbb{R}}^{1|1}\rtimes{\mathbb{Z}}/2,{\mathbb{A}}^{1|1})\hbox{-}{\sf
SMfld},\quad(Y^{c},Y)\mapsto(\overline{Y}{}^{c},\overline{Y})$

covering complex conjugation of supermanifolds, that sends an $S$-family of
$1|1$-Euclidean manifolds to an $\overline{S}$-family. There are natural
isomorphisms of functors

(72) $\displaystyle{\sf Or}^{2}\simeq{\sf Fl},\quad{\sf RG}_{\mu}\circ{\sf
RG}_{\mu^{\prime}}\simeq{\sf RG}_{\mu\mu^{\prime}},\quad{\sf R}\circ{\sf
Or}\simeq\overline{{\sf Fl}}\circ\overline{{\sf Or}}\circ{\sf
R},\quad\overline{{\sf R}}\circ{\sf R}\simeq{{\rm id}}$

coming from the equalities of maps

${\sf or}^{2}={\sf fl},\quad{\sf rg}_{\mu}\circ{\sf rg}_{\mu^{\prime}}={\sf
rg}_{\mu\mu^{\prime}},\quad\overline{{\sf or}}\circ{\sf r}=\overline{{\sf
fl}}\circ{\sf r}\circ{\sf or},\quad\overline{{\sf r}}\circ{\sf r}={{\rm
id}}_{{\mathbb{R}}^{1|1}}.$

###  3.3. Super Euclidean paths in a smooth manifold

Given a triple $(\phi,(t,\theta),(s,\eta))$ where $\phi\colon
S\times{\mathbb{R}}^{1|1}\to M$ is a map and
$(t,\theta),(s,\eta)\in{\mathbb{R}}^{1|1}(S)$, consider

(73)

for the maps defined in (66).

######  Definition 3.12.

Let $M$ be a smooth manifold and $S$ a supermanifold. An $S$-family of
_positively oriented super Euclidean paths_ in $M$ is a triple $(\phi,(t_{\rm
out},\theta_{\rm out}),(t_{\rm in},\theta_{\rm in}))$ determining a diagram
(73) with the property

$\displaystyle(t_{\rm out},\theta_{\rm out})\cdot(t_{\rm in},\theta_{\rm
in})^{-1}\in{\mathbb{R}}^{1|1}_{\geq 0}(S)\subset{\mathbb{R}}^{1|1}(S).$

Similarly, an $S$-family of _negatively oriented super Euclidean paths_ in $M$
is a triple where

$\displaystyle(t_{\rm out},\theta_{\rm out})\cdot(t_{\rm in},\theta_{\rm
in})^{-1}\in{\mathbb{R}}^{1|1}_{\leq 0}(S)\subset{\mathbb{R}}^{1|1}(S).$

An $S$-family of _super Euclidean paths_ in $M$ is a triple $(\phi,(t_{\rm
out},\theta_{\rm out}),(t_{\rm in},\theta_{\rm in}))$ that over each component
of $S$ is a family of positively or negatively oriented superpaths. The
_incoming_ and _outgoing_ boundaries of $(\phi,(t,\theta),(s,\eta))$ are the
compositions

$\phi_{\rm out}\colon S\times{\mathbb{R}}^{0|1}\xhookrightarrow{i_{(t_{\rm
out},\theta_{\rm
out})}}S\times{\mathbb{R}}^{1|1}\xrightarrow{\phi}M,\quad\phi_{\rm in}\colon
S\times{\mathbb{R}}^{0|1}\xhookrightarrow{i_{(t_{\rm in},\theta_{\rm
in})}}S\times{\mathbb{R}}^{1|1}\xrightarrow{\phi}M.$

An _isometry_ $(\phi,(t_{\rm out},\theta_{\rm out}),(t_{\rm in},\theta_{\rm
in}))\stackrel{{\scriptstyle\sim}}{{\to}}(\phi^{\prime},(t_{\rm
out}^{\prime},\theta_{\rm out}^{\prime}),(t_{\rm in}^{\prime},\theta_{\rm
in}^{\prime}))$ between a pair of $S$-families of superpaths is the data of
$(s,\eta)\in{\mathbb{R}}^{1|1}(S)$ such that

$\phi^{\prime}=\phi\circ T_{s,\eta},\quad T_{s,\eta}\circ\phi_{\rm
in}=\phi_{\rm in}^{\prime},\quad T_{s,\eta}\circ\phi_{\rm out}=\phi_{\rm
out}^{\prime}$

where $T_{s,\eta}$ is the super translation by
$(s,\eta)\in{\mathbb{R}}^{1|1}(S)$ defined in (67).

Super Euclidean paths in $M$ and their isometries form a stack: for each $S$
we have a groupoid of $S$-families of super Euclidean paths over $S$, and for
a map $f\colon S^{\prime}\to S$ of supermanifolds we obtain a functor between
groupoids by pulling back the triples $(\phi,(t_{\rm out},\theta_{\rm
out}),(t_{\rm in},\theta_{\rm in}))$ along $f$. Analogously to (59), we have
the following.

######  Lemma 3.13.

The groupoid of super Euclidean paths over $S$ is equivalent to the discrete
groupoid given by the value of the sheaf

(74) $\displaystyle{\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)\coprod{\mathbb{R}}^{1|1}_{\leq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)$

on the supermanifold $S$, and hence the stack of super Euclidean paths in $M$
is equivalent to the sheaf (74).

###### Proof.

Given an $S$-family of positively (respectively, negatively) oriented super
Euclidean paths $(\phi,(t_{\rm out},\theta_{\rm out}),(t_{\rm in},\theta_{\rm
in}))$, there is a unique isomorphism to a superpath with $(t_{\rm
in},\theta_{\rm in})=(0,0)\in{\mathbb{R}}^{1|1}(S)$. With this identification
fixed, the remaining data is

(75) $\displaystyle\phi\colon S\times{\mathbb{R}}^{1|1}\to M,\qquad(t_{\rm
out},\theta_{\rm out})=(t,\theta)\in({\mathbb{R}}^{1|1}_{\geq
0}\coprod{\mathbb{R}}^{1|1}_{\leq 0})(S).$

The result follows. ∎

###  3.4. The category of nearly constant superpaths

Concatenation of $S$-families of superpaths in $M$ runs into essentially the
same issues as in the case of ordinary paths: to get a (smooth) superpath one
requires agreement in a neighborhood of the gluing. Just as before, the
problem goes away when one only considers suitably constant maps to $M$.

######  Definition 3.14.

An $S$-family of _nearly constant super Euclidean paths in $M$_ is the data of
a super Euclidean path $(\phi,(t_{\rm out},\theta_{\rm out}),(t_{\rm
in},\theta_{\rm in}))$ for which $\phi$ is determined by a map $\phi_{0}$,

(76) $\displaystyle\phi\colon
S\times{\mathbb{R}}^{1|1}\xrightarrow{q}S\times{\mathbb{R}}^{1|1}/{\mathbb{R}}\simeq
S\times{\mathbb{R}}^{0|1}\xrightarrow{\phi_{0}}M,$

where the first arrow is the quotient of $S\times{\mathbb{R}}^{1|1}$ by the
${\mathbb{R}}$-action on ${\mathbb{R}}^{1|1}$.

######  Lemma 3.15.

The groupoid of $S$-families of nearly constant super Euclidean path in $M$ is
equivalent to the discrete groupoid given by the value of the sheaf

(77) $\displaystyle{\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{0|1},M)\coprod{\mathbb{R}}^{1|1}_{\leq 0}\times{\sf
Map}({\mathbb{R}}^{0|1},M),$

on the supermanifold $S$.

###### Proof.

The follows the same argument as the proof of Lemma 3.13, but the map $\phi$
in (75) is required to be of the form (76) and hence determined by
$\phi_{0}\colon S\times{\mathbb{R}}^{0|1}\to M$. ∎

######  Lemma 3.16.

Suppose that $\gamma=(\phi,(t_{\rm out},\theta_{\rm out}),(t_{\rm
in},\theta_{\rm in}))$ and $\gamma^{\prime}=(\phi^{\prime},(t_{\rm
out}^{\prime},\theta_{\rm out}^{\prime}),(t_{\rm in}^{\prime},\theta_{\rm
in}^{\prime}))$ are $S$-families of positively (respectively, negatively)
oriented nearly constant superpaths in $M$ and that $\phi_{\rm out}=\phi_{\rm
in}^{\prime}\colon S\times{\mathbb{R}}^{0|1}\to M$. Then there is a well-
defined concatenation $\gamma*\gamma^{\prime}$ as a positively (respectively,
negatively) oriented nearly constant superpath in $M$ that is unique up to
unique isomorphism.

###### Proof.

It suffices to work in the universal case (77), for which the concatenation of
positively oriented superpaths is given by

$\displaystyle({\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{0|1},M))\times_{{\sf
Map}({\mathbb{R}}^{0|1},M)}({\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{0|1},M))$ (78) $\displaystyle\simeq{\mathbb{R}}^{1|1}_{\geq
0}\times{\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{0|1},M)\stackrel{{\scriptstyle m\times{{\rm
id}}}}{{\longrightarrow}}{\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{0|1},M)$

where $m$ is the restriction of multiplication on ${\mathbb{R}}^{1|1}$. The
concatenation map in the negatively oriented case is completely analogous. ∎

Super Lie categories are reviewed in Definition B.9.

######  Definition 3.17.

The super Lie category of _nearly constant superpaths in $M$_, denoted ${\sf
sPath}_{0}(M)$, has objects and morphisms,

${\rm Ob}({\sf sPath}_{0}(M))={\sf Map}({\mathbb{R}}^{0|1},M)\coprod{\sf
Map}({\mathbb{R}}^{0|1},M),$ ${\rm Mor}({\sf
sPath}_{0}(M))={\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{0|1},M)\coprod{\mathbb{R}}_{\leq 0}^{1|1}\times{\sf
Map}({\mathbb{R}}^{0|1},M),$

where source, target, and unit data are given by the incoming and outgoing
boundaries of a superpath and the identity superpath (using Lemma 3.15) while
composition is the concatenation of superpaths from Lemma 3.16.

The above super Lie category can be understood as a subcategory of the
following super Lie groupoid.

######  Lemma 3.18.

There is an essentially surjective and faithful (but not full) functor

(79) $\displaystyle{\sf sPath}_{0}(M)\hookrightarrow\left({\sf
Map}({\mathbb{R}}^{0|1},M)\coprod{\sf
Map}({\mathbb{R}}^{0|1},M)\right)/\\!\\!/{\mathbb{R}}^{1|1},$

to the quotient super Lie groupoid for the ${\mathbb{R}}^{1|1}$-action
factoring through the homomorphism ${\mathbb{R}}^{1|1}\to{\mathbb{R}}^{0|1}$
(see (81)), with the functor (79) determined by the standard inclusions
${\mathbb{R}}^{1|1}_{\geq 0},{\mathbb{R}}^{1|1}_{\leq
0}\subset{\mathbb{R}}^{1|1}$.

###### Proof.

By definition, the source map on an $S$-point of ${\sf sPath}_{0}(M)$ is

$(\phi,(t_{\rm out},\theta_{\rm out}),(t_{\rm in},\theta_{\rm
in}))\mapsto\phi_{\rm in},\qquad\phi_{\rm in}\colon
S\times{\mathbb{R}}^{0|1}\xhookrightarrow{i_{0}}S\times{\mathbb{R}}^{1|1}\xrightarrow{\phi}M,$

with $\phi$ as in (76). Since $q\circ i_{0}={{\rm
id}}_{S\times{\mathbb{R}}^{0|1}}$, we find $\phi_{\rm in}=\phi_{0}$ in the
notation of (76). The source map is therefore the projection ${\sf
s}=p\colon{\rm Mor}({\sf sPath}_{0}(M))\to{\rm Ob}({\sf sPath}_{0}(M))$. The
target map on an $S$-point of ${\sf sPath}_{0}(M)$ is

(80) $\displaystyle(\phi,(t_{\rm out},\theta_{\rm out}),(t_{\rm
in},\theta_{\rm in}))\mapsto\phi_{\rm out},\qquad\phi_{\rm out}\colon
S\times{\mathbb{R}}^{0|1}\xhookrightarrow{T_{t,\theta}\circ
i_{0}}S\times{\mathbb{R}}^{1|1}\xrightarrow{\phi}M.$

Again using that $q\circ i_{0}={{\rm id}}_{S\times{\mathbb{R}}^{0|1}}$, the
target map is the restriction of the (left) action

(81) $\displaystyle{\rm act}\colon{\mathbb{R}}^{1|1}\times{\sf
Map}({\mathbb{R}}^{0|1},M)\to{\sf Map}({\mathbb{R}}^{0|1},M)$

coming from the precomposition with the ${\mathbb{R}}^{1|1}$-action on
${\mathbb{R}}^{0|1}$,

${\mathbb{R}}^{1|1}\times{\mathbb{R}}^{0|1}\to{\mathbb{R}}^{0|1},\qquad(t,\theta)\cdot\eta=(\theta+\eta),\quad(t,\theta)\in{\mathbb{R}}^{1|1}(S),\
\eta\in{\mathbb{R}}^{0|1}(S).$

The lemma is proved. ∎

We recall the isomorphism of supermanifolds ${\sf
Map}({\mathbb{R}}^{0|1},M)\simeq\Pi TM$ leading to the isomorphism between
functions and differential forms

(82) $\displaystyle C^{\infty}({\sf Map}({\mathbb{R}}^{0|1},M))\simeq
C^{\infty}(\Pi TM_{\mathbb{C}})\simeq\Omega^{\bullet}(M).$

Together with Lemma 3.18, this will allow us to give explicit formulas for the
structure maps in the super Lie category ${\sf sPath}_{0}(M)$, presented in
terms of differential forms on $M$.

######  Remark 3.19.

Using (82), functions on the quotient groupoid ${\sf
Map}({\mathbb{R}}^{0|1},M)/\\!\\!/{\mathbb{R}}^{0|1}$ give a supergeometric
interpretation of the de Rham complex of $M$, e.g., see [HKST11, §3]. Lemma
3.18 shows that the category of superpaths is closely related to this
groupoid.

The isomorphism (82) implies

(85) $\displaystyle\begin{array}[]{c}C^{\infty}({\mathbb{R}}^{1|1}_{\geq
0}\times{\sf
Map}({\mathbb{R}}^{0|1},M))\simeq\Omega^{\bullet}(M;C^{\infty}({\mathbb{R}}^{1|1}_{\geq
0})),\\\ C^{\infty}({\mathbb{R}}^{1|1}_{\leq 0}\times{\sf
Map}({\mathbb{R}}^{0|1},M))\simeq\Omega^{\bullet}(M;C^{\infty}({\mathbb{R}}^{1|1}_{\leq
0})).\end{array}$

######  Lemma 3.20.

Using (82) and (85), the source and target maps ${\sf s},{\sf t}\colon{\rm
Mor}({\sf sPath}_{0}(M))\rightrightarrows{\rm Ob}({\sf sPath}_{0}(M))$ are
determined by the super algebra maps

$\Omega^{\bullet}(M)\oplus\Omega^{\bullet}(M)\to\Omega^{\bullet}(M;C^{\infty}({\mathbb{R}}_{\geq
0}^{1|1}))\oplus\Omega^{\bullet}(M;C^{\infty}({\mathbb{R}}_{\leq 0}^{1|1}))$
${\sf s}^{*}(\alpha,\beta)=(\alpha,\beta),\qquad{\sf
t}^{*}(\alpha,\beta)=(\alpha-\theta d\alpha,\beta-\theta
d\beta),\quad\alpha,\beta\in\Omega^{\bullet}(M),$

where we use the canonical inclusions

$\Omega^{\bullet}(M)\hookrightarrow\Omega^{\bullet}(M;C^{\infty}({\mathbb{R}}_{\geq
0}^{1|1})),\qquad\Omega^{\bullet}(M)\hookrightarrow\Omega^{\bullet}(M;C^{\infty}({\mathbb{R}}_{\leq
0}^{1|1}))$

along the constant functions on ${\mathbb{R}}^{1|1}_{\geq 0}$ or
${\mathbb{R}}^{1|1}_{\leq 0}$

###### Proof.

Standard results (e.g., [HKST11, Lemma 3.8]) compute the
${\mathbb{R}}^{0|1}$-action on ${\sf Map}({\mathbb{R}}^{0|1},M)$ in terms of a
map on differential forms:

$\displaystyle\Omega^{\bullet}(M)\simeq C^{\infty}({\sf
Map}({\mathbb{R}}^{0|1},M))$ $\displaystyle\to$ $\displaystyle
C^{\infty}({\mathbb{R}}^{0|1}\times{\sf
Map}({\mathbb{R}}^{0|1},M))\simeq\Omega^{\bullet}(M;C^{\infty}({\mathbb{R}}^{0|1}))$
(86) $\displaystyle\alpha$ $\displaystyle\mapsto$ $\displaystyle\alpha-\theta
d\alpha,\quad\alpha\in\Omega^{\bullet}(M),\
C^{\infty}({\mathbb{R}}^{0|1})\simeq C^{\infty}({\mathbb{R}})[\theta]$

where $d$ is the de Rham differential on $\Omega^{\bullet}(M)$. From Lemma
3.18, the source and target maps in ${\sf sPath}_{0}(M)$ are therefore
determined by restrictions of (86). ∎

######  Remark 3.21.

The formula (86) differs from [HKST11, Lemma 3.8] by a sign; their formula is
for the right action of ${\mathbb{R}}^{0|1}$ on $\Pi TM_{\mathbb{C}}$, whereas
(86) is for the left action.

###  3.5. Automorphisms of the category of superpaths

######  Lemma 3.22.

The maps (68), (69), (70) and (71) determine smooth functors

(87) $\displaystyle{\sf Or}\colon{\sf sPath}_{0}(M)\to{\sf sPath}_{0}(M),$
$\displaystyle(t,\theta,\phi)\mapsto(-t,-i\theta,\phi\circ{\sf or}^{-1})$ (88)
$\displaystyle{\sf RG}\colon{\mathbb{R}}_{>0}\times{\sf sPath}_{0}(M)\to{\sf
sPath}_{0}(M),$
$\displaystyle(\mu,t,\theta,\phi)\mapsto(\mu^{2}t,\mu\theta,\phi\circ{\sf
rg}_{\mu}^{-1})$ (89) $\displaystyle{\sf Fl}\colon{\sf sPath}_{0}(M)\to{\sf
sPath}_{0}(M),$ $\displaystyle(t,\theta,\phi)\mapsto(t,-\theta,\phi\circ{\sf
fl}^{-1})$ (90) $\displaystyle{\sf R}\colon{\sf sPath}_{0}(M)\to\overline{{\sf
sPath}_{0}(M)},$
$\displaystyle(t,\theta,\phi)\mapsto(\overline{t},\overline{\theta},\overline{\phi}\circ{\sf
r})$

given by the indicated maps on $S$-points of morphisms. Furthermore, there are
equalities of smooth functors

(91) $\displaystyle{\sf Or}\circ{\sf Or}={\sf Fl},\qquad{\sf R}\circ{\sf
Or}=\overline{{\sf Fl}}\circ\overline{{\sf Or}}\circ{\sf R}$

and ${\sf RG}$ gives the structure of a strict ${\mathbb{R}}_{>0}$-action on
${\sf sPath}_{0}(M)$ in the sense that the diagram of super Lie categories
strictly commutes

(92)

where ${\mathbb{R}}_{>0}$ is regarded as a discrete super Lie category and
$m\colon{\mathbb{R}}_{>0}\times{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$ is
multiplication.

###### Proof.

The formulas (87)-(89) follow from Example 3.11. The equalities (91) and
commutativity (92) follow from (72). The real structure (90) is inherited from
the real structure on $1|1$-Euclidean supermanifolds in Example 3.11, together
with the canonical real structure on $M$, giving the $\bar{S}$-point of ${\sf
Map}({\mathbb{R}}^{0|1},M)$

$\overline{S}\times{\mathbb{R}}^{1|1}\xrightarrow{{{\rm
id}}_{\overline{S}}\times{\sf
r}}\overline{S\times{\mathbb{R}}^{1|1}}\xrightarrow{\overline{\phi}}\overline{M}\simeq
M$

determined by the image of $\phi\colon S\times{\mathbb{R}}^{0|1}\to M$ under
the conjugation functor on supermanifolds. ∎

We use the notation

${\sf RG}_{\mu}\colon{\sf sPath}_{0}(M)\to{\sf sPath}_{0}(M)$

to denote the restriction of the functor ${\sf RG}$ along
$\mu\in{\mathbb{R}}_{>0}$.

######  Corollary 3.23.

The functors (87)-(89) are determined by the maps on morphisms induced by the
super algebra maps

$\Omega^{\bullet}(M;C^{\infty}({\mathbb{R}}^{1|1}_{\geq
0}))\oplus\Omega^{\bullet}(M;C^{\infty}({\mathbb{R}}^{1|1}_{\leq
0}))\to\Omega^{\bullet}(M;C^{\infty}({\mathbb{R}}^{1|1}_{\geq
0}))\oplus\Omega^{\bullet}(M;C^{\infty}({\mathbb{R}}^{1|1}_{\leq 0}))$ (93)
$\displaystyle{\sf Or}^{*}(\alpha(t,\theta),\beta(t,\theta))$ $\displaystyle=$
$\displaystyle(i^{|\beta|}\beta(-t,-i\theta),i^{|\alpha|}\alpha(-t,-i\theta)),\quad
i=\sqrt{-1}$ (94) $\displaystyle{\sf
RG}_{\mu}^{*}(\alpha(t,\theta),\beta(t,\theta))$ $\displaystyle=$
$\displaystyle(\mu^{|\alpha|}\alpha(\mu^{2}t,\mu\theta),\mu^{|\beta|}\beta(\mu^{2}t,\mu\theta)$
(95) $\displaystyle{\sf Fl}^{*}(\alpha(t,\theta),\beta(t,\theta))$
$\displaystyle=$
$\displaystyle((-1)^{|\alpha|}\alpha(t,-\theta),(-1)^{|\beta|}\beta(t,-\theta))$

for $\alpha,\beta\in\Omega^{\bullet}(M;C^{\infty}({\mathbb{R}}^{1|1}_{\geq
0}))$, where $|\alpha|,|\beta|\in{\mathbb{Z}}$ denote the
${\mathbb{Z}}$-grading of $\alpha$ and $\beta$.

###### Proof.

This follows from (87)-(89) and standard results (e.g., [HKST11, Lemma 3.8])
that compute the ${\mathbb{R}}^{0|1}\rtimes{\mathbb{C}}^{\times}$-action on
${\sf Map}({\mathbb{R}}^{0|1},M)$. The above statements follow by specializing
to the subgroups ${\mathbb{Z}}/4$, ${\mathbb{R}}_{>0}$ and ${\mathbb{Z}}/2$ of
${\mathbb{C}}^{\times}$. ∎

###  3.6. Superpaths as super loops

######  Definition 3.24.

Given an $S$-point $(\ell,\lambda)\in{\mathbb{R}}^{1|1}_{>0}(S)$, define the
family of $1|1$-dimensional _super Euclidean circles_ as the quotient

(96) $\displaystyle
S^{1|1}_{\ell,\lambda}:=(S\times{\mathbb{R}}^{1|1})/{\mathbb{Z}}$

for the left ${\mathbb{Z}}$-action on $S\times{\mathbb{R}}^{1|1}$ given by

(97) $\displaystyle
n\cdot(t,\theta)=(t+n\ell+n\lambda\theta,n\lambda+\theta),\quad
n\in{\mathbb{Z}}(S),(t,\theta)\in{\mathbb{R}}^{1|1}(S).$

Let $S^{1|1}={\mathbb{R}}^{1|1}/{\mathbb{Z}}$ denote the quotient by the
standard inclusion ${\mathbb{Z}}\subset{\mathbb{R}}\subset{\mathbb{R}}^{1|1}$.
For any $(\ell,\lambda)$ there is an isomorphism

(98) $\displaystyle S^{1|1}_{\ell,\lambda}\xrightarrow{\sim}S\times S^{1|1}$

determined by the ${\mathbb{Z}}$-equivariant map

(99) $\displaystyle S\times{\mathbb{R}}^{1|1}$ $\displaystyle\to$
$\displaystyle S\times{\mathbb{R}}^{1|1}$
$\displaystyle(\ell,\lambda,t,\theta)$ $\displaystyle\mapsto$
$\displaystyle(\ell,\lambda,t(\ell+\lambda\theta),\theta+t\lambda),\qquad(\ell,\lambda)\in{\mathbb{R}}^{1|1}_{>0}(S),\
(t,\theta)\in{\mathbb{R}}^{1|1}(S).$

######  Remark 3.25.

The $S$-family of subgroups $S\times{\mathbb{Z}}\hookrightarrow
S\times{\mathbb{R}}^{1|1}$ generated by
$(\ell,\lambda)\in{\mathbb{R}}^{1|1}_{>0}(S)$ is normal if and only if
$\lambda=0$. Hence, the standard super circle
$S^{1|1}={\mathbb{R}}^{1|1}/{\mathbb{Z}}$ inherits a group structure from
${\mathbb{R}}^{1|1}$, but a generic $S$-family of super Euclidean circles
$S^{1|1}_{\ell,\lambda}$ does not.

######  Definition 3.26.

Define the _super Euclidean loop space_ as the generalized supermanifold (see
Example A.16)

$\mathcal{L}^{1|1}(M):={\mathbb{R}}^{1|1}_{>0}\times{\sf Map}(S^{1|1},M).$

We identify an $S$-point of $\mathcal{L}^{1|1}(M)$ with a map
$S^{1|1}_{\ell,\lambda}\to M$ given by the composition

(100) $\displaystyle S^{1|1}_{\ell,\lambda}\simeq S\times S^{1|1}\to M$

using the isomorphism (98). There is a left action of
${\mathbb{R}}^{1|1}\rtimes{\mathbb{Z}}/2$ on $\mathcal{L}^{1|1}(M)$ determined
by the diagram

(101)

where the horizontal arrows are the isomorphism (98) and the super Euclidean
isometry $f$ is inherited from the action of
${\mathbb{R}}^{1|1}\rtimes{\mathbb{Z}}/2$ on $S\times{\mathbb{R}}^{1|1}$. The
arrow $\phi^{\prime}$ is uniquely determined by these isomorphisms and the
input map $\phi$. Hence, given
$(\ell,\lambda,\phi)\in{\mathbb{R}}^{1|1}_{>0}(S)\times{\sf
Map}(S^{1|1},M)(S)$ and an $S$-point of
${\mathbb{R}}^{1|1}\rtimes{\mathbb{Z}}/2$, the action on
$\mathcal{L}^{1|1}(M)$ outputs
$(\ell^{\prime},\lambda^{\prime},\phi^{\prime})$ as in (101).

There is an evident $S^{1}$-action on $\mathcal{L}^{1|1}(M)$ coming from the
precomposition action of
$S^{1}={\mathbb{R}}/{\mathbb{Z}}<{\mathbb{R}}^{1|1}/{\mathbb{Z}}$ on ${\sf
Map}(S^{1|1},M)$. Since the quotient is given by
$S^{1|1}/S^{1}\simeq{\mathbb{R}}^{0|1}$, the $S^{1}$-fixed points are

(102)
$\displaystyle\mathcal{L}^{1|1}_{0}(M):={\mathbb{R}}^{1|1}_{>0}\times{\sf
Map}({\mathbb{R}}^{0|1},M)\subset{\mathbb{R}}^{1|1}_{>0}\times{\sf
Map}(S^{1|1},M)=\mathcal{L}^{1|1}(M).$

We identify an $S$-point of $\mathcal{L}^{1|1}_{0}(M)$ with a map
$S^{1|1}_{\ell,\lambda}\to M$ that factors as

(103) $\displaystyle S^{1|1}_{\ell,\lambda}\simeq S\times
S^{1|1}=S\times{\mathbb{R}}^{1|1}/{\mathbb{Z}}\stackrel{{\scriptstyle
p}}{{\to}}S\times{\mathbb{R}}^{0|1}\to M$

where the map $p$ is induced by the projection
${\mathbb{R}}^{1|1}\to{\mathbb{R}}^{0|1}$. The
${\mathbb{R}}^{1|1}\rtimes{\mathbb{Z}}/2$-action in (101) preserves this
factorization condition, and is computed explicitly in [BE23a, §2.3].

The equalizer of the source and target maps in ${\sf sPath}_{0}(M)$ is

${\mathbb{R}}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{0|1},M)\coprod{\mathbb{R}}_{\leq 0}\times{\sf
Map}({\mathbb{R}}^{0|1},M)\subset{\rm Mor}({\sf
sPath}_{0}(M))\rightrightarrows{\rm Ob}({\sf sPath}_{0}(M)).$

Geometrically, this equalizer parameterizes constant superpaths with the same
source and target. We obtain nearly constant super loops by restricting to a
further subspace.

######  Lemma 3.27.

There is a span of inclusions

${\rm Mor}({\sf sPath}_{0}(M))\hookleftarrow{\mathbb{R}}_{>0}\times{\sf
Map}({\mathbb{R}}^{0|1},M)\coprod{\mathbb{R}}_{<0}\times{\sf
Map}({\mathbb{R}}^{0|1},M)\hookrightarrow\mathcal{L}_{0}^{1|1}(M)$

that views a constant superpath with the same source and target and nonzero
super length as a constant super loop.

##  4\. Super Euclidean paths as bordisms

The main goal of this section is to construct an internal functor

(104) $\displaystyle\iota\colon{\sf sPath}_{0}(M)\to 1|1\hbox{-}{\sf
EBord}(M)$

from the super Lie category of constant superpaths in $M$ to Stolz and
Teichner’s $1|1$-Euclidean bordism category over $M$. We further show that
(104) is compatible with additional structures from Lemma 3.22: the spin flip,
orientation reversal, renormalization group action, and real structure. The
arguments are all phrased within the framework of [ST11] (see §B for a
review). However, (104) is a totally geometric construction that we expect to
hold in any reasonable definition of the $1|1$-Euclidean bordism category.

###  4.1. The $1|1$-Euclidean bordism category

######  Definition 4.1.

Let $1|1\hbox{-}{\sf EBord}(M)$ denote the geometric bordism category
associated with the super Euclidean geometry from Example 3.3. We call
$1|1\hbox{-}{\sf EBord}(M)$ the _$1|1$ -Euclidean bordism category over $M$_.

We refer to §B.3 for Stolz and Teichner’s definition of a geometric bordism
category associated to a rigid geometry. To briefly summarize,
$1|1\hbox{-}{\sf EBord}(M)$ is a category internal to the category of
symmetric monoidal stacks with involution determined by the spin flip (see
Definition 3.8 and Example B.26). Hence, part of the data of $1|1\hbox{-}{\sf
EBord}(M)$ are maps of symmetric monoidal stacks with involution

${\sf s},{\sf t}\colon{\rm Mor}(1|1\hbox{-}{\sf
EBord}(M))\rightrightarrows{\rm Ob}(1|1\hbox{-}{\sf EBord}(M)),\quad{\sf
u}\colon{\rm Ob}(1|1\hbox{-}{\sf EBord}(M))\to{\rm Mor}(1|1\hbox{-}{\sf
EBord}(M))$ ${\sf c}\colon{\rm Mor}(1|1\hbox{-}{\sf EBord}(M))^{[2]}\to{\rm
Mor}(1|1\hbox{-}{\sf EBord}(M))$

giving the source, target, unit and composition maps, where we use the
notation333By Lemma B.16, the strict fibered product agrees with the weak
fibered product.

${\rm Mor}(1|1\hbox{-}{\sf EBord}(M))^{[2]}={\rm Mor}(1|1\hbox{-}{\sf
EBord}(M))\times_{{\rm Ob}(1|1\hbox{-}{\sf EBord}(M))}{\rm
Mor}(1|1\hbox{-}{\sf EBord}(M)).$

for the fibered product relative to the source and target maps, i.e., ${\rm
Mor}(1|1\hbox{-}{\sf EBord}(M))^{[2]}$ is the stack that classifies pairs of
composable bordisms. The full definition of $1|1\hbox{-}{\sf EBord}(M)$
includes further associator and unitor data satisfying certain coherence
properties; see §B.2.

Structures in the $1|1$-Euclidean geometry discussed in §3.2 equip the
internal category $1|1\hbox{-}{\sf EBord}(M)$ with certain endofunctors.

######  Lemma 4.2.

The internal category $1|1\hbox{-}{\sf EBord}(M)$ admits internal
endofunctors:

  1. (1)

orientation reversal, ${\sf Or}\colon 1|1\hbox{-}{\sf EBord}(M)\to
1|1\hbox{-}{\sf EBord}(M)$,

  2. (2)

the renormalization group action, ${\sf RG}_{\mu}\colon 1|1\hbox{-}{\sf
EBord}(M)\to 1|1\hbox{-}{\sf EBord}(M)$, and

  3. (3)

a real structure, ${\sf R}\colon 1|1\hbox{-}{\sf EBord}(M)\to 1|1\hbox{-}{\sf
EBord}(M)$,

where ${\sf Or}$, and ${\sf RG}_{\mu}$ are maps of categories internal to
symmetric monoidal stacks with involution, whereas the functor ${\sf R}$ is an
internal functor that covers complex conjugation on supermanifolds.

###### Proof.

From Definition 3.5 and Example B.28, we obtain an orientation reversal
functor

(105) $\displaystyle{\sf Or}\colon 1|1\hbox{-}{\sf EBord}(M)\to
1|1\hbox{-}{\sf EBord}(M).$

Using Definition 3.7 and Example B.33, we obtain an internal functor

(106) $\displaystyle{\sf RG}_{\mu}\colon 1|1\hbox{-}{\sf EBord}(M)\to
1|1\hbox{-}{\sf EBord}(M).$

Finally, Definition 3.10 determines an internal functor (see Example B.29)

(107) $\displaystyle{\sf R}\colon 1|1\hbox{-}{\sf EBord}(M)\to 1|1\hbox{-}{\sf
EBord}(M)$

whose values are given by

$(S\leftarrow Y\to M)\stackrel{{\scriptstyle{\sf
R}_{0}}}{{\mapsto}}(\overline{S}\leftarrow\overline{Y}\to\overline{M}\simeq
M)\qquad(S\leftarrow\Sigma\to M)\stackrel{{\scriptstyle{\sf
R}_{1}}}{{\mapsto}}(\overline{S}\leftarrow\overline{\Sigma}\to\overline{M}\simeq
M)$

where $\overline{Y}\to\overline{S}$ and $\overline{\Sigma}\to\overline{S}$
have the canonical structure of an $\overline{S}$-family of super Euclidean
manifolds. We emphasize that the family parameter changes under the functor
${\sf R}$ to the conjugate supermanifold. ∎

A map of smooth manifolds $M\to N$ determines an internal functor

$1|1\hbox{-}{\sf EBord}(M)\to 1|1\hbox{-}{\sf EBord}(N)$

see Lemma B.17. In particular, the $1|1$-Euclidean bordism category over $M$
is always an internal category over $1|1\hbox{-}{\sf EBord}({\rm pt})$, the
$1|1$-Euclidean bordism category over a point.

###  4.2. Families of bordisms in $1|1\hbox{-}{\sf EBord}(M)$

Definition B.15 reviews the construction of the symmetric monoidal stacks with
involution ${\rm Ob}(1|1\hbox{-}{\sf EBord}(M))$ and ${\rm
Mor}(1|1\hbox{-}{\sf EBord}(M))$. In this subsection, we use the same notation
for the underlying stack, i.e., forgetting the monoidal structure and spin
flip. Throughout, the family $S\times{\mathbb{R}}^{1|1}\to S$ will always have
the canonical structure of a $1|1$-Euclidean family. When $M={\rm pt}$, the
definitions below specialize to the pictures (48), and we use analogous
notation to impart this intuition.

######  Definition 4.3 (Compare [HST10, Remark 6.40]).

Define maps of stacks

(108) $\displaystyle{\rm spt}^{+}_{M},{\rm spt}^{-}_{M}\colon{\sf
Map}({\mathbb{R}}^{1|1},M)\to{\rm Ob}(1|1\hbox{-}{\sf EBord}(M))$

as follows. Given an $S$-point $\phi\colon S\times{\mathbb{R}}^{1|1}\to M$ of
the source, define objects $({\rm spt}^{+},\phi),({\rm spt}^{-},\phi)$ in the
groupoid ${\rm Ob}(1|1\hbox{-}{\sf EBord}(M))(S)$ by

(113) $\displaystyle\begin{array}[]{c}Y^{c}=S\times{\mathbb{R}}^{0|1}\subset
S\times{\mathbb{R}}^{1|1}=Y\stackrel{{\scriptstyle\phi}}{{\to}}M,\\\
Y\setminus Y^{c}\simeq S\times{\mathbb{R}}^{1|1}_{>0}\coprod
S\times{\mathbb{R}}^{1|1}_{<0}=\left\\{\begin{array}[]{ll}Y^{+}\coprod
Y^{-}&({\rm spt}^{+},\phi)\\\ Y^{-}\coprod Y^{+}&({\rm
spt}^{-},\phi)\end{array}\right.\end{array}$

where $Y^{c}\hookrightarrow Y$ (the inclusion of the _core_) is determined by
the standard inclusion
$0\colon{\mathbb{R}}^{0|1}\hookrightarrow{\mathbb{R}}^{1|1}$. As indicated in
the second line, the $+$ or $-$ decoration on ${\rm spt}^{\pm}_{M}$ reflects
the choice of the partition of the collar. By pulling back families, the above
is natural in $S$ and hence leads to the claimed maps (108).

######  Remark 4.4.

A pair of $S$-points of ${\sf Map}({\mathbb{R}}^{1|1},M)$ determine isomorphic
objects under ${\rm spt}^{\pm}_{M}$ if the the maps
$S\times{\mathbb{R}}^{1|1}\to M$ agree on a half-open neighborhood of the core
$S\times{\mathbb{R}}^{0|1}\hookrightarrow S\times{\mathbb{R}}^{1|1}$; see
Definition B.15. Hence, the map of stacks (108) factors through the quotient
of ${\sf Map}({\mathbb{R}}^{1|1},M)$ that parameterizes half-open germs of
superpaths in $M$.

The following gives a map from super Euclidean paths in $M$ to bordisms.

######  Definition 4.5 (Compare [HST10, Definition 6.41]).

Define maps of stacks

(116) $\displaystyle\begin{array}[]{c}{\sf
I}^{+}_{M}\colon{\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)\to{\rm Mor}(1|1\hbox{-}{\sf EBord}(M))\\\ {\sf
I}^{-}_{M}\colon{\mathbb{R}}^{1|1}_{\leq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)\to{\rm Mor}(1|1\hbox{-}{\sf EBord}(M))\end{array}$

as follows. Given $\Phi\colon S\times{\mathbb{R}}^{1|1}\to M$ and
$(t,\theta)\colon S\to{\mathbb{R}}^{1|1}$, consider

(117) $\displaystyle S\times{\mathbb{R}}^{0|1}\coprod
S\times{\mathbb{R}}^{0|1}\xrightarrow{i_{0}\coprod
i_{t,\theta}}S\times{\mathbb{R}}^{1|1}=\Sigma\xrightarrow{\Phi}M,$

using the notation from (66). To promote (117) to an object in the groupoid
${\rm Mor}(1|1\hbox{-}{\sf EBord}(M))(S)$, we require a choice of source and
target data. Take

(120) $\displaystyle(Y_{{\rm in}}^{c},Y_{{\rm
out}}^{c})=\left\\{\begin{array}[]{lll}\Big{(}({\rm spt}^{+},\Phi),({\rm
spt}^{+},\Phi\circ
T_{t,\theta}^{-1})\Big{)}&\implies(I_{t,\theta}^{+},\Phi),&(t,\theta)\in{\mathbb{R}}^{1|1}_{\geq
0}(S)\\\ \Big{(}({\rm spt}^{-},\Phi),({\rm spt}^{-},\phi\circ
T_{t,\theta}^{-1})\Big{)}&\implies(I_{t,\theta}^{-},\Phi)&(t,\theta)\in{\mathbb{R}}^{1|1}_{\leq
0}(S)\end{array}\right.$

i.e., the ordering in the coproduct in (117) lists the source followed by the
target $1|1$-Euclidean pair. The resulting options and the restriction on
$(t,\theta)$ in (120) follow from requiring the existence of collars for the
source and target. Any pair of choices of collar provide isomorphic functors
(116); see Remark 4.8.

######  Remark 4.6.

In parallel to Remark 4.4, $S$-points of ${\mathbb{R}}^{1|1}_{\geq
0}\times{\sf Map}({\mathbb{R}}^{1|1},M)$ or ${\mathbb{R}}^{1|1}_{\leq
0}\times{\sf Map}({\mathbb{R}}^{1|1},M)$ determine isomorphic objects in ${\rm
Mor}(1|1\hbox{-}{\sf EBord}(M))$ under ${\sf I}^{\pm}_{M}$ if the associated
maps $S\times{\mathbb{R}}^{1|1}\to M$ agree on a neighborhood of the core of
the bordism, and so the family pulls back from the associated quotient.

Changing the source and target data give variations on (116).

######  Definition 4.7 (Compare [HST10, Definition 6.41]).

Define maps of stacks

(125) $\displaystyle\begin{array}[]{c}{\sf
L}^{+}_{M}\colon{\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)\to{\rm Mor}(1|1\hbox{-}{\sf EBord}(M))\\\ {\sf
L}^{-}_{M}\colon{\mathbb{R}}^{1|1}_{\leq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)\to{\rm Mor}(1|1\hbox{-}{\sf EBord}(M)),\\\ {\sf
R}^{+}_{M}\colon{\mathbb{R}}^{1|1}_{>0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)\to{\rm Mor}(1|1\hbox{-}{\sf EBord}(M))\\\ {\sf
R}^{-}_{M}\colon{\mathbb{R}}^{1|1}_{<0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)\to{\rm Mor}(1|1\hbox{-}{\sf EBord}(M))\end{array}$

as follows. Again for $\Phi\colon S\times{\mathbb{R}}^{1|1}\to M$ and
$(t,\theta)\colon S\to{\mathbb{R}}^{1|1}$ consider (117), but choose source
and target data

(130) $\displaystyle(Y_{{\rm in}}^{c},Y_{{\rm
out}}^{c})=\left\\{\begin{array}[]{lll}\Big{(}\big{(}({\rm
spt}^{+},\Phi),({\rm spt}^{-},\Phi\circ
T^{-1}_{t,\theta})\big{)},\emptyset\Big{)}&\implies(L_{t,\theta}^{+},\Phi)&(t,\theta)\in{\mathbb{R}}^{1|1}_{\geq
0}(S)\\\ \Big{(}\big{(}({\rm spt}^{-},\Phi),({\rm spt}^{+},\Phi\circ
T^{-1}_{t,\theta})\big{)},\emptyset\Big{)}&\implies(L_{t,\theta}^{-},\Phi)&(t,\theta)\in{\mathbb{R}}^{1|1}_{\leq
0}(S)\\\ \Big{(}\emptyset,\big{(}({\rm spt}^{-},\Phi),({\rm spt}^{+},\Phi\circ
T^{-1}_{t,\theta})\big{)}\Big{)}&\implies(R_{t,\theta}^{+},\Phi)&(t,\theta)\in{\mathbb{R}}^{1|1}_{>0}(S)\\\
\Big{(}\emptyset,\big{(}({\rm spt}^{+},\Phi),({\rm spt}^{-},\Phi\circ
T^{-1}_{t,\theta})\big{)}\Big{)}&\implies(R_{t,\theta}^{-},\Phi)&(t,\theta)\in{\mathbb{R}}^{1|1}_{<0}(S)\end{array}\right.$

where the restriction on $(t,\theta)$ is necessary for the existence of
collars for the source and target. Any pair of choices provide isomorphic
functors (125); see Remark 4.8.

######  Remark 4.8.

The requirements on $(t,\theta)\in{\mathbb{R}}^{1|1}(S)$ in (120) and (130)
come from demanding the existence of the collars $W_{{\rm in}/{\rm
out}}^{\pm}$ with properties demanded by Definition B.15; any pair of choices
of $W_{{\rm in}/{\rm out}}^{\pm}$ lead to isomorphic objects in ${\rm
Mor}(1|1\hbox{-}{\sf EBord}(M))$ by intersecting the choices of collar. We
emphasize that the families $(R_{t,\theta}^{\pm},\Phi)$ in (120) do not have a
limit in $1|1\hbox{-}{\sf EBord}(M)$ as $(t,\theta)\to 0$; compare [HST10,
Theorem 6.23].

######  Lemma 4.9.

The restriction of the spin flip on ${\rm Ob}(1|1\hbox{-}{\sf EBord}(M))$ or
${\rm Mor}(1|1\hbox{-}{\sf EBord}(M))$ along ${\rm spt}^{\pm}_{M}$, ${\sf
I}^{\pm}_{M}$, ${\sf L}^{\pm}_{M}$ or ${\sf R}^{\pm}_{M}$ is determined by the
maps

$\displaystyle{\sf Map}({\mathbb{R}}^{1|1},M)$ $\displaystyle\to$
$\displaystyle{\sf Map}({\mathbb{R}}^{1|1},M),\quad\phi\mapsto\phi\circ{\sf
fl}^{-1}$ $\displaystyle{\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)$ $\displaystyle\to$
$\displaystyle{\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M),\quad(t,\theta,\phi)\mapsto(t,-\theta,\phi\circ{\sf
fl}^{-1})$ $\displaystyle{\mathbb{R}}^{1|1}_{\leq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)$ $\displaystyle\to$
$\displaystyle{\mathbb{R}}^{1|1}_{\leq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M),\quad(t,\theta,\phi)\mapsto(t,-\theta,\phi\circ{\sf
fl}^{-1})$

###### Proof.

This follows from Definition 3.8 of the spin flip. ∎

###  4.3. Composition of bordisms

The source and target of the families of bordisms ${\sf I}^{\pm}_{M}$, ${\sf
L}^{\pm}_{M}$ and ${\sf R}^{\pm}_{M}$ are specified in their definitions. This
source and target data can equivalently be phrased in terms of 2-commuting
diagrams of stacks. For example, we have

(131)

where ${\rm proj}$ and ${\rm act}$ are the projection and action maps

$\displaystyle{\rm proj},\ {\rm act}\colon{\mathbb{R}}^{1|1}\times{\sf
Map}({\mathbb{R}}^{1|1},M)$ $\displaystyle\to$ $\displaystyle{\sf
Map}({\mathbb{R}}^{1|1},M)$ (132) $\displaystyle{\rm proj}(t,\theta,\Phi)$
$\displaystyle=$ $\displaystyle\Phi$ $\displaystyle{\rm act}(t,\theta,\Phi)$
$\displaystyle=$ $\displaystyle\Phi\circ T_{t,\theta}^{-1}.$

for $(t,\theta)\in{\mathbb{R}}^{1|1}(S)$ and $\Phi\colon
S\times{\mathbb{R}}^{1|1}\to M$, and ${\rm proj}_{+},{\rm act}_{+}$
(respectively, ${\rm proj}_{-},{\rm act}_{-}$) are the restrictions of ${\rm
proj},{\rm act}$ to ${\mathbb{R}}^{1|1}_{\geq 0}\subset{\mathbb{R}}^{1|1}$
(respectively, ${\mathbb{R}}^{1|1}_{\leq 0}\subset{\mathbb{R}}^{1|1}$). The
other structure maps in $1|1\hbox{-}{\sf EBord}(M)$ lead to similar
2-commuting diagrams.

######  Proposition 4.10.

Let ${\sf u}$ be the unit functor in $1|1\hbox{-}{\sf EBord}(M)$. There is a
2-commutative diagrams of stacks

(133)

where the left vertical arrow is inclusion along $0\in{\mathbb{R}}^{1|1}_{\geq
0}$. Let ${\sf c}$ be the composition functor. There is a 2-commutative
diagrams of stacks

(134)

where the top horizontal arrow is determined by the family ${\sf I}^{+}_{M}$
and the maps

$\displaystyle p_{i}\colon{\mathbb{R}}^{1|1}_{\geq
0}\times{\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)\to{\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M),$ (135) $\displaystyle
p_{1}\big{(}(s,\eta),(t,\theta),\Phi\big{)}=(t,\theta,\Phi),\qquad
p_{2}\big{(}(s,\eta),(t,\theta),\Phi\big{)}=(s,\eta,\Phi\circ
T^{-1}_{t,\theta})$

and ${\sf m}$ is inherited from multiplication in ${\mathbb{R}}^{1|1}$,

(136) $\displaystyle{\sf
m}\big{(}(s,\eta),(t,\theta),\Phi\big{)}=(s+t+\eta\theta,\eta+\theta,\Phi).$

There are analogous diagrams to (133) and (134) for the family of bordisms
classified by ${\sf I}^{-}_{M}$.

###### Proof.

The commutativity of the diagram (133) is almost immediate; the only subtlety
comes from the choice of collars in the construction of the family ${\sf
I}^{+}_{M}$. Indeed, rather than a strictly commutative diagram this choice
leads to 2-commutativity data gotten from the isomorphism between objects of
${\rm Mor}(1|1\hbox{-}{\sf EBord}(M))$ from shrinking the collar in (113).

Turning to the diagram (134), the definition of the $S$-family
$(I_{t,\theta}^{+},\Phi)$ of bordisms implies that the pair (135) is indeed in
the fibered product. The construction of 2-commutativity data comes from
unpacking the definition of the functor ${\sf c}$, i.e., composition of
bordisms. Consider the diagram

(137) $\displaystyle S\times{\mathbb{R}}^{0|1}\coprod
S\times{\mathbb{R}}^{0|1}\coprod
S\times{\mathbb{R}}^{0|1}\xhookrightarrow{i_{0}\coprod i_{t,\theta}\coprod
i_{(s,\eta)\cdot(t,\theta)}}S\times{\mathbb{R}}^{1|1}\xrightarrow{\Phi}M.$

By forgetting any of the three inclusions of $S\times{\mathbb{R}}^{0|1}$, one
obtains three possible $S$-points of the source of ${\sf I}^{+}_{M}$ via the
input datum (117): forgetting $i_{(s,\eta)\cdot(t,\theta)}$ gives the
$S$-family of bordisms $(I^{+}_{t,\theta},\Phi)$, forgetting $i_{t,\theta}$
gives the $S$-family of bordisms $(I^{+}_{(s,\eta)\cdot(t,\theta)},\Phi)$, and
forgetting $i_{0}$ gives a family that is isomorphic to
$(I^{+}_{s,\eta},\Phi\circ T^{-1}_{t,\theta})$ (where the isomorphism is
specified by $T_{t,\theta}$). These three $S$-families of bordisms gotten from
(137) coincide with the two projections out of the fibered product and the
value of the functor ${\sf c}$. The only subtle aspect in commutativity of the
diagram (134) is the choice of collars in the definitions of these families of
bordisms, i.e., the choices in the construction of the functor ${\sf
I}^{+}_{M}$. However, since any pair of choices lead to isomorphic bordisms
(by intersecting the collars), we obtain canonical 2-commutativity data in
(134).

The arguments involving ${\rm spt}^{-}_{M}$ and ${\sf I}^{-}_{M}$ are
identical. ∎

######  Remark 4.11.

In terms of $S$-families of bordisms, the top horizontal map in (134) is

$\big{(}(s,\eta),(t,\theta),\Phi\big{)}\mapsto\big{(}(I^{+}_{s,\eta},\Phi\circ
T^{-1}_{t,\theta}),(I^{+}_{t,\theta},\Phi)\big{)}.$

######  Notation 4.12.

Let ${\sf I}^{\pm}_{0}$ denote the composition ${\sf u}\circ{\rm
spt}^{\pm}_{M}$,

${\sf I}^{\pm}_{0}\colon{\sf Map}({\mathbb{R}}^{1|1},M)\to{\rm
Mor}(1|1\hbox{-}{\sf EBord}(M)).$

For an $S$-point $\Phi$ of the source above, let $(I_{0}^{\pm},\Phi)$ denote
the associated $S$-family of (identity) $1|1$-Euclidean bordisms.

The following generalizes the adjunction between super semigroups coming from
the last relation in (49).

######  Proposition 4.13.

There is a 2-commutative diagrams of stacks

(138)

where the top vertical arrow is determined by the families ${\sf L}^{+}_{M}$,
${\sf I}^{+}_{M}$ and ${\sf I}^{-}_{M}$ and the maps

$\displaystyle p_{i}\colon{\mathbb{R}}^{1|1}_{\geq
0}\times{\mathbb{R}}^{1|1}_{\geq 0}\times{\mathbb{R}}^{1|1}_{\leq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)\to{\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M),\quad i=1,2$ (139) $\displaystyle
p_{1}((t,\theta),(s,\eta),(r,\rho),\Phi)=(t,\theta,\Phi),\quad
p_{2}((t,\theta),(s,\eta),(r,\rho),\Phi)=(s,\eta,\Phi\circ T^{-1}_{t,\theta})$
$\displaystyle p_{3}\colon{\mathbb{R}}^{1|1}_{\geq
0}\times{\mathbb{R}}^{1|1}_{\geq 0}\times{\mathbb{R}}^{1|1}_{\leq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)\to{\mathbb{R}}^{1|1}_{\leq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)$ (140) $\displaystyle
p_{3}((t,\theta),(s,\eta),(r,\rho),\Phi)=(r,\rho,\Phi\circ
T^{-1}_{t,\theta}\circ T^{-1}_{s,\eta})$

and the left vertical arrow is determined by the multiplication in
${\mathbb{R}}^{1|1}$,

$((t,\theta),(s,\eta),(r,\rho),\Phi)\mapsto((r,\rho)^{-1}\cdot(s,\eta)\cdot(t,\theta),\Phi).$

There is a similar 2-commutative diagram for ${\sf L}^{-}_{M}$.

###### Proof.

From the definitions of source and target data, the top map in (138) does
indeed yield an object in the fibered product. The 2-commutativity data comes
from unpacking the definition of the functor ${\sf c}$. Similar to before,
consider the diagram

In this case, forgetting any pair of factors in the coproduct gives the input
data to an $S$-family of bordisms. For example, forgetting the middle two
factors corresponds to the $S$-family of bordisms
$(L^{+}_{(r,\rho)^{-1}\cdot(s,\eta)\cdot(t,\theta)},\Phi)$, which is the
claimed composition. The bordisms in (139) can similarly be extracted by
forgetting the appropriate pair of inclusions and applying an appropriate
isometry determined by a translation of $S\times{\mathbb{R}}^{1|1}$. ∎

######  Remark 4.14.

In terms of $S$-families of bordisms, the top horizontal map in (138) is

$((t,\theta),(s,\eta),(r,\rho),\Phi)\mapsto\Big{(}(L^{+}_{s,\eta},\Phi\circ
T^{-1}_{t,\theta}),\big{(}(I^{+}_{t,\theta},\Phi)\coprod(I^{-}_{r,\rho},\Phi\circ
T^{-1}_{t,\theta}\circ T^{-1}_{s,\eta})\big{)}\Big{)}.$

The following shows that the family of bordisms ${\sf I}^{\pm}_{M}$ factors as
a composition involving ${\sf L}^{\pm}_{M}$ and ${\sf R}^{\pm}_{M}$. The proof
uses similar techniques as above.

######  Proposition 4.15.

There is a 2-commutative diagrams of stacks

(141)

where the top vertical arrow is determined by the families ${\sf L}^{+}_{M}$,
${\sf R}^{+}_{M}$, ${\sf I}^{\pm}_{0}$, and the maps

$\displaystyle p_{i}\colon{\mathbb{R}}^{1|1}_{\geq
0}\times{\mathbb{R}}^{1|1}_{>0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)\to{\mathbb{R}}^{1|1}_{>0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)$ (142) $\displaystyle
p_{1}((t,\theta),(s,\eta),\Phi)=(t,\theta,\Phi),\quad
p_{2}((t,\theta),(s,\eta),\Phi)=(s,\eta,\Phi\circ T^{-1}_{t,\theta}),$
$\displaystyle q_{i}\colon\colon{\mathbb{R}}^{1|1}_{\geq
0}\times{\mathbb{R}}^{1|1}_{>0}\times{\sf Map}({\mathbb{R}}^{1|1},M)\to{\sf
Map}({\mathbb{R}}^{1|1},M)$ $\displaystyle\quad
q_{1}((t,\theta),(s,\eta),\Phi)=\Phi,\quad
q_{2}((t,\theta),(s,\eta),\Phi)=\Phi\circ T_{t,\theta}^{-1}\circ
T_{s,\eta}^{-1}$

and ${\sf m}$ is (136). There is a similar 2-commutative diagram factoring
${\sf I}^{-}_{M}$ as a composition.

######  Remark 4.16.

In terms of $S$-families of bordisms, the top horizontal arrow in (141) is

$((t,\theta),(s,\eta),\Phi)\mapsto\Big{(}(L^{+}_{t,\theta},\Phi)\coprod(I^{+}_{0},\Phi\circ
T^{-1}_{t,\theta}\circ
T^{-1}_{s,\eta})\Big{)},\Big{(}(I_{0}^{+},\Phi)\coprod(R_{s,\eta}^{+},\Phi\circ
T^{-1}_{t,\theta})\Big{)}.$

######  Proposition 4.17.

Let ${\sf L}^{\pm}_{M}\circ\sigma$ denote the functor ${\sf L}^{\pm}_{M}$
post-composed with the isomorphism in ${\rm Mor}(1|1\hbox{-}{\sf EBord}(M))$
that exchanges the order of components of the target, i.e.,

${\sf t}({\sf L}^{\pm}_{t,\theta}\circ\sigma,\Phi)=({\rm spt}^{\mp},\Phi\circ
T_{t,\theta}^{-1})\coprod({\rm spt}^{\pm},\Phi).$

There is a 2-commutative triangle

(143)

for the left vertical arrow

$\displaystyle{\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)$ $\displaystyle\to$
$\displaystyle{\mathbb{R}}^{1|1}_{\leq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M),$ (144) $\displaystyle(t,\theta,\phi)$
$\displaystyle\mapsto$ $\displaystyle(-t,-\theta,\phi\circ
T_{t,\theta}^{-1}).$

###### Proof.

Starting with the data that determines $(L_{t,\theta}^{+},\Phi)$,

$\displaystyle S\times{\mathbb{R}}^{0|1}\coprod
S\times{\mathbb{R}}^{0|1}\xhookrightarrow{i_{0}\coprod
i_{t,\theta}}S\times{\mathbb{R}}^{1|1}\xrightarrow{\Phi}M.$

exchanging the factors of the coproduct and acting on
$S\times{\mathbb{R}}^{1|1}$ by $T^{-1}_{t,\theta}=T_{(-t,-\theta)}$ provides
the data determining $(L_{-t,-\theta}^{-},\Phi\circ T_{t,\theta}^{-1})$. ∎

We recall the super Euclidean loop space $\mathcal{L}^{1|1}(M)$ from
Definition 3.26. By [BE23a, Lemma 2.12], there is a functor

$\mathcal{L}^{1|1}(M)\to{\rm Mor}(1|1\hbox{-}{\sf EBord}(M)),$

sending a super Euclidean loop in $M$ to a bordism over $M$ whose source and
target are the empty family of $1|1$-Euclidean manifolds.

######  Proposition 4.18.

There is 2-commutative diagrams of stacks

(145)

where the ${\mathbb{Z}}$-fixed subspace has $S$-points
$(t,\theta)\in{\mathbb{R}}^{1|1}_{\leq 0}(S)$,
$(s,\eta)\in{\mathbb{R}}^{1|1}_{>0}(S)$ and $\Phi\colon
S\times{\mathbb{R}}^{1|1}\to M$ satisfying

(146) $\displaystyle\Phi\circ T_{t,\theta}^{-1}=\Phi\circ
T_{s,\eta}^{-1}\iff\Phi=\Phi\circ(T_{(s,\eta)\cdot(t,\theta)^{-1}})^{n},\quad
n\in{\mathbb{Z}}$

the top vertical arrow is determined by the families ${\sf L}^{-}_{M}$ and
${\sf R}^{+}_{M}$ and the maps

(147) $\displaystyle p_{1}((t,\theta),(s,\eta),\Phi)=(t,\theta,\Phi),\qquad
p_{2}((t,\theta),(s,\eta),\Phi)=(s,\eta,\Phi)$

and the left vertical arrow is determined by

$((t,\theta),(s,\eta),\Phi)\mapsto((s,\eta)\cdot(t,\theta)^{-1},\Phi)\in\big{(}{\mathbb{R}}^{1|1}_{>0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)\big{)}(S)\supset\mathcal{L}^{1|1}(M)(S)$

which determines an $S$-point of $\mathcal{L}^{1|1}(M)$ by the condition
(146). There is a diagram analogous to (145) involving ${\sf L}^{-}_{M}$ and
${\sf R}^{+}_{M}$.

###### Proof.

The compatibility $\Phi\circ T_{t,\theta}^{-1}=\Phi\circ T_{s,\eta}^{-1}$ in
(146) guarantees that the target of $(R^{-}_{s,\eta},\Phi)$ agrees with the
source of $(L^{+}_{t,\theta},\Phi)$. Hence, the bordisms are indeed composable
and their composition is necessarily a bordism from the empty set to itself.
Unpacking the definition of the functor ${\sf c}$ and examining the
periodicity condition (146), one finds that this composition is indeed the
claimed super Euclidean loop in $M$ with circumference
$(\ell,\lambda)=(s,\eta)\cdot(t,\theta)^{-1}$. ∎

######  Remark 4.19.

In terms of $S$-families of bordisms, the top horizontal arrow in (145) is

$((t,\theta),(s,\eta),\Phi)\mapsto\Big{(}(L^{-}_{t,\theta},\Phi),(R^{+}_{s,\eta},\Phi)\Big{)}.$

###  4.4. Restriction of additional structures

Consider the restriction of extra structures from Lemma 4.2 to the families of
bordisms above.

######  Proposition 4.20.

The real structure on $1|1\hbox{-}{\sf EBord}(M)$ sends $S$-families of
bordisms to $\overline{S}$-families of bordisms specified by

$\displaystyle{\sf R}({\rm spt}^{\pm},\phi)$ $\displaystyle\simeq$
$\displaystyle({\rm spt}^{\pm},\overline{\phi}\circ{\sf r})$
$\displaystyle{\sf R}(I^{\pm}_{t,\theta},\Phi)$ $\displaystyle\simeq$
$\displaystyle(I^{\pm}_{\overline{t},\overline{\theta}},\overline{\Phi}\circ{\sf
r})$ $\displaystyle{\sf R}(L^{\pm}_{t,\theta},\Phi)$ $\displaystyle\simeq$
$\displaystyle(L^{\pm}_{\overline{t},\overline{\theta}},\overline{\Phi}\circ{\sf
r})$ $\displaystyle{\sf R}(R^{\pm}_{t,\theta},\Phi)$ $\displaystyle\simeq$
$\displaystyle(R^{\pm}_{\overline{t},\overline{\theta}},\overline{\Phi}\circ{\sf
r})$

where (in an abuse of notation) ${\sf
r}\colon\overline{S}\times{\mathbb{R}}^{1|1}\to\overline{S}\times\overline{{\mathbb{R}}}{}^{1|1}$
and ${\sf
r}\colon\overline{S}\times{\mathbb{R}}^{0|1}\to\overline{S}\times\overline{{\mathbb{R}}}{}^{0|1}$
are the maps determined by the real structure (71) on ${\mathbb{R}}^{1|1}$,
$(\overline{t},\overline{\theta})\colon\overline{S}\to\overline{{\mathbb{R}}}{}^{1|1}$
is value of the conjugation functor on the map $(t,\theta)\colon
S\to{\mathbb{R}}^{1|1}$, and
$\overline{\Phi}\colon\overline{S}\times\overline{{\mathbb{R}}}{}^{1|1}\to M$,
$\overline{\phi}\colon\overline{S}\times\overline{{\mathbb{R}}}{}^{0|1}\to M$
are the maps conjugate to $\Phi$ and $\phi$ that use the canonical real
structure $\overline{M}\simeq M$ on the ordinary manifold $M$.

###### Proof.

The real structure on $1|1\hbox{-}{\sf EBord}(M)$ evaluated on any of the
above families is determined by the diagram

where the middle row is the value of the conjugation functor on a bordism over
$M$, and the isomorphisms with the lower row use the real structure on the
$1|1$-Euclidean geometry. Together with the fact that the real structure
preserves the subspaces ${\mathbb{R}}^{1|1}_{\geq 0},{\mathbb{R}}^{1|1}_{\leq
0}\subset{\mathbb{R}}^{1|1}$, one can choose isomorphisms as indicated in the
proposition. These isomorphisms are unique up to the choice of collar data for
the source and target. ∎

######  Proposition 4.21.

The orientation reversal automorphism of $1|1\hbox{-}{\sf EBord}(M)$ takes
values

$\displaystyle{\sf Or}({\rm spt}^{\pm},\phi)$ $\displaystyle\simeq$
$\displaystyle({\rm spt}^{\mp},\phi\circ{\sf or}^{-1})$ $\displaystyle{\sf
Or}(I^{\pm}_{t,\theta},\Phi)$ $\displaystyle\simeq$
$\displaystyle(I^{\mp}_{-t,-i\theta},\Phi\circ{\sf or}^{-1})$
$\displaystyle{\sf Or}(L^{\pm}_{t,\theta},\Phi)$ $\displaystyle\simeq$
$\displaystyle(L^{\mp}_{-t,-i\theta},\Phi\circ{\sf or}^{-1})$
$\displaystyle{\sf Or}(R^{\pm}_{t,\theta},\Phi)$ $\displaystyle\simeq$
$\displaystyle(R^{\mp}_{-t,-i\theta},\Phi\circ{\sf or}^{-1})$

where (in an abuse of notation) ${\sf or}\colon S\times{\mathbb{R}}^{1|1}\to
S\times{\mathbb{R}}^{1|1}$ is determined by the map (68).

###### Proof.

The orientation reversal automorphism of $1|1\hbox{-}{\sf EBord}(M)$ evaluated
on any of the above families is determined by the diagram

where the inclusions in the lower row are the unique ones making the diagram
commute. Together with the fact that orientation reversal exchanges the
subspaces ${\mathbb{R}}^{1|1}_{\geq 0},{\mathbb{R}}^{1|1}_{\leq
0}\subset{\mathbb{R}}^{1|1}$, one can choose isomorphisms as in the statement.
These isomorphisms are unique up to the choice of collar data for the source
and target. ∎

######  Proposition 4.22.

For $\mu\in{\mathbb{R}}_{>0}$, the renormalization group automorphism of
$1|1\hbox{-}{\sf EBord}(M)$ takes values

$\displaystyle{\sf RG}_{\mu}({\rm spt}^{\pm},\phi)$ $\displaystyle\simeq$
$\displaystyle({\rm spt}^{\pm},\phi\circ{\sf rg}_{\mu}^{-1})$
$\displaystyle{\sf RG}_{\mu}(I^{\pm}_{t,\theta},\Phi)$ $\displaystyle\simeq$
$\displaystyle(I^{\pm}_{\mu^{2}t,\mu\theta},\Phi\circ{\sf rg}_{\mu}^{-1})$
$\displaystyle{\sf RG}_{\mu}(L^{\pm}_{t,\theta},\Phi)$ $\displaystyle\simeq$
$\displaystyle(L^{\pm}_{\mu^{2}t,\mu\theta},\Phi\circ{\sf rg}_{\mu}^{-1})$
$\displaystyle{\sf RG}_{\mu}(R^{\pm}_{t,\theta},\Phi)$ $\displaystyle\simeq$
$\displaystyle(R^{\pm}_{\mu^{2}t,\mu\theta},\Phi\circ{\sf rg}_{\mu}^{-1})$

where (in an abuse of notation) ${\sf rg}_{\mu}\colon
S\times{\mathbb{R}}^{1|1}\to S\times{\mathbb{R}}^{1|1}$ is determined by the
map (69).

###### Proof.

The argument is analogous to the proof of Proposition 4.21, using this time
that ${\sf rg}_{\mu}$ preserves the subspaces ${\mathbb{R}}^{1|1}_{\geq
0},{\mathbb{R}}^{1|1}_{\leq 0}\subset{\mathbb{R}}^{1|1}$. ∎

###  4.5. Small bordisms and nearly constant superpaths

######  Definition 4.23.

Consider the families of bordisms associated to the maps of stacks

(155) $\displaystyle\begin{array}[]{c}{\rm spt}^{\pm}_{M}\colon{\sf
Map}({\mathbb{R}}^{0|1},M)\to{\rm Ob}(1|1\hbox{-}{\sf EBord}(M))\\\ {\sf
i}^{+}_{M}\colon{\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{0|1},M)\to{\rm Mor}(1|1\hbox{-}{\sf EBord}(M))\\\ {\sf
i}^{-}_{M}\colon{\mathbb{R}}^{1|1}_{\leq 0}\times{\sf
Map}({\mathbb{R}}^{0|1},M)\to{\rm Mor}(1|1\hbox{-}{\sf EBord}(M))\\\ {\rm
l}^{+}_{M}\colon{\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{0|1},M)\to{\rm Mor}(1|1\hbox{-}{\sf EBord}(M))\\\ {\rm
l}^{-}_{M}\colon{\mathbb{R}}^{1|1}_{\leq 0}\times{\sf
Map}({\mathbb{R}}^{0|1},M)\to{\rm Mor}(1|1\hbox{-}{\sf EBord}(M))\\\ {\sf
r}^{+}_{M}\colon{\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{0|1},M)\to{\rm Mor}(1|1\hbox{-}{\sf EBord}(M))\\\ {\sf
r}^{-}_{M}\colon{\mathbb{R}}^{1|1}_{\leq 0}\times{\sf
Map}({\mathbb{R}}^{0|1},M)\to{\rm Mor}(1|1\hbox{-}{\sf EBord}(M))\end{array}$

by restriction of (108), (116) and (125) to the (finite-dimensional) subspace

${\sf Map}({\mathbb{R}}^{0|1},M)\subset{\sf
Map}({\mathbb{R}}^{1|1},M),\qquad\phi_{0}\mapsto(S\times{\mathbb{R}}^{1|1}\xrightarrow{p}S\times{\mathbb{R}}^{0|1}\xrightarrow{\phi_{0}}M).$

We refer to the families (155) as _small bordisms_ in $M$.

In particular, the families (155) determine maps of stacks

(156) $\displaystyle{\rm Ob}({\sf sPath}_{0}(M))\to{\rm Ob}(1|1\hbox{-}{\sf
EBord}(M)),\quad{\rm Mor}({\sf sPath}_{0}(M))\to{\rm Mor}(1|1\hbox{-}{\sf
EBord}(M)).$

######  Proposition 4.24.

The maps of stacks (156) are part of the data of an internal functor (104)
from the super Lie category of nearly constant superpaths in $M$ (viewed as a
category internal to stacks) to the $1|1$-Euclidean bordism category over $M$.
Under this functor, the restriction of the structures ${\sf Or},{\sf
RG}_{\mu}$ and ${\sf R}$ in Lemma 4.2 are compatible with the structures on
${\sf sPath}_{0}(M)$ in Lemma 3.22.

###### Proof.

The data of an internal functor is reviewed in Definition B.7. In addition to
the maps of stacks (156), we require compatibility data for composition and
units. This is given by Proposition 4.10. Since the 2-isomorphism data in this
proposition is canonical (coming from intersecting collars of bordisms), the
coherence property demanded by these 2-isomorphisms is automatic. This
completes the construction of the internal functor (104).

The compatibility of this functor with the additional structures follows from
Propositions 4.20, 4.21 and 4.22: the isomorphisms in these propositions
restricted to the appropriate families in (155) supply 2-commutativity data,

(157)

witnessing the claimed compatibility. ∎

######  Remark 4.25.

Composition and disjoint union of small bordisms generates a subcategory
$1|1\hbox{-{\sf ebord}}(M)\subset 1|1\hbox{-}{\sf EBord}(M)$ of small bordisms
in $M$. Morally, the categories $1|1\hbox{-{\sf eft}}^{n}(M)$ in Theorems 1.1
and 1.2 considers degree $n$ field theories defined on these subcategories.
However, it is difficult to formulate a precise notion of generators for a
(sub)category in the framework from [ST11], and so instead we formulate our
results in terms of the further subcategory ${\sf sPath}_{0}(M)\subset
1|1\hbox{-{\sf ebord}}(M)$.

##  5\. $1|1$-Euclidean field theories of degree $n$

In this section we review Stolz and Teichner’s degree $n$ twist and degree $n$
twisted field theories over $M$ from [ST11]. In brief, these are internal
natural transformations (31) for a $\otimes$-invertible functor $\mathscr{T}$
called the _degree twist_. A complete construction of $\mathscr{T}$ has yet to
appear, but expected properties of $\mathscr{T}$ are described in [ST04,
HST10, ST11] for three variations on the definition of the bordism category
$1|1\hbox{-}{\sf EBord}(M)$ and Morita bicategory ${\sf{Alg}}$. With all of
these slightly differing past formalisms and others under current development,
our goal below is provide a conceptual description of the degree $n$ twist
that can be parsed in any framework for field theories. Our description relies
on the cobordism hypothesis for 2-dimensional extended topological field
theories. Assuming this hypothesis, one can calculate the values of the degree
$n$ twist and twisted field theories for the families of bordisms constructed
in the previous section. The punchline is that degree $n$ field theories are
(non-topological) boundary conditions for the 2-dimensional topological field
theory determined by the Morita invertible (and hence fully-dualizable)
algebra ${\mathbb{C}}{\rm l}_{n}$, the $n$th Clifford algebra. The Morita
equivalences between ${\mathbb{C}}{\rm l}_{n}\otimes{\mathbb{C}}{\rm l}_{m}$
and ${\mathbb{C}}{\rm l}_{n+m}$ then give the desired (graded) multiplication
on degree $n$ field theories via the tensor product.

######  Remark 5.1.

The values of the degree $n$ twist for a $d$-dimensional Euclidean field
theories without supersymmetry are listed in [ST11, §5.3]. One possible way to
extend this construction to (supersymmetric) $1|1$-Euclidean bordisms involves
a generators and relations presentation for the bordism category. Such
presentations have been described in [HST10, 6.7], [ST11, 3.2], and the proof
of [ST11, Proposition 4.16]. However, complete statements and proofs have yet
to appear. In terms of these expected generators and relations,
$1|1$-Euclidean field theories of degree $n$ are equivalent to a quantum
mechanical theory with the $n$th Clifford algebra ${\rm Cl}_{n}$ acting by
symmetries in the sense of §2.4, see [ST04, §3.2] and [HST10, Main Theorem].
The construction of the degree $n$ twist outlined below is compatible with
these prior expectations.

###  5.1. The values of twisted field theories

Twisted geometric field theories are defined as internal natural
transformations (258) for an internal functor $\mathscr{T}$ from a geometric
bordism category to the internal Morita category; see §B.4 for a review. We
unpack the values of a twisted $1|1$-Euclidean field theory on the families of
bordisms from the previous section; this expands upon the discussion [ST11,
page 52].

######  Proposition 5.2.

Given an internal functor $\mathscr{T}\colon 1|1\hbox{-}{\sf
EBord}(M)\to{\sf{Alg}}$, a $\mathscr{T}$-twisted field theory $E$ determines
data

  1. (1)

bundles of left modules

$\displaystyle\begin{array}[]{c}_{\mathscr{T}({\rm spt}^{+}_{M})}E({\rm
spt}^{+}_{M})\to{\sf Map}({\mathbb{R}}^{1|1},M),\quad_{\mathscr{T}({\rm
spt}^{-}_{M})}E({\rm spt}^{-}_{M})\to{\sf
Map}({\mathbb{R}}^{1|1},M)\end{array}$

where $\mathscr{T}({\rm spt}^{\pm}_{M})$ are super algebra bundles over ${\sf
Map}({\mathbb{R}}^{1|1},M)$;

  2. (2)

maps of bimodule bundles over ${\mathbb{R}}^{1|1}_{\geq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)$

$\displaystyle E({\sf I}^{+}_{M})\colon\mathscr{T}({\sf
I}^{+}_{M})\otimes_{{\rm proj}_{+}^{*}\mathscr{T}({\sf I}^{+}_{M})}{\rm
proj}_{+}^{*}E({\rm spt}^{+}_{M})\to{\rm act}_{+}^{*}E({\rm spt}^{+}_{M}),$
$\displaystyle E({\sf L}^{+}_{M})\colon\mathscr{T}({\sf
L}^{+}_{M})\otimes_{{\rm proj}_{+}^{*}\mathscr{T}({\rm
spt}^{+}_{M})\otimes{\rm act}_{+}^{*}\mathscr{T}({\rm spt}^{-}_{M})}{\rm
proj}_{+}^{*}E({\rm spt}^{+}_{M})\otimes{\rm act}_{+}^{*}E({\rm
spt}^{-}_{M})\to\underline{{\mathbb{C}}},$

maps of bimodule bundles over ${\mathbb{R}}^{1|1}_{>0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)$

$\displaystyle E({\sf R}^{+}_{M})\colon\mathscr{T}({\sf R}^{+}_{M})\to{\rm
proj}_{+}^{*}E({\rm spt}^{-}_{M})\otimes{\rm act}_{+}^{*}E({\rm
spt}^{+}_{M}),$

maps of bimodule bundles over ${\mathbb{R}}^{1|1}_{\leq 0}\times{\sf
Map}({\mathbb{R}}^{1|1},M)$,

$\displaystyle E({\sf I}^{-}_{M})\colon\mathscr{T}({\sf
I}^{-}_{M})\otimes_{{\rm proj}_{-}^{*}\mathscr{T}({\sf I}^{-}_{M})}{\rm
proj}_{-}^{*}E({\rm spt}^{-}_{M})\to{\sf t}^{*}E({\rm spt}^{-}_{M}),$
$\displaystyle E({\sf L}^{-}_{M})\colon\mathscr{T}({\sf