# Bordism for the 2-group symmetries of the heterotic and CHL strings

Arun Debray

###### Abstract.

In the presence of a nonzero B-field, the symmetries of the
$\mathrm{E}_{8}\times\mathrm{E}_{8}$ heterotic string form a $2$-group, or a
categorified group, as do the symmetries of the CHL string. We express the
bordism groups of the corresponding tangential structures as twisted string
bordism groups, then compute them through dimension $11$ modulo a few
unresolved ambiguities. Then, we use these bordism groups to study anomalies
and defects for these two string theories.

###### Contents

  1. 0 Introduction
  2. 1 Tangential structures for heterotic and CHL string theories
  3. 2 Bordism computations
  4. 3 Consequences in string theory

##  0\. Introduction

String theory has long been a place where higher-categorical structures in
mathematics meet their applications. This is true for a few different reasons,
but one crucial reason is that many fields in superstring and supergravity
theories have mathematical incarnations that are higher-categorical objects,
and so even precisely setting up mathematical questions coming out of string
theory, let alone solving them, often requires engaging with or developing the
foundations of various kinds of geometric objects with higher structure. This
paper is concerned with the appearance of a higher structure called a
_2-group_ in two specific string theories, and how including this structure
affects computations of bordism groups for the tangential structures of these
theories. These bordism groups control anomalies and extended objects for
these theories. The main results of this paper are computations of bordism
groups and their generating manifolds through dimension $11$, except for a few
ambiguities we did not addres, for the tangential structures underlying these
two string theories.

For the higher structures we investigate in this paper, the story begins with
the _Kalb-Ramond field_ , or the _B-field_. This is an analogue of the field
strength of an electromagnetic field, represented as a closed differential
$2$-form with a quantization condition. Locality of quantum field theory means
expressing the field strength of the electromagnetic field as a section of a
sheaf, specifically as a connection on a principal $\mathbb{T}$-bundle, where
$\mathbb{T}$ is the circle group. For the B-field, everything is one degree
higher: it comes to us as a closed differential $3$-form with a quantization
condition, which we would like to express as a geometric object that
sheafifies. This cannot be a connection on a principal $G$-bundle for a
finite-dimensional Lie group $G$; instead, one models the B-field as a
connection on a _$\mathbb{T}$ -gerbe_, which is a categorification of a
principal $\mathbb{T}$-bundle. A $\mathbb{T}$-gerbe on a manifold $M$ is,
roughly speaking, a bundle of groupoids on $M$ which is locally equivalent to
$\mathrm{pt}/\mathbb{T}$. There are several ways to make this precise; we
discuss one, Murray’s _bundle gerbes_ [Mur96], in Definition 1.1.

In this article, we consider higher structures in two string theories: the
$\mathrm{E}_{8}\times\mathrm{E}_{8}$ heterotic string, and the Chaudhuri-
Hockney-Lykken (CHL) string. The former is a ten-dimensional superstring
theory whose low-energy limit is ten-dimensional $\mathcal{N}=1$ supergravity,
and the latter is a nine-dimensional theory obtained from the
$\mathrm{E}_{8}\times\mathrm{E}_{8}$ heterotic string theory by compactifying
on a circle. Both of these theories have B-fields, but Green and Schwarz
[GS84] showed that in order to cancel an anomaly, the B-field and the gauge
field must satisfy a relation known as a _Bianchi identity_. Fiorenza-
Schreiber-Stasheff [FSS12] and Sati-Schreiber-Stasheff [SSS12] describe how
the Bianchi identity mixes the data of the B-field and the gauge field into
data that can be interpreted as a connection on a principal bundle for a
$2$-group $\mathbb{G}$, specifically a _string $2$-group_
$\mathit{\mathcal{S}tr}(G,\mu)$ associated to the data of a compact Lie group
$G$ and a class $\mu\in H^{4}(BG;\mathbb{Z})$; typically, $G$ is the gauge
group and $\mu$ is determined by the anomaly polynomial.

$2$-groups have been used in the theoretical and mathematical physics
literature for some time now. This program began in earnest with work of Baez,
Crans, Lauda, Stevenson, and Schreiber [Bae02, BC04, BL04, BSCS07, BS07]; more
recently, $2$-groups, their symmetries, and their anomalies have made a
resurgence in quantum field theory following work of Córdova-Dumitriescu-
Intrilligator [CDI19] and Benini-Córdova-Hsin [BCH19] identifying many
examples of $2$-group symmetries in commonly studied QFTs. See also Sharpe
[Sha15] and the references therein.

In the first part of this article, we introduce the Bianchi identity and
$2$-groups, then review work of Fiorenza-Schreiber-Stasheff [FSS12] and Sati-
Schreiber-Stasheff [SSS12] mentioned above. These authors work in the setting
of stacks on the site $\mathit{\mathscr{M}an}$ of smooth manifolds; the data
of the B-field $(Q,\Theta_{Q})$ and the principal $G$-bundle with connection
$(P,\Theta_{P})$ on a manifold $M$ refine to maps from $M$ to classifying
stacks of these data. The data of an identification of two differential
characteristic classes associated to $\Theta_{P}$ and $\Theta_{Q}$ gives rise
to

  1. (1)

a principal $\mathit{\mathcal{S}tr}(G,\mu)$-bundle lifting $P$ for a specified
choice of $\mu$ (Proposition 1.35), and

  2. (2)

local data of solutions to the Bianchi identity (Proposition 1.37, [FSS12,
§6.3]).

Inspired by this, we introduce the tangential structures
${\xi^{\mathrm{het}}}$ and ${\xi^{\mathrm{CHL}}}$, which are special cases of
a general construction of Sati-Schreiber-Stasheff [SSS12, Definition 2.8]: a
${\xi_{n}^{\mathrm{het}}}$-structure on a spin manifold $M$ is data of a
principal $\mathbb{G}^{\mathrm{het}}_{n}$-bundle, where
$\mathbb{G}^{\mathrm{het}}_{n}\coloneqq\mathit{\mathcal{S}tr}(\mathrm{Spin}_{n}\times(\mathrm{E}_{8}\times\mathrm{E}_{8}\rtimes\mathbb{Z}/2),c_{1}+c_{2}-\lambda)$
(1.42), whose associated $\mathrm{Spin}_{n}$-bundle via the quotient
$\mathbb{G}^{\mathrm{het}}_{n}\to\mathrm{Spin}_{n}$ is the principal
$\mathrm{Spin}_{n}$-bundle of spin frames (Definition 1.41). This is
compatible as $n$ varies, allowing us to stabilize and define a
${\xi^{\mathrm{het}}}$-structure as usual. The definition of
${\xi^{\mathrm{CHL}}}$ in Definition 1.52, which coincides with
$B\mathrm{String}^{2a}$ in [SSS12, (2.18), §2.2.3], is analogous. Related
tangential structures appear in [Sat11b, FSS15a, FSS15b, FSS21].

Given a tangential structure, we can compute bordism groups, and indeed the
point of this paper is to compute ${\xi^{\mathrm{het}}}$ and
${\xi^{\mathrm{CHL}}}$ bordism groups in low dimensions. These bordism groups
can then be used to learn more about the $\mathrm{E}_{8}\times\mathrm{E}_{8}$
heterotic and CHL strings. We have two primary applications in mind.

  1. (1)

The _cobordism conjecture_ of McNamara-Vafa [MV19] is an application to the
question of what kinds of spacetime backgrounds are summed over in quantum
gravity. Such backgrounds are often taken to be manifolds or something closely
related equipped with data of a tangential structure $\xi$. The cobordism
conjecture says that if $\xi$ is the most general tangential structure which
can appear in this way in any particular $d$-dimensional theory of quantum
gravity, then $\Omega_{k}^{\xi}=0$ for $3\leq k\leq d-1$. We will see that
$\Omega_{k}^{{\xi^{\mathrm{het}}}}$ and $\Omega_{k}^{{\xi^{\mathrm{CHL}}}}$
are often nonzero in that range. This is consistent with the cobordism
conjecture: it suggests that ${\xi^{\mathrm{het}}}$ and ${\xi^{\mathrm{CHL}}}$
are not the most general tangential structures that can be summed over.
Typically these bordism groups are killed by allowing singular manifolds
corresponding to considering the theory with branes or other defects, so one
can use bordism computations to predict new defects in string theories.

  2. (2)

A broad class of $n$-dimensional quantum field theories come with data of an
_anomaly_ , which in many cases can roughly be described an
$(n+1)$-dimensional invertible field theory $\alpha$. In some cases one wants
to trivialize $\alpha$, meaning exhibiting an isomorphism from $\alpha$ to the
trivial field theory. By work of Freed-Hopkins-Teleman [FHT10] and Freed-
Hopkins [FH21b], invertible field theories can be classified using bordism
group computations. For both the $\mathrm{E}_{8}\times\mathrm{E}_{8}$
heterotic string and the CHL string, the bordism groups indicating a potential
anomaly are nonzero, and it would be interesting to check whether the
corresponding anomalies are nontrivial.

See §3, as well as Questions 0.1, 0.2, and 0.3 below, for more on these
applications and what we can learn from our bordism computations.

Our main theorems are the following two computations of the
${\xi^{\mathrm{het}}}$ and ${\xi^{\mathrm{CHL}}}$ bordism groups in low
dimensions.

######  Theorem A.

For $k\leq 10$, the ${\xi^{\mathrm{het}}}$-bordism groups are:

$\displaystyle\Omega_{0}^{{\xi^{\mathrm{het}}}}$
$\displaystyle\cong\mathbb{Z}$
$\displaystyle\qquad\qquad\Omega_{6}^{\xi^{\mathrm{het}}}$
$\displaystyle\cong\mathbb{Z}/2$
$\displaystyle\Omega_{1}^{{\xi^{\mathrm{het}}}}$
$\displaystyle\cong\mathbb{Z}/2\oplus\mathbb{Z}/2$
$\displaystyle\Omega_{7}^{\xi^{\mathrm{het}}}$
$\displaystyle\cong\mathbb{Z}/16$
$\displaystyle\Omega_{2}^{{\xi^{\mathrm{het}}}}$
$\displaystyle\cong\mathbb{Z}/2\oplus\mathbb{Z}/2$
$\displaystyle\Omega_{8}^{\xi^{\mathrm{het}}}$
$\displaystyle\cong\mathbb{Z}^{3}\oplus(\mathbb{Z}/2)^{\oplus i}$
$\displaystyle\Omega_{3}^{{\xi^{\mathrm{het}}}}$
$\displaystyle\cong\mathbb{Z}/8$
$\displaystyle\Omega_{9}^{\xi^{\mathrm{het}}}$
$\displaystyle\cong(\mathbb{Z}/2)^{\oplus j}$
$\displaystyle\Omega_{4}^{{\xi^{\mathrm{het}}}}$
$\displaystyle\cong\mathbb{Z}\oplus\mathbb{Z}/2$
$\displaystyle\Omega_{10}^{\xi^{\mathrm{het}}}$
$\displaystyle\cong(\mathbb{Z}/2)^{\oplus k}.$
$\displaystyle\Omega_{5}^{{\xi^{\mathrm{het}}}}$ $\displaystyle\cong 0$

Here, either $i=1$, $j=4$, and $k=4$, or $i=2$, $j=6$, and $k=5$.

$\Omega_{11}^{\xi^{\mathrm{het}}}$ is an abelian group of order $64$
isomorphic to one of $\mathbb{Z}/8\oplus\mathbb{Z}/8$,
$\mathbb{Z}/16\oplus\mathbb{Z}/4$, $\mathbb{Z}/32\oplus\mathbb{Z}/2$, or
$\mathbb{Z}/64$.

This is a combination of Theorems 2.62 and 2.74. In §2.2.1, we find manifold
representatives for all classes in $\Omega_{k}^{{\xi^{\mathrm{het}}}}$ for
$k\leq 10$ except potentially for two missing classes $X_{8}$ and $X_{9}$ of
dimensions $8$, resp. $9$ and their products with $S_{\mathit{nb}}^{1}$. These
classes may or may not be zero depending on the fate of an Adams differential.
In §2.2.2, we find a manifold representing $X_{8}$: if the unaddressed Adams
differential vanishes, $X_{8}$ should be added to the list of generators in
§2.2.1, and if the differential does not vanish, then $X_{8}$ bounds as a
${\xi^{\mathrm{het}}}$-manifold.

Our calculation of ${\xi^{\mathrm{CHL}}}$-bordism builds on work of Hill
[Hil09, Theorem 1.1], who computes
$\Omega_{*}^{\mathrm{String}}(B\mathrm{E}_{8})$ in dimensions $14$ and below.

######  Theorem B.

For $k\leq 11$, there is an abstract isomorphism from
$\Omega_{*}^{{\xi^{\mathrm{CHL}}}}$ to the free and $2$-torsion summands of
$\Omega_{*}^{\mathrm{String}}(B\mathrm{E}_{8})$. Therefore, by Hill’s
computation [Hil09], there are isomorphisms

$\displaystyle\Omega_{0}^{{\xi^{\mathrm{CHL}}}}$
$\displaystyle\cong\mathbb{Z}$
$\displaystyle\Omega_{6}^{{\xi^{\mathrm{CHL}}}}$
$\displaystyle\cong\mathbb{Z}/2$
$\displaystyle\Omega_{1}^{{\xi^{\mathrm{CHL}}}}$
$\displaystyle\cong\mathbb{Z}/2\qquad\qquad$
$\displaystyle\Omega_{7}^{{\xi^{\mathrm{CHL}}}}$ $\displaystyle\cong 0$
$\displaystyle\Omega_{2}^{{\xi^{\mathrm{CHL}}}}$
$\displaystyle\cong\mathbb{Z}/2$
$\displaystyle\Omega_{8}^{{\xi^{\mathrm{CHL}}}}$
$\displaystyle\cong\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}/2$
$\displaystyle\Omega_{3}^{{\xi^{\mathrm{CHL}}}}$
$\displaystyle\cong\mathbb{Z}/8$
$\displaystyle\Omega_{9}^{{\xi^{\mathrm{CHL}}}}$
$\displaystyle\cong\mathbb{Z}/2\oplus\mathbb{Z}/2\oplus\mathbb{Z}/2$
$\displaystyle\Omega_{4}^{{\xi^{\mathrm{CHL}}}}$
$\displaystyle\cong\mathbb{Z}$
$\displaystyle\Omega_{10}^{{\xi^{\mathrm{CHL}}}}$
$\displaystyle\cong\mathbb{Z}/2\oplus\mathbb{Z}/2$
$\displaystyle\Omega_{5}^{{\xi^{\mathrm{CHL}}}}$ $\displaystyle\cong 0$
$\displaystyle\Omega_{11}^{{\xi^{\mathrm{CHL}}}}$
$\displaystyle\cong\mathbb{Z}/8.$

This is a combination of Theorems 2.90 and 2.92. We also obtain some
information about manifold representatives of generators of these groups.

The computational tool we use to prove Theorems A and B is standard: the Adams
spectral sequence. This spectral sequence has seen plenty of applications in
the mathematical physics literature, and there is a standard procedure
reviewed by Beaudry-Campbell [BC18] for simplifying the $E_{2}$-page for a
wide class of tangential structures, namely those which can be described as
oriented, spin$c$, spin, or string bordism twisted by a virtual vector bundle.
For example, the twisted string bordism computations of [FK96, Fan99, FW10]
make use of this simplifying technique. Unfortunately, this procedure is
unavailable to us: in Lemma 2.2, we prove that ${\xi^{\mathrm{het}}}$ and
${\xi^{\mathrm{CHL}}}$ cannot be described as twists of this sort. However, we
are still able to describe them as twists in a more general sense due to Ando-
Blumberg-Gepner-Hopkins-Rezk [ABG+14a, ABG+14b]: adapting an argument of
Hebestreit-Joachim [HJ20], one learns that the Thom spectra for
${\xi^{\mathrm{het}}}$ and ${\xi^{\mathrm{CHL}}}$ can be produced as the
$\mathit{MTString}$-module Thom spectra associated to certain maps to
$B\mathrm{GL}_{1}(\mathit{MTString})$. Using this structure, in upcoming joint
work with Matthew Yu, we are able to prove a theorem simplifying the
calculation of the $E_{2}$-page:

######  Theorem C (Debray-Yu [DY]).

In topological degrees $15$ and below, the $E_{2}$-pages of the Adams spectral
sequences computing $2$-completed twisted string bordism for a class of twists
including those for ${\xi^{\mathrm{het}}}$ and ${\xi^{\mathrm{CHL}}}$ can be
computed as $\operatorname{Ext}$ over the subalgebra $\mathcal{A}(2)$ of the
Steenrod algebra.

What we prove is more precise and holds in more generality; see Theorems 2.20
and 2.22 for that version of the result.111Since [DY] is not available yet, we
provide a proof sketch of the case we need in Remark 2.26.

The $\mathcal{A}(2)$-module Ext groups we have to compute are simpler than
what one a priori has to work with over the entire Steenrod algebra
$\mathcal{A}$. We do not need this simplification at odd primes; there the
full Adams spectral sequence is easier to work with, and the absence of a
simplification does not hinder us (though see also [DY, §3.2]).

The reason we computed these bordism groups in this paper is with applications
to physics, specifically to anomalies and the cobordism conjecture, in mind.
We discuss some implications of our calculations in §3; for example, one of
the $\mathbb{Z}/2$ summands of $\Omega_{1}^{\xi^{\mathrm{het}}}$ corresponds
to the non-supersymmetric $7$-brane recently discovered by Kaidi-Ohmori-
Tachikawa-Yonekura [KOTY23]. We end this section of the introduction with some
questions related to these physics predictions.

######  Question 0.1.

What does the Kaidi-Ohmori-Tachikawa-Yonekura $7$-brane correspond to in
Hořava-Witten theory, and what does this look like in bordism? Hořava-Witten
[HW96a, HW96b, Wit96] proposed that the $\mathrm{E}_{8}\times\mathrm{E}_{8}$
heterotic string can be identified with a certain limit of M-theory
compactified on an interval; thus this ought to correspond to a notion of
bordism of manifolds with boundary. Conner-Floyd [CF66, §16] define a notion
bordism of compact manifolds with boundary — is this the correct kind of
bordism for applications to McNamara-Vafa’s conjecture?

We discuss some additional extended objects predicted by our bordism
computations to exist in the $\mathrm{E}_{8}\times\mathrm{E}_{8}$ heterotic
and CHL strings in §3.1.

######  Question 0.2.

Is the $\mathbb{Z}/2$ symmetry exchanging the two $\mathrm{E}_{8}$-bundles in
$\mathrm{E}_{8}\times\mathrm{E}_{8}$ heterotic string theory anomalous?
Because $\Omega_{11}^{{\xi^{\mathrm{het}}}}$ is nonzero, we were unable to
rule out this anomaly.

Witten [Wit86, §4] and Tachikawa-Yonekura [TY21] show that the
$\mathrm{E}_{8}\times\mathrm{E}_{8}$ heterotic string is anomaly-free in
certain cases, but they do not address the $\mathbb{Z}/2$ symmetry.

######  Question 0.3.

Does the CHL string have an anomaly? This anomaly could be nontrivial, because
$\Omega_{10}^{{\xi^{\mathrm{CHL}}}}\cong\mathbb{Z}/2\oplus\mathbb{Z}/2$.

There is another application of twisted string bordism to physics that we did
not address in this paper: studying elliptic genera, the Witten genus and
related invariants, along the lines of, e.g., Bunke-Naumann [BN14], McTague
[McT14], Han-Huang-Duan [HHD21], and Berwick-Evans [BE23]. It would be
interesting to study whether the calculations in this paper could be applied
in similar contexts.

### Outline

We begin in §1.1 by introducing the fields present in 10d $\mathcal{N}=1$
supergravity, the low-energy limit of heterotic string theory. We discuss how
the Green-Schwarz anomaly cancellation condition imposes an equation called
the _Bianchi identity_ (1.10) on the fields in this theory. We then generalize
this to a _twisted Bianchi identity_ (1.12) associated to data of a Lie group
$G$ and a class $\mu\in H^{4}(BG;\mathbb{Z})$. In §1.2, we relate these
Bianchi identities to the presence of a $2$-group symmetry in this field
theory. We begin by reviewing $2$-groups, their principal bundles, and their
connections, and in Example 1.22 define the string cover $\mathcal{S}(G,\mu)$
corresponding to a group $G$ and a class $\mu\in H^{4}(BG;\mathbb{Z})$. Then
we review work of Fiorenza-Schreiber-Stasheff [FSS12] and Sati-Schreiber-
Stasheff [SSS12] relating the Bianchi identity to twisted string structures.
Using this, we define the heterotic tangential structure in Definition 1.41,
which is the topological part of the structure necessary for defining
$\mathcal{N}=1$ supergravity. Then, in §1.3, we introduce the CHL string and
define the CHL tangential structure using what we learned in §1.2.

In §2, we compute the bordism groups $\Omega_{*}^{{\xi^{\mathrm{het}}}}$ and
$\Omega_{*}^{{\xi^{\mathrm{CHL}}}}$ in low degrees. For the latter we are able
to completely compute them in dimensions $11$ and below, but for the former,
we have only partial information above dimension $7$, occluded by Adams
differentials and an extension problem we could not solve. We begin in §2.1 by
discussing how to simplify the Thom spectra $\mathit{MT{\xi^{\mathrm{het}}}}$
and $\mathit{MT{\xi^{\mathrm{CHL}}}}$; we prove in Lemma 2.2 that a standard
approach does not work, and so we use a different idea: construct
$\mathit{MT{\xi^{\mathrm{het}}}}$ and $\mathit{MT{\xi^{\mathrm{CHL}}}}$ as
$\mathit{MTString}$-module Thom spectra using machinery developed by Ando-
Blumberg-Gepner-Hopkins-Rezk. We review this machinery and discuss how it
leads to Corollary 2.22, a special case of the main theorem of our work [DY]
joint with Matthew Yu, simplifying the calculation of the $E_{2}$-page of the
Adams spectral sequence at $2$ for a wide class of twisted string bordism
groups. Next, in §2.2, we undertake this computation for
${\xi^{\mathrm{het}}}$. We do not have such a simplification at odd primes, so
in §2.3 we press ahead directly with the Adams spectral sequence for
${\xi^{\mathrm{het}}}$, proving in Theorem 2.74 that
$\Omega_{*}^{{\xi^{\mathrm{het}}}}$ lacks odd-primary torsion in degrees $11$
and below. Finally, in §2.4 we run the analogous calculations for the CHL
string, again using Corollary 2.22 at $p=2$ and arguing more directly at odd
primes.

The final section, §3, is about applications to string theory. We first
discuss the cobordism conjecture of McNamara-Vafa [MV19] in §3.1, and go over
a few predictions that follow from the bordism group computations in §2. In
§3.2, we briefly introduce anomalies of quantum field theories and their
bordism-theoretic classification, and touch on questions raised by our bordism
computations.

### Acknowledgements

I especially want to thank Miguel Montero both for suggesting this project and
for many helpful conversations about the material in this paper, and Matthew
Yu for many helpful discussions relating to [DY] and other ideas related to
this paper. I also want to thank Markus Dierigl and the anonymous referee for
helpful comments on a draft. In addition, this paper benefited from
conversations with Ivano Basile, Matilda Delgado, Jacques Distler, Dan Freed,
Jonathan J. Heckman, Justin Kaidi, Jacob McNamara, Yuji Tachikawa, and Roberto
Telléz Domínguez; thank you to all!

Part of this project was completed while AD visited the Perimeter Institute
for Theoretical Physics; research at Perimeter is supported by the Government
of Canada through Industry Canada and by the Province of Ontario through the
Ministry of Research & Innovation.

##  1\. Tangential structures for heterotic and CHL string theories

The goal of this section is to define the tangential structures
${\xi^{\mathrm{het}}}$ and ${\xi^{\mathrm{CHL}}}$ that are necessary to
formulate the (low-energy limits of) the $\mathrm{E}_{8}\times\mathrm{E}_{8}$
heterotic string and the CHL string. By “tangential structure” we mean the
topological part of the structure needed on a manifold to define a given field
theory; see Definition 1.40 for the precise definition. The presence of a
B-field in both theories means that these tangential structures arise as
classifying spaces of higher groups. First, we introduce the heterotic string
in §1.1, and see what data and conditions are told to us by Green-Schwarz
anomaly cancellation; then in §1.2, we reinterpret that data as combining the
gauge field and the B-field into a connection for a principal bundle for a
higher group. Finally, in §1.3, we use the general theory from §1.2 to
determine the tangential structure for the CHL string.

The material in this section is not new, though it was not always stated in
this form before. The fact that a Bianchi identity/Green-Schwarz mechanism is
expressing a lift to a connection for a higher-group principal bundle is well-
known; see Fiorenza-Schreiber-Stasheff [FSS12] and Sati-Schreiber-Stasheff
[SSS12].

###  1.1. The $\mathrm{E}_{8}\times\mathrm{E}_{8}$ heterotic string

Heterotic string theories are ten-dimensional superstring theories whose low-
energy limits are 10d $\mathcal{N}=1$ supergravity theories. These
supergravity theories can have Yang-Mills terms, and so are parametrized by
the data of the gauge group $G$, a compact Lie group. However, not all choices
of $G$ yield valid supergravity theories; there is the potential for an
anomaly that must be trivialized, and this is quite a strong constraint,
implying that the connected component of the identity in $G$ must be either
$\mathrm{E}_{8}\times\mathrm{E}_{8}$ or $G=\mathrm{SemiSpin}_{32}$222The
center of $\mathrm{Spin}_{4k}$ is isomorphic to
$\mathbb{Z}/2\times\mathbb{Z}/2$. Quotienting by one copy of $\mathbb{Z}/2$
yields $\mathrm{SO}_{4k}$; the quotients by the two other $\mathbb{Z}/2$
subgroups are isomorphic, and are called $\mathrm{SemiSpin}_{4k}$. See
[McI99]. [GS84, ATD10]. The anomaly cancellation mechanism itself, due to
Green-Schwarz [GS84], combines the different fields in the theory into a
connection for a principal $\mathbb{G}$-bundle, where $\mathbb{G}$ is a higher
group;333Green-Schwarz’ work only cancels the perturbative part of the
anomaly; see §3.2 for more information. we use this subsection to discuss the
fields and the Green-Schwarz condition, and the next subsection to discuss the
role of higher group. In this paper, we will focus solely on the
$\mathrm{E}_{8}\times\mathrm{E}_{8}$ case; it would be interesting to study
the analogues of the computations and applications in this paper in the
$\mathrm{SemiSpin}_{32}$ case.

The group $\mathbb{Z}/2$ acts on $\mathrm{E}_{8}\times\mathrm{E}_{8}$ by
exchanging the two factors, and the setup of heterotic string theory,
including the low-energy supergravity limit and Green-Schwarz’ anomaly
cancellation, is invariant under this symmetry, so we can expand the gauge
group to
$G\coloneqq(\mathrm{E}_{8}\times\mathrm{E}_{8})\rtimes\mathbb{Z}/2$.444Though
we often use the standard name “the $\mathrm{E}_{8}\times\mathrm{E}_{8}$
heterotic string” to refer to this theory, we will always consider the larger
gauge group $(\mathrm{E}_{8}\times\mathrm{E}_{8})\rtimes\mathbb{Z}/2$. This
appears to have first been noticed by McInnes [McI99, §I]; see also [dBDH+00,
§2.2.1].

The fields of 10d $\mathcal{N}=1$ supergravity on a manifold $M$ include:

  * •

a metric $g$,

  * •

a spin structure on $M$,

  * •

a principal $G$-bundle $P\to M$ with connection $\Theta_{P}$,

  * •

a _B-field_ or _Kalb-Ramond field_ , a gerbe $Q\to M$ with connection
$\Theta_{Q}$, and

  * •

several additional fields (the dilaton, dilatino, gravitino, and gaugino)
which will not be directly relevant to this paper.

Let us say more about the B-field, since its model as a gerbe with connection
may be less familiar. A _gerbe_ is a categorification of the idea of a
principal $\mathbb{T}$-bundle; here $\mathbb{T}$ is the circle group. Thus,
for example, a principal $\mathbb{T}$-bundle $P\to M$ is classified by its
first Chern class $c_{1}(P)\in H^{2}(M;\mathbb{Z})$, and a gerbe $Q\to M$ is
classified by its _Dixmier-Douady class_ $\mathrm{DD}(Q)\in
H^{3}(M;\mathbb{Z})$ [DD63, Bry93]. A connection on a principal
$\mathbb{T}$-bundle has holonomy around loops; a connection on a gerbe has
holonomy on closed surfaces. And so on.

Gerbes were first introduced by Giraud [Gir71]. There are several different
and equivalent ways to precisely define gerbes and their connections;
heuristically you can think of a gerbe on $M$ as a sheaf of groupoids on $M$
locally equivalent to the trivial sheaf with fiber $\mathrm{pt}/\mathbb{T}$.
One way to make this precise is the following.

If $f\colon Y\to X$ is a map, we let $Y^{[n]}\coloneqq
Y\times_{X}Y\times_{X}\dotsm\times_{X}Y$; $Y^{[n]}$ is the space of
$n$-simplices in the Čech nerve for $f$.

######  Definition 1.1 (Murray [Mur96]).

A _bundle gerbe_ over a manifold $M$ is a surjective submersion $\pi\colon
Y\to M$, a $\mathbb{T}$-bundle $P\to Y^{[2]}$, and an isomorphism
$\mu\colon\pi_{12}^{*}P\otimes\pi_{23}^{*}P\overset{\cong}{\to}\pi_{13}^{*}P$
of $\mathbb{T}$-bundles over $Y^{[3]}$ satisfying the natural associativity
condition (see below) over $Y^{[4]}$.

Given two $\mathbb{T}$-bundles $P_{1},P_{2}\to X$, their _tensor product_
$P_{1}\otimes P_{2}$ is the unit circle bundle inside the tensor product of
the Hermitian line bundles $L_{1},L_{2}\to X$ associated to $P_{1}$, resp.
$P_{2}$. The maps $\pi_{12},\pi_{23},\pi_{13}\colon
Y^{[3]}\mathrel{\vbox{\halign{#\cr\nointerlineskip\cr$\rightarrow$\crcr\nointerlineskip\cr$\rightarrow$\crcr\nointerlineskip\cr$\rightarrow$\crcr}}}Y^{[2]}$
are the three face maps in the Čech nerve $Y^{\bullet}$ associated to $f$,
given explicitly by contracting two of the three copies of $Y$ via
$Y\times_{X}Y\to Y$.

The associativity condition in Definition 1.1 is a little unwieldy to state
explicitly, but can be found in in [Mur10, Definition 4.1(2)].

######  Definition 1.2 ([Mur96]).

A _connection_ $\Theta_{Q}$ on a bundle gerbe $Q=(Y,P,\mu)$ is data of a
$2$-form $B\in\Omega^{2}(Y)$ and a connection $\Theta_{P}$ on $P$ such that if
$\Omega_{P}\in\Omega^{2}(P)$ denotes the curvature of $P$ and
$\pi_{1},\pi_{2}\colon Y^{[2]}\to Y$ are the two projections, then

(1.3) $\Omega_{P}=\pi_{2}^{*}B-\pi_{1}^{*}B.$

The _curvature_ of $\Theta_{Q}$ is $\Omega_{Q}\coloneqq\mathrm{d}B$, which is
a closed $3$-form.

The key thing to know about this definition is that, just like a principal
$\mathbb{T}$-bundle $P\to M$ with connection locally has a connection $1$-form
$A$ and globally has a curvature $2$-form $\Omega_{P}$ which locally satisfies
$\Omega_{P}=\mathrm{d}A$, a gerbe with connection $Q$ locally has a connection
$2$-form $B$ and globally has a curvature $3$-form $\Omega_{Q}$ which locally
satisfies $\Omega_{Q}=\mathrm{d}B$. For more information, see, e.g., Brylinski
[Bry93, §5.3].

######  Definition 1.4.

Because $\mathrm{E}_{8}$ is a simple, connected, simply connected, compact Lie
group, there is a canonical isomorphism
$H^{4}(B\mathrm{E}_{8};\mathbb{Z})\overset{\cong}{\to}\mathbb{Z}$. Let $c$
denote the generator corresponding to $1\in\mathbb{Z}$. In
$B(\mathrm{E}_{8}\times\mathrm{E}_{8})\simeq B\mathrm{E}_{8}\times
B\mathrm{E}_{8}$, let $c_{1}$ and $c_{2}$ denote the copies of $c$ coming from
the first, resp. second copies of $B\mathrm{E}_{8}$ via the Künneth map.

The class $c_{1}+c_{2}$ is invariant under the $\mathbb{Z}/2$ swapping action,
so descends via the Serre spectral sequence to a class in
$H^{4}(B((\mathrm{E}_{8}\times\mathrm{E}_{8})\rtimes\mathbb{Z}/2);\mathbb{Z})$,
which we also call $c_{1}+c_{2}$.

######  Definition 1.5.

$\mathrm{Spin}_{n}$ is also a compact, connected, simply connected simple Lie
group when $n\geq 3$, and the generator of
$H^{4}(B\mathrm{Spin}_{n};\mathbb{Z})\overset{\cong}{\to}\mathbb{Z}$
corresponding to $1$ is denoted $\lambda$.

The class $\lambda$ is preserved under the standard embeddings
$\mathrm{Spin}_{n}\hookrightarrow\mathrm{Spin}_{n+k}$, so we often work with
its stabilized avatar $\lambda\in H^{4}(B\mathrm{Spin};\mathbb{Z})$. We use
this to define $\lambda$ for $\mathrm{Spin}_{n}$ when $n<3$. Because
$2\lambda=p_{1}$, the class $\lambda$ is often denoted $\tfrac{1}{2}p_{1}$.
The mod $2$ reduction of $\lambda$ is the Stiefel-Whitney class $w_{4}$.

######  Lemma 1.6 (Whitney sum formula).

Let $X$ be a topological space and $E_{1},E_{2}\to X$ be two vector bundles
with spin structure. Then $\lambda(E_{1}\oplus
E_{2})=\lambda(E_{1})+\lambda(E_{2})$.

###### Proof.

It suffices to prove the universal case, which amounts to the calculation of
the pullback of $\lambda$ by the map

(1.7) $\oplus\colon B\mathrm{Spin}_{k_{1}}\times
B\mathrm{Spin}_{k_{2}}\longrightarrow B\mathrm{Spin}_{k_{1}+k_{2}}.$

For $n\geq 3$, $\mathrm{Spin}_{n}$ is a connected, simply connected, compact
simple Lie group, so $H^{\ell}(B\mathrm{Spin}_{n};\mathbb{Z})$ vanishes for
$\ell=1,2,3$ and is isomorphic to $\mathbb{Z}$ for $\ell=0,4$. For $n<3$,
$H^{\ast}(B\mathrm{Spin}_{n};\mathbb{Z})$ is still trivial or free abelian in
degrees $4$ and below. Therefore by the Künneth formula, for all
$k_{1},k_{2}$, $H^{4}(B\mathrm{Spin}_{k_{1}}\times
B\mathrm{Spin}_{k_{2}};\mathbb{Z})$ is a free abelian group, meaning that if
we can show $2\lambda(E_{1}\oplus E_{2})=2\lambda(E_{1})+2\lambda(E_{2})$,
then we can deduce $\lambda(E_{1}\oplus E_{2})=\lambda(E_{1})+\lambda(E_{2})$.

As $2\lambda=p_{1}$, we have reduced to the Whitney sum formula for $p_{1}$.
The Whitney sum formula $p_{1}(E_{1}\oplus E_{2})=p_{1}(E_{1})+p_{1}(E_{2})$
does not actually hold for all vector bundles, but Brown [Bro82, Theorem 1.6]
(see also Thomas [Tho62]) showed that the difference $p_{1}(E_{1}\oplus
E_{2})-p_{1}(E_{1})-p_{1}(E_{2})$ vanishes when $E_{1}$ and $E_{2}$ are
orientable, so in our setting of spin vector bundles, we can conclude. ∎

######  Remark 1.8.

There are other ways to prove Lemma 1.6: for example, it follows immediately
from a result of Jenquin [Jen05, Corollary 4.9] in a simple generalized
cohomology theory. Johnson-Freyd and Treumann [JFT20, §1.4] sketch another
proof of Lemma 1.6.

Next, we introduce the Chern-Weil homomorphism. Let $G$ be a Lie group with
Lie algebra $\mathfrak{g}$, and let
$f\in\mathrm{Sym}^{k}(\mathfrak{g}^{\vee})$, i.e. $f$ is a degree-$k$
polynomial function on $\mathfrak{g}$ which is invariant under the adjoint
$G$-action on $\mathfrak{g}$. Given a manifold $M$, a principal $G$-bundle
$P\to M$, and a connection $\Theta$ on $P$, let
$\Omega\in\Omega_{P}^{2}(\mathfrak{g})$ denote the curvature $2$-form. Then
one can evaluate $f$ on $\Omega^{\wedge
k}\in\Omega_{P}^{2k}(\mathfrak{g}^{\otimes k})$, producing a form
$f(\Omega^{\wedge k})\in\Omega_{P}^{2k}$; because $f$ is
$\mathrm{Ad}$-invariant, $f(\Omega^{\wedge k})$ descends to a form
$w(\Theta)\in\Omega_{M}^{2k}$, which is always closed. This defines a ring
homomorphism, called the _Chern-Weil homomorphism_ [Car50, Che52],

(1.9a) $w\colon\mathrm{Sym}^{\bullet}(\mathfrak{g}^{\vee})\longrightarrow
H_{\mathrm{dR}}^{*}(M),$ which doubles the degree and is natural in $M$;
moreover, the de Rham class of $w(\Theta)$ depends on $P$ but not on the
connection. Using de Rham’s theorem and naturality, $w$ upgrades to a ring
homomorphism (1.9b)
$w\colon\mathrm{Sym}^{\bullet}(\mathfrak{g}^{\vee})\longrightarrow
H^{*}(BG;\mathbb{R}),$

which Chern and Weil showed is an isomorphism when $G$ is compact [Che52,
Wei49]. Thus, when $G$ is compact, a class $x\in H^{2*}(BG;\mathbb{Z})$
defines a polynomial
$\mathrm{CW}_{x}\in\mathrm{Sym}^{*}(\mathfrak{g}^{\vee})$, the $w$-preimage of
the de Rham class of $x$. We will also write $\mathrm{CW}_{x}(\Theta)$ to
denote the form defined by evaluating the polynomial $\mathrm{CW}_{x}$ on the
curvature form of $\Theta$.

Returning to 10d $\mathcal{N}=1$ supergravity, Green-Schwarz [GS84] noticed
that in order to trivialize an anomaly, one has to impose a relation between
$P$ and $Q$ and their connections, so that $Q$ is not quite a gerbe, but
instead something twisted. Specifically, the curvature $\Omega_{Q}$ is no
longer closed, but instead satisfies the equation

(1.10)
$\mathrm{d}\Omega_{Q}=\mathrm{CW}_{c_{1}+c_{2}}(\Theta_{P})-\mathrm{CW}_{\lambda}(\Theta^{\mathrm{LC}}),$

where $\Theta^{\mathrm{LC}}$ is the Levi-Civita connection on the principal
$\mathrm{Spin}_{n}$-bundle of frames of $M$.555Before Green-Schwarz, it was
already known that $\mathrm{CW}_{c_{1}+c_{2}}(\Theta_{P})$ and
$\mathrm{d}\Omega_{Q}$ had to mix in order to preserve supersymmetry, thanks
to work of Bergshoeff-de Roo-de Wit-van Nieuwenhuizen [BdRdWvN82] and
Chapline-Manton [CM83]. This is called a _Bianchi identity_ in the physics
literature, motivating the following definition.

######  Definition 1.11.

Given data of a compact Lie group $G$ and a class $\mu\in
H^{4}(BG;\mathbb{Z})$, the _twisted Bianchi identity_ is the equation

(1.12) $\mathrm{d}H=\mathrm{CW}_{\mu}(\Theta_{P}),$

where $H$ is a $3$-form and $\Theta_{P}$ is a connection on a principal
$G$-bundle.

As in the case of (1.10), we think of this as mixing the data of two
connections, one on a principal $G$-bundle and one on a gerbe. In the next
section, we interpret twisted Bianchi identities as coming from connections on
higher groups.

###  1.2. From the Bianchi identity to higher groups

In this section, we show that the twisted Bianchi identity (1.12) is a natural
consequence of combining a principal $G$-bundle and a gerbe, each with
connections, into a principal $\mathbb{G}$-bundle, where $\mathbb{G}$ is a
certain Lie $2$-group built from $G$ and $\mu$, together with additional data
that we think of as a connection on $\mathbb{G}$. First we introduce
$2$-groups and their principal bundles; then, following [FSS12, SSS12], we
recover the twisted Bianchi identity. As a result, we can precisely define the
tangential structure for the $\mathrm{E}_{8}\times\mathrm{E}_{8}$ heterotic
string, i.e. the topological part of the data which, when put on a manifold
$M$, allows one to study $\mathrm{E}_{8}\times\mathrm{E}_{8}$ heterotic string
theory on that manifold.

######  Definition 1.13.

A _$2$ -group_ $\mathbb{G}$ is a group object in the bicategory of small
categories.

######  Definition 1.14.

A _Lie $2$-group_ is a $2$-group $\mathbb{G}$ whose underlying category has
been given the structure of a category object in smooth manifolds.

This means that the sets of objects and morphisms are smooth manifolds, and
assignments such as the source of a map or the composition of two maps are
smooth. $2$-groups were first introduced by Hoàng Xuân Sính in her thesis
[Hoá75], and Lie $2$-groups were introduced by Baez [Bae02, §2].

We call a $2$-group _strict_ if it is strict as a monoidal category, i.e. its
associators and unitors are all identity maps. Mac Lane’s coherence theorem
[Mac71, Chapter 7] implies every $2$-group is equivalent to a strict
$2$-group, but the analogous statement is false for Lie $2$-groups; see Remark
1.25.

######  Example 1.15.

If $G$ is a group, it defines a monoidal groupoid with $G$ as its set of
objects, tensor product $g\otimes h\coloneqq gh$, and only the identity
morphisms. This is a $2$-group, and inherits the structure of a Lie $2$-group
if $G$ is a Lie group.

This procedure embeds the bicategory of groups, group homomorphisms, and
identity $2$-morphisms into the bicategory of $2$-groups, and we will
therefore abuse notation and call this $2$-group $G$ again.

######  Example 1.16.

Let $A$ be an abelian group, and let $A[1]$ denote the monoidal groupoid with
a single object $*$ and $\operatorname{Hom}_{A[1]}(*,*)\coloneqq A$. This is a
$2$-group, and if $A$ is Lie, $A[1]$ is a Lie $2$-group.

It turns out every $2$-group $\mathbb{G}$ factors as an extension of these
examples. Let $e$ be the identity object of $\mathbb{G}$ and
$\pi_{0}(\mathbb{G})$ be the group of isomorphism classes of objects in
$\mathbb{G}$. Then there is a short exact sequence of $2$-groups

(1.17)
${0}$${{\mathrm{Aut}_{\mathbb{G}}(e)[1]}}$${{\mathbb{G}}}$${{\pi_{0}(\mathbb{G})}}$${{0.}}$

The Eckmann-Hilton theorem guarantees $\mathrm{Aut}_{\mathbb{G}}(e)$ is
abelian. Extensions (1.17) are classified by the data of:

  1. (1)

an action of $\pi_{0}(\mathbb{G})$ on $\mathrm{Aut}_{\mathbb{G}}(e)$, and

  2. (2)

a cohomology class $k\in
H^{3}(B\pi_{0}(\mathbb{G});\mathrm{Aut}_{\mathbb{G}}(e))$, called the _$k$
-invariant_ of $\mathbb{G}$.

When $\mathbb{G}$ has the discrete topology, this is unambiguous, but when
$\mathbb{G}$ is a Lie $2$-group, one must be careful what kind of cohomology
is used here. The correct notion of cohomology is the _Segal-Mitchison
cohomology_ [Seg70, Seg75] of $\pi_{0}(\mathbb{G})$ valued in the abelian Lie
group $\mathrm{Aut}_{\mathbb{G}}(e)$, as shown by Schommer-Pries [SP11,
Theorem 1].

Now we want to discuss principal $\mathbb{G}$-bundles. The idea is that if $G$
is a group, a principal $G$-bundle is a submersion which is locally trivial,
and whose fibers are $G$-torsors. For a Lie $2$-group $\mathbb{G}$, we need
the fibers to locally look like $\mathbb{G}$, meaning they must be
categorified somehow.

######  Definition 1.18 (Bartels [Bar06], Nikolaus-Waldorf [NW13, Definition
6.1.5]).

Let $\mathbb{G}$ be a Lie $2$-group. A _principal $\mathbb{G}$-bundle_ over a
smooth manifold $M$ is a Lie groupoid $\mathcal{P}$ with a surjective
submersion $\mathrm{obj}(P)\to M$ and a smooth right action $\rho$ of
$\mathbb{G}$ on $\mathcal{P}$ such that the map

(1.19)
$(\mathrm{pr_{1}},\rho)\colon\mathcal{P}\times\mathbb{G}\longrightarrow\mathcal{P}\times_{M}\mathcal{P}$

is a weak equivalence of Lie groupoids.

See Nikolaus-Waldorf [NW13, §6] for more details. The principal
$\mathbb{G}$-bundles on a manifold $M$ form a $2$-groupoid
$\mathit{\mathscr{B}un}_{\mathbb{G}}(X)$ [NW13, Theorem 6.2.1].

######  Definition 1.20.

Let $\mathbb{G}$ be a $2$-group, and let $C_{\mathbb{G}}$ be the bicategory
with a single object $*$ and morphism category
$\operatorname{Hom}_{C_{\mathbb{G}}}(*,*)\coloneqq\mathbb{G}$. The
_classifying space_ of $\mathbb{G}$, denoted $B\mathbb{G}$, is the geometric
realization of the nerve of $C_{\mathbb{G}}$.666There are many different
definitions of the nerve of a bicategory; the fact that their geometric
realizations are canonically homotopy equivalent is a theorem of Carrasco-
Cegarra-Garzón [CCG10], allowing us to speak about $B\mathbb{G}$ without
specifying which kind of bicategorical nerve to use.

When $\mathbb{G}$ is a Lie $2$-group, we make the same definition. This time
$C_{\mathbb{G}}$ is a topological bicategory, so its nerve is a simplicial
space, and geometrically realizing, we obtain the space $B\mathbb{G}$.

######  Theorem 1.21 (Nikolaus-Waldorf [NW13, Theorems 4.6, 5.3.2, 7.1]).

If $\mathbb{G}$ is a strict Lie $2$-group, then there is a natural equivalence
$[X,B\mathbb{G}]\overset{\simeq}{\to}\pi_{0}(\mathit{\mathscr{B}un}_{\mathbb{G}}(X))$.

Nikolaus-Waldorf’s proof builds on Baez-Stevenson’s related but distinct
characterization of $[X,B\mathbb{G}]$ [BS09, Theorem 1] in terms of nonabelian
Čech cohomology.

When $G$ is an ordinary group, if $G$ has the discrete topology, $BG$ has only
one nonzero homotopy group, which is $\pi_{1}(BG)=G$; likewise if $\mathbb{G}$
is a discrete $2$-group, $\pi_{i}(B\mathbb{G})$ is nontrivial only for
$i=1,2$; $\pi_{1}(B\mathbb{G})=\pi_{0}(\mathbb{G})$ and
$\pi_{2}(B\mathbb{G})=\mathrm{Aut}_{\mathbb{G}}(e)$. When $\mathbb{G}$ is a
Lie $2$-group, we have no control over its homotopy groups in general, just
like $BG$ when $G$ is positive-dimensional.

If $\mathbb{G}$ has the discrete topology, the data classifying (1.17), namely
the action of $\pi_{0}(\mathbb{G})$ on $\mathrm{Aut}_{\mathbb{G}}(e)$ and the
$k$-invariant, is equivalent to the Postnikov data of $B\mathbb{G}$, worked
out by Mac Lane-Whitehead [MLW50]: this data classifies fibrations over $BG$
with fiber the Eilenberg-Mac Lane space $K(\mathrm{Aut}_{\mathbb{G}}(e),2)$.
The total space of the fibration with this Postnikov data is homotopy
equivalent to $B\mathbb{G}$.

######  Example 1.22.

Let $G$ be a compact Lie group; then, the Segal-Mitchison cohomology group
$H^{3}_{\mathrm{SM}}(G;\mathbb{T})$ classifying Lie $2$-group extensions of
$G$ by $\mathbb{T}[1]$ is naturally isomorphic to $H^{4}(BG;\mathbb{Z})$
[SP11, Corollary 97]. Therefore given a class $\mu\in H^{4}(BG;\mathbb{Z})$,
we obtain a Lie $2$-group $\mathit{\mathcal{S}tr}(G,\mu)$ fitting into a
central extension

(1.23)
${0}$${{\mathbb{T}[1]}}$${{\mathit{\mathcal{S}tr}(G,\mu)}}$${{G}}$${{0,}}$

which is sometimes called the _string $2$-group_ or _string cover_ associated
to $G$ and $\lambda$. Of all the string covers, the most commonly studied one
is
$\mathrm{String}_{n}\coloneqq\mathit{\mathcal{S}tr}(\mathrm{Spin}_{n},\lambda)$,
which is called _the_ string $2$-group.

This class of $2$-groups was first studied by Baez-Lauda [BL04, §8.5].

The sequence (1.23) implies that upon taking classifying spaces,

(1.24) $B\mathbb{G}\longrightarrow
BG\overset{\mu}{\longrightarrow}K(\mathbb{Z},4)$

is a fibration.

######  Remark 1.25.

Theorem 1.21 classified principal $\mathbb{G}$-bundles when $\mathbb{G}$ is a
strict $2$-group, but it is a theorem of Baez-Lauda [BL04, Corollary 60] that
there is no strict Lie $2$-group model for $\mathit{\mathcal{S}tr}(G,\mu)$
when $G$ is simply connected and $\mu\neq 0$. However, there is a fix: in the
setting of _Fréchet Lie $2$-groups_, where we allow the spaces of objects and
morphisms of $\mathbb{G}$ to be Fréchet manifolds, there is a strict model for
$\mathit{\mathcal{S}tr}(G,\mu)$ [BSCS07, LW23], so
$B\mathit{\mathcal{S}tr}(G,\mu)$ actually classifies principal
$\mathit{\mathcal{S}tr}(G,\mu)$-bundles. This suffices for studying bordism
groups.

Following Sati-Schreiber-Stasheff [SSS12], we now relate
$\mathit{\mathcal{S}tr}(G,\mu)$ to the twisted Bianchi identity for $G$ and
$\mu$. To do so, we use the language of stacks and differential cohomology,
following [HS05, FH13, Sch13, BNV16, ADH21]. Make the category
$\mathit{\mathscr{M}an}$ into a site by defining the covers to be surjective
submersions, and define a _stack_ to be a functor of $\infty$-categories
$\mathit{\mathscr{M}an}^{\mathit{op}}\to\mathit{\mathscr{T}op}$ which
satisfies descent for hypercovers. This defines a presentable
$\infty$-category $\mathit{\mathscr{S}t}$ of stacks [Lur09, Proposition
6.5.2.14], and the Yoneda embedding
$h\colon\mathit{\mathscr{M}an}\to\mathit{\mathscr{S}t}$ embeds
$\mathit{\mathscr{M}an}$ as a full subcategory. We will often simply write $M$
for the stack $h(M)$; we never compare these two notions directly, so this
will not introduce confusion.

For any space $X$, the functor
$\mathrm{Map}(\text{--},X)\colon\mathit{\mathscr{M}an}\to\mathit{\mathscr{T}op}$
is a sheaf, and this procedure defines a functor of $\infty$-categories
$\Gamma^{*}\colon\mathit{\mathscr{T}op}\to\mathit{\mathscr{S}t}$. The values
of the stacks produced by $\Gamma^{*}$ evaluated on manifolds $M$ are
homotopy-invariant in $M$. $\Gamma^{*}$ has a left adjoint
$\Gamma_{\sharp}\colon\mathit{\mathscr{S}t}\to\mathit{\mathscr{T}op}$ (see
Dugger [Dug01, Proposition 8.3], Morel-Voevodsky [MV99, Proposition 3.3.3],
and [ADH21, Proposition 4.3.1]); $\Gamma_{\sharp}(\mathbf{X})$ for a stack
$\mathbf{X}$ can be thought of as the best approximation to $\mathbf{X}$ by a
stack whose values on manifolds are homotopy-invariant.

Let $\Delta_{\mathrm{alg}}^{n}\coloneqq\\{(t_{0},\dotsc,t_{n})\mid
t_{0}+\dotsb+t_{n}=1\\}\subset\mathbb{R}^{n+1}$. These “algebraic
$n$-simplices” assemble into a cosimplicial manifold
$\Delta_{\mathrm{alg}}^{\bullet}$, and [ADH21, Corollary 5.1.4] there is a
natural homotopy equivalence
$\Gamma_{\sharp}(\mathbf{X})\simeq\lvert\mathbf{X}(\Delta_{\mathrm{alg}}^{\bullet})\rvert$,
where as usual $\lvert\text{--}\rvert$ denotes geometric realization.

Thus, for a manifold $M$, there is a natural homotopy equivalence
$\Gamma_{\sharp}(M)\overset{\simeq}{\to}M$, so a map $M\to\mathbf{X}$
naturally induces a map $M\to\Gamma_{\sharp}(\mathbf{X})$.

######  Lemma 1.26.

Suppose $\mathbf{X}\to\mathbf{Y}\leftarrow\mathbf{Z}$ is a diagram in
$\mathit{\mathscr{S}t}$, and that $\mathbf{Y}(\Delta_{\mathrm{alg}}^{n})$ and
$\mathbf{Z}(\Delta_{\mathrm{alg}}^{n})$ are connected for all $n$. Then

(1.27)
$\Gamma_{\sharp}(\mathbf{X}\times_{\mathbf{Y}}\mathbf{Z})\simeq\Gamma_{\sharp}(\mathbf{X})\times_{\Gamma_{\sharp}(\mathbf{Y})}\Gamma_{\sharp}(\mathbf{Z}).$

###### Proof.

Pullbacks of sheaves can be computed pointwise, then sheafifying, so given a
pullback $\mathbf{X}\to\mathbf{Y}\leftarrow\mathbf{Z}$ in
$\mathit{\mathscr{S}t}$, for each $n\geq 0$ the pullback of

(1.28)
$\mathbf{X}(\Delta_{\mathrm{alg}}^{n})\longrightarrow\mathbf{Y}(\Delta_{\mathrm{alg}}^{n})\longleftarrow\mathbf{Z}(\Delta_{\mathrm{alg}}^{n})$

is $(\mathbf{X}\times_{\mathbf{Y}}\mathbf{Z})(\Delta_{\mathrm{alg}}^{n})$. The
Bousfield-Friedlander theorem [BF78, Bou01] implies that, given the hypotheses
on $\mathbf{Y}$ and $\mathbf{Z}$ in the theorem statement, the homotopy
pullback of the geometric realizations of $\mathbf{X}$, $\mathbf{Y}$, and
$\mathbf{Z}$ is the geometric realization of the levelwise homotopy pullback
(1.28) (see [War20, p. 14-9] for this specific consequence of the Bousfield-
Friedlander theorem). ∎

######  Example 1.29.

For $G$ a Lie group, there is a stack $B_{\nabla}G$ whose value on a manifold
$M$ is the geometric realization of the nerve of the groupoid of principal
$G$-bundles on $M$ with connection [FH13]. This object is denoted
$\mathbf{B}G_{\mathrm{conn}}$ in [FSS12, SSS12, Sch13], $\mathbb{B}G^{\nabla}$
in [BNV16, §5], and $\mathrm{Bun}_{G}^{\nabla}$ in [ADH21].

There is a natural homotopy equivalence
$\Gamma_{\sharp}(B_{\nabla}G)\overset{\simeq}{\to}BG$ [ADH21, Corollary
13.3.29], which can be interpreted as forgetting from a principal bundle with
connection to a principal bundle.

######  Example 1.30.

For $k\geq 0$, there is a stack $B_{\nabla}^{k}\mathbb{T}$ whose value on a
manifold $M$ is the geometric realization of the nerve of the
$\infty$-groupoid of cocycles for the differential cohomology group
$\check{H}^{k+1}(M;\mathbb{Z})$. This object is studied in [FSS12, SSS12,
Sch13], where it is denoted $\mathbf{B}^{k}U(1)_{\mathrm{conn}}$.

######  Lemma 1.31.

There is a homotopy equivalence
$\Gamma_{\sharp}(B_{\nabla}^{k}\mathbb{T})\simeq K(\mathbb{Z},k+1)$.

###### Proof.

Schreiber [Sch13, Observation 1.2.134] produces the following pullback square
in $\mathit{\mathscr{S}t}$:

(1.32)
${{B_{\nabla}^{k}\mathbb{T}}}$${{\Omega_{\mathit{c\ell}}^{k+1}}}$${{K(\mathbb{Z},k+1)}}$${{K(\mathbb{R},k+1),}}$$\scriptstyle{\lrcorner}$

where $\Omega_{\mathit{c\ell}}^{k+1}$ is the stack of closed $(k+1)$-forms.
For that stack and $K(\mathbb{R},n+1)$, the values on each
$\Delta_{\mathrm{alg}}^{n}$ are connected spaces, so Lemma 1.26 identifies
$\Gamma_{\sharp}(B_{\nabla}^{k}\mathbb{T})\simeq
K(\mathbb{Z},k+1)\times_{K(\mathbb{R},k+1)}\Gamma_{\sharp}(\Omega_{\mathit{c\ell}}^{k+1})$.
To finish, observe that, essentially by the de Rham theorem, the map
$\Omega_{\mathit{c\ell}}^{k+1}\to K(\mathbb{R},k+1)$ passes to a homotopy
equivalence after applying $\Gamma_{\sharp}$. This follows from [BNV16, Lemma
7.15] together with the Dold-Kan theorem. ∎

These stacks are the universal setting for the Chern-Weil map.

######  Theorem 1.33 (Cheeger-Simons [CS85, Theorem 2.2], Bunke-Nikolaus-Völkl
[BNV16, §5.2]).

Let $G$ be a compact Lie group and $c\in H^{k}(BG;\mathbb{Z})$, where $k$ is
even. Then there is a map $\check{c}\colon B_{\nabla}G\to
B^{k-1}_{\nabla}\mathbb{T}$ natural in $(G,c)$ such that for any manifold $M$
and map $f\colon M\to B_{\nabla}G$, interpreted as a principal $G$-bundle
$P\to M$ with connection $\Theta$,

  1. (1)

if $\mathit{char}\colon\check{H}^{*}(\text{--};\mathbb{Z})\to
H^{*}(\text{--};\mathbb{Z})$ denotes the characteristic class map, then
$\mathit{char}(\check{c}\circ f)=c(P)$, and

  2. (2)

if
$\mathit{curv}\colon\check{H}^{*}(\text{--};\mathbb{Z})\to\Omega^{*}_{\mathit{c\ell}}$
denotes the curvature map, then $\mathit{curv}(\check{c}\circ
f)=\mathrm{CW}_{c}(\Theta)$.

Cheeger-Simons lifted the Chern-Weil map to differential cohomology; Bunke-
Nikolaus-Völkl recast it in terms of $B_{\nabla}G$. The map $\mathit{char}$ in
Theorem 1.33 is the map down the left of the square (1.32); $\mathit{curv}$ is
the map across the top of (1.32).

######  Definition 1.34 (Fiorenza-Schreiber-Stasheff [FSS12, §6.2]).

Given a compact Lie group $G$ and a class $\mu\in H^{4}(BG;\mathbb{Z})$, let
$\mathbf{BStr}(G,\mu)$ denote the fiber of the map $\check{\mu}\colon
B_{\nabla}G\to B_{\nabla}^{3}\mathbb{T}$.

We will see momentarily that maps to $\mathbf{BStr}(G,\mu)$ lead to solutions
to the twisted Bianchi identity for $G$ and $\mu$.

######  Proposition 1.35.

There is a natural homotopy equivalence
$\Gamma_{\sharp}(\mathbf{BStr}(G,\mu))\simeq B\mathit{\mathcal{S}tr}(G,\mu)$.

For this reason we think of $\mathbf{BStr}(G,\mu)$ as the classifying stack of
principal $\mathit{\mathcal{S}tr}(G,\mu)$-bundles with connection, though this
is only a heuristic.777There are at least _five_ notions of a connection on
principal $\mathbb{G}$-bundles for $\mathbb{G}$ a $2$-group: three are
discussed by Waldorf [Wal18, §5], a fourth by Rist-Saemann-Wolf [RSW22], and a
fifth, defined only for $\mathbb{G}=\mathrm{String}_{n}$, by Waldorf [Wal13].

###### Proof.

Apply Lemma 1.26 to the diagram

(1.36)
$B_{\nabla}G\overset{\check{\mu}}{\longrightarrow}B^{3}_{\nabla}\mathbb{T}\longleftarrow*,$

as the values of both $*$ and $B^{3}_{\nabla}\mathbb{T}$ are connected on
$\Delta_{\mathrm{alg}}^{n}$ for each $n$. This implies that
$\Gamma_{\sharp}(\mathbf{BStr}(G,\mu))$ is the fiber of $\mu\colon BG\to
K(\mathbb{Z},4)$, which we identified with $B\mathbb{G}$ in (1.24). ∎

######  Proposition 1.37 (Fiorenza-Schreiber-Stasheff [FSS12, §6.3]).

Let $G$ be a compact Lie group, $U\subset\mathbb{R}^{n}$ be an open set, and
$P\to U$ be a principal $G$-bundle with connection $\Theta$. A lift of the
corresponding map $f_{P,\Theta}\colon U\to B_{\nabla}G$ to a map
$\widetilde{f}_{P,\Theta}\colon U\to\mathbf{BStr}(G,\mu)$ induces a form
$H\in\Omega^{3}(U)$ such that $H$ and $\Theta$ satisfy the twisted Bianchi
identity (1.12).

The idea here is that we have specified a trivialization of the differential
characteristic class $\check{\mu}(P,\theta)$. Applying the curvature map
$\mathit{curv}\colon B_{\nabla}^{3}\mathbb{T}\to\Omega^{4}_{\mathit{c\ell}}$,
we have also specified a trivialization of $\mathrm{CW}_{\mu}(\Theta)$, which
locally is the data $H$ showing that $\mathrm{CW}_{\mu}(\Theta)$ is exact.

A map to $\mathbf{BStr}(G,\mu)$ is more data than what we get from Proposition
1.37, as we have trivialized not just the Chern-Weil form, but also the
differential characteristic class. This can be interpreted as saying the data
$H$ specifying the trivialization is quantized to form a twisted version of a
gerbe with connection.

To summarize, given a map $M\to\mathbf{BStr}(G,\mu)$, the stack which we think
of as modeling $\mathit{\mathcal{S}tr}(G,\mu)$-bundles with connection, we
obtain:

  1. (1)

a principal $\mathit{\mathcal{S}tr}(G,\mu)$-bundle $\mathcal{P}\to M$ by
Proposition 1.35, and

  2. (2)

a “twisted gerbe with connection,” i.e. local data of a gerbe $Q\to M$ such
that $\Omega_{Q}$ and the $G$-connection $\Theta$ induced by the map
$\mathbf{BStr}(G,\mu)\to B_{\nabla}G$ satisfy the twisted Bianchi identity
(1.12) by Proposition 1.37.

Motivated by this, we define of the tangential structure for the
$\mathrm{E}_{8}\times\mathrm{E}_{8}$ heterotic string. This first appears in
[SSS12, §3.2], with [Sat11b, FSS15a] considering some related examples.

######  Definition 1.38.

Let $G\coloneqq(\mathrm{E}_{8}\times\mathrm{E}_{8})\rtimes\mathbb{Z}/2$. A
_differential ${\xi_{n}^{\mathrm{het}}}$-structure_ on a manifold $M$ is the
following data:

  1. (1)

a Riemannian metric and spin structure on $M$,

  2. (2)

a principal $G$-bundle $P\to M$ with connection $\Theta$, and

  3. (3)

a lift of

(1.39) $((B_{\mathrm{Spin}}(M),\Theta^{\mathrm{LC}}),(P,\Theta))\colon
M\longrightarrow B_{\nabla}(\mathrm{Spin}_{n}\times G)$

to a map $M\to\mathbf{BStr}(\mathrm{Spin}_{n}\times G,c_{1}+c_{2}-\lambda)$.

Here $B_{\mathrm{Spin}}(M)\to M$ is the principal $\mathrm{Spin}_{n}$-bundle
of frames of $M$, and $\Theta^{\mathrm{LC}}$ denotes its Levi-Civita
connection.

For bordism groups we want the topological version of this.

######  Definition 1.40.

A _tangential structure_ is a space $B$ and a map $\xi\colon B\to
B\mathrm{O}$. Given a tangential structure $\xi$, a _$\xi$ -structure_ on a
virtual vector bundle $E\to X$ is a lift of the classifying map $f_{E}\colon
X\to B\mathrm{O}$ to a map $\widetilde{f}_{E}\colon X\to B$ such that
$\xi\circ\widetilde{f}_{E}=f_{E}$. A _$\xi$ -structure_ on a manifold $M$ is a
$\xi$-structure on its tangent bundle.

We make the analogous definition with maps $\xi_{n}\colon B_{n}\to
B\mathrm{O}_{n}$; in this case, we only refer to $\xi_{n}$-structures on
$n$-manifolds.

Lashof [Las63] defined bordism groups $\Omega_{*}^{\xi}$ of manifolds with
$\xi$-structure, and Boardman [Boa65, §V.1] defined a Thom spectrum
$\mathit{MT\xi}$ whose homotopy groups are naturally isomorphic to
$\Omega_{*}^{\xi}$ via the Pontrjagin-Thom construction.888In homotopy theory,
it is common to study the Thom spectra $\mathit{M\xi}$ representing
$\xi$-structures on the stable _normal_ bundle $\nu_{M}$ of a manifold $M$,
and indeed many of the results we cite about $\mathit{MTSO}$,
$\mathit{MTString}$, etc. are stated for $\mathit{MSO}$, $\mathit{MString}$,
etc., or about Thom spectra $\mathit{M\xi}$ in general. This is not a problem:
for any tangential structure $\xi$, there is a tangential structure
$\xi^{\perp}$ such that a $\xi$-structure on $TM$ is equivalent data to a
$\xi^{\perp}$-structure on $\nu_{M}$ and vice versa, so that
$\mathit{MT\xi}\simeq\mathit{M\xi^{\perp}}$, so the general theory is the
same. And for $\xi=\mathrm{O}$, $\mathrm{SO}$, $\mathrm{Spin}$,
$\mathrm{Spin}^{c}$, and $\mathrm{String}$, $\xi\simeq\xi^{\perp}$ and in
those cases we can ignore the difference between $\mathit{M\xi}$ and
$\mathit{MT\xi}$. We think of the category of tangential structures as the
category of spaces over $B\mathrm{O}$, and bordism groups and Thom spectra are
functorial in this category. That is, taking bordism groups and Thom spectra
is functorial as long as one commutes with the map down to $B\mathrm{O}$.

The following definition is a special case of a definition due to Sati-
Schreiber-Stasheff [SSS12, Definition 2.8]. See [Sat11b, FSS15a, FSS21] for
other related examples.

######  Definition 1.41.

Let
$G_{n}\coloneqq\mathrm{Spin}_{n}\times(\mathrm{E}_{8}\times\mathrm{E}_{8})\rtimes\mathbb{Z}/2$
and

(1.42)
$\mathbb{G}^{\mathrm{het}}_{n}\coloneqq\mathit{\mathcal{S}tr}(G_{n},c_{1}+c_{2}-\lambda).$

The _$\mathrm{E}_{8}\times\mathrm{E}_{8}$ heterotic tangential structure_ is
the tangential structure

(1.43) ${\xi_{n}^{\mathrm{het}}}\colon
B\mathbb{G}^{\mathrm{het}}_{n}\longrightarrow
B\mathrm{Spin}_{n}\longrightarrow B\mathrm{O}_{n},$

where the first map comes from the quotient of $\mathbb{G}^{\mathrm{het}}$ by
$\mathbb{T}[1]$, followed by projection onto the $\mathrm{Spin}_{n}$ factor in
$G_{n}$. We also define $\mathbb{G}^{\mathrm{het}}$ and ${\xi^{\mathrm{het}}}$
analogously by stabilizing in $n$.

In other words: a differential ${\xi_{n}^{\mathrm{het}}}$-structure is a lift
of a map to $B_{\nabla}(\mathrm{Spin}\times G)$ to $\mathbf{BStr}(G,\mu)$; by
Proposition 1.35, a topological ${\xi_{n}^{\mathrm{het}}}$-structure is the
image of this data under $\Gamma_{\sharp}$. In particular, a
${\xi_{n}^{\mathrm{het}}}$-structure on an $n$-manifold $M$ includes data of a
principal $\mathbb{G}^{\mathrm{het}}_{n}$-bundle $\mathcal{P}\to M$.

Taking the quotient of $\mathbb{G}^{\mathrm{het}}$ by $\mathbb{T}[1]$ induces
a map of tangential structures

(1.44) $\phi\colon B\mathbb{G}^{\mathrm{het}}\longrightarrow
B\mathrm{Spin}\times B(\mathrm{E}_{8}^{2}\rtimes\mathbb{Z}/2).$

Thus, much like a spin$c$ manifold $M$ has an associated $\mathbb{T}$-bundle
$P$ with $c_{1}(P)\bmod 2=w_{2}(M)$, a ${\xi^{\mathrm{het}}}$-manifold has
associated $(\mathrm{E}_{8}^{2}\rtimes\mathbb{Z}/2)$-bundle $P$. From this
perspective, a ${\xi^{\mathrm{het}}}$-structure on a manifold $M$ is the
following data:

  * •

a spin structure on $M$,

  * •

a double cover $\pi\colon\widetilde{M}\to M$,

  * •

two principal $\mathrm{E}_{8}$-bundles $P,Q\to\widetilde{M}$ which are
exchanged by the nonidentity deck transformation of $\pi$, and

  * •

a trivialization of the class $\lambda(M)-(c(P)+c(Q))\in H^{4}(M;\mathbb{Z})$.

By a _trivialization_ of a cohomology class $\alpha\in H^{n}(M;A)$ we mean a
null-homotopy of the classifying map $f_{\alpha}\colon M\to K(A,n)$. Thus
orientations are identified with trivializations of $w_{1}$, etc. To make the
trivialization of $\lambda(M)-(c(P)+c(Q))$ precise, we have to descend the
class $c(P)+c(Q)$, a priori an element of $H^{4}(\widetilde{M};\mathbb{Z})$,
to $H^{4}(M;\mathbb{Z})$. We can do this because, as noted in Definition 1.4,
the class $c_{1}+c_{2}$ descends through the Serre spectral sequence to the
base.

######  Remark 1.45.

We can combine some the data of a ${\xi^{\mathrm{het}}}$ structure on $M$ into
a twisted characteristic class. Let $\mathbb{Z}^{\sigma}$ be the
$\mathbb{Z}[\mathbb{Z}/2]$-module isomorphic to $\mathbb{Z}^{2}$ as an abelian
group, and in which the nontrivial element of $\mathbb{Z}/2$ swaps the two
factors. Then, let $\mathbb{Z}^{\sigma}_{\pi}$ denote the local system on $M$
which is the associated bundle
$\widetilde{M}\times_{\mathbb{Z}/2}\mathbb{Z}^{\sigma}$. A pair of classes
$x,y\in H^{k}(\widetilde{M};\mathbb{Z})$ exchanged by the deck transformation
thus define a class in $H^{k}(M;\mathbb{Z}^{\sigma}_{\pi})$, so, the classes
$c(P)$ and $c(Q)$ in $H^{4}(\widetilde{M};\mathbb{Z})$ together define a class
$\widetilde{c}(P,Q)\in H^{4}(M;\mathbb{Z}^{\sigma}_{\pi})$, which is a
characteristic class of an
$((\mathrm{E}_{8}\times\mathrm{E}_{8})\rtimes\mathbb{Z}/2)$-bundle.

If $\mathbb{Z}$ denotes the $\mathbb{Z}[\mathbb{Z}/2]$-module isomorphic to
$\mathbb{Z}$ as an abelian group and with trivial $\mathbb{Z}/2$-action, then
taking the quotient of $\mathbb{Z}^{\sigma}$ by the submodule generated by
$(1,-1)$ defines a map of $\mathbb{Z}[\mathbb{Z}/2]$-modules
$q\colon\mathbb{Z}^{\sigma}\to\mathbb{Z}$, hence also a map between the
corresponding twisted cohomology groups, and this map sends
$\widetilde{c}(P,Q)\mapsto c(P)+c(Q)$. Therefore one could recast a
${\xi^{\mathrm{het}}}$-structure on a spin manifold $M$ as the data of a
principal $((\mathrm{E}_{8}\times\mathrm{E}_{8})\rtimes\mathbb{Z}/2)$-bundle
$(P,Q,\pi)$ together with a trivialization of
$\lambda(M)-q(\widetilde{c}(P,Q))$.

Bott-Samelson [BS58, Theorems IV, V(e)] showed that the map
$B\mathrm{E}_{8}\to K(\mathbb{Z},4)$ defined by the characteristic class $c$
is $15$-connected. This implies that up to isomorphism, a principal
$((\mathrm{E}_{8}\times\mathrm{E}_{8})\rtimes\mathbb{Z}/2)$-bundle on a
manifold of dimension $15$ or lower is equivalent data to its characteristic
class $\widetilde{c}$.

######  Remark 1.46.

One might want to simplify by restricting to the special case where
$\pi\colon\widetilde{M}\to M$ is trivial (as done in, e.g., [Wit86]), in which
case the data of a ${\xi^{\mathrm{het}}}$-structure simplifies to the data of
a spin structure on $M$, two principal $\mathrm{E}_{8}$-bundles $P,Q\to M$,
and a trivialization of $\lambda(M)-c(P)-c(Q)$. This corresponds to the
tangential structure $\xi^{r,\mathrm{het}}\colon
B\mathit{\mathcal{S}tr}(\mathrm{Spin}\times\mathrm{E}_{8}\times\mathrm{E}_{8},c_{1}+c_{2}-\lambda)\to
B\mathrm{Spin}\to B\mathrm{O}$.

###  1.3. The CHL string

Eleven-dimensional $\mathcal{N}=1$ supergravity admits a time-reversal
symmetry, allowing it to be defined on pin$+$ $11$-manifolds.999In addition to
the pin$+$ structure, one needs the additional data of a lift of $w_{4}(TM)$
to $w_{1}(TM)$-twisted integral cohomology. See [Wit97, Wit16, FH21a].
Therefore we can compactify it on a Möbius strip with certain boundary data to
obtain a nine-dimensional supergravity theory; the goal of this subsection is
to determine the tangential structure of this theory. Eleven-dimensional
$\mathcal{N}=1$ supergravity is expected to be the low-energy limit of a
theory called M-theory,101010M-theory is expected to require additional data
on top of the tangential structure described above for $11$-dimensional
$\mathcal{N}=1$ supergravity. See [FSS20, Table 1] and the references listed
there. and compactifying M-theory on the Möbius strip is expected to produce a
string theory called the _Chaudhuri-Hockney-Lykken (CHL) string_ [CHL95] whose
low-energy limit is the $9$-dimensional supergravity theory described above;
we study the tangential structure of this supergravity theory in this
subsection with the aim of also learning about the CHL string.

However, we do not want our perspective on the CHL string to be overly one-
sided. There is another way to produce the CHL string by compactifying:
consider the circle with its nontrivial principal $\mathbb{Z}/2$-bundle $P\to
S^{1}$. Via the map
$\mathbb{Z}/2\hookrightarrow\mathrm{Spin}\times((\mathrm{E}_{8}\times\mathrm{E}_{8})\rtimes\mathbb{Z}/2)$,
this bundle defines a
$\mathrm{Spin}\times((\mathrm{E}_{8}\times\mathrm{E}_{8})\rtimes\mathbb{Z}/2)$-structure
on $S^{1}$ for which $\lambda$ and $c_{1}+c_{2}$ are both trivial, so this
structure lifts to define a ${\xi^{\mathrm{het}}}$-structure on $S^{1}$. We
will call the circle with this ${\xi^{\mathrm{het}}}$-structure
$\mathbb{RP}^{1}$, as $S^{1}\cong\mathbb{RP}^{1}$ as manifolds and the
${\xi^{\mathrm{het}}}$-structure comes from the double cover
$S^{1}\to\mathbb{RP}^{1}$. The CHL string is precisely what one obtains by
compactifying the $\mathrm{E}_{8}^{2}$ heterotic string on $\mathbb{RP}^{1}$.

We want to determine the tangential structure ${\xi^{\mathrm{CHL}}}$ such that
the product of $\mathbb{RP}^{1}$ with a manifold with
${\xi^{\mathrm{CHL}}}$-structure has an induced
${\xi^{\mathrm{het}}}$-structure. In general, keeping track of how the
tangential structure changes under compactification can be subtle; for a
careful analysis, see Schommer-Pries [SP18, §9]. But for the CHL string, we
can get away with a more ad hoc approach: following Chaudhuri-Polchinski
[CP95] (see also [dBDH+00, §2.2.1]) we restrict to the case where the
principal $\mathbb{Z}/2$-bundle on $\mathbb{RP}^{1}\times M$ obtained by the
quotient map (1.44) is the pullback of the Möbius bundle
$S^{1}\to\mathbb{RP}^{1}$ along the projection
$\mathrm{pr}_{1}\colon\mathbb{RP}^{1}\times M\to\mathbb{RP}^{1}$.

######  Proposition 1.47.

Let $M$ be a spin manifold and $P\to M$ be a principal
$\mathrm{E}_{8}$-bundle. The data of a trivialization $\mathfrak{s}$ of
$\lambda(M)-2c(P)$ induces a ${\xi^{\mathrm{het}}}$-structure on
$\mathbb{RP}^{1}\times M$ whose associated principal $\mathbb{Z}/2$-bundle is
the Möbius bundle $S^{1}\times M\to\mathbb{RP}^{1}\times M$. Moreover, if
$\dim(M)\leq 14$, this assignment is a natural bijection from the set of
isomorphism classes of data $(P,\mathfrak{s})$ to the set of
${\xi^{\mathrm{het}}}$-structures on $\mathbb{RP}^{1}\times M$ whose
associated $\mathbb{Z}/2$-bundle is $S^{1}\times M\to\mathbb{RP}^{1}\times M$.

###### Proof.

Let $\pi\colon S^{1}\times M\to M$ be the projection onto the second factor.
Given $P\to M$ and $\mathfrak{s}$, the pair of $\mathrm{E}_{8}$-bundles
$(\pi^{*}P,\pi^{*}P)\to S^{1}\times M$ are exchanged by the deck
transformation for $S^{1}\times M\to\mathbb{RP}^{1}\times M$, and
$(c_{1}+c_{2})$ evaluated on the pair $(\pi^{*}P,\pi^{*}P)$ is $2c(P)\in
H^{4}(\mathbb{RP}^{1}\times M;\mathbb{Z}/2)$. Choosing the string structure on
$\mathbb{RP}^{1}$ induced from the bounding framing, we obtain a canonical
trivialization of $\lambda(\mathbb{RP}^{1}\times M)-\lambda(M)\in
H^{4}(\mathbb{RP}^{1}\times M;\mathbb{Z})$ from the two-out-of-three property
of string structures. Putting all of this together, we see that we have data
of two $\mathrm{E}_{8}$-bundles on $S^{1}\times M$ exchanged by the deck
transformation, and a trivialization of $\lambda-(c_{1}+c_{2})$ on
$\mathbb{RP}^{1}\times M$, thus defining a ${\xi^{\mathrm{het}}}$-structure as
claimed.

To see that this produces all ${\xi^{\mathrm{het}}}$-structures associated
with $S^{1}\times M\to\mathbb{RP}^{1}\times M$, recall from Remark 1.45 that
the $((\mathrm{E}_{8}\times\mathrm{E}_{8})\rtimes\mathbb{Z}/2)$-bundle
associated to a ${\xi^{\mathrm{het}}}$-structure is classified by a
characteristic class in twisted cohomology. The assumption that the associated
$\mathbb{Z}/2$-bundle is $S^{1}\times M\to\mathbb{RP}^{1}\times M$ implies
this class belongs to $H^{4}(\mathbb{RP}^{1}\times
M;\underline{\mathbb{Z}\oplus\mathbb{Z}})$, where a generator of
$\pi_{1}(\mathbb{RP}^{1})$ acts on $\underline{\mathbb{Z}\oplus\mathbb{Z}}$ by
swapping the two factors, and $\pi_{1}(M)$ acts trivially. The twisted Künneth
formula [Gre06, Theorem 1.7] gives us an isomorphism

(1.48) $H^{4}(\mathbb{RP}^{1}\times
M;\underline{\mathbb{Z}\oplus\mathbb{Z}})\overset{\cong}{\longrightarrow}H^{4}(M;\mathbb{Z}),$

meaning that the pair of $\mathrm{E}_{8}$-bundles on the orientation double
cover $S^{1}\times M$ pull back from bundles on $M$, which must be isomorphic
in order to be exchanged by the $\mathbb{Z}/2$-action. ∎

The Bianchi identity corresponding to this data can therefore be simplified to
use a single bundle $P\to M$ and the class $c(P)+c(P)$: we obtain

(1.49)
$\mathrm{d}H=\mathrm{CW}_{2c}(\Theta_{P})-\mathrm{CW}_{\lambda}(\Theta^{\mathrm{LC}}),$

i.e. the twisted Bianchi identity for $G=\mathrm{Spin}\times\mathrm{E}_{8}$
and $\mu=2c-\lambda$. Then, following Definitions 1.41 and 1.38, we make the
following definitions.

######  Definition 1.50.

A _differential $\xi_{n}^{\mathrm{CHL}}$-structure_ on a manifold $M$ is the
following data:

  1. (1)

a Riemannian metric and spin structure on $M$,

  2. (2)

a principal $\mathrm{E}_{8}$-bundle $P\to M$ with connection $\Theta$, and

  3. (3)

a lift of

(1.51) $((B_{\mathrm{Spin}}(M),\Theta^{\mathrm{LC}}),(P,\Theta))\colon
M\longrightarrow B_{\nabla}(\mathrm{Spin}_{n}\times\mathrm{E}_{8})$

to a map
$M\to\mathbf{BStr}(\mathrm{Spin}_{n}\times\mathrm{E}_{8},2c-\lambda)$.

What we call $B\mathbb{G}^{\mathrm{CHL}}_{n}$ coincides with Sati-Schreiber-
Stasheff’s $B\mathrm{String}^{2a}$ [SSS12, (2.18), §2.3.3] and also appears in
work of Fiorenza-Sati-Schreiber [FSS15a, Remark 4.1.1], though those papers do
not discuss its relationship with the CHL string.

######  Definition 1.52 (Sati-Schreiber-Stasheff [SSS12, (2.18), §2.3.3]).

Let

(1.53)
$\mathbb{G}^{\mathrm{CHL}}_{n}\coloneqq\mathit{\mathcal{S}tr}(\mathrm{Spin}_{n}\times\mathrm{E}_{8},2c-\lambda).$

The _CHL tangential structure_ is the tangential structure

(1.54) $\xi_{n}^{\mathrm{CHL}}\colon
B\mathbb{G}^{\mathrm{CHL}}_{n}\longrightarrow
B\mathrm{Spin}_{n}\longrightarrow B\mathrm{O}_{n},$

where the first map comes from the quotient of $\mathbb{G}^{\mathrm{CHL}}$ by
$\mathbb{T}[1]$, followed by projection onto the $\mathrm{Spin}_{n}$ factor.
Stabilizing in $n$, we also obtain $\mathbb{G}^{\mathrm{CHL}}$ and a
tangential structure ${\xi^{\mathrm{CHL}}}$.

A ${\xi^{\mathrm{CHL}}}$-structure on an $n$-manifold $M$ in particular comes
with data of a principal $\mathbb{G}^{\mathrm{CHL}}_{n}$-bundle
$\mathcal{P}\to M$, and can be formulated as the data of a principal
$\mathrm{E}_{8}$-bundle $P\to M$ and a trivialization of $\lambda(M)-2c(P)\in
H^{4}(M;\mathbb{Z})$.

######  Remark 1.55.

Since a ${\xi^{\mathrm{CHL}}}$ structure includes data identifying $\lambda$
as twice another class, it induces a trivialization of the mod $2$ reduction
of $\lambda$, which is $w_{4}$. That is, a ${\xi^{\mathrm{CHL}}}$ structure
induces a $\mathrm{Spin}\langle w_{4}\rangle$ structure, where
$B\mathrm{Spin}\langle w_{4}\rangle$ is the homotopy fiber of $w_{4}\colon
B\mathrm{Spin}\to K(\mathbb{Z},4)$. This structure has been studied in, e.g.
[Wit97, KS04, FH21a] for applications to M-theory.

######  Remark 1.56 (Variation of the tangential structure along the moduli
space).

There is a moduli space of CHL string theories, not just one, and the gauge
group depends on where in the moduli space one is; this moduli space was first
studied by Chaudhuri-Polchinski [CP95]. At a generic point, the gauge group is
broken to $\mathbb{T}^{8}$, and at various special points the gauge group
enhances to $\mathrm{E}_{8}$ or other nonabelian groups: see [FFG+21, Table
3]. We work only at the $\mathrm{E}_{8}$ point of the moduli space in this
paper; it would be interesting to apply the techniques in this paper to other
points in the CHL moduli space.

There has been quite a bit of recent research studying the moduli spaces of
compactifications of the $\mathrm{E}_{8}\times\mathrm{E}_{8}$ heterotic string
and the CHL string, and investigating which gauge groups can occur [FGN18,
CDLZ20, FFG+20, CDLZ21, FFG+21, FPDF21, MV21, CDLZ22, CGH22, CMM22, FPDF22,
PDF23, MPDF23].

##  2\. Bordism computations

Now it is time to compute. We will use the Adams spectral sequence to compute
$\Omega_{*}^{\xi^{\mathrm{het}}}$ and $\Omega_{*}^{\xi^{\mathrm{CHL}}}$; this
is a standard tool in computational homotopy theory and more recently appears
frequently in the mathematical physics literature, and we point the interested
reader to Beaudry-Campbell’s introductory article [BC18].

Applications of the Adams spectral sequence to mathematical physics questions
tend to follow the same formula. Suppose that we want to compute
$\Omega_{*}^{\xi}$ for some tangential structure $\xi$.

  1. (1)

First, express $\xi$ as a “twisted $\xi^{\prime}$-structure,” where
$\xi^{\prime}$ is one of $\mathrm{SO}$, $\mathrm{Spin}$, $\mathrm{Spin}^{c}$,
or $\mathrm{String}$: prove that a $\xi$-structure on a vector bundle $E\to M$
is equivalent data to an auxiliary vector bundle $V\to M$ and a
$\xi^{\prime}$-structure on $E\oplus V$.

This implies that $\mathit{MT\xi}\simeq\mathit{MT\xi^{\prime}}\wedge X$ for
some Thom spectrum $X$ that is usually not too complicated.

  2. (2)

Next, invoke a change-of-rings theorem to greatly simplify the calculation of
the $E_{2}$-page for $\xi^{\prime}$-bordism of spaces or spectra. Then run the
Adams spectral sequence, taking advantage of the extra structure afforded by
the change-of-rings theorem.

This recipe goes back to work of Anderson-Brown-Peterson [ABP69] and Giambalvo
[Gia73b, Gia73a, Gia76] computing twisted spin bordism. It is most commonly
used in the case $\xi^{\prime}=\mathrm{Spin}$, where it has been frequently
used to compute bordism groups for tangential structures representing field
theories with fermions; $\xi^{\prime}=\mathrm{String}$ is less common but
still appears in physically motivated examples, including the tangential
structure of the Sugimoto string [Sug99] and $\xi=\mathrm{String}^{c}$ [CHZ11,
Sat11b].

Unfortunately, ${\xi^{\mathrm{het}}}$ and ${\xi^{\mathrm{CHL}}}$ do not belong
to this class of examples: we will see in Lemma 2.2 that there is no way to
write these tangential structures as twisted string structures in the sense
above.111111The presence of the B-field, and how the Bianchi identity mixes it
with the principal $\mathrm{Spin}_{n}$-bundle of frames, rules out
$\xi^{\prime}=\mathrm{SO}$, $\mathrm{Spin}$, or
$\mathrm{Spin}^{c}$.,121212This problem also happens to the tangential
structures studied in [FH21a, DY22]. So we have to do something different.

At odd primes, we plow ahead with the unsimplified Adams spectral sequence,
though since we only care about dimensions $11$ and below the computations are
very tractable. At $p=2$, though, we can modify the above strategy to simplify
the computation: in §2.1, we generalize the notion of “twisted string bordism”
for which the change-of-rings trick works to include string covers (in the
sense of Example 1.22) of groups of the form $\mathrm{Spin}\times G$. This
applies to both ${\xi^{\mathrm{het}}}$ and ${\xi^{\mathrm{CHL}}}$, and so we
are off to the races.

######  Remark 2.1.

We are far from the first to compute bordism groups for a tangential structure
$\xi\colon B\to B\mathrm{O}$ where $B$ is the classifying space of a
$2$-group. For example, $\Omega_{*}^{\mathrm{String}}$ has been calculated in
a range of degrees by [Gia71, HR95, MG95, Hov08]; other examples include
[Hil09, KT17, WW19a, WW19b, WWZ19, Tho20, LT21, Yu21, DL23].

###  2.1. Twists of string bordism

> “Started out with a twist, how did it end up like this?
> It was only a twist, it was only a twist…”

Once the tangential structure for a bordism question is known, the next step
is typically to prove a “shearing” theorem simplifying the tangential
structure. For example, the usual route to computing pin$-$ bordism [Pet68,
§7] first establishes an isomorphism between pin$-$ bordism and the spin
bordism of the Thom spectrum $\Sigma^{-1}\mathit{MO}_{1}$, and then computes
the latter groups using something like the Adams or Atiyah-Hirzebruch spectral
sequence.

There are a few different approaches to shearing theorems, such as those in
[FH21b, DDHM23], but generally they work with Thom spectra of vector bundles;
for example, the above simplification of pin$-$ bordism begins with the
observation that a pin$-$ structure on a bundle $E\to M$ is equivalent data to
a real line bundle $L\to M$ and a spin structure on $E\oplus L$, which follows
from a characteristic class computation, and then passes the data of “$L$ and
a spin structure on $E\oplus L$” through the Pontrjagin-Thom theorem.

This approach does not work for the heterotic and CHL tangential structures.

######  Lemma 2.2.

There is no spin vector bundle $V$ on
$B((\mathrm{E}_{8}\times\mathrm{E}_{8})\rtimes\mathbb{Z}/2)$ such that
$\lambda(V)=c_{1}+c_{2}$, and there is no spin vector bundle $W$ on
$B\mathrm{E}_{8}$ such that $\lambda(V)=2c$.

This means there is no way to express a ${\xi^{\mathrm{het}}}$-structure as “a
$G$-bundle and a string structure on $E$ plus some associated bundle,” and
likewise for ${\xi^{\mathrm{CHL}}}$.

###### Proof.

Let $G$ be a compact, simple, simply connected Lie group and $\rho\colon
G\to\mathrm{SU}_{n}$ be a representation. $H^{4}(BG;\mathbb{Z})$ and
$H^{4}(B\mathrm{SU}_{n};\mathbb{Z})$ are both canonically isomorphic to
$\mathbb{Z}$, so the pullback map $\rho^{*}$ on $H^{4}$ is a map
$\mathbb{Z}\to\mathbb{Z}$, necessarily multiplication by some integer
$\delta(\rho)$. Because $\mathrm{SU}_{n}$ is compact, connected, and simply
connected, the standard inclusion
$\mathrm{SU}_{n}\to\mathrm{GL}_{2n}(\mathbb{R})$ lifts to a map
$\mathrm{SU}_{n}\to\mathrm{Spin}_{2n}$. Choices of this lift are a torsor over
$H^{1}(B\mathrm{SU}_{n};\mathbb{Z}/2)=0$, meaning that the characteristic
class $\lambda$ is uniquely defined for $\mathrm{SU}_{n}$-representations.
Moreover, $\lambda$ of the defining representation is a generator of
$H^{4}(B\mathrm{SU}_{n};\mathbb{Z})$; because
$H^{4}(B\mathrm{SU}_{n};\mathbb{Z})$ is torsion-free, it suffices to show
$2\lambda=p_{1}$ is twice a generator, which is standard. The _Dynkin index_
of $G$ is the minimum value of $\lvert\delta(\rho)\rvert$ over all such
representations $\rho$. Laszlo-Sorger [LS97, Proposition 2.6] show that the
Dynkin index of $\mathrm{E}_{8}$ is $60$, meaning that for any vector bundle
$V\to B\mathrm{E}_{8}$ with $\mathrm{SU}$-structure induced from a
representation, $\lambda(V)$ is at least $60$ times a generator.

We would like to generalize to real representations.

######  Lemma 2.3.

The complexification map $\mathrm{Spin}_{n}\to\mathrm{O}_{n}\to\mathrm{U}_{n}$
has image contained in $\mathrm{SU}_{n}$.

###### Proof.

A lift of a representation $\rho\colon G\to\mathrm{U}_{n}$ has image contained
in $\mathrm{SU}_{n}$ if and only if $c_{1}$ of the complex vector bundle
associated to $\rho$ vanishes. When one pulls back across the complexification
map $B\mathrm{O}_{n}\to B\mathrm{U}_{n}$, $c_{1}$ is sent to the image of
$w_{1}$ under the Bockstein map $\beta\colon
H^{1}(B\mathrm{O}_{n};\mathbb{Z}/2)\to H^{2}(B\mathrm{O}_{n};\mathbb{Z})$;
when we pull back further to $B\mathrm{Spin}_{n}$, $w_{1}\mapsto 0$, so
$c_{1}=\beta w_{1}\mapsto 0$ too. ∎

Thus the Dynkin index fact we mentioned above applies to complexifications of
representations landing in $\mathrm{Spin}_{n}$.

If $V$ is a real representation of a group $G$, $V\otimes\mathbb{C}\cong
V\oplus V$ as real representations, so using the Whitney sum formula for
$\lambda$ (Lemma 1.6), $\lambda(V\otimes\mathbb{C})=2\lambda(V)$. Therefore if
$V$ is any real spin representation of $\mathrm{E}_{8}$,
$\lambda(V\otimes\mathbb{C})$ is at least $60$ times a generator, so
$\lambda(V)$ is at least $30$ times a generator. Thus the class defining
$\mathbb{G}^{\mathrm{CHL}}$, which is twice a generator, is not $\lambda$ of
any spin representation of $\mathrm{E}_{8}$; likewise for
$\mathbb{G}^{\mathrm{het}}$, as one could restrict to either factor of
$\mathrm{E}_{8}$ inside $\mathrm{E}_{8}^{2}\rtimes\mathbb{Z}/2$ and obtain a
representation with $\lambda$ equal to the generator.

Finally, the Atiyah-Segal completion theorem extends this from representations
to all vector bundles. Because $\lambda$ is additive (Lemma 1.6), it factors
through the Grothendieck group $\mathit{KSpin}(BG)$ of spin vector bundles on
$BG$, and similarly, evaluated on spin representations, $\lambda$ factors
through the corresponding Grothendieck group $\mathit{RSpin}(G)$. Atiyah-Segal
[AS69, §7, §8] show that taking the associated bundle of an arbitrary
representation exhibits the Grothendieck ring $\mathit{KO}^{0}(BG)$ of all
vector bundles on $BG$ as the completion of the representation ring
$\mathit{RO}(G)$ at its augmentation ideal. Thus given a $\mathbb{Z}$-valued
characteristic class $c$ of arbitrary vector bundles of $G$ which satisfies
the Whitney sum formula, passing from representations of $G$ to vector bundles
on $BG$ does not decrease the minimal value of $\lvert c\rvert$.

In order to use the Atiyah-Segal theorem, we need to get from spin
representations and vector bundles to arbitrary ones. We will do so, at the
cost of lowering the minimum value of $\lambda$ a little bit. For any vector
bundle $V$, $V^{\oplus 4}$ admits a canonical spin structure: the Whitney sum
formula for Stiefel-Whitney classes shows a spin structure exists; then choose
a spin structure universally over $B\mathrm{O}$. Therefore we can define
$\lambda$ of an arbitrary representation of $\mathrm{E}_{8}$ or vector bundle
on $B\mathrm{E}_{8}$ by $\lambda(V)\coloneqq\tfrac{1}{4}\lambda(V^{\oplus
4})$, valued in $\tfrac{1}{4}\mathbb{Z}$. Therefore passing from
$\mathit{RO}(\mathrm{E}_{8})\to\mathit{KO}^{0}(B\mathrm{E}_{8})$ to
$\mathit{RSpin}(\mathrm{E}_{8})\to\mathit{KSpin}(B\mathrm{E}_{8})$ divides the
minimal value of $\lambda$ by at most $4$, and now we can invoke Atiyah-Segal,
so it is still not possible to get $2c$ and ${\xi^{\mathrm{CHL}}}$; and
likewise for $\mathrm{E}_{8}^{2}\rtimes\mathbb{Z}/2$ in place of
$\mathrm{E}_{8}$ to show that the characteristic class for
${\xi^{\mathrm{het}}}$ cannot be achieved. ∎

So we take a different approach: we cannot get Thom spectra corresponding to
vector bundles, but we can still obtain $\mathit{MTString}$-module Thom
spectra. We accomplish this using the theory of Ando-Blumberg-Gepner-Hopkins-
Rezk [ABG+14a, ABG+14b] (ABGHR), which we briefly summarize.

The idea behind the ABGHR perspective on Thom spectra is to generalize the
notion of local coefficients to generalized cohomology theories. Given a
based, connected space $X$ and a homomorphism
$\rho\colon\pi_{1}(X)\to\mathrm{GL}_{1}(\mathbb{Z})\cong\\{\pm 1\\}$, one
obtains a local coefficient system $\mathbb{Z}_{\rho}$ on $X$: this is a
bundle on $X$ with fiber $\mathbb{Z}$, and whose monodromy around a loop
$\gamma\in\pi_{1}(X)$ is precisely $\rho(\gamma)$. Given $\mathbb{Z}_{\rho}$,
we can take twisted cohomology groups: if $\widetilde{X}\to X$ denotes the
universal cover, then the cochain complex $C^{\ast}(\widetilde{X};\mathbb{Z})$
has a $\pi_{1}(X)$-action induced from the $\pi_{1}(X)$-action on
$\widetilde{X}$. If $C^{\ast}(X;\mathbb{Z}_{\rho})$ denotes the subcomplex of
$C^{\ast}(\widetilde{X};\mathbb{Z})$ of cochains which transform under this
$\pi_{1}(X)$-action by $\rho$, then $H^{\ast}(X;\mathbb{Z}_{\rho})\coloneqq
H^{\ast}(C^{\ast}(X;\mathbb{Z}_{\rho}))$.

Another way to say this is that if $\mathrm{pt}/G$ denotes the category with
one object $\ast$ and $\operatorname{Hom}(\ast,\ast)=G$, $\rho$ defines a
$\mathrm{pt}/\pi_{1}(X)$-shaped diagram of chain complexes of abelian groups:

(2.4) $\mathrm{pt}/\pi_{1}(X)\overset{\rho}{\longrightarrow}\mathrm{pt}/\\{\pm
1\\}\longrightarrow\mathit{\mathscr{C}h}_{\mathbb{Z}},$

sending $\mathrm{pt}$ to $C^{\ast}(\widetilde{X};\mathbb{Z})$, and sending
$g\in\pi_{1}(X)$ to the action by $\rho(g)$. The subcomplex of cochains that
transform by $\rho$ is precisely the limit of this diagram. For functoriality
reasons, we envision this complex as cochains on some object $\mathcal{X}$
which is a _co_ limit of a diagram akin to (2.4).

To summarize, twisted cohomology, i.e. cohomology of the Thom spectrum, is
expressed as a colimit of a diagram of chain complexes of $\mathbb{Z}$-modules
induced from a map $X\to B\mathrm{Aut}(\mathbb{Z})$. Ando-Blumberg-Gepner-
Hopkins-Rezk lift this to spectra. Specifically, given a ring spectrum $R$,
Ando-Blumberg-Gepner-Hopkins-Rezk naturally associate a topological
group131313$\mathrm{GL}_{1}(R)$ is not exactly a topological group, but the
homotopy-coherent version thereof: a grouplike $A_{\infty}$-space.
$\mathrm{GL}_{1}(R)$, thought of as the group of units or group of
automorphisms of $R$. The classifying space $B\mathrm{GL}_{1}(R)$ carries the
universal local system of $R$-lines; a local system of $R$-lines over $X$ is
equivalent data to a map $X\to B\mathrm{GL}_{1}(R)$.

######  Definition 2.5 (Ando-Blumberg-Gepner-Hopkins-Rezk [ABG+14a, Definition
2.20]).

The _Thom spectrum_ $Mf$ associated to a map $f\colon X\to
B\mathrm{GL}_{1}(R)$ is the colimit of the diagram $X\to
B\mathrm{GL}_{1}(R)\to\mathit{\mathscr{M}od}_{R}$, where we think of $X$ as
its fundamental $\infty$-groupoid.

When $R=\mathbb{S}$, this is due to Lewis [LMSM86, Chapter IX]. In Definition
2.5, we have to consider the fundamental $\infty$-groupoid, rather than just
$\pi_{1}$, because $R$ can have higher automorphisms, because spectra are
derived objects.

The Thom spectrum of a map to $B\mathrm{GL}_{1}(R)$ is an $R$-module.

######  Example 2.6 (Twisted ordinary cohomology).

It turns out $B\mathrm{GL}_{1}(H\mathbb{Z})\simeq K(\mathbb{Z}/2,1)$, so the
ABGHR viewpoint recovers $\mathrm{Aut}(\mathbb{Z})$ and the usual notion of
cohomology twisted by a local system. To prove this homotopy equivalence, use
the homotopy pullback square of $E_{\infty}$-spaces [ABG+14b, Definition 2.1]

(2.7)
${{\mathrm{GL}_{1}(H\mathbb{Z})}}$${{\Omega^{\infty}H\mathbb{Z}}}$${{(\pi_{0}(H\mathbb{Z}))^{\times}}}$${{\pi_{0}(H\mathbb{Z}).}}$$\scriptstyle{\varphi}$$\scriptstyle{\psi}$

$\Omega^{\infty}H\mathbb{Z}\simeq\mathbb{Z}$ as $E_{\infty}$-spaces, and
$\psi$ is a homotopy equivalence of $E_{\infty}$-spaces. Therefore $\varphi$
is also a homotopy equivalence of $E_{\infty}$-spaces, and we conclude.

######  Example 2.8 (Thom spectra from vector bundles).

Boardman’s original definition of Thom spectra [Boa65, §V.1] associates them
to virtual vector bundles $V\to X$. Let us connect this to the ABGHR
definition. Virtual vector bundles are classified by maps $f_{V}\colon X\to
B\mathrm{O}$, and one avatar of the $J$-homomorphism [Whi42] is a map
$J\colon\mathrm{O}\to\mathrm{GL}_{1}(\mathbb{S})$ [ABG10, Example 3.15], which
deloops to a map of spaces $BJ\colon B\mathrm{O}\to
B\mathrm{GL}_{1}(\mathbb{S})$. A map with this signature is a natural
assignment from virtual vector bundles $V\to X$ to local systems of invertible
$\mathbb{S}$-modules, and $BJ$ assigns to $V$ the local system with fiber
$\mathbb{S}^{V_{x}}$ at each $x\in X$. Putting these maps together, we have an
$X$-shaped diagram

(2.9)
$X\overset{f_{V}}{\longrightarrow}B\mathrm{O}\overset{BJ}{\longrightarrow}B\mathrm{GL}_{1}(\mathbb{S})\longrightarrow\mathit{\mathscr{S}p},$

and the colimit of this diagram, which is a Thom spectrum in the ABGHR sense,
coincides with the Thom spectrum $X^{V}$ in the usual sense. This is a
combination of theorems of Lewis [LMSM86, Chapter IX] and Ando-Blumberg-
Gepner-Hopkins-Rezk [ABG+14a, Corollary 3.24].

This approach to Thom spectra plays well with multiplicative structures. If
$R$ is an $E_{\infty}$-ring spectrum, then the grouplike
$A_{\infty}$-structure on $\mathrm{GL}_{1}(R)$ refines to a grouplike
$E_{\infty}$-structure, making $\mathrm{GL}_{1}(R)$ and therefore
$B\mathrm{GL}_{1}(R)$ into infinite loop spaces. For $1\leq k\leq\infty$, if
$X$ is a $k$-fold loop space and $f\colon X\to B\mathrm{GL}_{1}(R)$ is a
$k$-fold loop map, then the Thom spectrum $Mf$ inherits the structure of an
$E_{k}$-ring spectrum. This is a theorem of Lewis [LMSM86, Theorem IX.7.1] for
$R=\mathbb{S}$ and Ando-Blumberg-Gepner [ABG18, Theorem 1.7] for more general
$R$.

$B\mathrm{O}$ has an infinite loop space structure coming from the addition-
like operation on $B\mathrm{O}$ of direct sum of vector bundles. The
$J$-homomorphism $BJ\colon B\mathrm{O}\to B\mathrm{GL}_{1}(\mathbb{S})$ is an
infinite loop map, so we get an $E_{\infty}$-ring structure on
$\mathit{MT}\xi$ if $\xi$ is a tangential structure satisfying a _2-out-of-3
property_ , i.e. whenever any two of $E$, $F$, and $E\oplus F$ have a
$\xi$-structure, the third has an induced $\xi$-structure. The idea is that
the 2-out-of-3 property implies that $\xi\colon B\to B\mathrm{O}$ is an
infinite loop map, so passing to $B\mathrm{GL}_{1}(\mathbb{S})$ and taking the
Thom spectrum, we obtain an $E_{\infty}$-ring spectrum. This applies to
$\mathit{MTO}$, $\mathit{MTSO}$, $\mathit{MTSpin}^{c}$, $\mathit{MTSpin}$, and
$\mathit{MTString}$; however, some commonly considered tangential structures
appearing in physics do not have this property, including
$B\mathrm{Pin}^{\pm}$.

######  Proposition 2.10.

Let $B$ and $X$ be infinite loop spaces and $\xi\colon B\to B\mathrm{O}$ and
$f\colon B\to X$ be infinite loop maps, so that the fiber $\eta\colon F\to B$
of $f$ is also a map of infinite loop spaces. This data naturally defines
twists of the Thom spectrum $M(\xi\circ\eta)$ over $X$, i.e. a map $X\to
B\mathrm{GL}_{1}(M(\xi\circ\eta))$.

###### Proof.

The fiber of $\eta\colon F\to B$ is another infinite loop map
$\zeta\colon\Omega X\to F$, so the induced map of Thom spectra (where the maps
down to $B\mathrm{O}$ are $\xi\circ\eta\circ\zeta$ and $\xi\circ\eta$
respectively) is a map of $E_{\infty}$-ring spectra. Because
$\xi\circ\eta\circ\zeta$ is nullhomotopic, its Thom spectrum is a suspension
spectrum, so we have a map of $E_{\infty}$-ring spectra
$\Sigma_{+}^{\infty}\Omega X\to M(\xi\circ\eta)$.

Ando-Blumberg-Gepner-Hopkins-Rezk [ABG+14b, (1.4), (1.7)] prove that
$\Sigma^{\infty}_{+}$ and $\mathrm{GL}_{1}$ are an adjoint pair on the
categories of infinite loop spaces and $E_{\infty}$-ring spectra. Applying
this adjunction, we have a map of infinite loop spaces $\Omega
X\to\mathrm{GL}_{1}(M(\xi\circ\eta))$; deloop to obtain the map in the theorem
statement. ∎

######  Theorem 2.11 (Beardsley [Bea17, Theorem 1]).

With notation as in Proposition 2.10, the Thom spectrum of the “universal
twist” $X\to B\mathrm{GL}_{1}(M(\xi\circ\eta))$ is canonically equivalent to
$M\xi$.

######  Corollary 2.12.

  1. (1)

There is a map $\widehat{w}_{1}\colon K(\mathbb{Z}/2,1)\to
B\mathrm{GL}_{1}(\mathit{MTSO})$ which, after taking the quotient
$\mathit{MTSO}\to H\mathbb{Z}$, passes to the usual homotopy equivalence
$K(\mathbb{Z}/2,1)\to B\mathrm{GL}_{1}(H\mathbb{Z})$ from Example 2.6.

  2. (2)

There is a map $\widehat{w}_{2}\colon K(\mathbb{Z}/2,2)\to
B\mathrm{GL}_{1}(\mathit{MTSpin})$ which, after composing with the Atiyah-
Bott-Shapiro map $\mathit{MTSpin}\to\mathit{ko}$ [ABS64, Joa04], is the usual
map $K(\mathbb{Z}/2,2)\hookrightarrow B\mathrm{GL}_{1}(\mathit{ko})$ [DK70,
HJ20].

  3. (3)

There is a map $\widehat{\beta w_{2}}\colon K(\mathbb{Z},3)\to
B\mathrm{GL}_{1}(\mathit{MTSpin}^{c})$ which, after composing with the Atiyah-
Bott-Shapiro map $\mathit{MTSpin}^{c}\to\mathit{ku}$ [ABS64, Joa04, AHR10], is
the usual twist of $K$-theory by degree-$3$ classes $K(\mathbb{Z},3)\to
B\mathrm{GL}_{1}(\mathit{ku})$ [DK70, Ros89, AS04, ABG10].

  4. (4)

There is a map $\widehat{\lambda}\colon K(\mathbb{Z},4)\to
B\mathrm{GL}_{1}(\mathit{MTString})$ which, when composed with the Ando-
Hopkins-Rezk orientation $\mathit{MTString}\to\mathit{tmf}$ [AHR10], is the
Ando-Blumberg-Gepner map $K(\mathbb{Z},4)\to B\mathrm{GL}_{1}(\mathit{tmf})$
[ABG10, Proposition 8.2].

Part (3) is a theorem of Hebestreit-Joachim [HJ20, Appendix C]. The other
parts are surely known, though we were unable to find them in the literature.

###### Proof.

Apply Proposition 2.10 to the four maps

  1. (1)

$w_{1}\colon B\mathrm{O}\to K(\mathbb{Z}/2,1)$, whose fiber is $B\mathrm{SO}$;

  2. (2)

$w_{2}\colon B\mathrm{SO}\to K(\mathbb{Z}/2,2)$, whose fiber is
$B\mathrm{Spin}$;

  3. (3)

$\beta\circ w_{2}\colon B\mathrm{SO}\to K(\mathbb{Z},3)$, whose fiber is
$B\mathrm{Spin}^{c}$, where $\beta\colon H^{k}(\text{--};\mathbb{Z}/2)\to
H^{k+1}(\text{--};\mathbb{Z})$ is the Bockstein; and

  4. (4)

$\lambda\colon B\mathrm{Spin}\to K(\mathbb{Z},4)$, whose fiber is
$B\mathrm{String}$.

All four of these are infinite loop maps, because these characteristic classes
are additive in direct sums. For compatibility with preexisting twists, we use
the fact that in the spin$c$ and string cases, Ando-Blumberg-Gepner [ABG10,
§7, §8] construct the desired twists $K(\mathbb{Z},3)\to
B\mathrm{GL}_{1}(\mathit{ku})$ and $K(\mathbb{Z},4)\to
B\mathrm{GL}_{1}(\mathit{tmf})$ in the same way as we construct the twists of
$\mathit{MTSpin}^{c}$ and $\mathit{MTString}$, so compatibility follows from
functoriality. The cases of $\mathit{ko}$ and $H\mathbb{Z}$ are analogous. ∎

The homotopy groups of the Thom spectra of the twists Corollary 2.12 have
bordism interpretations. Looking at $\widehat{w}_{2}$ for example, a spin
structure on an oriented manifold is a trivialization of $w_{2}(TM)$, but
given a space $X$ and a degree-$2$ cohomology class $B$, thought of as a map
$f_{B}\colon X\to K(\mathbb{Z}/2,2)$, the homotopy groups of
$\mathit{MT}(\widehat{w}_{2}\circ f_{B})$ are the bordism groups of oriented
manifolds $M$ together with a map $g\colon M\to X$ and a trivialization of
$w_{2}(TM)+g^{*}B$, as was shown by Hebestreit-Joachim [HJ20, Corollary
3.3.8]. The other three cases are analogous; in particular, we have described
the Thom spectra for ${\xi^{\mathrm{het}}}$ and ${\xi^{\mathrm{CHL}}}$ as
$\mathit{MTString}$-module Thom spectra.

These kinds of twisted bordism have been studied before: spin$c$ structures
twisted by a degree-$3$ cohomology class were first studied by Douglas [Dou06,
§5], and they appear implicitly in work of Freed-Witten [FW99] on anomaly
cancellation. Twisted spin and string structures of the sort appearing in
Corollary 2.12 were first considered by B.L. Wang [Wan08, Definitions 8.2,
8.4]. See [DFM11a, DFM11b, Sat11a, Sat11c, Sat12, Sat15, SW15, LSW20, SY21]
for more examples of twisted generalized cohomology theories from a similar
point of view and some applications in physics.

The first case, involving twists of $\mathit{MTSO}$ by degree-$1$
$\mathbb{Z}/2$-cohomology classes, is the notion of a twisted orientation from
the beginning of this section: given a real line bundle $L\to X$, we ask for
data of a map $g\colon M\to X$ and an orientation on $TM\oplus g^{*}(L)$. In
the ABGHR perspective this says that the map $\widehat{w}_{1}$ factors through
$B\mathrm{O}_{1}$ as

(2.13) $K(\mathbb{Z}/2,1)\overset{\simeq}{\to}B\mathrm{O}_{1}\hookrightarrow
B\mathrm{O}\to B\mathrm{GL}_{1}(\mathbb{S})\to
B\mathrm{GL}_{1}(\mathit{MTSO}).$

But the others do not factor this way.

######  Remark 2.14.

There is a complex version of (2.13). Let $\mathcal{W}$ denote _Wall’s bordism
spectrum_ [Wal60], whose homotopy groups are the bordism groups of manifolds
with an integral lift of $w_{1}$. Explicitly, if $\xi\colon F\to B\mathrm{O}$
is the fiber of $\beta w_{1}\colon B\mathrm{O}\to K(\mathbb{Z},2)$, then
$\mathcal{W}\coloneqq\mathit{MT\xi}$. Proposition 2.10 then produces a map
$\widehat{\beta w_{1}}\colon K(\mathbb{Z},2)\to
B\mathrm{GL}_{1}(\mathcal{W})$, but degree-$2$ cohomology classes are
equivalent to complex line bundles, and $\widehat{\beta w_{1}}$ factors as

(2.15) $K(\mathbb{Z},2)\overset{\simeq}{\to}B\mathbb{T}\to B\mathrm{O}_{2}\to
B\mathrm{O}\to B\mathrm{GL}_{1}(\mathbb{S})\to B\mathrm{GL}_{1}(\mathcal{W}).$

######  Remark 2.16.

One consequence of the fact that $\widehat{w}_{1}$ (resp. $\widehat{\beta
w}_{1}$) factors as in (2.13) (resp. (2.15)), i.e. as a twist associated to a
real (resp. complex) line bundle $L\to X$ is that the associated
$\mathit{MTSO}$-module (resp. $\mathcal{W}$-module) Thom spectrum splits as
$\mathit{MTSO}\wedge X^{L-1}$ (resp. $\mathcal{W}\wedge X^{L-2}$). Working
universally over $B\mathrm{O}_{1}$ and $B\mathbb{T}$, Theorem 2.11 gives us
homotopy equivalences
$\mathit{MTSO}\wedge(B\mathrm{O}_{1})^{L-1}\simeq\mathit{MTO}$ and
$\mathcal{W}\wedge(B\mathbb{T})^{L-2}\simeq\mathit{MTO}$; the former is a
theorem of Atiyah [Ati61, Proposition 4.1].

We will apply Corollary 2.12 to the degree-$4$ characteristic classes that the
Bianchi identity told us for the heterotic and CHL tangential structures.
Given a space $X$ with a class $\mu\in H^{4}(X;\mathbb{Z})$, let
$\mathcal{B}(X)$ denote the homotopy fiber of $\lambda+\mu\colon
B\mathrm{Spin}\times X\to K(\mathbb{Z},4)$, and let $\xi^{\mu}$ denote the
tangential structure

(2.17) $\xi^{\mu}\colon\mathcal{B}(X)\longrightarrow B\mathrm{Spin}\times
X\longrightarrow B\mathrm{O}.$

$\mathit{MT\xi}^{\mu}$ is equivalent to the $\mathit{MTString}$-module Thom
spectrum associated to the twist $\widehat{\lambda}\circ\mu\colon X\to
B\mathrm{GL}_{1}(\mathit{MTString})$. If $X=BG$ for a Lie group $G$,
$\mathcal{B}(X)$ is the classifying space of the string $2$-group
$\mathcal{S}(\mathrm{Spin}\times G,\lambda+\mu)$. Let $\mathcal{A}$ denote the
$2$-primary Steenrod algebra and for $n\geq 0$, let $\mathcal{A}(n)$ denote
the subalgebra of $\mathcal{A}$ generated by
$\mathrm{Sq}^{1},\dotsc,\mathrm{Sq}^{2^{n}}$. In work to appear joint with
Matthew Yu [DY], we compute the $\mathcal{A}$-module structure on
$H^{*}(\mathit{MT\xi}^{\mu};\mathbb{Z}/2)$.

######  Definition 2.18.

Let $R$ denote the $\mathbb{Z}/2$-algebra $\mathcal{A}(1)[S]$, i.e. the
algebra with generators $\mathrm{Sq}^{1}$, $\mathrm{Sq}^{2}$, and $S$, and
with Adem relations for $\mathrm{Sq}^{1}$ and $\mathrm{Sq}^{2}$. Given $X$ and
$\mu$ as above, define the $\mathcal{A}(1)$-module $T(X,\mu)\coloneqq
H^{*}(X;\mathbb{Z}/2)$, and give $T(X,\mu)$ an $R$-module structure by
defining

(2.19) $S(x)\coloneqq\mu x+\mathrm{Sq}^{4}(x).$

We want to think of $S$ as $\mathrm{Sq}^{4}$ and $T(X,\mu)$ as an
$\mathcal{A}(2)$-module, but a priori it is not clear that this $S$-action
satisfies the Adem relations.

######  Theorem 2.20 ([DY]).

  1. (1)

The $R$-module structure on $T(X,\mu)$ satisfies the Adem relations for
$\mathrm{Sq}^{1}$, $\mathrm{Sq}^{2}$, and $\mathrm{Sq}^{4}=S$, hence induces
an $\mathcal{A}(2)$-module structure on $T(X,\mu)$.

  2. (2)

There is an map of $\mathcal{A}$-modules

(2.21)
$H^{*}(\mathit{MT\xi}^{\mu};\mathbb{Z}/2)\longrightarrow\mathcal{A}\otimes_{\mathcal{A}(2)}T(X,\mu),$

natural in the data $(X,\mu)$, which is an isomorphism in degrees $15$ and
below.

As [DY] is not yet available, we describe a proof of this theorem in Remark
2.26.

######  Corollary 2.22.

For $t-s\leq 15$, the $E_{2}$-page of the Adams spectral sequence computing
$2$-completed $\xi^{\mu}$-bordism is

(2.23)
$E_{2}^{t,s}=\operatorname{Ext}_{\mathcal{A}(2)}^{s,t}(T(X,\mu),\mathbb{Z}/2).$

As $\mathcal{A}(2)$ is much smaller than $\mathcal{A}$, this is much easier to
work with.

###### Proof.

This follows from the change-of-rings formula: if $\mathcal{B}$ is a graded
Hopf algebra, $\mathcal{C}$ is a graded Hopf subalgebra of $\mathcal{B}$, and
$M$ and $N$ are graded $\mathcal{B}$-modules, then there is a natural
isomorphism

(2.24)
$\operatorname{Ext}_{\mathcal{B}}^{s,t}(\mathcal{B}\otimes_{\mathcal{C}}M,N)\overset{\cong}{\longrightarrow}\operatorname{Ext}_{\mathcal{C}}^{s,t}(M,N).$

This you can think of as the derived version of a maybe more familiar
isomorphism

(2.25)
$\operatorname{Hom}_{\mathcal{B}}(\mathcal{B}\otimes_{\mathcal{C}}M,N)\overset{\cong}{\longrightarrow}\operatorname{Hom}_{\mathcal{C}}(M,N).$

In our example, $\mathcal{B}$ is the Steenrod algebra, which is a Hopf
algebra, and $\mathcal{C}$ is $\mathcal{A}(2)$, which is indeed a Hopf
subalgebra of $\mathcal{A}$, so we can invoke (2.24) and conclude. ∎

We will use this simplification in the cases
$\xi^{\mu}={\xi^{\mathrm{het}}},{\xi^{\mathrm{CHL}}}$ to run the Adams
spectral sequences computing $\Omega_{*}^{{\xi^{\mathrm{het}}}}$ and
$\Omega_{*}^{{\xi^{\mathrm{CHL}}}}$ at $p=2$.

######  Remark 2.26 (Proof sketch of Theorem 2.20).

To prove (1), check the Adem relations for $\mathcal{A}(2)$ directly. The
first step in proving part (2) is to establish a Thom isomorphism for mod $2$
cohomology. We make use of the _Thom diagonal_ , a map of
$\mathit{MTString}$-modules

(2.27)
$\mathit{MT\xi}^{\mu}\overset{\Delta^{t}}{\longrightarrow}\mathit{MT\xi}^{\mu}\wedge\mathit{MTString}\wedge\Sigma_{+}^{\infty}X$

defined as follows: the diagonal map $\Delta\colon X\to X\times X$ is a map of
spaces over $B\mathrm{GL}_{1}(\mathit{MTString})$, if we give $X$ the map
$\widehat{\lambda}\circ\mu$ to $B\mathrm{GL}_{1}(\mathit{MTString})$ and we
give $X\times X$ the map $(\widehat{\lambda}\circ\mu,\ast)$. Applying the
$\mathit{MTString}$-module Thom spectrum functor to $\Delta$ produces (2.27).
Smash (2.27) with $H\mathbb{Z}/2$. The result is the Thom diagonal for a twist
of $H\mathbb{Z}/2$, but all such twists are trivializable (i.e. all
$H\mathbb{Z}/2$-bundles admit an orientation). Therefore by [ABG+14b,
Proposition 3.26] the following composition is an equivalence:

(2.28) $\mathit{MT\xi}^{\mu}\wedge
H\mathbb{Z}/2\overset{\Delta^{t}}{\longrightarrow}\mathit{MT\xi}^{\mu}\wedge\Sigma_{+}^{\infty}X\wedge
H\mathbb{Z}/2\longrightarrow\mathit{MTString}\wedge\Sigma_{+}^{\infty}X\wedge
H\mathbb{Z}/2,$

which is the $\mathbb{Z}/2$-homology Thom isomorphism. The analogous fact is
true for mod $2$ cohomology.

The Thom diagonal makes $H^{*}(\mathit{MT\xi}^{\mu};\mathbb{Z}/2)$ into a
free, rank-$1$ module over $H^{*}(\mathcal{B}(X);\mathbb{Z}/2)$, generated by
the Thom class $U$. As the Thom diagonal is a map of spectra, we may use the
Cartan formula to compute the Steenrod squares of an arbitrary element of
$H^{*}(\mathit{MT\xi}^{\mu};\mathbb{Z}/2)$ in terms of Steenrod squares in
$\mathcal{B}(X)$ and $\mathrm{Sq}(U)$. As both $\mathrm{Sq}(U)$ and our
desired isomorphism in (2.21) are natural in $X$ and $\mu$, it suffices to
understand the universal case, where $X=K(\mathbb{Z},4)$ and $\mu$ is the
tautological class $\tau\in H^{4}(K(\mathbb{Z},4);\mathbb{Z})$. In this case,
Theorem 2.11 implies $\mathit{MT\xi}^{\mu}\simeq\mathit{MTSpin}$. By work of
Anderson-Brown-Peterson [ABP67], if $J$ is the $\mathcal{A}(1)$-module
$\mathcal{A}(1)/\mathrm{Sq}^{3}$ and $M$ is the $\mathcal{A}(1)$-module
$\mathbb{Z}/2\oplus\Sigma^{8}\mathbb{Z}/2\oplus\Sigma^{10}J$, then there is a
map of $\mathcal{A}$-modules

(2.29)
$H^{*}(\mathit{MTSpin};\mathbb{Z}/2)\longrightarrow\mathcal{A}\otimes_{\mathcal{A}(1)}M$

which is an isomorphism in degrees $15$ and below. And Giambalvo [Gia71,
Corollary 2.3] shows that there is a map
$H^{*}(\mathit{MTString};\mathbb{Z}/2)\to\mathcal{A}\otimes_{\mathcal{A}(2)}\mathbb{Z}/2$
which is also an isomorphism in degrees $15$ and below. Therefore by the
change-of-rings theorem (2.24) it suffices to exhibit a map of
$\mathcal{A}(2)$-modules

(2.30)
$T(K(\mathbb{Z},4),\tau)\longrightarrow\mathcal{A}(2)\otimes_{\mathcal{A}(1)}M$

which is an isomorphism in degrees $15$ and below. This can be verified
directly, using as input the $\mathcal{A}(2)$-module structure on
$H^{*}(K(\mathbb{Z},4);\mathbb{Z}/2)$ calculated by Serre [Ser53, §10].

###  2.2. ${\xi^{\mathrm{het}}}$ bordism at $p=2$

In this section we will first compute $H^{\ast}(BG;\mathbb{Z}/2)$ as an
$\mathcal{A}(2)$-module in low degrees, where
$G\coloneqq\mathrm{E}_{8}^{2}\rtimes\mathbb{Z}/2$; then, using Corollary 2.22,
we run the Adams spectral sequence computing $2$-completed
${\xi^{\mathrm{het}}}$ bordism in degrees $11$ and below.

First, though, we reformulate the problem slightly. Consider the tangential
structure ${\xi^{\mathrm{het}}}^{\prime}\colon B^{\mathrm{het}^{\prime}}\to
B\mathrm{O}$ defined in the same manner as ${\xi^{\mathrm{het}}}$, but with
$K(\mathbb{Z},4)$ replacing $B\mathrm{E}_{8}$. In a little more detail,
$\mathbb{Z}/2$ acts on $K(\mathbb{Z},4)\times K(\mathbb{Z},4)$ by swapping the
two factors; taking the Borel construction

(2.31) $B\coloneqq(K(\mathbb{Z},4)\times
K(\mathbb{Z},4))\times_{\mathbb{Z}/2}E\mathbb{Z}/2$

produces a fiber bundle

(2.32) $K(\mathbb{Z},4)\times K(\mathbb{Z},4)\longrightarrow B\longrightarrow
B\mathbb{Z}/2.$

For $i=1,2$, let $c_{i}\in H^{4}(K(\mathbb{Z},4)\times
K(\mathbb{Z},4);\mathbb{Z})$ be the tautological class for the
$i^{\mathrm{th}}$ $K(\mathbb{Z},4)$ factor. The class $c_{1}+c_{2}$ is
invariant under the $\mathbb{Z}/2$-action, so we can follow it through the
Serre spectral sequence to learn that it defines a nonzero class
$c_{1}+c_{2}\in H^{4}(B;\mathbb{Z}/2)$. Define $f\colon
B^{\mathrm{het}^{\prime}}\to B\mathrm{Spin}\times B$ to be the fiber of
$\lambda-(c_{1}+c_{2})\colon B\mathrm{Spin}\times B\to K(\mathbb{Z},4)$; then
the tangential structure ${\xi^{\mathrm{het}}}^{\prime}$ is the composition

(2.33) ${{B^{\mathrm{het}^{\prime}}}}$${{B\mathrm{Spin}\times
B}}$${B\mathrm{Spin}}$${B\mathrm{O}.}$$\scriptstyle{\mathrm{pr}_{1}}$$\scriptstyle{f}$$\scriptstyle{{\xi^{\mathrm{het}}}^{\prime}}$

That is, a ${\xi^{\mathrm{het}}}^{\prime}$ structure on a manifold $M$ is a
spin structure, a principal $\mathbb{Z}/2$-bundle $P\to M$, two classes
$c_{1},c_{2}\in H^{4}(P;\mathbb{Z})$ which are exchanged under the deck
transformation, and a trivialization of $\lambda(M)-(c_{1}+c_{2})$ (where the
latter class is descended to $M$). This is the same data as a
${\xi^{\mathrm{het}}}$ structure, except that we do not ask for $c_{1}$ or
$c_{2}$ to come from principal $\mathrm{E}_{8}$-bundles; therefore there is a
map of tangential structures
$\widetilde{c}\colon{\xi^{\mathrm{het}}}\to{\xi^{\mathrm{het}}}^{\prime}$,
i.e. a map of spaces $B\mathbb{G}^{\mathrm{het}}\to B^{\mathrm{het}^{\prime}}$
commuting with the maps down to $B\mathrm{O}$. Like for
${\xi^{\mathrm{het}}}$, a ${\xi^{\mathrm{het}}}^{\prime}$-structure is a
twisted string structure in the sense of Corollary 2.12, via the class
$\lambda-(c_{1}+c_{2})\colon B\to K(\mathbb{Z},4)$.

Bott-Samelson [BS58, Theorems IV, V(e)] showed that the characteristic class
$c\in H^{4}(B\mathrm{E}_{8};\mathbb{Z})$ we defined in Definition 1.4,
interpreted as a map $c\colon B\mathrm{E}_{8}\to K(\mathbb{Z},4)$, is
$15$-connected. This means that the homomorphism $\widetilde{c}$ induces on
bordism groups,
$\widetilde{c}\colon\Omega_{k}^{{\xi^{\mathrm{het}}}}\to\Omega_{k}^{{\xi^{\mathrm{het}}}^{\prime}}$,
is an isomorphism in degrees $14$ and below. For our string-theoretic
purposes, we only care about $k\leq 12$, so we may as well compute
${\xi^{\mathrm{het}}}^{\prime}$-bordism. In the rest of this subsection, we
often blur the distinction between ${\xi^{\mathrm{het}}}$ and
${\xi^{\mathrm{het}}}^{\prime}$; we will point out where it matters which one
we are looking at.

######  Remark 2.34.

Turning off the $\mathbb{Z}/2$ symmetry switching the two $\mathrm{E}_{8}$
factors, i.e. passing to a $\xi^{r,\mathrm{het}}$-structure as in Remark 1.46,
simplifies this story considerably: the bordism groups were known decades ago.
Specifically, replace $B\mathrm{E}_{8}$ with $K(\mathbb{Z},4)$ in the
definition of $\xi^{r,\mathrm{het}}$ to define a tangential structure
$\xi^{r,\mathrm{het}}{}^{\prime}$, which on a manifold $M$ consists of a spin
structure on $M$, two classes $c_{1},c_{2}\in H^{4}(M;\mathbb{Z})$, and a
trivialization of $\lambda(M)-c_{1}-c_{2}$. As Witten [Wit86, §4] noticed,
this data is equivalent to a spin structure and the single class $c_{1}$,
which may be freely chosen; then $c_{2}$ must be $\lambda(M)-c_{1}$. Therefore
the tangential structure $\xi^{r,\mathrm{het}}{}^{\prime}$-structure is simply
$B\mathrm{Spin}\times K(\mathbb{Z},4)\to B\mathrm{O}$, and just as for
${\xi^{\mathrm{het}}}$, the map
$\mathit{MT\xi}^{r,\mathrm{het}}\to\mathit{MT\xi}^{r,\mathrm{het}}{}^{\prime}\simeq\mathit{MTSpin}\wedge
K(\mathbb{Z},4)_{+}$ is an isomorphism on homotopy groups in degrees $14$ and
below. Stong [Sto86] computes $\Omega_{*}^{\mathrm{Spin}}(K(\mathbb{Z},4))$ in
degrees $12$ and below.

As we discussed in §1.2, the data of a trivial principal $\mathbb{Z}/2$-bundle
on a manifold $M$ and two principal $\mathrm{E}_{8}$-bundles $P,Q\to M$ define
a principal $\mathrm{E}_{8}^{2}\rtimes\mathbb{Z}/2$-bundle on $M$ with
$c_{1}+c_{2}$ equal to $c(P)+c(Q)$; data trivializing $c(P)+c(Q)-\lambda(M)$
therefore defines a ${\xi^{\mathrm{het}}}$ structure. Analogously, the trivial
$\mathbb{Z}/2$-bundle and a pair $c_{1},c_{2}\in H^{4}(M;\mathbb{Z})$ with a
trivialization of $c_{1}+c_{2}-\lambda$ define a
${\xi^{\mathrm{het}}}^{\prime}$ structure.

######  Lemma 2.35.

A spin manifold $M$ has a canonical ${\xi^{\mathrm{het}}}^{\prime}$ structure
specified as above by the trivial principal $\mathbb{Z}/2$-bundle, the
cohomology classes $c_{1}=\lambda$ and $c_{2}=0$, and the canonical
trivialization of $\lambda-\lambda=0\in H^{4}(M;\mathbb{Z})$.

This defines a map of tangential structures and therefore a map of Thom
spectra
$s_{1}\colon\mathit{MTSpin}\to\mathit{MT{\xi^{\mathrm{het}}}}^{\prime}$. A
${\xi^{\mathrm{het}}}^{\prime}$-structure includes data of a spin structure;
forgetting the rest of the ${\xi^{\mathrm{het}}}^{\prime}$-structure defines a
map $s_{2}\colon\mathit{MT{\xi^{\mathrm{het}}}}^{\prime}\to\mathit{MTSpin}$.
The composition of $s_{1}$ and $s_{2}$ is homotopy equivalent to the identity,
because the underlying spin structure of the ${\xi^{\mathrm{het}}}^{\prime}$
manifold built in Lemma 2.35 is the same spin structure we began with.

######  Corollary 2.36.

There is a spectrum $\mathcal{Q}$ and a splitting

(2.37)
$(s_{2},q)\colon\mathit{MT{\xi^{\mathrm{het}}}}^{\prime}\overset{\simeq}{\longrightarrow}\mathit{MTSpin}\vee\mathcal{Q}.$

We will use this later to reduce the amount of spectral sequence computations
we have to make.

Both Lemmas 2.35 and 2.36 require us to use ${\xi^{\mathrm{het}}}^{\prime}$
and not ${\xi^{\mathrm{het}}}$, though of course the consequence on low-degree
bordism groups is true for both.

When $K$ is a finite group, Nakaoka [Nak61, Theorem 3.3] proved that there is
a ring isomorphism from the mod $2$ cohomology of
$B(\mathbb{Z}/2\ltimes(K\times K))$ to the $E_{2}$-page of the Serre spectral
sequence

(2.38) $E_{2}^{p,q}=H^{p}(B\mathbb{Z}/2;\underline{H^{q}(BK\times
BK;\mathbb{Z}/2)})\Longrightarrow H^{p+q}(B(\mathbb{Z}/2\ltimes(K\times
K));\mathbb{Z}/2).$

Here the underline denotes the local coefficient system arising from the
$\mathbb{Z}/2$-action on $BK\times BK$ by switching the two factors. Since
this local coefficient system can be nontrivial, one has to be careful
defining the multiplicative structure on the $E_{2}$-page of (2.38), but here
it can be made explicit. As a $\mathbb{Z}/2[\mathbb{Z}/2]$-module,
$\underline{H^{\ast}(BK\times BK;\mathbb{Z}/2)}$ is a direct sum of:

  * •

the subalgebra $\mathcal{H}_{1}$ of classes fixed by $\mathbb{Z}/2$, which are
of the form $x\otimes x$ for $x\in H^{\ast}(BK;\mathbb{Z}/2)$; and

  * •

the submodule $\mathcal{H}_{2}$ spanned by classes of the form $x\otimes y$
where $x$ and $y$ are linearly independent.

Since $\mathbb{Z}/2$ acts trivially on $\mathcal{H}_{1}$ and $\mathcal{H}_{1}$
is a ring, $H^{\ast}(B\mathbb{Z}/2;\mathcal{H}_{1})$ has a ring structure. And
as a $\mathbb{Z}/2[\mathbb{Z}/2]$-module, $\mathcal{H}_{2}$ is of the form
$M\oplus M$ where $\mathbb{Z}/2$ acts by swapping the two factors, so
$H^{\ast}(B\mathbb{Z}/2;\mathcal{H}_{2})$ vanishes in positive
degrees.141414To see this, first observe that mod $2$ group cohomology for $G$
is additive in the $\mathbb{Z}/2[G]$-module of coefficients, so it suffices to
prove that $H^{*}(B\mathbb{Z}/2;M\oplus M)$ vanishes in positive degrees when
$M=\mathbb{Z}/2$. But $\mathbb{Z}/2\oplus\mathbb{Z}/2$ is isomorphic to
$\mathbb{Z}/2[\mathbb{Z}/2]$ as $\mathbb{Z}/2[\mathbb{Z}/2]$-modules (i.e. as
vector spaces with $\mathbb{Z}/2$-representations,
$\mathbb{Z}/2\oplus\mathbb{Z}/2$ is isomorphic to the vector space of
functions on the group $\mathbb{Z}/2$), and group cohomology valued in the
group ring is trivial, e.g. because the group ring is its own free resolution.
In degree zero, we obtain invariants, spanned by elements of the form
$x\otimes y+y\otimes x$, with $x,y\in H^{\ast}(BK;\mathbb{Z}/2)$.
$\mathcal{H}_{1}\oplus(\mathcal{H}_{2})^{\mathbb{Z}/2}=E_{2}^{0,\bullet}$ is a
subalgebra of $H^{\ast}(BK\times BK;\mathbb{Z}/2)$.

So far we have specified ring structures on
$H^{\ast}(B\mathbb{Z}/2;\mathcal{H}_{1})\supsetneq E_{2}^{>0,\bullet}$ and
$\mathcal{H}_{1}\oplus(\mathcal{H}_{2})^{\mathbb{Z}/2}=E_{2}^{0,\bullet}$, and
these ring structures agree where they overlap. Therefore to specify a ring
structure on the entirety of the $E_{2}$-page, it suffices to write down the
product of an element in $(\mathcal{H}_{2})^{\mathbb{Z}/2}$ and an element in
positive $p$-degree. We say that all such products vanish; this is the ring
structure that appears in Nakaoka’s theorem.

Of course, $\mathrm{E}_{8}$ is not a finite group. Nakaoka’s theorem is true
in quite great generality [Eve65, Kah84, Lea97]; the version we need is proven
by Evens [Eve65], who proves the same ring isomorphism when $K$ is a compact
Lie group. Thus this applies to ${\xi^{\mathrm{het}}}$, and not necessarily to
${\xi^{\mathrm{het}}}^{\prime}$, but since their cohomology rings are
isomorphic in degrees $14$ and below, it does not matter which one we use in
this calculation.

Now we make this ring structure and $\mathcal{A}(2)$-module structure
explicit. Since $c\colon B\mathrm{E}_{8}\to K(\mathbb{Z},4)$ is
$15$-connected, it induces an isomorphism in cohomology in degrees $14$ and
below, so we can use the cohomology of $K(\mathbb{Z},4)$ as a stand-in for the
cohomology of $B\mathrm{E}_{8}$. Serre [Ser53, §10] computed the mod $2$
cohomology of $K(\mathbb{Z},4)$. It is an infinitely generated polynomial
algebra; in degrees $12$ and below the generators are: the tautological class
$D\in H^{4}(K(\mathbb{Z},4);\mathbb{Z}/2)$, $F\coloneqq\mathrm{Sq}^{2}D$,
$G\coloneqq\mathrm{Sq}^{3}D$, $J\coloneqq\mathrm{Sq}^{4}F$, and
$K\coloneqq\mathrm{Sq}^{5}F$.

If $C$ is one of $D$, $F$, $G$, $J$, or $K$, we let $C_{1}$ denote the class
coming from the first copy of $B\mathrm{E}_{8}$ and $C_{2}$ denote the class
coming from the second copy. Thus we have the following additive basis for the
low-degree cohomology of $BG$:

  1. (1)

In $\mathcal{H}_{1}$, $D_{1}D_{2}x^{k}$ and $F_{1}F_{2}x^{k}$ for $k\geq 0$.

  2. (2)

In $(\mathcal{H}_{2})^{\mathbb{Z}/2}$, $D_{1}+D_{2}$, $F_{1}+F_{2}$,
$G_{1}+G_{2}$, $D_{1}^{2}+D_{2}^{2}$, $J_{1}+J_{2}$, $D_{1}F_{1}+D_{2}F_{2}$,
$D_{1}F_{2}+D_{2}F_{1}$, $D_{1}G_{1}+D_{2}G_{2}$, $D_{1}G_{2}+D_{2}G_{1}$,
$K_{1}+K_{2}$, $F_{1}^{2}+F_{2}^{2}$, $D_{1}^{3}+D_{2}^{3}$, and
$D_{1}^{2}D_{2}+D_{1}D_{2}^{2}$.

Next, we determine the $\mathcal{A}(2)$-module structure using a theorem of
Quillen.

######  Theorem 2.39 (Quillen’s detection theorem [Qui71, Proposition 3.1]).

Let $X$ be a space and let $\mathbb{Z}/k$ act on $X^{k}$ by cyclic
permutations. Let $Y\coloneqq E\mathbb{Z}/k\times_{\mathbb{Z}/k}X^{k}$, which
is a fiber bundle over $B\mathbb{Z}/k$ with fiber $X^{k}$. Let $i_{1}\colon
X^{k}\to Y$ be inclusion of the fiber at the basepoint and $i_{2}\colon
B\mathbb{Z}/k\times X\to Y$ be induced by the diagonal map; then

(2.40) $(i_{1}^{\ast},i_{2}^{\ast})\colon
H^{\ast}(Y;\mathbb{Z}/k)\longrightarrow H^{\ast}(X^{k};\mathbb{Z}/k)\oplus
H^{\ast}(B\mathbb{Z}/k\times X;\mathbb{Z}/k)$

is injective.

For us, $k=2$, $X=B\mathrm{E}_{8}$, and $Y=BG$. Thus, to compute Steenrod
squares for classes in $H^{\ast}(BG;\mathbb{Z}/2)$, we can assume we are in
$B\mathrm{E}_{8}^{2}$ if the class is in $(\mathcal{H}_{2})^{\mathbb{Z}/2}$;
for $\mathcal{H}^{1}$, we also need to know $\mathrm{Sq}(x)$, and
$i_{2}^{\ast}$ tells us $\mathrm{Sq}(x)=x+x^{2}$. Thus we can compute the
$\mathcal{A}(2)$-module structure on $H^{*}(BG;\mathbb{Z}/2)$, hence also on
$T(-(c_{1}+c_{2}))$; we focus on the latter. Like most calculations of this
form, it is a little tedious but straightforward, and can be done by hand in a
reasonable length of time. After working through the calculation, we have
learned the following.

######  Proposition 2.41.

Let $\mathcal{M}$ be the quotient of $T(-(c_{1}+c_{2}))$ by all elements in
degrees $14$ and higher. Then $\mathcal{M}$ is the direct sum of the following
submodules.

  1. (1)

${\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color<EMAIL_ADDRESS>the summand containing the Thom class $U$.

  2. (2)

${\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}M_{2}}\coloneqq\widetilde{H}^{\ast}(\mathbb{RP}^{\infty};\mathbb{Z}/2)$
modulo those elements in degrees $13$ and above.

  3. (3)

${\color[rgb]{0.67,0.603,0.1072}\definecolor[named]{pgfstrokecolor}{rgb}{0.67,0.603,0.1072}\pgfsys@color@cmyk@stroke{0}{0.067}{0.5628}{0.33}\pgfsys@color<EMAIL_ADDRESS>the summand containing $U(D_{1}^{2}+D_{2}^{2})$.

  4. (4)

${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}M_{4}}$,
the summand containing $UD_{1}D_{2}$.

  5. (5)

${\color[rgb]{0,0.75,0.16}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.75,0.16}\pgfsys@color@cmyk@stroke{0.92}{0}{0.59}{0.25}\pgfsys@color<EMAIL_ADDRESS>the summand containing $UD_{1}D_{2}x$.

  6. (6)

${\color[rgb]{0,0.44,0.57}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44,0.57}\pgfsys@color@cmyk@stroke{0.98}{0.13}{0}{0.43}\pgfsys@color<EMAIL_ADDRESS>the summand containing $U(D_{1}F_{1}+D_{2}F_{2})$.

  7. (7)

${\color[rgb]{0.45,0.01,0.92}\definecolor[named]{pgfstrokecolor}{rgb}{0.45,0.01,0.92}\pgfsys@color@cmyk@stroke{0.47}{0.91}{0}{0.08}\pgfsys@color<EMAIL_ADDRESS>the summand containing $U(D_{1}D_{2}^{2}+D_{1}^{2}D_{2})$.

We draw this decomposition in Figure 1.

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$U$$Ux$$U(D_{1}^{2}+D_{2}^{2})$$UD_{1}D_{2}$$UD_{1}D_{2}x$$U(D_{1}F_{1}+D_{2}F_{2})$$U(D_{1}D_{2}^{2}+D_{1}^{2}D_{2})$
Figure 1. The $\mathcal{A}(2)$-module $T(-(c_{1}+c_{2}))$ in low degrees. The
pictured submodule contains all classes in degrees $12$ and below.

Recall from Corollary 2.36 that $\mathit{MT{\xi^{\mathrm{het}}}}^{\prime}$
splits as $\mathit{MTSpin}\vee\mathcal{Q}$. Since
$\Omega_{*}^{{\xi^{\mathrm{het}}}}\cong\Omega_{*}^{{\xi^{\mathrm{het}}}^{\prime}}$
in the range we need and $\Omega_{*}^{\mathrm{Spin}}$ is known thanks to work
of Anderson-Brown-Peterson [ABP67], we focus on $\pi_{*}(\mathcal{Q})$. To do
so, we will identify the submodule of the $E_{2}$-page of the Adams spectral
sequence for ${\xi^{\mathrm{het}}}^{\prime}$ coming from spin bordism via
$s_{1}\colon\mathit{MTSpin}\to\mathit{MT{\xi^{\mathrm{het}}}}^{\prime}$; the
$E_{2}$-page for $\mathcal{Q}$ is then a complementary submodule.

The canonical ${\xi^{\mathrm{het}}}^{\prime}$-structure on a spin manifold
from Lemma 2.35 can be rephrased as follows: a spin structure on a manifold
$M$ is equivalent data to: a spin structure on $M$, a map $c\colon M\to
K(\mathbb{Z},4)$, and a trivialization of $c-\lambda(M)$. Thus spin structures
are twisted string structures in the sense of Corollary 2.12 (in fact the
universal twist in the sense of Remark 2.16), so the map

(2.42) $(1,0)\colon K(\mathbb{Z},4)\longrightarrow(K(\mathbb{Z},4)\times
K(\mathbb{Z},4))\times_{\mathbb{Z}/2}E\mathbb{Z}/2=B$

lifts to a map of $\mathit{MTString}$-module Thom spectra
$s_{1}\colon\mathit{MTSpin}\to\mathit{MT{\xi^{\mathrm{het}}}}^{\prime}$.
Naturality of Theorem 2.20 then tells us the image of $s_{1}^{*}$ on mod $2$
cohomology, allowing us to determine which of the summands in Proposition 2.41
correspond to $\mathit{MTSpin}$ and which correspond to $\mathcal{Q}$.
Specifically, the pullback map sends $x\mapsto 0$, is nonzero on $D_{1}$,
$F_{1}$, $G_{1}$, etc., and sends $D_{2}$, $F_{2}$, $G_{2}$, etc., to zero.
This implies that in the direct-sum decomposition
$\mathit{MT{\xi^{\mathrm{het}}}}^{\prime}\simeq\mathit{MTSpin}\vee\mathcal{Q}$,
the summands
${\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color<EMAIL_ADDRESS>${\color[rgb]{0.67,0.603,0.1072}\definecolor[named]{pgfstrokecolor}{rgb}{0.67,0.603,0.1072}\pgfsys@color@cmyk@stroke{0}{0.067}{0.5628}{0.33}\pgfsys@color<EMAIL_ADDRESS>and
${\color[rgb]{0,0.44,0.57}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44,0.57}\pgfsys@color@cmyk@stroke{0.98}{0.13}{0}{0.43}\pgfsys@color<EMAIL_ADDRESS>come from the cohomology of $\mathit{MTSpin}$, and the remaining summands come
from the cohomology of $\mathcal{Q}$.

In order to run the Adams spectral sequence for $\mathcal{Q}$, we need to
compute the Ext of
${\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}M_{4}}$,
${\color[rgb]{0,0.75,0.16}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.75,0.16}\pgfsys@color@cmyk@stroke{0.92}{0}{0.59}{0.25}\pgfsys@color<EMAIL_ADDRESS>and
${\color[rgb]{0.45,0.01,0.92}\definecolor[named]{pgfstrokecolor}{rgb}{0.45,0.01,0.92}\pgfsys@color@cmyk@stroke{0.47}{0.91}{0}{0.08}\pgfsys@color<EMAIL_ADDRESS>over $\mathcal{A}(2)$. After we compute this, we will display the $E_{2}$-page
in Figure 3. For an $\mathcal{A}(2)$-module $M$,
$\operatorname{Ext}_{\mathcal{A}(2)}^{*,*}(M,\mathbb{Z}/2)$, which we will
usually denote $\operatorname{Ext}_{\mathcal{A}(2)}(M)$ or
$\operatorname{Ext}(M)$, is a bigraded module over the bigraded
$\mathbb{Z}/2$-algebra $\operatorname{Ext}_{\mathcal{A}(2)}(\mathbb{Z}/2)$;
both the algebra and module structures arise from the _Yoneda product_ [Yon54,
§4] (see [BC18, §4.2] for a review). This module structure is helpful for
determining differentials in the Adams spectral sequence: differentials are
equivariant with respect to the action. The module structure also constrains
extensions on its $E_{\infty}$-page.

May (unpublished) and Shimada-Iwai [SI67, §8] determined the algebra
$\operatorname{Ext}_{\mathcal{A}(2)}(\mathbb{Z}/2)$. We will only need to
track the actions of three elements:
$h_{0}\in\operatorname{Ext}_{\mathcal{A}(2)}^{1,1}(\mathbb{Z}/2)$,
$h_{1}\in\operatorname{Ext}_{\mathcal{A}(2)}^{1,2}(\mathbb{Z}/2)$, and
$h_{2}\in\operatorname{Ext}_{\mathcal{A}(2)}(\mathbb{Z}/2)$. These elements
are in the image of the map
$\operatorname{Ext}_{\mathcal{A}}(\mathbb{Z}/2)\to\operatorname{Ext}_{\mathcal{A}(2)}(\mathbb{Z}/2)$
induced by the quotient $\mathcal{A}\to\mathcal{A}(2)$, so we do not have to
worry about whether Corollary 2.22 is compatible with the
$\operatorname{Ext}_{\mathcal{A}(2)}(\mathbb{Z}/2)$-action on the $E_{2}$-page
of the Adams spectral sequence. (It is, though.) When we draw Ext charts as in
Figure 3, we denote $h_{0}$-actions as vertical lines, $h_{1}$-actions as
diagonal lines with slope $1$, and $h_{2}$-actions as diagonal lines with
slope $1/3$. When one of these lines is not present, the corresponding $h_{i}$
acts as $0$.

Often one computes Ext groups of $\mathcal{A}(2)$-modules using computer
programs developed by Bruner [Bru18] and Chatham-Chua [CC21], or tools such as
the May spectral sequence [May66] or the Davis-Mahowald spectral sequence
[DM82, MS87] (see also [BR21, Chapter 2]) to compute Ext groups of
$\mathcal{A}(2)$-modules, but for the four modules we care about, we can get
away using simpler calculations by hand and computations already in the
literature.

  1. (1)

Davis-Mahowald [DM78, Table 3.2] compute
$\operatorname{Ext}_{\mathcal{A}(2)}({\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}M_{2}})$
in the degrees we need.

  2. (2)

In degrees $13$ and below,
${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}M_{4}}$
is isomorphic to
$\Sigma^{8}(\mathcal{A}(2)\otimes_{\mathcal{A}(0)}\mathbb{Z}/2)$; therefore
the Ext groups of these two $\mathcal{A}(2)$-modules, as algebras over
$\operatorname{Ext}_{\mathcal{A}(2)}(\mathbb{Z}/2)$, are isomorphic in
topological degrees $12$ and below. Thus we can compute with the change-of-
rings theorem (2.24): as
$\operatorname{Ext}_{\mathcal{A}(2)}(\mathbb{Z}/2)$-algebras,

(2.43)
$\operatorname{Ext}_{\mathcal{A}(2)}(\mathcal{A}(2)\otimes_{\mathcal{A}(0)}\mathbb{Z}/2)\cong\operatorname{Ext}_{\mathcal{A}(0)}(\mathbb{Z}/2)\cong\mathbb{Z}/2[h_{0}],$

with $h_{0}\in\operatorname{Ext}^{1,1}$. This identification of
$\operatorname{Ext}_{\mathcal{A}(0)}(\mathbb{Z}/2)$ follows from Koszul
duality [BC18, Example 4.5.5].

  3. (3)

${\color[rgb]{0,0.75,0.16}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.75,0.16}\pgfsys@color@cmyk@stroke{0.92}{0}{0.59}{0.25}\pgfsys@color<EMAIL_ADDRESS>looks a lot like
${\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>which gives us a technique to compute
$\operatorname{Ext}_{\mathcal{A}(2)}({\color[rgb]{0,0.75,0.16}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.75,0.16}\pgfsys@color@cmyk@stroke{0.92}{0}{0.59}{0.25}\pgfsys@color@cmyk@fill{0.92}{0}{0.59}{0.25}M_{5}})$.
Specifically, if $\tau_{\leq k}M$ denotes the quotient of an
$\mathcal{A}(2)$-module $M$ by the submodule of elements in degrees greater
than $k$, then there is a short exact sequence of $\mathcal{A}(2)$-modules

(2.44)
${0}$${{{\color[rgb]{1,0,0.87}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0.87}\pgfsys@color@cmyk@stroke{0}{1}{0.13}{0}\pgfsys@color<EMAIL_ADDRESS>13}{\color[rgb]{0,0.75,0.16}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.75,0.16}\pgfsys@color@cmyk@stroke{0.92}{0}{0.59}{0.25}\pgfsys@color<EMAIL_ADDRESS>13}{\color[rgb]{0.43,0.45,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.43,0.45,1}\pgfsys@color@cmyk@stroke{0.57}{0.55}{0}{0}\pgfsys@color<EMAIL_ADDRESS>
We draw this sequence in Figure 2, left. (2.44) induces a long exact sequence
in Ext groups; passage between $M$ and $\tau_{\leq 13}M$ does not change Ext
groups in degrees $12$ and below, and since we only care about degrees $12$
and below, we can and do pass between $\tau_{\leq 13}M$ and $M$ without
comment.

$9$$10$$11$$12$$13$$\Sigma^{13}\mathbb{Z}/2$$\tau_{\leq 13}M_{5}$$\tau_{\leq
13}\Sigma^{8}M_{2}$

Figure 2. Left: the short exact sequence (2.44) of $\mathcal{A}(2)$-modules.
Right: the associated long exact sequence in Ext. See the discussion after
(2.45) for why the pictured boundary map (black arrow) is nonzero.

We already know
$\operatorname{Ext}_{\mathcal{A}(2)}({\color[rgb]{1,0,0.87}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0.87}\pgfsys@color@cmyk@stroke{0}{1}{0.13}{0}\pgfsys@color@cmyk@fill{0}{1}{0.13}{0}\mathbb{Z}/2})$
and
$\operatorname{Ext}_{\mathcal{A}(2)}({\color[rgb]{0.43,0.45,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.43,0.45,1}\pgfsys@color@cmyk@stroke{0.57}{0.55}{0}{0}\pgfsys@color@cmyk@fill{0.57}{0.55}{0}{0}M_{2}})$,
so we can run the long exact sequence associated to (2.44) to compute
$\operatorname{Ext}_{\mathcal{A}(2)}({\color[rgb]{0,0.75,0.16}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.75,0.16}\pgfsys@color@cmyk@stroke{0.92}{0}{0.59}{0.25}\pgfsys@color@cmyk@fill{0.92}{0}{0.59}{0.25}M_{5}})$
in degrees $12$ and below; we draw this long exact sequence in Figure 2,
right. In the range we care about, there is exactly one boundary map that is
not forced to be zero for degree reasons, namely

(2.45)
$\partial\colon\operatorname{Ext}_{\mathcal{A}(2)}^{0,13}({\color[rgb]{1,0,0.87}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0.87}\pgfsys@color@cmyk@stroke{0}{1}{0.13}{0}\pgfsys@color@cmyk@fill{0}{1}{0.13}{0}\Sigma^{13}\mathbb{Z}/2})\longrightarrow\operatorname{Ext}_{\mathcal{A}(2)}^{1,13}({\color[rgb]{0.43,0.45,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.43,0.45,1}\pgfsys@color@cmyk@stroke{0.57}{0.55}{0}{0}\pgfsys@color@cmyk@fill{0.57}{0.55}{0}{0}\Sigma^{8}M_{2}});$

it must be nonzero, because that is the only way to obtain