#  Combinatorial 2d higher topological quantum field theory from a local
cyclic $A_{\infty}$ algebra

Justin Beck  University of Notre Dame, Notre Dame, IN 46556, USA
<EMAIL_ADDRESS>,  Andrey Losev  Shanghai Institute for Mathematics and
Interdisciplinary Sciences, Building 3, 62 Weicheng Road, Yangpu District,
Shanghai, 200433, China<EMAIL_ADDRESS>and  Pavel Mnev  University of
Notre Dame, Notre Dame, IN 46556, USA  St. Petersburg Department of V. A.
Steklov Institute of Mathematics of the Russian Academy of Sciences, 27
Fontanka, St. Petersburg, Russia, 191023<EMAIL_ADDRESS>
###### Abstract.

We construct combinatorial analogs of 2d higher topological quantum field
theories. We consider triangulations as vertices of a certain CW complex
$\Xi$. In the “flip theory,” cells of $\Xi_{\mathrm{flip}}$ correspond to
polygonal decompositions obtained by erasing the edges in a triangulation.
These theories assign to a cobordism $\Sigma$ a cochain $Z$ on
$\Xi_{\mathrm{flip}}$ constructed as a contraction of structure tensors of a
cyclic $A_{\infty}$ algebra $V$ assigned to polygons. The cyclic $A_{\infty}$
equations imply the closedness equation $(\delta+Q)Z=0$. In this context we
define combinatorial BV operators and give examples with coefficients in
$\mathbb{Z}_{2}$.

In the “secondary polytope theory,” $\Xi_{\mathrm{sp}}$ is the secondary
polytope (due to Gelfand-Kapranov-Zelevinsky) and the cyclic $A_{\infty}$
algebra has to be replaced by an appropriate refinement that we call an
$\widehat{A}_{\infty}$ algebra.

We conjecture the existence of a good Pachner CW complex $\Xi$ for any
cobordism, whose local combinatorics is descibed by secondary polytopes and
the homotopy type is that of Zwiebach’s moduli space of complex structures.

###### Contents

  1. 1 Introduction
  2. 2 Warm up: a combinatorial 2d TQFT (strict case)
    1. 2.1 Local data
    2. 2.2 Partition function
    3. 2.3 Functoriality (cutting-gluing property)
    4. 2.4 Invariance with respect to triangulations
      1. 2.4.1 Space of states for a non-triangulated circle
    5. 2.5 A famous example: 2d Dijkgraaf-Witten model
  3. 3 HTQFTs vs. strict TQFTs
    1. 3.1 HTQFT setup
    2. 3.2 Examples
    3. 3.3 Example: 2d topological conformal field theory
      1. 3.3.1 TCFT in genus zero: framed little disk operad and the BV algebra structure
    4. 3.4 HTQFT on triangulated cobordisms – a preliminary discussion
  4. 4  First attempt at 2d HTQFT on triangulated cobordisms: HTQFT on the flip complex from a cyclic $A_{\infty}$ algebra (“Model 1”)
    1. 4.1 Local data: cyclic $A_{\infty}$ algebra
    2. 4.2 Flip complex
      1. 4.2.1 Aside: polygonal decompositions, ribbon graphs and the moduli space $\mathcal{M}_{h,N}$
    3. 4.3 HTQFT on the flip complex
      1. 4.3.1 The analogy between combinatorial HTQFT and continuum TCFT
    4. 4.4 Naive BV operator and its problem
  5. 5 Combinatorial BV operators
    1. 5.1 The simplest BV cycle: $k=2$
    2. 5.2 Generalization – a “long” BV cycle
    3. 5.3 A combinatorial $\mathrm{BV}_{\infty}$ operator that squares to zero exactly (or: combinatorial replacement of the operator $Q+uG_{0,-}$)
      1. 5.3.1 Dual language: $\mathrm{BV}_{\infty}$ chains as Maurer-Cartan elements in the dga $\mathrm{CYL}_{k}$
    4. 5.4 Questions/wish list for the flip theory
  6. 6 Towards the full 2d HTQFT on triangulated cobordisms
    1. 6.1 A quick recollection of secondary polytopes
      1. 6.1.1 Admissible subsets of $\mathbb{R}^{d}$
    2. 6.2 The definition of an $\widehat{A}_{\infty}$ algebra
    3. 6.3  Combinatorial HTQFT on a convex polygon based on secondary polytopes (“Model 2”)
    4. 6.4 A toy model: combinatorial 1d HTQM based on secondary polytopes
      1. 6.4.1 The definition of an $\mathsf{SP}_{1}$ algebra
      2. 6.4.2 Combinatorial HTQM
    5. 6.5  Tentative Model 3 on triangulated surfaces. Conjecture on existence of the proper Pachner CW complex
      1. 6.5.1 Pachner complex
      2. 6.5.2 HTQFT on the Pachner complex
      3. 6.5.3 Chain-level BV algebra structure
      4. 6.5.4 Reduction by passing to $Q$-cohomology and by attaching triangulated cylinders. The strict BV algebra.

##  1\. Introduction

Simplest TQFTs are defined as quantum field theories that do not depend on
geometrical data on the manifold. Refined versions contain a homological
symmetry (i.e. the target of the QFT functor is the category of chain
complexes rather than vector spaces) such that this dependence is a exact. We
propose to study a new class of models where the manifold is equipped with a
triangulation playing the role of geometry. Changes of this triangulation
(generated by Pachner moves) are required to be exact with respect to the
differential, see Figure 1.

Figure 1. Exactness with respect to Pachner’s moves, schematically.

In this paper we study the 2d case (and 1d case as a toy model). Here
partition functions are assigned to triangulated surfaces. The space of
triangulations $\Xi$ itself may be considered as a CW complex (we also call it
the Pachner complex), where:

  * •

Vertices of $\Xi$ are triangulations of the surface.

  * •

Edges of $\Xi$ correspond to Pachner moves (Figure 3). An edge corresponding
to a Pachner flip can be seen as an insertion of one square (where the flip
occurs) in a triangulation, cf. Figure 11. An edge corresponding to the second
Pachner move – a stellar subdivision/aggregation – can be seen as having one
triangle with a “floating point” in the triangulation (see the discussion of
secondary polytopes in the paper and Figure 26).

  * •

There are also higher cells of $\Xi$ corresponding to relations between
Pachner’s moves (see Figures 12, 29, 30).

In this paper we present two models of such a theory, and conjecture the
existence of the ideal model combining the features of the two.

Model 1 (“flip theory”): Only Pachner flips are considered, in particular the
set of vertices is fixed. We denote the corresponding CW complex
$\Xi_{\mathrm{flip}}$. The cells of $\Xi_{\mathrm{flip}}$ correspond to
polygonal decomposition of the surface, where the dimension of the cell $e$ is

(1) $\dim(e)=\sum_{i}(n_{i}-3),$

where the sum is over the polygons and $n_{i}$ is the number of sides of
$i$-the polygon. The cell $e$ can be identified with a product of Stasheff’s
associahedra $\prod_{i}K_{n_{i}-1}$.

Model 2 (“secondary polytope theory”): For $A\subset\mathbb{R}^{2}$ a
collection of points in general position, one has a convex polytope
$\Xi_{\mathrm{sp}}$ – the “secondary polytope” introduced in [4]. Vertices of
$\Xi_{\mathrm{sp}}$ correspond to _regular_ 111 A triangulation is regular if
one can find a continuous convex function linear on the triangles and breaking
(non-differentiable) on the edges.  triangulations of the polygon $\Sigma$ –
the convex hull of $A$, with vertices in a subset of $A$. Edges of
$\Xi_{\mathrm{sp}}$ correspond to Pachner moves between regular triangulations
and there are certain higher cells. This is a partial model as it is defined
only on a disk, it relies on convex geometry of $\mathbb{R}^{2}$ and not only
on topology, and the maximal set of vertices $A$ is fixed (one can remove
vertices via stellar aggregations but one cannot add vertices not in $A$).

The ideal Model 3: We conjecture the existence of an ideal model combining the
features of Models 1 and 2. Namely, it should be defined for all surfaces and
allow both Pachner moves. We also conjecture that the homotopy type of $\Xi$
in the ideal model is that of the moduli space of complex structures decorated
with jets of local coordinates at punctures (Zwiebach’s moduli space
$\widetilde{\mathcal{M}}_{h,n}$). In this correspondence, boundary circles of
the surface (“polygonal holes”) correspond to the punctures.

After we described the space of triangulations, we proceed to the construction
of functors from cobordism category to the category of chain complexes. The
value of the functor on a cobordism is a cochain on the “space of
triangulations” $\Xi$ taking values in linear maps between spaces of in- and
out-states. In particular, in Model 1, we fix a cyclic $A_{\infty}$ algebra
$V$ with differential $Q=m_{1}$ and other operations $m_{2},m_{3},\ldots$ and
bilinear pairing $g$. Cyclicity of an $A_{\infty}$ algebra with respect to the
pairing $g$ means that operations

(2) $c_{n+1}=g(\bullet,m_{n}(\bullet,\ldots,\bullet))\colon
V^{n+1}\rightarrow\mathbbm{k}$

are invariant with respect to cyclic permutations of inputs. The space of
states for a circle triangulated into $k$ intervals is just $V^{\otimes k}$
considered as a cochain complex. The partition function for a surface equipped
with a polygonal decomposition $\alpha$ is

(3)
$Z(\alpha)=\left\langle\bigotimes_{\mathrm{polygons}\,p\,\mathrm{of}\,\alpha}c_{|p|},\bigotimes_{\mathrm{edges\,of\,}\alpha}g^{-1}\right\rangle$

– the contraction of elements $c_{|p|}\in(V^{*})^{\otimes|p|}$ assigned to
polygons ($|p|$ is the number of sides of the polygon $p$) – the cyclic
$A_{\infty}$ operations – and inverse pairings attached to the edges. Since
polygonal decompositions correspond to cells of $\Xi_{\mathrm{flip}}$, the
formula (3) determines the value of a cochain on $\Xi_{\mathrm{flip}}$ on
these cells.

By virtue of higher associativity relations in the $A_{\infty}$ algebra $V$,
the cochain $Z$ satisfies the universal closedness equation

(4) $(\delta+Q)Z=0,$

with $\delta$ the coboundary operator on cochains!

The equation (4) in degree zero means that $Z$ for a triangulated surface is
$Q$-closed. In degree one it means that for $\alpha_{1},\alpha_{2}$ two
triangulations that differ by a Pachner flip, and for $\alpha$ the same
triangulation with a square inserted where the flip is occurring (see Figure
11), one has

(5) $Z(\alpha_{1})-Z(\alpha_{2})=QZ(\alpha).$

If we think of the flip as an analog of a change of complex structure on the
surface then the l.h.s. of (5) corresponds to insertion of the stress-energy
tensor $T_{zz}$ while the $Z(\alpha)$ in the r.h.s. corresponds to insertion
of its superpartner $G_{zz}$. The meaning of (5) is just

(6) $T_{zz}=Q(G_{zz}).$

In Section 5, in the context of Model 1 we define a combinatorial version of
the BV operator $G_{0,-}$ as the value of $Z$ on a “BV-cycle” – a 1-cycle of
$\Xi_{\mathrm{flip}}$ on a cylinder that squares to a boundary. We construct
examples of such BV cycles with coefficients in $\mathbb{Z}_{2}$.

In Model 2, the spaces of states are the same and the partition function is
defined by the same formula (3). However, $\alpha$ (the decoration of a cell
in $\Xi_{\mathrm{sp}}$) now is a polygonal decomposition with floating points
(with a prescribed “configuration chamber” describing their geometry rather
than topology). Thus, instead of cyclic $A_{\infty}$ operations on $V$ we need
a more refined “$\widehat{A}_{\infty}$” algebra structure (see Section 6.2)
where multilinear operations correspond to polygons with floating points in a
configuration chamber. Together with new operations (which include the usual
$A_{\infty}$ operations), we have new relations, coming from the adjacency of
cells in secondary polytopes. These relations imply that the partition
function seen as a cochain on the secondary polytope again satisfies the
universal closedness equation (4).

In the toy model of dimension one, Model 2 corresponds to quantum mechanics
with $Q$-exact Hamiltonian, see Section 6.4. On the other hand, in dimension
two the stellar subdivision equation is an analog of $Q$-exactness of the
trace of the stress-energy tensor.

In the table below we summarize the analogy between the continuum and
combinatorial 2d field theory.

Combinatorial HTQFT | Continuum TCFT
---|---
Pachner flip | deformation of complex structure
stellar subdivision | Weyl transformation of metric
Pachner complex $\Xi$ | Zwiebach’s moduli space $\widetilde{\mathcal{M}}_{h,n}$
Pachner complex in genus zero | framed little disk operad $E_{2}^{\mathrm{fr}}$
state on a polygon | quantum field $/$ point observable
partition function paired with states | correlator of observables
on the boundary of polygonal holes |
partition function | partition function
is a $(\delta+Q)$-closed | is a $(d+Q)$-closed
cochain on $\Xi$ | form on the moduli space
cyclic product $c_{3}$ | vacuum vector
| (partition function of a disk)
cyclic $A_{\infty}$ operation $c_{4}$ | field $G_{zz}(dz)^{2}+G_{\bar{z}\bar{z}}(d\bar{z})^{2}$
$\widehat{A}_{\infty}$ operation $\mu$ | field $G_{z\bar{z}}dzd\bar{z}$
(triangle with a point inside) |
defect of associativity $Q(c_{4})$ | stress-energy tensor (traceless part)
defect of stellar subdivision $Q(\mu)$ | trace of stress-energy tensor

### Acknowledgements

We thank Nikolai Mnev and Lev Soukhanov for inspiring discussions. We also
thank Kirill Salmagambetov for discussions at an early stage of the project
(in 2017).

##  2\. Warm up: a combinatorial 2d TQFT (strict case)

Throughout the paper $\mathbbm{k}$ will stand for the ground field. By
default, $\mathbbm{k}=\mathbb{C}$ or $\mathbb{R}$. In Section 5 we will have
results on combinatorial BV operators which require
$\mathbbm{k}=\mathbb{Z}_{2}$.

###  2.1. Local data

Fix a finite-dimensional associative algebra $V$ over $\mathbbm{k}$ with
multiplication $m_{2}\colon V\otimes V\rightarrow V$ and with nondegenerate
invariant pairing ${g\colon V\otimes V\rightarrow\mathbbm{k}}$. Denote
$c_{3}\colon V^{\otimes 3}\rightarrow\mathbbm{k}$ the corresponding 3-to-1
operation $c_{3}(x,y,z)=g(x,m_{2}(y,z))$; it is invariant under cyclic
permutations of inputs.

###  2.2. Partition function

Let $\Sigma$ be an oriented compact surface, possibly with boundary, equipped
with a triangulation $T$ (Figure 2). Assume that $\Sigma$ has $n$ boundary
circles $S^{1}_{1},\ldots,S^{1}_{n}$ subdivided by $T$ into
$k_{1},\ldots,k_{n}$ intervals.

Figure 2. Triangulated surface.

We define the spaces of states for the boundary circles as

(7) $\mathcal{H}(S^{1}_{i},T|_{S^{1}_{i}})=V^{\otimes k_{i}}$

and the partition function for the surface as the contraction of tensors
$c_{3}$ assigned to the triangles (2-simplices of $T$), using tensors
$g^{-1}\in V\otimes V$ assigned to the edges (1-simplices of $T$):

(8)
$Z(\Sigma,T)=\left\langle\bigotimes_{\mathrm{triangles}}c_{3},\bigotimes_{\mathrm{edges}}g^{-1}\right\rangle\qquad\in\bigotimes_{i=1}^{n}V^{\otimes
k_{i}}=\bigotimes_{i=1}^{n}\mathcal{H}(S^{1}_{i},T|_{S^{1}_{i}}).$

Here we note that the edges of $T$ which lie on the boundary of $\Sigma$ are
not contracted, so the result is a vector in the tensor product of spaces
assigned to all boundary edges. If $\Sigma$ is a closed surface, $Z(\Sigma,T)$
is a number.

If some boundary circles are assigned as “in-boundary” and some as “out-
boundary,” one thinks of $\Sigma$ as a cobordism. Then one can reinterpret
(using the inner product on spaces of states induced by $g$) the partition
function (8) as a linear operator – an element of
$\mathrm{Hom}(\mathcal{H}_{\mathrm{in}},\mathcal{H}_{\mathrm{out}})$.

###  2.3. Functoriality (cutting-gluing property)

The construction above gives rise to a QFT as a monoidal functor from the
category of triangulated 2-cobordisms (the objects in this category are
disjoint unions of triangulated circles) to the category $\mathrm{Vect}$ of
vector spaces. In particular, if one has two composable triangulated
cobordisms $((S^{1})^{\sqcup
n_{1}},\tau_{1})\xrightarrow{(\Sigma^{\prime},T^{\prime})}((S^{1})^{\sqcup
n_{2}},\tau_{2})$ and $((S^{1})^{\sqcup
n_{2}},\tau_{2})\xrightarrow{(\Sigma^{\prime\prime},T^{\prime\prime})}((S^{1})^{\sqcup
n_{3}},\tau_{3})$, then their for gluing $((S^{1})^{\sqcup
n_{1}},\tau_{1})\xrightarrow{(\Sigma,T)}((S^{1})^{\sqcup n_{3}},\tau_{3})$ one
has

(9) $Z(\Sigma,T)=Z(\Sigma^{\prime\prime},T^{\prime\prime})\circ
Z(\Sigma^{\prime},T^{\prime})\qquad\in\mathrm{Hom}(\mathcal{H}_{1},\mathcal{H}_{3}).$

###  2.4. Invariance with respect to triangulations

By the celebrated Pachner’s theorem [18],222 In fact, here we are using its
relative version, see [1].  one can transition between two triangulations
$T,T^{\prime}$ of the same surface with boundary (assume that $T,T^{\prime}$
induce the same triangulation of the boundary) by a sequence of “bistellar
moves,” a.k.a. “Pachner moves” (Figure 3):

  1. (1)

flips (switching the diagonal in a square);

  2. (2)

stellar subdivisions/aggregations – subdividing a triangle into three
triangles, or the inverse (merging) move.

Figure 3. 2d Pachner moves.

Invariance of partition the function $Z(\Sigma,T)$ w.r.t. flip on $T$ follows
from associativity of $m_{2}$:

(10) $\underbrace{\left\langle
g(\bullet_{1},m_{2}(\bullet_{2},m_{2}(\bullet_{3},\bullet_{4})))\otimes
Z(\widetilde{\Sigma},\widetilde{T}),(g^{-1})^{\otimes
4}\right\rangle}_{Z(\Sigma,T)}=\\\ =\underbrace{\left\langle
g(\bullet_{1},m_{2}(m_{2}(\bullet_{2},\bullet_{3}),\bullet_{4})))\otimes
Z(\widetilde{\Sigma},\widetilde{T}),(g^{-1})^{\otimes
4}\right\rangle}_{Z(\Sigma,T_{\mathrm{flipped}})},$

see Figure 4. Here $\widetilde{\Sigma},\widetilde{T}$ is the complement of the
square where the flip is performed, and its triangulation.

Figure 4. Associativity as a move on ribbon graphs dual to triangulations.

To have invariance w.r.t. stellar subdivisions/aggregations, one needs to have

(11) $\mathrm{tr}_{V}m_{2}(x,m_{2}(y,m_{3}(z,\bullet)))=c_{3}(x,y,z),\quad
x,y,z\in V,$

see Figure 5.

Figure 5. Relation (11).

This property is not automatic. However it is true if the metric is related to
$m_{2}$ by

(12) $\mathrm{tr}_{V}m_{2}(x,m_{2}(y,\bullet))=g(x,y),$

see Figure 6.

Figure 6. Metric $g$ in terms of $m_{2}$.

To summarize, partition functions $Z(\Sigma,T)$ are invariant w.r.t. Pachner
flips. Assuming the extra property (12) for the algebra $V$, we have that
partition functions $Z(\Sigma,T)$ are independent of the triangulation in the
bulk. In this case the model defines a 2d TQFT as a monoidal functor from the
category of non-triangulated cobordisms to the category of vector spaces.

####  2.4.1. Space of states for a non-triangulated circle

The space of states for a non-triangulated circle can be constructed as the
space of equivalence classes of tensors in $V^{\otimes k}$ for $k=1,2,\ldots$
where the equivalence relation is generated by partition functions for
triangulated cylinders.333 Put another way, it is the colimit of the diagram
in $\mathrm{Vect}$ given by all triangulated cylinders.  Equivalently,

(13) $\mathcal{H}(S^{1})=HH_{0}(V,V)=V/[V,V]\simeq\mathcal{Z}$

is the zeroth Hochschild homology of $V$ with coefficients in $V$, or,
equivalently, the center of $V$.

Explanation of (13): First, note that the algebra $V$ admits an orthogonal
decomposition (w.r.t. the metric (12)) into the commutant and the center,444
Indeed, $\lambda\in[V,V]^{\perp}\Leftrightarrow g(\lambda,[x,y])=0\;\forall
x,y\in V\Leftrightarrow g([\lambda,x],y)=0\;\forall x,y\in
V\Leftrightarrow[\lambda,x]=0\;\forall x\in
V\Leftrightarrow\lambda\in\mathcal{Z}$. Thus, $[V,V]^{\perp}=\mathcal{Z}$.
Note that this argument only uses that $g$ is an invariant nondegenerate
metric on $V$, it does not use the particular form (12).

(14) $V=[V,V]\oplus\mathcal{Z}.$

Consider the simplest triangulated cylinder (Figure 7).

Figure 7. Triangulated cylinder (left and right sides of the square are
glued).

Treating both boundaries as “in,” its value is the pairing

(15) $\begin{array}[]{cccc}\gamma\colon&V\otimes V&\rightarrow&\mathbbm{k}\\\
&(x,y)&\mapsto&\mathrm{tr}_{V}m_{2}(x,m_{2}(\bullet,y))\end{array}$

The pairing (15) is symmetric, has $[V,V]$ as its kernel and coincides with
$g$ when restricted to $\mathcal{Z}$. Hence, the partition function of this
cylinder, with one boundary circle seen as “in” and the other as “out,” is the
projection to $\mathcal{Z}$ in (14). The equivalence relation induced by this
cylinder says that two elements $x,y\in V$ are equivalent if their projections
to $\mathcal{Z}=V/[V,V]$ coincide. More generally, the equivalence relation
is:

(16) $\otimes_{i=1}^{k}x_{i}\sim\otimes_{j=1}^{l}y_{j}\quad\mbox{if}\quad
P(x_{1}\cdots x_{k})=P(y_{1}\cdots y_{l})$

where $P$ is the projection onto $\mathcal{Z}$ in the decomposition (14) and
the products inside are the iterated $m_{2}$-products. E.g. by considering the
triangulated cylinder

we see the equivalence relation (16) for $k$ any and $l=1$; by composing such
a cylinders with an upside-down one, we obtain the full relation (16). By
triangulation invariance of partition functions relative to the boundary, the
triangulation of the bulk of the cylinder does not matter.

###  2.5. A famous example: 2d Dijkgraaf-Witten model

Fix $G$ a finite group and $V=\mathbb{C}[G]$ its group algebra (in this
subsection we set $\mathbbm{k}=\mathbb{C}$). The corresponding TQFT is known
as 2d Dijkgraaf-Witten model [2]. Its partition functions give the volume of
the groupoid of $G$-coverings of $\Sigma$. In particular, the partition
function for a closed orientable surface $\Sigma$ of genus $h$ is

(17)
$Z(\Sigma)=\frac{|\mathrm{Hom}(\pi_{1}(\Sigma),G)|}{|G|}=|G|^{2h-2}\sum_{R\in\mathrm{irrep}(G)}\dim(R)^{2-2h}.$

######  Remark 2.1.

A finite group $G$ can be replaced by a compact Lie group, with $V=L^{2}(G)$ –
the space of square-integrable functions of $G$ equipped with convolution
product. This is the 2d Yang-Mills theory at zero coupling constant, a.k.a. 2d
$BF$ theory, cf. [16, 22]. Its partition function for a closed surface is the
symplectic volume of the moduli space of flat $G$-bundles on $\Sigma$ and is
again given by the sum in the r.h.s. of (17) without $|G|^{2h-2}$ prefactor.
Note the space of irreducible representations in this case is infinite and the
sum over irreducible representations in (17) is divergent for genus $h=0,1$
(but converges for $h\geq 2$).

######  Remark 2.2 (Area deformation).

One can deform the construction of this section by equipping triangles with
areas (seen as a geometric structure) and by allowing the product $m_{2}$ on
$V$ to depend on the area $A\geq 0$ of a triangle, with associativity relation
$m_{2}^{A_{1}}(m_{2}^{A_{2}}(x,y),z)=m_{2}^{A_{3}}(x,m_{2}^{A_{4}}(y,z))$ for
any $x,y,z\in V$ and areas of triangles satisfying $A_{1}+A_{2}=A_{3}+A_{4}$.
In this setting the partition function (8) of a triangulated surface depends
on the total area – the sum of areas of the triangles.555See [20].

For $V=\mathbb{C}[G]$ such a deformation corresponds to weighing the summand
in the r.h.s. of (17) with the factor $e^{-C(R)\mathrm{Area}(\Sigma)}$, with
$\mathrm{Area}(\Sigma)$ the total area of the surface, and $C(R)$ some
constants assigned to irreducible representations $R$ (these constants
parametrize the possible area deformations).666 In this example, the deformed
product in the group algebra is defined on generators by
$m_{2}^{A}(g_{1},g_{2})=\sum_{g_{3}\in
G}\sum_{R\in\mathrm{irrep}(G)}e^{-AC(R)}\dim(R)\chi_{R}(g_{1}g_{2}g_{3}^{-1})g_{3}$,
where $g_{1},g_{2}$ are elements in $G$, $\chi_{R}$ is the character of the
representation $R$ and $A\geq 0$ is the area of the triangle.

In the setting of Remark 2.1, such a deformation with $C(R)$ being the
quadratic Casimir of the Lie algebra $\mathrm{Lie}(G)$ in the representation
$R$ yields the 2d Yang-Mills theory (at nonzero coupling constant).

##  3\. HTQFTs vs. strict TQFTs

###  3.1. HTQFT setup

In a functorial QFT we equip a cobordism $\Sigma$ with extra geometric data
such as metric, complex structure, triangulation, etc.

E.g. in dimension 1 the cobordism is an interval and we can equip it with
length. The image of the functor applied to an interval (the partition
function of the interval) is
$Z(t)\in\mathrm{Map}(\mathbb{R}_{+},\mathrm{End}(\mathcal{H}))$, where
$\mathcal{H}$ is the space of states for a point. Functoriality implies

${Z(t_{1}+t_{2})=Z(t_{1})\circ Z(t_{2})}.$

That implies

(18) $Z(t)=\exp(tH)$

for some linear operator $H\in\mathrm{End}(\mathcal{H})$ – the Hamiltonian.777
In high energy physics, the signature of the metric is Lorentzian, $H$ is
accompanied by an $i$ factor, and $Z(t)=\exp(itH)$ is known as the evolution
operator.

In a general functorial QFT, the partition function for a cobordism is

(19)
$Z(\Sigma)\in\mathrm{Map}(\mathrm{Geom}(\Sigma),\mathrm{Hom}(\mathcal{H}_{\mathrm{in}},\mathcal{H}_{\mathrm{out}})),$

where $\mathrm{Geom}$ is the space of possible geometric data on $\Sigma$. It
is understood that $\mathcal{H}_{\mathrm{in,out}}$ can depend on the geometric
data restricted to the boundary. So, more appropriately, $Z(\Sigma)$ is a
section of the bundle over $\mathrm{Geom}(\Sigma)$ with fiber over
$\gamma\in\mathrm{Geom}(\Sigma)$ being
$\mathrm{Hom}(\mathcal{H}_{\mathrm{in}}(\gamma|_{\mathrm{in}}),\mathcal{H}_{\mathrm{out}}(\gamma|_{\mathrm{out}}))$.

In HTQFT,888H stands for “Homotopical” or “Higher” or “chain-level version of
coHomological QFT.” the codomain of the functor is the category of cochain
complexes (with differential denoted $Q$); $\mathrm{Geom}$ is replaced by a
differential graded manifold $\mathrm{sGeom}$ (“s” for “super”) with
differential $q$. We impose an extra condition of total closedness of the
functor

(20) $(q+Q)Z=0.$

###  3.2. Examples

######  Example 3.1.

The case of a (strict) TQFT corresponds to $\mathrm{sGeom}$ being a single
point and $Q=0$.

For instance, a one-dimensional TQFT – (strict) topological quantum mechanics
– assigns to an interval an element $Z\in\mathrm{End}(\mathcal{H})$ subject to
the identity

(21) $Z^{2}=Z,$

since the gluing of two intervals is diffeomorphic to a single interval. Note
that (21) means that $Z$ is a projector.

######  Remark 3.2.

A typical case for a HTQFT is when $\mathrm{sGeom}$ has the form of an odd
tangent bundle,

(22) $\mathrm{sGeom}=T[1]\mathrm{Geom}$

of some manifold $\mathrm{Geom}$ of geometric data on the cobordism, with
$q=d_{\mathrm{Geom}}$ the de Rham differential. Then the partition function is
a nonhomogeneous differential form on $\mathrm{Geom}$,

$Z(\Sigma)\in\Omega^{\bullet}(\mathrm{Geom})\otimes\mathrm{Hom}(\mathcal{H}_{\mathrm{in}},\mathcal{H}_{\mathrm{out}}),$

with the equation (20) taking the form

(23) $(d_{\mathrm{Geom}}+Q)Z=0.$

Denote $Z^{(p)}$, with $p=0,1,2,\ldots$, the component of $Z$ of de Rham
degree $p$ along $\mathrm{Geom}$. Then equation (23) restricted to de Rham
degrees $0$ and $1$ yields

(24) $QZ^{(0)}=0,\quad d_{\mathrm{Geom}}Z^{(0)}=-QZ^{(1)}.$

I.e., (a) $Z^{(0)}$ gives a $Q$-closed state for any geometric data and (b)
changing the geometric data infinitesimally changes $Z^{(0)}$ by a $Q$-exact
term. In other words, $Z^{(0)}$ defines an element in the cohomology of $Q$,
locally constant along $\mathrm{Geom}$. In particular, if $\mathrm{Geom}$ is
connected, $Z^{(0)}$ induces a strict TQFT, with spaces of states given by
$Q$-cohomology of the spaces of states of the original HTQFT.

We emphasize that there is more information in an HTQFT than in the strict
TQFT induced by $Z^{(0)}$ on $Q$-cohomology. For instance, one can integrate a
higher form component $Z^{(p)}$ over some distinguished $p$-cycle in
$\mathrm{Geom}$ and then project to $Q$-cohomology of the spaces of states.
Examples of this construction are

  * •

Gromov-Witten invariants (arising from the A model) and

  * •

the BV algebra structure on $Q$-cohomology in a 2d topological conformal field
theory (see Section 3.3.1 below).

######  Example 3.3 (HTQM).

In higher topological quantum mechanics, for an interval,
${\mathrm{sGeom}=T[1]\mathbb{R}_{+}}$ with de Rham differential
$q=d_{t}=dt\wedge\frac{d}{dt}$. The space of states for a point is a cochain
complex $\mathcal{H},Q$ Then the value of the functor on an interval takes the
form

$Z(t,dt)=\exp\left((d_{t}+Q)(tG)\right)=\exp(t[Q,G]+dt\,G)$

where $G\in\mathrm{End}(\mathcal{H})$ is of degree $-1$ where $G^{2}=0$.

In particular, the Hamiltonian

(25) $H=[Q,G]=QG+GQ$

is $\mathrm{ad}_{Q}$-exact. Also, note that $G^{2}=0$ implies $[H,G]=0$.

A strict topological quantum mechanics is the case $Q=G=0$.

###  3.3. Example: 2d topological conformal field theory

In 2-dimensional topological conformal field theory (TCFT), the space of local
quantum fields (which is the same as the space of states for a tiny circle)
$\mathcal{H}=\mathcal{H}_{S^{1}}$ is a cochain complex with differential $Q$
and the stress-energy tensor is $Q$-exact: $T=Q(G),\bar{T}=Q(\bar{G})$ for
some degree $-1$ fields in $\mathcal{H}$ (and $T_{z\bar{z}}$ vanishes
strictly). Here we assume that $\bar{\partial}G$ and $\partial\bar{G}$ vanish
strictly and that $G,\bar{G}$ have regular OPE [24, pp.6–7]. For a surface
$\Sigma$ of genus $h$ with $n$ punctures,999 One thinks of punctures as tiny
disks removed from the surface, each creating an $S^{1}$ boundary component.
We treat all these boundary components as in-boundary. the space of geometric
data is $\mathrm{sGeom}=T[1]\widetilde{\mathcal{M}}_{h,n}$. Here
$\widetilde{\mathcal{M}}_{h,n}$ is the moduli space of complex structures
(non-compactified, with ordered punctures), equipped with a formal coordinate
chart on $\Sigma$ at each puncture, see [25]. We will refer to
$\widetilde{\mathcal{M}}_{h,n}$ as _Zwiebach’s moduli space_. Thus, the
partition function

(26)
$Z(\Sigma)\in\Omega^{\bullet}(\widetilde{\mathcal{M}}_{h,n},\mathrm{Hom}(\mathcal{H}^{\otimes
n},\mathbb{C}))$

is a nonhomogeneous differential form on $\widetilde{\mathcal{M}}_{h,n}$ with
values in $\mathrm{Hom}(\mathcal{H}^{\otimes n},\mathbb{C})$.101010 For the
purpose of this section, since we are talking about differential forms, we
have to fix the ground field to be $\mathbbm{k}=\mathbb{C}$ (or $\mathbb{R}$).
The $p$-form component of $Z$ is defined as a correlator involving $p$
insertions of the field $G^{\mathrm{tot}}=G(dz)^{2}+\bar{G}(d\bar{z})^{2}$:

(27)
$Z^{(p)}(\Phi_{1}\otimes\cdots\otimes\Phi_{n})=\Big{\langle}\underbrace{G^{\mathrm{tot}}\cdots
G^{\mathrm{tot}}}_{p}\Phi_{1}(z_{1})\cdots\Phi_{n}(z_{n})\Big{\rangle}_{\Sigma}.$

With $\Phi_{1},\ldots,\Phi_{n}\in\mathcal{H}$ a collection of fields
(observables111111Here “observables” does not imply that $\Phi_{i}$ are
$Q$-closed.) plugged in the punctures $z_{1},\ldots,z_{n}$. More explicitly,
if we have $p$ tangent vectors to $\widetilde{\mathcal{M}}_{h,n}$ at a given
complex structure, described by Beltrami differentials
$\mu_{i}+\bar{\mu}_{i}$, then the contraction of (27) with these tangent
vectors is

(28)
$\iota_{\mu_{1}+\bar{\mu}_{1}}\cdots\iota_{\mu_{p}+\bar{\mu}_{p}}Z^{(p)}(\Phi_{1}\otimes\cdots\otimes\Phi_{n})=\Big{\langle}\prod_{i=1}^{p}\int_{\Sigma}\iota_{\mu_{i}+\bar{\mu}_{i}}G^{\mathrm{tot}}\cdot\Phi_{1}(z_{1})\cdots\Phi_{n}(z_{n})\Big{\rangle}_{\Sigma}.$

Equation (20) in this case has the form

(29)
$d_{\mathcal{M}}Z^{(p-1)}(\Phi_{1}\otimes\cdots\otimes\Phi_{n})+\sum_{j=1}^{n}\pm
Z^{(p)}(\Phi_{1}\otimes\cdots\otimes
Q(\Phi_{j})\otimes\cdots\otimes\Phi_{n})=0.$

In the left term $d_{\mathcal{M}}$ is the de Rham differential on the moduli
of complex structures. In that term, the action of $d_{\mathcal{M}}$ is
tantamount to replacing one of $G^{\mathrm{tot}}$ insertions by the stress-
energy tensor $T^{\mathrm{tot}}$. The equation (29) is standard and is proven
by a contour-trading trick.

An important special case: if the field insertions $\Phi_{i}$ are $Q$-closed,
(27) is a closed form on the moduli space. This restriction induces a reduced
version of a TCFT, assigning to a surface a de Rham cohomology class of the
moduli space

(30) $Z_{\mathrm{red}}\in
H^{\bullet}(\widetilde{\mathcal{M}}_{h,n})\otimes\mathrm{Hom}(\mathcal{H}_{\mathrm{red}}^{\otimes
n},\mathbb{C}),$

where

(31) $\mathcal{H}_{\mathrm{red}}\colon=H_{Q}(\mathcal{H})$

is the $Q$-cohomology of the space of states/local quantum fields.

In a class of TCFTs (e.g. in the A model) the correlators of some $Q$-closed
observables (“evaluation observables” in the case of the A model, see [3])
descend from $\widetilde{\mathcal{M}}_{h,n}$ to $\mathcal{M}_{h,n}$ and then
extend to the Deligne-Mumford compactification $\overline{\mathcal{M}}_{h,n}$
of the moduli space $\mathcal{M}_{h,n}$. In this case, in (30) one can replace
$H^{\bullet}(\widetilde{\mathcal{M}}_{h,n})$ with
$H^{\bullet}(\overline{\mathcal{M}}_{h,n})$. In this case, by virtue of
cutting-gluing property (functoriality) of the original TCFT, the reduced
partition functions (30) factorize on nodal surfaces, which results in the
famous WDVV equation. This reduction is known as “cohomological field theory,”
see [11], also [14]. The value of this cohomology class paired with
fundamental class of $\overline{\mathcal{M}}_{h,n}$ is known as the “$h$-loop
string amplitude.”

####  3.3.1. TCFT in genus zero: framed little disk operad and the BV algebra
structure

Recall (see [5] for details) that the framed operad of little 2-disks
$E_{2}^{\mathrm{fr}}$ has as its $n$-th space $E_{2}^{\mathrm{fr}}(n)$ the
configuration space of $n\geq 0$ disjoint disks (or “holes”) inside the
standard unit disk in $\mathbb{C}$, where all disks are equipped with a marked
point on the boundary circle; the outer circle has the standard marked point
at $1$. The operadic compositions $\circ_{i}\colon
E_{2}^{\mathrm{fr}}(n)\times E_{2}^{\mathrm{fr}}(m)\rightarrow
E_{2}^{\mathrm{fr}}(n+m-1)$ are given by scaling and rotating one disk
configuration and replacing with it the $i$-th disk of the second
configuration (Figure 8).

Figure 8. Composition in the framed little disk operad.

Homology of $E_{2}^{\mathrm{fr}}$ is generated (via operadic composition) by
the following classes (Figure 9):

  1. (i)

Unit121212 By abuse of notations, here we use the same notations and names for
generators of homology of $E_{2}^{\mathrm{fr}}$ as those commonly used for the
BV algebra operations representing these homology classes on
$H_{Q}(\mathcal{H})$.  $1$ – the generator of $H_{0}(E_{2}^{\mathrm{fr}}(0))$.

  2. (ii)

Product $\cdot\in H_{0}(E_{2}^{\mathrm{fr}}(2))$ (two inner disks “sitting
still”).

  3. (iii)

BV operator $\Delta\in H_{1}(E_{2}^{\mathrm{fr}}(1))$ – corresponds to the
inner circle making a full rotation. We denote the BV operator $\Delta$ or
$G_{0,-}$.

  4. (iv)

BV bracket (a.k.a. odd Poisson bracket) $\\{,\\}\in
H_{1}(E_{2}^{\mathrm{fr}}(2))$ – corresponds to one inner disks moving a full
circle around the other inner disk.

Figure 9. Generators of homology of the framed little disk operad.

These generators satisfy the following relations:

  1. (a)

The product is supercommutative and associative with $1$ the unit;
$\Delta(1)=0$.

  2. (b)

BV operator squares to zero:

(32) $\Delta^{2}=0.$

  3. (c)

$\Delta$ is a second-order derivation of the product:

(33)
$\Delta(xyz)-(-1)^{|x|}x\Delta(yz)-(-1)^{|y||z|}\Delta(xz)y-\Delta(xy)z+\\\
+(-1)^{|x|+|y|}xy\Delta(z)+(-1)^{|x|}x\Delta(y)z+\Delta(x)yz=0$

– the “7-term relation.”

  4. (d)

The BV bracket is the “defect” of the first-order Leibniz identity for
$\Delta$:

(34) $\Delta(xy)=\Delta(x)y+(-1)^{|x|}x\Delta(y)+(-1)^{|x|}\\{x,y\\}.$

(As a consequence, $\\{,\\}$ is a biderivation of the product and satisfies
Jacobi; $\Delta$ is a derivation of $\\{,\\}$.)

Homology of $E_{2}^{\mathrm{fr}}$ is known as the BV (Batalin-Vilkovisky)
operad.

Restricting a TCFT to genus zero cobordisms – disks with holes, with the outer
circle seen as “out” and inner circles seen as “ins” – yields a sequence of
differential forms on the configuration space of disks,

(35)
$Z_{[n]}\in\Omega^{\bullet}(E_{2}^{\mathrm{fr}}(n))\otimes\mathrm{Hom}(\mathcal{H}^{\otimes
n},\mathcal{H})$

satisfying the equation (20), which in this case becomes

(36) $(d_{E_{2}^{\mathrm{fr}}}+Q)Z_{[n]}=0$

with $d_{E_{2}^{\mathrm{fr}}}$ the de Rham operator on the configurations
space of disks. Equation (36) together with functoriality of the original
field theory implies that the space of states for the circle $\mathcal{H}$
carries a dg representation of the operad $E_{2}^{\mathrm{fr}}$. In other
words, $\mathcal{H}$ is an $E_{2}^{\mathrm{fr}}$-algebra.

As a consequence, the reduced space of states – the $Q$-cohomology
$\mathcal{H}_{\mathrm{red}}=H_{Q}(\mathcal{H})$ – has the structure of a BV
algebra.

######  Remark 3.4.

For a TCFT where the correlators of $Q$-closed observables extend to the
Deligne-Mumford compactification of the moduli space
$\overline{\mathcal{M}}_{h,n}$, the BV algebra on $H_{Q}(\mathcal{H})$ has
vanishing BV operator (and hence also the BV bracket, cf. (34)). Thus, the BV
operator plays the role of an obstruction for extendability of the correlators
to the Deligne-Mumford compactified moduli space. See e.g. [13] for an
explicit example of a TCFT with nontrivial BV algebra on $Q$-cohomology.

######  Remark 3.5.

To shed more light on the relation between the TCFT as a form on the moduli
space $\widetilde{\mathcal{M}}_{h,n}$ (26) and genus zero TCFT as a form on
the framed little disk operad (35), we note the following. The moduli space
$\widetilde{\mathcal{M}}_{0,n+1}$ contracts onto
$\mathcal{M}^{\prime}_{0,n+1}$ – a $(\mathbb{C}^{*})^{n+1}$-bundle over
$\mathcal{M}_{0,n+1}$ (since the group of formal pointed diffeomorphisms of
the tangent space to $\Sigma$ at a puncture contracts onto $\mathbb{C}^{*}$),
which in turn is homotopy equivalent to $E_{2}^{\mathrm{fr}}(n)$.131313 More
precisely, one has a homeomorphism $\mathcal{M}^{\prime}_{0,n+1}\simeq
E_{2}^{\mathrm{fr}}(n)/\gamma$ where $\gamma$ is the subgroup of the group of
Möbius transformations $PSL_{2}(\mathbb{C})$ mapping the unit circle to itself
and the preserving the point $1$. The subgroup $\gamma$ is conjugate to the
subgroup $\\{z\mapsto az+b\;|\;a>0,b\in\mathbb{R}\\}$ – conformal
automorphisms of the upper half-plane.

###  3.4. HTQFT on triangulated cobordisms – a preliminary discussion

We would like to consider the triangulations of a cobordism $\Sigma$ as
vertices of a certain CW complex $\Xi(\Sigma)$ (we will refer to it as the
“Pachner complex”), with edges corresponding to Pachner moves between
triangulations. We will discuss higher cells later. In this setting, the
partition function is a cellular cochain on $\Xi$ with coefficients in
$\mathcal{H}$, $Z(\Sigma)\in C^{\bullet}(\Xi,\mathcal{H})$ that should satisfy
the equation

(37) $(\delta_{\Xi}+Q)Z=0,$

with $\delta_{\Xi}$ the coboundary operator on cochains. Thus, here the
Pachner complex $\Xi$ plays the role of $\mathrm{sGeom}$, and cellular
cochains on $\Xi$ play the role of functions on $\mathrm{sGeom}$ (or
equivalently differential forms on $\mathrm{Geom}$, cf. (22)). Equation (37)
is a cellular analog of equation (23).

Denote $Z^{(p)}$ the degree $p$ component of $Z$ as a cochain on $\Xi$. Then
equation (37) says

(38) $\delta_{\Xi}Z^{(p-1)}+QZ^{(p)}=0.$

In particular, for $p=0$ and $p=1$ we have

(39) $QZ^{(0)}=0,\quad\delta_{\Xi}(Z^{(0)})=-QZ^{(1)}.$

I.e., the partition function on a triangulated cobordism is $Q$-closed and
changes by $Q$-exact terms under Pachner moves.

##  4\.  First attempt at 2d HTQFT on triangulated cobordisms: HTQFT on the
flip complex from a cyclic $A_{\infty}$ algebra (“Model 1”)

Our strategy now is to construct a 2d HTQFT on triangulated cobordisms by
relaxing the construction of Section 2: we will require the associativity of
the product $m_{2}$ in $V$ and the relation (11) to hold only up to homotopy
(i.e. up to a $Q$-exact term $Q(\cdots)$, where the homotopy $\cdots$ will be
taken as a part of the structure), ensuring that invariance with respect to
Pachner moves holds up to homotopy. In this section we will focus only on
Pachner flips and relaxing associativity up to homotopy. In Section 6 we will
incorporate the stellar subdivisions/aggregations and invariance up to
homotopy with respect to them.

A part of the discussion in both cases is the appropriate treatment of higher
homotopies. In the case of flips, this results in having a homotopy for
associativity assigned to squares and higher homotopies assigned to polygons
with $\geq 5$ vertices, with the overall local structure being that of a
cyclic $A_{\infty}$ algebra (replacing an associative algebra with an inner
product, in the setup of Section 2). In the case of stellar
subdivisions/aggregations, that structure needs to be enhanced with further
homotopies associated with polygons with “floating vertices” inside; this new
structure comes from the construction of “secondary polytopes” by Gelfand-
Kapranov-Zelevinsky [4], see also [8].

###  4.1. Local data: cyclic $A_{\infty}$ algebra

Let $V$ be a $\mathbb{Z}$-graded vector space over $\mathbbm{k}$ endowed with
the structure of an $A_{\infty}$ algebra141414See e.g. [15] and [12], see also
[7]. – a sequence of multilinear operations $m_{n}\colon V^{\otimes
n}\rightarrow V$ of degree $2-n$, with $n=1,2,3\ldots$ Here $Q=m_{1}$ is a
differential151515 By abuse of notations, we also denote by $Q$ the induced
(via Leibniz identity) differential on tensors – elements of
$\mathrm{Hom}(V^{\otimes k},V^{\otimes l})$. For instance in (41), $m_{2}$ is
an element of $\mathrm{Hom}(V^{\otimes 2},V)$ and so
$Q(m_{2})\in\mathrm{Hom}(V^{\otimes 2},V)$ is defined by
$(Q(m_{2}))(x,y)={Q(m_{2}(x,y))-m_{2}(Qx,y)-(-1)^{|x|}m_{2}(x,Qy)}$.  and the
operations are understood to satisfy $A_{\infty}$ relations:

(40) $\displaystyle Q^{2}$ $\displaystyle=$ $\displaystyle 0$ (41)
$\displaystyle{Q(m_{2})}$ $\displaystyle=$ $\displaystyle 0$ $\displaystyle
m_{2}(m_{2}(x,y),z)-m_{2}(x,m_{2}(y,z))$ $\displaystyle=$
$\displaystyle-Q(m_{3})(x,y,z)$ $\displaystyle\vdots$
$\displaystyle{Q(m_{n})}$ $\displaystyle=$
$\displaystyle\sum_{k+l=n+1,\;k,l\geq 2}m_{k}\circ m_{l}$
$\displaystyle\vdots$

where $m_{k}\circ m_{l}=\sum_{i=1}^{k}\pm m_{k}\circ_{i}m_{l}$ and $\circ_{i}$
is the $i$-th composition, putting the output of $m_{l}$ as $i$-th input of
$m_{k}$.161616 Another description: one can extend a multilinear map
$A\in\mathrm{Hom}(V^{\otimes k},V)$ to a coderivation of the tensor coalgebra
of $V[1]$, $\widehat{A}\in\mathrm{Coder}(\bigoplus_{i\geq 1}(V[1])^{\otimes
i})$. Then the composition $A\circ B$ is defined via $\widehat{A\circ
B}=\widehat{A}\circ\widehat{B}$, where on the right one has the ordinary
composition of coderivations. In the language of coderivations, the
$A_{\infty}$ relations jointly are equivalent to the equation
${(\widehat{m}_{1}+\widehat{m}_{2}+\cdots)^{2}=0}$.  In particular,

  * •

(40) says that $Q$ is a differential,

  * •

(41) is Leibniz identity for $m_{2}$,

  * •

(4.1) is associativity of $m_{2}$ up to homotopy, with $m_{3}$ being the
homotopy,

  * •

subsequent relations (4.1) are coherencies on higher homotopies.

We will assume additionally that $V$ is equipped with a nondegenerate
symmetric pairing $g\colon V\otimes V\rightarrow\mathbbm{k}$ of degree zero,
such that operations

(44) $\begin{array}[]{cccc}c_{n+1}=g\circ(\mathrm{id}\otimes
m_{n})\colon&V^{\otimes(n+1)}&\rightarrow&\mathbbm{k}\\\ &x_{0}\otimes
x_{1}\ldots\otimes
x_{n}&\mapsto&(-1)^{n|x_{0}|}g(x_{0},m_{n}(x_{1},\ldots,x_{n}))\end{array}$

are invariant under cyclic permutations of inputs for $n=1,2,\ldots$.171717
With explicit signs:
$c_{n+1}(x_{r},\ldots,x_{n},x_{0},\ldots,x_{r-1})=(-1)^{\sigma}c_{n+1}(x_{0},\ldots,x_{r-1},x_{r},\ldots,x_{n})$
with $\sigma=nr+(\sum_{i=r}^{n}|x_{i}|)(\sum_{j=0}^{r-1}|x_{j}|)$, for
$r=0,\ldots,n$.  We will refer to $c_{n+1}$ as “cyclic operations.” The data
of an $A_{\infty}$ algebra $(V;m_{1},m_{2},\ldots)$ together with such pairing
$g$ is known as a “cyclic $A_{\infty}$ algebra.”181818See e.g. [21].

###  4.2. Flip complex

Let $\Sigma$ be an oriented surface, possibly with boundary. Let $P$ be a
finite collection of points (“vertices”) in $\Sigma$, either in the bulk or on
the boundary. We assume that each boundary circle contains at least one point
of $P$. We consider polygonal decompositions $\alpha$ (cell decompositions
with 2-cells being $n$-gons with $n\geq 3$) of $\Sigma$ with vertices at $P$,
up to cellular homeomorphism relative to $P$ and relative to the
boundary.191919We do not require the polygonal decompositions to be _regular_
CW complexes: edges or vertices of a closure of a cell are allowed to be glued
together.

We consider the “flip complex” $\Xi_{\mathrm{flip}}(\Sigma,P)$ – a CW complex
where:

  * •

$0$-cells are associated with triangulations of $\Sigma$ with vertices at $P$.

  * •

A polygonal decomposition $\alpha$ of $\Sigma$ with vertices at $P$ whose
$2$-cells are $n_{i}$-gons, is assigned a cell $e_{\alpha}$ in
$\Xi_{\mathrm{flip}}$ of dimension $\sum_{i}(n_{i}-3)$. This cell can be
identified with (the interior of) a product of Stasheff’s associahedra
$\prod_{i}K_{n_{i}-1}$ (Figure 10). The boundary of this cell is the union of
cells one dimension lower corresponding to subdividing one polygon in $\alpha$
by some diagonal into two polygons (they should both be at least triangles).

Figure 10. First few associahedra.

######  Example 4.1.

1-cells of $\Xi_{\mathrm{flip}}$ correspond to triangulations of $\Sigma$ with
one square. The boundary of this 1-cell is the difference of two
triangulations obtained by cutting that square by a diagonal in two possible
ways, see Figure 11.

Figure 11. 1-cell $e$ in $\Xi_{\mathrm{flip}}$. We show the polygonal
decompositions of $\Sigma$ decorating the strata (sub-cells of $\bar{e}$).
Shaded region stands for the triangulated part of $\Sigma$.

2-cells of $\Xi_{\mathrm{flip}}$ correspond to either (a) triangulations with
two squares, or (b) triangulations with one pentagon. Coincidentally, in the
case (a) the cell in $\Xi_{\mathrm{flip}}$ itself looks like a square (its
boundary corresponds to splitting one of the two squares in $\alpha$ by a
diagonal), and in the case (b) the corresponding cell in $\Xi_{\mathrm{flip}}$
looks like a pentagon, see Figure 12.

Figure 12. 2-cells in $\Xi_{\mathrm{flip}}$.

######  Example 4.2.

For $\Sigma$ a disk with $P$ an $n$-tuple of points on the boundary,
$\Xi_{\mathrm{flip}}$ is the associahedron $K_{n-1}$.

######  Example 4.3.

For $\Sigma$ a cylinder with $P$ consisting of two points – one on each
boundary circle – $\Xi_{\mathrm{flip}}$ is homeomorphic to a circle, see
Figure 13.

Figure 13. $\Xi_{\mathrm{flip}}$ for a cylinder with one vertex on each
boundary. Dashed line means that the two sides of a square are identified (to
obtain the cylinder). Note that the two triangulated squares with identified
opposite sides are cellular homeomorphic (related by Dehn twist) and hence
correspond to the same vertex of $\Xi_{\mathrm{flip}}$.

######  Example 4.4.

For $\Sigma$ a disk with $P$ consisting of three points on the boundary and
one point in the bulk, $\Xi_{\mathrm{flip}}$ is itself homeomorphic to a disk,
with cell decomposition into three pentagons, see Figure 14.

Figure 14. $\Xi_{\mathrm{flip}}$ for a triangle with an extra vertex in the
bulk. Note that “singular” polygons (with identified edges) occur abundantly
in this example. E.g. the 2-cells of $\Xi_{\mathrm{flip}}$ correspond to
representations of $\Sigma$ as a pentagon with glued consecutive edges.

Note that one can also has conventional ribbon graphs dual to the polygonal
decompositions of $\Sigma$ with singular polygons, see e.g. Figure 15.

Figure 15. A polygonal decomposition with a singular polygon and the dual
ribbon graph (shown in dashed lines).

####  4.2.1. Aside: polygonal decompositions, ribbon graphs and the moduli
space $\mathcal{M}_{h,N}$

In this section we compare $\Xi_{\mathrm{flip}}$ with the moduli space
$\mathcal{M}_{h,N}$. Here, crucially, $N$ is the number of vertices in $P$,
_not_ the number of holes in the surface $\Sigma$ (which we denote $n$). On
the other hand, the relevant moduli space for comparison with TCFT is
$\mathcal{M}_{h,n}$, see Section 4.3.1 below. I.e., in this section vertices
of polygonal decompositions correspond to marked points on a Riemann surface,
while in comparison to field theory, polygonal holes become marked
points/punctures.

For a polygonal decomposition $\alpha$ of a surface $\Sigma$ with $P$ the set
of vertices, one has the dual ribbon graph $\Gamma$ on $\Sigma$ with $P$ the
set of “borders” or “faces.” If $\Sigma$ has boundary, then $\Gamma$ is a
ribbon graph with “leaves” (loose half-edges), dual to the boundary edges of
$\alpha$.

If $\Sigma$ is a closed surface of genus $h$ and $P$ is a set of $N$ points,
one has a natural bijection $\rho$ between cells $e_{\alpha}$ of
$\Xi_{\mathrm{flip}}$ and building blocks (“orbi-cells”202020
$RG^{\mathrm{met}}$ is a noncompact orbi-cell complex, and $C_{\Gamma}$ are
its strata – the noncompact orbi-cells. )

(45) $C_{\Gamma}=\mathbb{R}_{+}^{E(\Gamma)}/\mathrm{Aut}(\Gamma)$

of the space of metric ribbon graphs $RG^{\mathrm{met}}_{h,N}$, with $\Gamma$
the ribbon graph dual to $\alpha$. The latter is the well-known combinatorial
model for the “decorated” moduli space of complex structures212121 We are
considering the non-compactified moduli space with ordered marked points.
$\mathcal{M}_{h,N}\times\mathbb{R}_{+}^{N}$ due to Harer-Mumford-Thurston and
Penner, see [6, 19, 10, 17].

For the rest of this subsection we will assume that $\mathrm{Aut}(\Gamma)=1$
for all ribbon graphs involved; it is true if $h=0,N\geq 3$ or if $h>0$ and
$N$ is large enough.222222 E.g., for genus $h=1$ one needs $N\geq 5$ to kill
the automorphisms of ribbon graphs. As an evidence, note that the orbifold
Euler characteristic $\chi(\mathcal{M}_{1,N})=(-1)^{N}\frac{(N-1)!}{12}$ (cf.
[10]) is an integer for $N\geq 5$.

The bijection $\rho$ is compatible with attaching maps but reverses the
incidence and reverses the dimensions of cells:

  * •

A triangulation $\alpha$ corresponds to a $0$-cell in $\Xi_{\mathrm{flip}}$.
The corresponding ribbon graph $\Gamma$ is trivalent and has

(46) $E=\underbrace{6h-6+3N}_{=\colon
D}=\dim(\mathcal{M}_{h,N}\times\mathbb{R}_{+}^{N})$

edges and gives a top-dimension building block of $RG^{\mathrm{met}}$.

  * •

For $\alpha$ a general polygonal decomposition, the corresponding ribbon graph
contains vertices of valence $\geq 4$ (corresponding to squares, pentagons
etc. in $\alpha$) and has $E=D-\dim e_{\alpha}$ edges.

  * •

The lowest dimension blocks of $RG^{\mathrm{met}}$ correspond to graphs
$\Gamma$ with a single vertex and $E=2h-1+N$ edges. They correspond to top-
dimension cells of $\Xi_{\mathrm{flip}}$, with $\alpha$ containing a single
polygon; the dimension of these cells in $\Xi_{\mathrm{flip}}$ is $\dim
e_{\alpha}=4h-5+2N$.

In particular, for $N$ sufficiently large, one has the following.

  * •

$\Xi_{\mathrm{flip}}(\Sigma,P)$ and $RG^{\mathrm{met}}$ are Poincaré dual cell
complexes (one of them noncompact), see Figure 16.

Figure 16. Stratification of $\mathcal{M}_{0,3}\times\mathbb{R}_{+}^{3}$
colored by ribbon graphs (shown as a stratification of the cross-section of
the octant) and the corresponding $\Xi_{\mathrm{flip}}$ (in blue). Numbers are
labels of vertices (resp., borders of ribbon graphs). Gluing of edges in
polygonal decomposition is shown by little dashes.

  * •

One has a canonical isomorphism $\rho_{*}$ between $p$-chains of
$\Xi_{\mathrm{flip}}$ and $(D-p)$-cochains of $RG^{\mathrm{met}}$ for any $p$,
acting on generators by

(47) $\rho_{*}(e_{\alpha})(C_{\Gamma})=\left\\{\begin{array}[]{cc}1&\mbox{if
$\Gamma$ is dual to $\alpha$,}\\\ 0&\mbox{otherwise}\end{array}\right.$

  * •

One has an isomorphism between homology of $\Xi_{\mathrm{flip}}$ and compactly
supported cohomology (and, via Poincaré duality, homology) of the moduli
space:

(48)
$H_{p}(\Xi_{\mathrm{flip}})\stackrel{{\scriptstyle\rho_{*}}}{{\simeq}}H^{D-p}_{c}(\mathcal{M}_{h,N})\simeq
H_{p}(\mathcal{M}_{h,N})$

An important consequence is that adding extra vertices in $P$ makes the
homotopy type of $\Xi_{\mathrm{flip}}$ more complicated, in particular
increases the rank of first homology. E.g., for $h=0$ ($\Sigma$ a sphere),
$N=3$ points, $\Xi_{\mathrm{flip}}$ is contractible, while for $N=4$ it has
nontrivial $H_{1}$: $H_{1}(\Xi_{\mathrm{flip}})\simeq
H_{1}(\mathcal{M}_{0,4})\simeq\mathbb{Z}^{2}$.

######  Question 4.5.

Does cellular Poincaré duality above give a homotopy equivalence
$\Xi_{\mathrm{flip}}\sim\mathcal{M}_{h,N}$?

######  Question 4.6.

Does the discussion above extend to surfaces with boundary? I.e., does one
have a ribbon graph model for the moduli space of complex structures on a
surface with boundary, with marked points in the bulk and on boundary circles,
and does one have a comparison of homology of $\Xi_{\mathrm{flip}}$ with
(co)homology of the moduli space going through the appropriate ribbon graph
complex?

###  4.3. HTQFT on the flip complex

We define our HTQFT (we will refer to it as “flip theory”) as follows. Given a
surface $\Sigma$ with set of vertices $P$, the space of states for $j$-th
boundary circle is

(49) $\mathcal{H}(S^{1}_{j},P\cap{S^{1}_{j}})=V^{\otimes k_{j}}$

where $k_{j}$ is the number of points of $P$ on $S^{1}_{j}$; we understand
factors $V$ as associated to the _arcs_ into which these points divide the
circle.

The partition function is a cellular cochain on $\Xi_{\mathrm{flip}}$,

(50) $Z\in
C^{\bullet}(\Xi_{\mathrm{flip}})\otimes\mathrm{Hom}(\mathcal{H}_{\mathrm{in}},\mathcal{H}_{\mathrm{out}})$

defined as follows: its value on the cell $e_{\alpha}$ of
$\Xi_{\mathrm{flip}}$ corresponding to a polygonal decomposition $\alpha$ of
$\Sigma$ with vertices at $P$ is

(51)
$Z(e_{\alpha})=\Big{\langle}\bigotimes_{\mathrm{polygons}\,p\,\mathrm{of}\,\alpha}c_{|p|},\bigotimes_{\mathrm{edges}\setminus\\{\mathrm{in-
edges}\\}}g^{-1}\Big{\rangle}.$

Here $|p|$ is the number of edges of the polygon $p$; $c_{\cdots}$ are the
cyclic $A_{\infty}$ operations (44). We will also denote (51) by $Z(\alpha)$.
Note that formula (51) is a natural generalization of the strict example (8),
allowing for polygons that are not triangles and decorated by higher
$A_{\infty}$ operations.

######  Proposition 4.7.

Properties of the construction:

  1. (i)

Functoriality232323Cf. Section 2.3. (compatibility with gluing): given two
composable cobordisms endowed with polygonal decompositions

(52)
$(\gamma_{1},\tau_{1})\xrightarrow{(\Sigma^{\prime},\alpha^{\prime})}(\gamma_{2},\tau_{2})\xrightarrow{(\Sigma^{\prime\prime},\alpha^{\prime\prime})}(\gamma_{3},\tau_{3})$

(with $\gamma_{i}$ collections of circles and $\tau_{i}$ the triangulations of
boundaries induced by the polygonal decompositions
$\alpha^{\prime},\alpha^{\prime\prime}$ of surfaces), for their composition

(53) $(\gamma_{1},\tau_{1})\xrightarrow{(\Sigma,\alpha)}(\gamma_{3},\tau_{3})$

one has

(54) $Z(\Sigma,\alpha)=Z(\Sigma^{\prime\prime},\alpha^{\prime\prime})\circ
Z(\Sigma^{\prime},\alpha^{\prime})\qquad\in\mathrm{Hom}(\mathcal{H}_{1},\mathcal{H}_{3}).$

  2. (ii)

The partition function defined by (51) satisfies the main equation

(55) $(\delta+Q)Z=0$

with $\delta$ the cellular coboundary operator on $\Xi_{\mathrm{flip}}$ and
$Q$ acting on the boundary spaces of states.

###### Proof.

(i) is immediate by construction (51). (ii) follows from the $A_{\infty}$
relation (4.1): $Qc_{n}$ is the sum of contractions of pairs of cyclic
$A_{\infty}$ operations corresponding to ways to cut an $n$-gon by a single
diagonal, see Figure 17.

Figure 17. $A_{\infty}$ relation as a sum over cuttings of a polygon. Dashed
lines depict the dual ribbon graphs.

For $\alpha$ some polygonal decomposition of $\Sigma$, we have

(56)
$QZ(\alpha)=-\sum_{\mathrm{polygons}\;p\;\mathrm{of}\;\alpha}\;\;\sum_{\mathrm{cutting}\;p\;\mathrm{by\;a\;diagonal}}Z(\alpha^{\prime})=-(\delta
Z)(\alpha),$

where $\alpha^{\prime}$ is the polygonal decomposition resulting from cutting
one polygon $p$ in $\alpha$ by a diagonal. In (56), the first equality is the
$A_{\infty}$ relation while the second equality is due to the combinatorics of
attachment of cells in $\Xi_{\mathrm{flip}}$. ∎

######  Example 4.8.

  * •

For $\alpha$ a triangulation, $Z(\alpha)$ is $Q$-closed. This is restriction
of the equation (55) to $0$-cochains. Or one can see it directly, from (41)
and cyclicity of $c_{2}$ (or equivalently, from $Q(c_{3})=0$ and $Q(g)=0$).

  * •

Let $\alpha$ be a polygonal decomposition where one polygon $p$ is a square
and all others are triangles, and let $\alpha_{1},\alpha_{2}$ be the two
triangulations resulting from dividing $p$ by a diagonal (in two possible
ways). Then $\alpha_{1}$ and $\alpha_{2}$ are related by a Pachner flip
performed in the square $p$ and one has

(57) $Z(\alpha_{1})-Z(\alpha_{2})=QZ(\alpha).$

This is the up-to-homotopy version of flip-invariance of the strict TQFT (10).

  * •

For $\mathsf{c}\in C_{\bullet}(\Xi_{\mathrm{flip}})$ any _cycle_ on
$\Xi_{\mathrm{flip}}$ (i.e., $\partial\mathsf{c}=0$), the value of $Z$ on it
is $Q$-closed:

(58) $QZ(\mathsf{c})=0.$

######  Remark 4.9.

Given two composable cobordisms
$\gamma_{1}\xrightarrow{\Sigma^{\prime}}\gamma_{2}$,
$\gamma_{2}\xrightarrow{\Sigma^{\prime\prime}}\gamma_{3}$ each equipped with a
set of points $P^{\prime}$, $P^{\prime\prime}$, such that
$P^{\prime}\cap\gamma_{2}=P^{\prime\prime}\cap\gamma_{2}$, one has the
composed cobordism $\gamma_{1}\xrightarrow{\Sigma}\gamma_{3}$ equipped with a
set of points $P=P^{\prime}\cup P^{\prime\prime}$. Gluing of polygonal
decompositions on $\Sigma^{\prime},\Sigma^{\prime\prime}$ induces a map

(59)
$\circ_{c}\colon\Xi_{\mathrm{flip}}(\Sigma^{\prime\prime},P^{\prime\prime})\times\Xi_{\mathrm{flip}}(\Sigma^{\prime},P^{\prime})\rightarrow\Xi_{\mathrm{flip}}(\Sigma,P).$

In terms of this operation, functoriality (54) can be stated as

(60)
$Z(\zeta^{\prime\prime}\circ_{c}\zeta^{\prime})=Z(\zeta^{\prime\prime})\circ
Z(\zeta^{\prime}),$

for $\zeta^{\prime}\in
C_{\bullet}(\Xi_{\mathrm{flip}}(\Sigma^{\prime},P^{\prime}))$,
$\zeta^{\prime\prime}\in
C_{\bullet}(\Xi_{\mathrm{flip}}(\Sigma^{\prime\prime},P^{\prime\prime}))$ any
two cellular chains. Here on the left one has the sewing operation (59) and on
the right one has a composition of linear maps between spaces of states.

######  Remark 4.10.

The construction of this section admits the following generalization. As the
local data, instead of a cyclic $A_{\infty}$ algebra $V$ one can take a
_cyclic $A_{\infty}$ category_ $\mathcal{V}$. Then vertices of polygonal
decompositions are decorated by objects of $\mathcal{V}$, edges – by morphisms
and polygons – by higher compositions. The partition function is defined by
the by the same formula (51) seen as a contraction of higher compositions in
$\mathcal{V}$.

######  Remark 4.11 (Degree of the metric on $V$).

In the considerations above we were assuming that the metric $g\colon V\otimes
V\rightarrow\mathbbm{k}$ has degree zero. However one can consider flip theory
with $V$ a cyclic $A_{\infty}$ algebra with metric $g$ of nonzero degree $q$.
Then the partition function for a triangulated cobordism $(\Sigma,T)$ has
degree $q(\\#\\{\mathrm{triangles}\\}-\\#\\{\mathrm{edges}\\})$. In
particular, for a closed triangulated surface the partition function
automatically vanishes for $q\neq 0$. Later, in the context of secondary
polytopes, we have to insist that $g$ has degree zero – otherwise, e.g., the
r.h.s. of equation (103) (compatibility with stellar subdivision/aggregation)
is not a homogenenous expression.

####  4.3.1. The analogy between combinatorial HTQFT and continuum TCFT

  * •

A triangulated surface corresponds to a “space” of a continuum TCFT – a
surface with complex structure.

  * •

A triangulated surface $\Sigma,\alpha$ with boundary (polygonal holes)
corresponds to a surface with punctures. Given a choice of states
$\Phi_{j}\in\mathcal{H}(S^{1}_{j},\alpha|_{S^{1}_{j}})$ on the boundary
polygons, their pairing with the partition function

(61) $\langle Z(\alpha),\bigotimes_{j}\Phi_{j}\rangle$

corresponds to a correlator of point observables
$\langle\prod_{j}\Phi_{j}(z_{j})\rangle$ in continuum theory.

  * •

We think of squares, pentagons etc. in a polygonal decomposition as “defects”
in a triangulation, where special observables $c_{4},\,c_{5},\ldots$ are put.
Moreover, we think of $c_{4}$ as $G^{\mathrm{tot}}$ – the superpartner (or
“$Q$-primitive”) of the stress-energy tensor, cf. Section 3.3 – in continuum
theory. So, e.g., evaluation of $Z$ on a surface with $n$ holes, with a
polygonal decomposition which is a triangulation everywhere except $p$ square
2-cells, paired with states $\Phi_{1},\ldots,\Phi_{n}$ placed in the polygonal
holes, corresponds to the $p$-form correlator (27) in continuum TCFT, see
Figure 18. From this viewpoint, $c_{3}$ corresponds to the “vacuum vector” or
“unit field” in TCFT (its insertion in a puncture erases the puncture). Cyclic
$A_{\infty}$ maps $c_{5},c_{6},\ldots$ should be thought of as “higher
components” of the combinatorial field $G^{\mathrm{tot}}$, not present present
in a usual TCFT.242424 One can imagine $c_{\geq 5}$ appearing as an artifact
of discretization. Another way these components might appear in continuum
theory is if one relaxes the conservation of $G^{\mathrm{tot}}$ to
conservation modulo $Q(\widetilde{G})$. In this way $\widetilde{G}$ would
correspond to $c_{5}$ etc.

Figure 18. A correlator in combinatorial HTQFT (on the left; shaded region is
the triangulated part of the surface) vs. continuuum TCFT (on the right).

###  4.4. Naive BV operator and its problem

Consider the setup of Example 4.3: $\Sigma$ a cylinder with $P$ consisting of
two points, one on each boundary circle. Let $\mathsf{c}_{\Delta}$ be the
1-cycle comprised of the only 1-cell of $\Xi_{\mathrm{flip}}$ (cf. Figure 13).
Remark that $\Xi_{\mathrm{flip}}$ is homotopy equivalent to the component
$E_{2}^{\mathrm{fr}}(1)$ of the framed little disk operad, cf. Section 3.3.1;
under this homotopy equivalence the homology class of $\mathsf{c}_{\Delta}$
corresponds to the homology class representing the BV operator $\Delta$ that
in the case of topological conformal field theory equals $G_{0,-}$. In
particular, we think of the value of our HTQFT on $\mathsf{c}_{\Delta}$ as a
chain-level representative of the BV operator, $Z(c_{\Delta})``="\Delta$:

(62)
$g(x,Z(\mathsf{c}_{\Delta})y)=(-1)^{|x|}\mathrm{Str}_{V}\,m_{3}(x,\bullet,y)\qquad\mathrm{for}\;x,y\in
V,$

with $\mathrm{Str}_{V}$ the supertrace over $V$. Note that
$Z(\mathsf{c}_{\Delta})$ is $Q$-closed, cf. (58).

The problem of the operator $Z(c_{\Delta})$ is that it is not guaranteed to
square to zero modulo $Q$-exact terms:

(63) $Z(\mathsf{c}_{\Delta})\circ Z(\mathsf{c}_{\Delta})\neq Q(\cdots),$

while the BV operator coming from the framed little disk operad does square to
zero. The reason is that the cycle
$\mathsf{c}_{\Delta}\circ_{c}\mathsf{c}_{\Delta}\in
C_{2}(\Xi_{\mathrm{flip}}(\Sigma^{\prime},P^{\prime}))$ is _not null-
homologous_ in $\Xi_{\mathrm{flip}}$:

(64) $\mathsf{c}_{\Delta}\circ_{c}\mathsf{c}_{\Delta}\neq\partial(\cdots),$

see Figures 19, 20. Here $\circ_{c}$ means composition of cobordisms equipped
with polygonal decompositions (59); $\Sigma^{\prime}$ is again a cylinder
(thought of as a composition of $\Sigma$ with itself) with $P^{\prime}$
comprised of three points – one on each boundary and one in the bulk. More
precisely, homology $H_{p}(\Xi_{\mathrm{flip}}(\Sigma^{\prime},P^{\prime}))$
is concentrated in degrees $p=0,1,2$ and has rank $1$ in these degrees;
$\mathsf{c}_{\Delta}\circ_{c}\mathsf{c}_{\Delta}$ is the generator of homology
in degree $p=2$.

Figure 19. 3-cells (larger rectangles) and 2-cells (smaller rectangles) of
$\Xi_{\mathrm{flip}}$ of a cylinder with one point on each boundary and one
bulk point. For each rectangle, left and right side are glued. Dashed lines
show cell attachment. The leftmost 2-cell is in fact a cycle and is
$\mathsf{c}_{\Delta}\circ_{c}\mathsf{c}_{\Delta}$. Figure 20. Generator of
$H_{2}(\Xi_{\mathrm{flip}}(\Sigma^{\prime},P^{\prime}))$.

The core of the problem here is that taking a composition-square is an
endomorphism in $E_{2}^{\mathrm{fr}}(1)$ (which has vanishing $H_{2}$) but in
the setting of combinatorial HTQFT is a map
$\Xi_{\mathrm{flip}}(\Sigma,P)\rightarrow\Xi_{\mathrm{flip}}(\Sigma^{\prime},P^{\prime})$
where the right hand side has nontrivial $H_{2}$. Put another way, the problem
is that changing the set $P$ for a cylinder (adding a bulk point) changes the
homotopy type of $\Xi_{\mathrm{flip}}$. Conjecturally, this is remedied by
transitioning from the flip complex $\Xi_{\mathrm{flip}}$ to the appropriate
“full” Pachner complex $\Xi$, see Remark 6.17.

##  5\. Combinatorial BV operators

In this section we introduce the notion of a BV cycle on $\Xi_{\mathrm{flip}}$
and the associated notion of a combinatorial BV operator that squares to zero
modulo $Q$-exact terms, and show a family of examples. While the definition
makes sense for any coefficient field $\mathbbm{k}$, in our examples we fix
$\mathbbm{k}=\mathbb{Z}_{2}$ (somewhat mysteriously, for $\mathbbm{k}$ of
characteristic $\neq 2$ we obtain signs which invalidate the main equation
(65)).

######  Definition 5.1.

Let us define a _BV cycle_ as a 1-cycle $\mathsf{c}_{\Delta}$ on
$\Xi_{\mathrm{flip}}$ of a cylinder with $P$ consisting of $k\geq 1$ points on
each boundary circle and $m\geq 0$ bulk points, such that
$\mathsf{c}_{\Delta}\circ\mathsf{c}_{\Delta}$ is null-homologous in
$\Xi_{\mathrm{flip}}$:

(65) $\mathsf{c}_{\Delta}\circ\mathsf{c}_{\Delta}=\partial(\cdots)$

for some 3-chain $\cdots$. We will say that a BV cycle $\mathsf{c}_{\Delta}$
is _nontrivial_ if it is not null-homologous, i.e., represents a nontrivial
class in homology $H_{1}(\Xi_{\mathrm{flip}})$.

Then the value of HTQFT on $\mathsf{c}_{\Delta}$,

(66)
$Z(\mathsf{c}_{\Delta})\in\mathrm{Hom}_{-1}(\underbrace{\mathcal{H}(S^{1}_{(k)})}_{V^{\otimes
k}},\underbrace{\mathcal{H}(S^{1}_{(k)})}_{V^{\otimes k}})$

is $Q$-closed but (generally252525For a general $A_{\infty}$ algebra $V$ and
assuming that $\mathsf{c}_{\Delta}$ is nontrivial. On the other hand, if
$\mathsf{c}_{\Delta}$ is null-homologous, then $Z(\mathsf{c}_{\Delta})$ is
automatically $Q$-exact.) not $Q$-exact and satisfies

(67) $Z(\mathsf{c}_{\Delta})^{2}=Q(\cdots).$

Here $S^{1}_{(k)}$ stands for a circle triangulated into $k$ intervals. We
call $Z(\mathsf{c}_{\Delta})$ a _combinatorial BV operator_.

Note that (67) implies that the operator that $Z(\mathsf{c}_{\Delta})$ induces
in $Q$-cohomology squares to zero on the nose.

A non-example is the 1-cycle we considered in Section 4.4: it _fails_ the
condition (ii) above.

###  5.1. The simplest BV cycle: $k=2$

Next, consider the 1-cycle $\mathsf{c}_{\Delta}$ in the case $k=2$, $m=0$ (two
points on each boundary circle, no bulk points) shown in Figure 21.

Figure 21. 1-cycle in $\Xi_{\mathrm{flip}}$ of a cylinder with two points on
each boundary, shown by thick lines. For each rectangle, left and right side
are glued. Large rectangles are 1-cells and small ones are 0-cells.

######  Proposition 5.2.

The 1-cycle of Figure 21 is a BV cycle for coefficients
$\mathbbm{k}=\mathbb{Z}_{2}$.

###### Proof.

Consider the following 3-chain $B$ in $\Xi_{\mathrm{flip}}$ shown in Figure
22.

Figure 22. 3-chain $B$ in $\Xi_{\mathrm{flip}}$.

By inspection, the boundary of $B$ in chains on $\Xi_{\mathrm{flip}}$ with
$\mathbb{Z}_{2}$ coefficients is exactly the 2-cycle
$c_{\Delta}\circ_{c}c_{\Delta}$. ∎

The combinatorial BV operator corresponding to the BV cycle of Figure 21 is:

(68) $Z(\mathsf{c}_{\Delta})\colon t_{i}\otimes
t_{j}\mapsto\Big{(}(c_{3})_{kca}(c_{3})_{iba}(c_{4})_{jclb}+(c_{4})_{iakc}(c_{3})_{jba}(c_{3})_{lbc}+\\\
+(c_{3})_{iac}(c_{4})_{jbka}(c_{3})_{lbc}+(c_{3})_{kca}(c_{4})_{ibla}(c_{3})_{jcb}\Big{)}t_{k}\otimes
t_{l},$

where $\\{t_{i}\\}$ is some orthonormal basis262626 Note that an orthonormal
basis in $V$ will necessarily mix degree $k$ and $-k$ components of $V$, so it
cannot be homogeneous w.r.t. $\mathbb{Z}$-grading. However it can be chosen to
be homogeneous w.r.t. the induced $\mathbb{Z}_{2}$-grading.  in $V$ and
$(c_{n})_{i_{1}\cdots i_{n}}$ are the structure constants of the cyclic
$A_{\infty}$ operations $c_{n}$ in this basis, see Figure 23.272727 For
comparison, the operator (62) in these notations is simply
$t_{i}\mapsto(c_{4})_{iaja}t_{j}$.

Figure 23. Combinatorial BV operator.

###  5.2. Generalization – a “long” BV cycle

One has a variant $\mathsf{c}_{\Delta}^{k}$ of the BV cycle above with an even
number $k=2p$, with $p=1,2,\ldots$, of points on each boundary circle (and no
bulk points).

For instance, for $k=4$ it is shown in Figure 24.

Figure 24. “Long” BV cycle $\mathsf{c}_{\Delta}^{k}$ for $k=4$. Numbers
$1,2,3,4$ are the labels of boundary vertices on in/out boundary circles
(vertically aligned in the picture). Square cell on the cylinder is shaded to
make the picture more readable.

Generally, the cycle $\mathsf{c}_{\Delta}^{k}$ consists of $k^{2}$ edges of
$\Xi_{\mathrm{flip}}$ and has the form

(69)
$\mathsf{c}_{\Delta}^{k}=\sum_{i=0}^{k-1}\sum_{j=1}^{k}T_{i}\circ_{c}[R^{j-1}SL^{k-j}]=\sum_{i=0}^{k-1}\sum_{j=1}^{k}\raisebox{-17.07182pt}{
\includegraphics[scale={0.8}]{c_n_1.eps} }$

where we introduced the following notations:

  * •

$S$ is a square (shaded in the picture above), $R$ is a square subdivided into
two triangles by a SW–NE (“right”) diagonal, $L$ is a square subdivided by a
SE–NW (“left”) diagonal.

  * •

$[R^{a}SL^{b}]$ is a strip consisting (going left-to-right) of $a$ right-
triangulated squares, a non-triangulated square, and $b$ left-triangulated
squares. Leftmost and rightmost edges of the strip are identified.

  * •

$T_{i}$ is the operation of cyclically relabeling the vertices of a
triangulated circle, shifting the labels by $i\bmod k$. One can think of
$T_{i}$ as a “degenerate” morphism in the category of cobordisms with
polygonal decompositions, which justifies the notation
$T_{i}\circ_{c}[\cdots]$, meaning the cobordism $[\cdots]$ with the labels of
vertices on the out-boundary (top, in our pictures) shifted by $i\bmod k$.
Likewise (we will need it below), $[\cdots]\circ_{c}T_{-i}$ stands for
shifting the labels of vertices on the in-boundary of the cobordism $[\cdots]$
by $i\bmod k$.

To lighten notations, in the remainder of this section we will be mostly
suppressing $\circ_{c}$ in computations.

Note that for $k=2$, the cycle (69) becomes the cycle of Fig. 21.

######  Proposition 5.3.

Formula (69) indeed defines a BV cycle with coefficients in
$\mathbbm{k}=\mathbb{Z}_{2}$.

###### Proof.

The fact that $\mathsf{c}_{\Delta}^{k}$ is a cycle is checked as follows.
Denote

(70) $\mathcal{A}\colon=\sum_{j=1}^{k}[R^{j-1}SL^{k-j}].$

Note that we have (a) $\partial\mathcal{A}=[R^{k}]-[L^{k}]$ and (b)
$[R^{k}]=T_{1}[L^{k}]$. This implies

(71)
$\partial\mathsf{c}_{\Delta}^{k}=\partial\left(\sum_{i\in\mathbb{Z}_{k}}T_{i}\mathcal{A}\right)=\sum_{i\in\mathbb{Z}_{k}}T_{i}(T_{1}[L^{k}]-[L^{k}])=\sum_{i\in\mathbb{Z}_{k}}T_{i+1}[L^{k}]-T_{i}[L^{k}]=0.$

Intuitively, $\mathcal{A}$ corresponds in $E_{2}^{\mathrm{fr}}(1)$ to rotation
of the in-circle282828 Rather, in $\mathsf{c}_{\Delta}^{k}$ the out- (top)
circle is rotated, but it is equivalent to the opposite rotation of the in-
circle.  by $2\pi/k$ (seen as a 1-chain on the space $E_{2}^{\mathrm{fr}}(1)$)
and $\mathsf{c}_{\Delta}^{k}$ corresponds to the full rotation of the in-
circle, which is a 1-cycle.

The square of $\mathsf{c}_{\Delta}^{k}$ is null-homologous – it is the
boundary of the 3-cycle

(72) $B=D\circ_{c}\mathsf{c}_{\Delta}^{k}$

with

(73)
$D=\sum_{l=0}^{p-1}\sum_{j=2}^{k}T_{2l}[SR^{j-2}SL^{k-j}]=\sum_{l=0}^{p-1}\sum_{j=2}^{k}\raisebox{-17.07182pt}{
\includegraphics[scale={0.7}]{D_2.eps} }$

Indeed, one has

(74) $\partial
D=\sum_{l=0}^{p-1}T_{2l}\Big{(}[SL^{k-1}]+[SR^{k-1}]+\sum_{j=2}^{k}([R^{j-1}SL^{k-j}]+[LR^{j-2}SL^{k-j}])\Big{)}\\\
=\sum_{l=0}^{p-1}T_{2l}(\sum_{j=1}^{k}[R^{j-1}SL^{k-j}]+[SR^{k-1}]+\sum_{j=2}^{k}[LR^{j-2}SL^{k-j}])\\\
=\sum_{l=0}^{p-1}T_{2l}(\mathcal{A}+T_{1}\mathcal{A}T_{-1}).$

We have

(75) $\partial B=\partial(D\mathsf{c}^{k}_{\Delta})=(\partial
D)\mathsf{c}^{k}_{\Delta}=\sum_{l=0}^{p-1}\sum_{i=0}^{k-1}T_{2l}(\mathcal{A}+T_{1}\mathcal{A}T_{-1})T_{i}\mathcal{A}\\\
=\sum_{l=0}^{p-1}\sum_{i=0}^{k-1}(T_{2l}+T_{2l+1})\mathcal{A}T_{i}\mathcal{A}=\sum_{i^{\prime},i=0}^{k-1}T_{i^{\prime}}\mathcal{A}T_{i}\mathcal{A}=\mathsf{c}_{\Delta}^{k}\circ_{c}\mathsf{c}_{\Delta}^{k}$

as claimed. ∎

######  Remark 5.4.

Note that the construction of the 1-cycle (69) makes sense for even or odd $k$
(and yields a cycle). However, in our construction of the cobounding 3-chain
for $\mathsf{c}_{\Delta}^{k}\circ_{c}\mathsf{c}_{\Delta}^{k}$ we do use that
$k$ is even.

######  Remark 5.5.

Repeating the computation for $\mathbbm{k}$ a field of characteristic $\neq
2$, we see that $\mathsf{c}_{\Delta}^{k}$ is still a cycle, but our proof that
it squares to a boundary breaks down due to signs: one obtains $\partial
D=\sum_{l=0}^{p-1}T_{2l}(\mathcal{A}-T_{1}\mathcal{A}T_{-1})$ and hence

(76)
$\partial(D\mathsf{c}_{\Delta}^{k})=\sum_{i^{\prime},i=0}^{k-1}\boxed{(-1)^{i^{\prime}}}T_{i^{\prime}}\mathcal{A}T_{i}\mathcal{A}\neq\mathsf{c}_{\Delta}^{k}\circ_{c}\mathsf{c}_{\Delta}^{k}.$

It is a part of Conjecture 6.14 below that $(\mathsf{c}_{\Delta}^{k})^{2}$ is
null-homologous (not just modulo 2 and not just for even $k$) in the
appropriate “full” Pachner complex $\Xi$.

######  Conjecture 5.6.

The BV cycle $\mathsf{c}_{\Delta}^{k}$ defined by (69) is nontrivial.

The intuition, why $\mathsf{c}_{\Delta}^{k}$ corresponds to the standard
generator of $H_{1}(E_{2}^{\mathrm{fr}}(1))$, was spelled out in the proof
above. A possible way to prove the Conjecture would be to show that
$Z(\mathsf{c}_{\Delta}^{k})$ is nonzero for some explicit minimal cyclic
$A_{\infty}$ algebra $V$ (“minimal” means with $Q=0$).

######  Remark 5.7.

For $V$ a cyclic $A_{\infty}$ algebra of Massey operations on cohomology
$H^{\bullet}(M,\mathbb{Z}_{2})$ of a compact manifold $M$,
$Z(\mathsf{c}_{\Delta}^{k})$ is zero. Indeed, by a degree argument, in each
term there will be a $c_{4}$ factor with $1$ as one of the inputs, hence the
full expression vanishes. A similar vanishing statement holds for the algebra
$H^{\bullet}(M,\mathbb{Z}_{2})$ extended by its coadjoint $A_{\infty}$ module.

In particular, for a proof of the Conjecture above, one needs to look for some
other examples of cyclic $A_{\infty}$ algebras – e.g. those coming from Fukaya
$A_{\infty}$ categories.

######  Remark 5.8 (Dressing the BV cycle by triangulated cobordisms).

Assume that $a,b$ are two triangulated cylinders with $k$ vertices on each
boundary (or more generally $0$-cycles on $\Xi_{\mathrm{flip}}$) and assume
additionally that the composition $b\circ_{c}a$ is “rotation-invariant,” i.e.,

(77) $T_{-1}baT_{1}=ba.$

Then

(78) $c\colon=a\circ_{c}\mathsf{c}_{\Delta}^{k}\circ_{c}b$

is also a BV cycle (for $\mathbbm{k}=\mathbb{Z}_{2}$). The fact that it is a
cycle is obvious: $\partial c=a(\partial c_{\Delta}^{k})b=0$. For its square,
similar to the computation above, we have

(79)
$\partial(aDba\mathsf{c}_{\Delta}^{k}b)=\sum_{l\in\mathbb{Z}_{p}}aT_{2l}\mathcal{A}ba\sum_{i^{\prime}\in\mathbb{Z}_{k}}T_{i^{\prime}}\mathcal{A}b+\sum_{l\in\mathbb{Z}_{p}}aT_{2l+1}\mathcal{A}\underbrace{T_{-1}baT_{1}}_{ba}\sum_{i^{\prime}\in\mathbb{Z}_{k}}T_{i^{\prime}}\mathcal{A}b\\\
=\underbrace{a\sum_{i\in\mathbb{Z}_{k}}T_{i}\mathcal{A}b}_{c}\underbrace{a\sum_{i^{\prime}\in\mathbb{Z}_{k}}T_{i^{\prime}}\mathcal{A}b}_{c}=c\circ_{c}c.$

Thus, $c\circ_{c}c$ is null-homologous.

###  5.3. A combinatorial $\mathrm{BV}_{\infty}$ operator that squares to zero
exactly (or: combinatorial replacement of the operator $Q+uG_{0,-}$)

Notation: Let $\mathrm{cyl}_{k\xrightarrow{m}k}$ stand for a cylinder with the
set $P$ consisting of $k$ vertices on each boundary circle and $m$ bulk
vertices. In particular, we think of $\mathrm{cyl}_{k\xrightarrow{m}k}$ as a
cobordism $S^{1}_{(k)}\rightarrow S^{1}_{(k)}$. In the case $m=0$, we will
just write $\mathrm{cyl}_{k\rightarrow k}$. Note that composing $s$ copies of
$\mathrm{cyl}_{k\rightarrow k}$ yields

(80) $\mathrm{cyl}_{k\rightarrow
k}^{\circ_{c}s}\colon=\underbrace{\mathrm{cyl}_{k\rightarrow
k}\circ_{c}\cdots\circ_{c}\mathrm{cyl}_{k\rightarrow
k}}_{s}=\mathrm{cyl}_{k\xrightarrow{k(s-1)}k}.$

The properties of a combinatorial BV operator

(81) $QZ(\mathsf{c}_{\Delta})=0,\quad Z(\mathsf{c}_{\Delta})^{2}=QZ(B)$

for some 3-chain $B$ can be summarized by saying that the operator

(82) $Q+uZ(\mathsf{c}_{\Delta})-u^{2}Z(B)$

squares to zero modulo $u^{3}$. Here and in the remainder of this section, $u$
is a formal parameter of degree $+2$.

This leads us to the following refinement of the notion of a combinatorial BV
operator.

######  Definition 5.9.

We define a combinatorial $\mathrm{BV}_{\infty}$ operator as a degree $+1$
element

(83) $\Delta_{\infty}\in\mathrm{End}(\mathcal{H}(S^{1}_{(k)}))[[u]]$

of the form

(84) $\Delta_{\infty}=Q+uZ(B_{1})+u^{2}Z(B_{2})+u^{3}Z(B_{3})+\cdots$

with $B_{s}\in C_{2s-1}(\Xi_{\mathrm{flip}}(\mathrm{cyl}_{k\rightarrow
k}^{\circ_{c}s}))$, with $s=1,2,\ldots$, satisfying the equation292929Note
that the degree of $s$-th term in (84) is indeed $2s+(1-2s)=1$.

(85) $(\Delta_{\infty})^{2}=0.$

Given a $\Delta_{\infty}$ operator, $\mathsf{c}_{\Delta}\colon=B_{1}$
automatically satisfies the identities (81), with $B=-B_{2}$.

######  Proposition 5.10.

For $k=2$ one has a $\mathrm{BV}_{\infty}$ operator (84) with coefficients in
$\mathbbm{k}=\mathbb{Z}_{2}$ with

(86) $B_{s}=D^{\circ_{c}(s-1)}\circ_{c}\mathsf{c}_{\Delta},\quad
s=1,2,\ldots,$

with $\mathsf{c}_{\Delta}=[SL]+[RS]+T_{1}[SL]+T_{1}[RS]$ the BV cycle of
Figure 21 and $D=[SS]$ – the 2-cycle (73) for $k=2$.

In particular, the $\mathrm{BV}_{\infty}$ operator can be written as

(87) $\Delta_{\infty}=Q+\frac{u}{1+uZ(D)}Z(\mathsf{c}_{\Delta})=\\\
=Q+uZ\left(\vbox{\hbox{ \includegraphics[scale={0.5}]{BV_infty_term1.eps}
}}\right)+u^{2}Z\left(\vbox{\hbox{
\includegraphics[scale={0.5}]{BV_infty_term2.eps}
}}\right)+u^{3}Z\left(\vbox{\hbox{
\includegraphics[scale={0.5}]{BV_infty_term3.eps} }}\right)+\cdots$

Pictures in the formula above are drawn on a cylinder: left and right sides
are always identified.

###### Proof.

We only have to check that (85) holds. From (84) we have

(88) $\Delta_{\infty}^{2}=\sum_{s\geq 1}u^{s}\left(QZ(B_{s})+\sum_{1\leq
s^{\prime}<s}Z(B_{s^{\prime}})Z(B_{s-s^{\prime}})\right)=\\\ =\sum_{s\geq
1}u^{s}Z\left(\partial B_{s}+\sum_{1\leq
s^{\prime}<s}B_{s^{\prime}}B_{s-s^{\prime}}\right).$

Note that from (74) we have

(89) $\partial D=\mathcal{A}+T_{1}\mathcal{A}T_{-1}.$

From (86), using the fact that $T_{i}$ commutes with $D$ and that
$T_{i}\mathsf{c}_{\Delta}=\mathsf{c}_{\Delta}$, we get

(90) $\partial B_{s}=\sum_{s^{\prime}=1}^{s-1}D^{s^{\prime}-1}(\partial
D)D^{s-1-s^{\prime}}\mathsf{c}_{\Delta}=\sum_{s^{\prime}=1}^{s-1}D^{s^{\prime}-1}\underbrace{(\mathcal{A}+T_{1}\mathcal{A})}_{\mathsf{c}_{\Delta}}D^{s-1-s^{\prime}}\mathsf{c}_{\Delta}\\\
=\sum_{s^{\prime}=1}^{s-1}D^{s^{\prime}-1}\mathsf{c}_{\Delta}D^{s-1-s^{\prime}}\mathsf{c}_{\Delta}=\sum_{s^{\prime}=1}^{s-1}B_{s^{\prime}}B_{s-s^{\prime}}.$

Thus, all terms in the r.h.s. of (88) vanish. ∎

######  Remark 5.11.

In continuum TCFT one has an important operator

(91) $Q_{S^{1}}\colon=Q+uG_{0,-}$

– the $S^{1}$-equivariant BRST operator acting on local fields. The
$\mathrm{BV}_{\infty}$ operators discussed in this section can be thought of
as a combinatorial replacement of $Q_{S^{1}}$.

Note that the operator (91) squares to $uL_{0,-}$ whereas $\Delta_{\infty}$
squares to zero. This may be interpreted as corresponding to the combinatorial
HTQFT having only fields of zero spin (thus, $L_{0,-}$ vanishes).

####  5.3.1. Dual language: $\mathrm{BV}_{\infty}$ chains as Maurer-Cartan
elements in the dga $\mathrm{CYL}_{k}$

Fix $k=1,2,\ldots$ and consider the differential graded algebra

(92) $\mathrm{CYL}_{k}^{\bullet}\colon=\bigoplus_{s\geq
1}C_{-\bullet}(\Xi_{\mathrm{flip}}(\mathrm{cyl}_{k\rightarrow
k}^{\circ_{c}s}))$

with differential given by the boundary operator $\partial$ on cellular chain
on $\Xi_{\mathrm{flip}}$ and with associative multiplication $\circ_{c}$
induced by composition of cobordisms (stacking of cylinders). Then the
Definition 5.9 implies that for a $\mathrm{BV}_{\infty}$ operator
$\Delta_{\infty}$, the constituent chains $B_{s}$ arrange into an element

(93) $\mathcal{C}\colon=uB_{1}+u^{2}B_{2}+\cdots\quad\in\mathrm{CYL}_{k}[[u]]$

satisfying the Maurer-Cartan equation

(94) $\partial\mathcal{C}+\mathcal{C}\circ_{c}\mathcal{C}=0.$

In other words, $\mathcal{C}$ is a Maurer-Cartan element of the dga
$\mathrm{CYL}_{k}[[u]]$.

###  5.4. Questions/wish list for the flip theory

  1. (i)

Construct explicit examples of BV cycles in $\Xi_{\mathrm{flip}}$ with
coefficients in $\mathbb{Z}$ rather than $\mathbb{Z}_{2}$. Do these examples
satisfy the 7-term relation (33) in $\Xi_{\mathrm{flip}}$ up to a boundary,
with product realized by a triangulated pair of pants?

  2. (ii)

Does the 7-term relation hold in $\Xi_{\mathrm{flip}}$ for the BV cycles
$\mathsf{c}_{\Delta}^{k}$ (with coefficients in $\mathbb{Z}_{2}$)?

  3. (iii)

Find examples of cyclic $A_{\infty}$ algebras for which the combinatorial BV
operator associated with the BV cycle $\mathsf{c}_{\Delta}^{k}$ is nontrivial
in $Q$-cohomology. In particular, this would prove Conjecture 5.6.

  4. (iv)

Understand the homotopy type of $\Xi_{\mathrm{flip}}$ for $\Sigma$ a surface
with boundary, cf. Question 4.6.

In particular, for $\Sigma$ a disk, does $\Xi_{\mathrm{flip}}$ have nontrivial
$H_{1}$ for sufficiently many bulk vertices, or does it stay contractible for
any set of vertices?

For $\Sigma$ a cylinder and $P$ containing no bulk points, is
$\Xi_{\mathrm{flip}}$ homotopy equivalent to $E_{2}^{\mathrm{fr}}(1)\sim
S^{1}$? And if so, does $[c_{\Delta}^{k}]$ map to the standard generator of
$H_{1}$ under this equivalence?

  5. (v)

One may consider a class of cyclic $A_{\infty}$ algebras arising as an
$A_{\infty}$ deformation of the commutative associative ring
$\mathbb{A}=O_{X}$ of functions on a manifold $X$ by a Maurer-Cartan element
of the Hochschild dgLa of $\mathbb{A}$. One can interpret the corresponding
combinatorial HTQFT as a model of topological string theory with target $X$.
One can also consider $X$ to be a noncommutative space. Note that
noncommutativity can in particular play the role of a regulator, making
$\mathbb{A}$ finite-dimensional (e.g. taking $X$ to be the fuzzy sphere).

  6. (vi)

It would be interesting to study flip theory for $V$ a _minimal_ cyclic
$A_{\infty}$ algebra, i.e., with $Q=0$ (but with nontrivial $m_{3}$). In this
case, partition functions for triangulated cobordisms are strictly (not modulo
$Q$) independent of the triangulation.

##  6\. Towards the full 2d HTQFT on triangulated cobordisms

In this section we explain the idea that making an appropriate replacement for
$\Xi_{\mathrm{flip}}$ coming from the secondary polytope construction, may
resolve the problems in the construction of combinatorial 2d HTQFT (namely,
one wants compatibility with stellar subdivisions/aggregations, non-fixed set
of vertices of triangulations, and a homotopy equivalence between $\Xi$ and
Zwiebach’s moduli space $\widetilde{\mathcal{M}}_{h,n}$ arising in continuum
TCFT [25]).

In particular, we construct Model 2 of combinatorial 2d HTQFT – “secondary
polytope theory.” It is a partial model: it is only defined on a disk,
realized as a convex hull of a set of points $A\subset\mathbb{R}^{2}$. This
model is invariant up to homotopy w.r.t. both Pachner moves. The appropriate
local algebraic input for the model is an “$\widehat{A}_{\infty}$ algebra” – a
refinement of a cyclic $A_{\infty}$ algebra by certain extra homotopies
(defined in Section 6.2). As an aside, we present a 1d toy model of the
construction yielding a generalization of the standard setting for topological
quantum mechanics, which may be of independent interest.

We also discuss the ideal Model 3, conjecturally defined on arbitrary
surfaces, compatible with both Pachner moves and having the expected homotopy
type of $\Xi$. In particular, in this model one has a natural BV algebra
structure on $Q$-cohomology.

###  6.1. A quick recollection of secondary polytopes

Fix $d\geq 1$ (we will be interested primarily in the case $d=2$) and a finite
collection of points in general linear position $A\subset\mathbb{R}^{d}$. We
assume that the affine span of $A$ is $\mathbb{R}^{d}$. The construction of
[4] (see [8, Section 1] for a detailed review) assigns to $A$ the “secondary
polytope” – a convex polytope
$\mathrm{sp}(A)\subset\mathbb{R}^{A}/\mathrm{Aff}(\mathbb{R}^{d})\simeq\mathbb{R}^{|A|-d-1}$
(with $\mathrm{Aff}(\mathbb{R}^{d})$ the space of affine linear functions on
$\mathbb{R}^{d}$) such that:

  * •

Vertices of $\mathrm{sp}(A)$ correspond to _regular_ triangulations of the
convex hull $\mathrm{Conv}(A)\subset\mathbb{R}^{d}$ with vertices in $A$
(there is no requirement that all points of $A$ have to be used as vertices of
the triangulation). A triangulation is “regular” if there exists a convex
continuous function on $\mathrm{Conv}(A)$, linear on each simplex and
“breaking” (non-differentiable) on the codimension one faces of the
triangulation. More precisely, the vertex $e_{T}$ corresponding to a
triangulation $T$ is

(95) $e_{T}=\pi(a\mapsto\mathrm{vol}(\mathrm{Star}(a)))$

– the function assigning to a point $a\in A$ the volume of its star in $T$,
seen as an element of $\mathbb{R}^{A}$ and then projected by the quotient map
$\pi\colon\mathbb{R}^{A}\mapsto\mathbb{R}^{A}/\mathrm{Aff}(\mathbb{R}^{d})$.
Thus, the secondary polytope is the convex hull of points (95):

(96) $\mathrm{sp}(A)=\mathrm{Conv}(\\{e_{T}\\}).$

  * •

Faces of $\mathrm{sp}(A)$ correspond to polyhedral subdivisions of
$\mathrm{Conv}(A)$. More precisely, a face $e_{\alpha}$ of $\mathrm{sp}(A)$
corresponds to a collection $\alpha$ of subsets $A_{i}$ of $A$ such that

(97) $\mathrm{Conv}(A)=\cup_{i}\mathrm{Conv}(A_{i})$

is a _regular_ polyhedral subdivision of $\mathrm{Conv}(A)$, i.e., a
presentation as a union of convex $d$-dimensional polyhedra (with non-empty
interior) overlapping over faces. Here “regular” again means that there exists
a convex continuous function on $\mathrm{Conv}(A)$, linear on the constituent
polyhedra and breaking along faces. The face $e_{\alpha}$ is combinatorially
equivalent303030 I.e., has the same poset of faces.  to the product of
secondary polytopes

(98) $e_{\alpha}\simeq\prod_{i}\mathrm{sp}(A_{i}).$

In particular for the dimension of the face $e_{\alpha}$ one has

(99) $\dim e_{\alpha}=\sum_{i}(|A_{i}|-d-1).$

We will refer to points of $A_{i}$ in the interior of $\mathrm{Conv}(A_{i})$
as “floating points.”

######  Remark 6.1.

In the terminology of [8], for $A\subset\mathbb{R}^{d}$ a set of points in
general position and $\mathsf{p}=\mathrm{Conv}(A)$ a convex
polytope,313131When we say “convex polytope,” we understand that it should
have non-empty interior. the pair $(\mathsf{p},A)$ is called a “marked
polytope.” For $A^{\prime}\subset A$,
$\mathsf{p}^{\prime}=\mathrm{Conv}(A^{\prime})$ a convex polytope, the pair
$(\mathsf{p}^{\prime},A^{\prime})$ is called a “marked subpolytope” of
$(\mathsf{p},A)$. A polyhedral subdivision of $(\mathsf{p},A)$ is a collection
of marked subpolytopes $\\{(\mathsf{p}_{i},A_{i})\\}$ such that
$\mathsf{p}=\cup_{i}\mathsf{p}_{i}$ is a polyhedral subdivision, i.e.,
$\mathsf{p}_{i}\cap\mathsf{p}_{j}$ is either empty or a common face of
$\mathsf{p}_{i}$ and $\mathsf{p}_{j}$.

One has a poset $R(\mathsf{p},A)$ of polyhedral subdivisions of
$(\mathsf{p},A)$, with order given by refinement. The minimal elements of
$R(\mathsf{p},A)$ are regular triangulations. The unique maximal element is
the tautological polyhedral subdivision, with $\\{(\mathsf{p}_{i},A_{i})\\}$
consisting of just $(\mathsf{p},A)$ itself. The face poset of the secondary
polytope $\mathrm{sp}(A)$ coincides with $R(\mathsf{p},A)$.

######  Example 6.2.

For $d=2$, $A\subset\mathbb{R}^{2}$ the set of vertices of a convex polygon,
$\mathrm{sp}(A)$ is combinatorially equivalent323232 In the following we will
usually omit “combinatorially equivalent” when describing a model for the
secondary polytope.  to Stasheff’s associahedron $K_{|A|-1}$.

For instance, for $|A|=4$ and $|A|=5$ the pictures of secondary polytopes are
those of Figure 11 and the right part of Figure 12, ignoring the shaded
regions.

######  Example 6.3.

For any $d$ and $A=\\{x_{0},\ldots,x_{d},y\\}$ a $(d+2)$-tuple of points such
that $y$ is in the interior of the $d$-simplex
$\sigma=\mathrm{Conv}(\\{x_{0},\ldots,x_{d}\\})$, the secondary polytope
$\mathrm{sp}(A)$ is an interval, with one endpoint decorated by tautological
triangulation of $\sigma$, the other – with stellar subdivision of $\sigma$
with vertex at $y$. The interval itself is decorated with tautological
triangulation of $\sigma$ with $y$ a floating point, see Figures 25, 26.

Figure 25. Secondary polytope for three points $A=\\{x_{0},x_{1},y\\}$ in
$\mathbb{R}$ with $x_{0}<y<x_{1}$. For the left endpoint of $\mathrm{sp}$,
$\\{A_{i}\\}=\\{\\{x_{0},x_{1}\\}\\}$. For the bulk,
$\\{A_{i}\\}=\\{\\{x_{0},x_{1},y\\}\\}$ with $y$ the floating point, shown as
a hollow vertex. For the right endpoint,
$\\{A_{i}\\}=\\{\\{x_{0},y\\},\\{x_{1},y\\}\\}$. Figure 26. Secondary polytope
for four points in $\mathbb{R}^{2}$ in non-convex position.

######  Remark 6.4.

For any $d\geq 1$, the edges of a secondary polytope correspond to Pachner
moves between regular triangulations. In particular, for $d=2$, both Pachner
flips and stellar subdivisions/aggregations can appear as edges of
$\mathrm{sp}(A)$.

######  Example 6.5.

For $d=1$ and any finite set of distinct points $A\subset\mathbb{R}$, the
secondary polytope $\mathrm{sp}(A)$ is a cube of dimension $|A|-2$, see
Figures 27, 28. More explicitly, assuming $A$ is the ordered set
$x_{0}<x_{1}<\cdots<x_{n}<x_{n+1}$ in $\mathbb{R}$, faces of $\mathrm{sp}(A)$
correspond to making for each $j=1,\ldots,n$ a choice whether

  1. (i)

$x_{j}$ appears in $\cup_{i}A_{i}$ as a vertex of the triangulation
$\mathrm{Conv}(A)=\cup_{i}\mathrm{Conv}(A_{i})$, or

  2. (ii)

$x_{j}$ appears as a floating point – interior point of $\mathrm{Conv}(A_{i})$
for some $i$, or

  3. (iii)

$x_{j}$ is not in $\cup_{i}A_{i}$.

The dimension of the face is the number of floating points among
$x_{1},\ldots,x_{n}$.

Figure 27. Secondary polytope for $A$ consisting of four points in
$\mathbb{R}$. Figure 28. Secondary polytope for $A$ consisting of five points
in $\mathbb{R}$. Decorations of vertices and 2-faces are shown.

Figure 29. Example: secondary polytope for $A$ consisting of five points in
$\mathbb{R}^{2}$, with $\mathrm{Conv}(A)$ a triangle. The secondary polytope
depends on convex geometry of the configuration; the situation shown
corresponds to the two floating points in a certain open set (“chamber”) in
the configuration space of two points. Figure 30. Example: secondary polytope
for a square with one floating point in $\mathbb{R}^{2}$. Again, combinatorics
of the face poset depends on the “chamber” in which the floating point is
located. Chambers in this case are the connected components of the square
interior with the two diagonals removed.

####  6.1.1. Admissible subsets of $\mathbb{R}^{d}$

Let us denote $\mathrm{Adm}_{m}(\mathbb{R}^{d})$ the set of “admissible
configurations of $m$ points” – subsets $A\subset\mathbb{R}^{d}$ with $|A|=m$
such that:

  1. (i)

$A$ is in general linear position and the affine span of $A$ is
$\mathbb{R}^{d}$.

  2. (ii)

$A$ is not in the locus of “exceptional wall-crossing,” see [9, Section
5.2].333333Configurations satisfying condition (i) form an open dense subset
of $(\mathbb{R}^{d})^{m}/S_{m}$. However, as one travels inside a connected
component of this subset, the combinatorial type of the secondary polytope
$\mathrm{sp}(A)$ might change (“jump”). Locus where such change happens are
the “exceptional walls.”

The set $\mathrm{Adm}_{m}(\mathbb{R}^{d})$ is an open dense subset of
$(\mathbb{R}^{d})^{m}/S_{m}$. We will call its connected components
“configuration chambers.” For $d=2$,
$\mathrm{Adm}_{m}(\mathbb{R}^{2})\subset(\mathbb{R}^{2})^{m}/S_{m}$ is known
(see [9]) to be $\mathbb{R}$-Zarisski open.

For an admissible configuration $A\subset\mathbb{R}^{d}$, we denote $\langle
A\rangle$ the number of codimension one faces of the polytope
$\mathrm{Conv}(A)$. We will also denote $\mathrm{Adm}_{m,n}(\mathbb{R}^{d})$
the set of admissible configurations of $m$ points in $\mathbb{R}^{d}$ such
that $\langle A\rangle=n$.

For instance for $d=2$, elements of $\mathrm{Adm}_{m,n}(\mathbb{R}^{2})$
correspond (via transitioning to the marked polygon
$(\mathsf{p}=\mathrm{Conv}(A),A)$) to convex $n$-gons in $\mathbb{R}^{2}$ with
$m-n$ “floating points” inside.

###  6.2. The definition of an $\widehat{A}_{\infty}$ algebra

Operations. We will call an $\widehat{A}_{\infty}$ algebra (or
“$\mathsf{SP}_{2}$ algebra”) a cochain complex $V$ over $\mathbbm{k}$ with
differential $Q$ equipped with a compatible metric $g$ and a family of
multilinear maps (“operations”) enumerated by “configuration chambers” –
elements of $\pi_{0}\mathrm{Adm}_{m,n}(\mathbb{R}^{2})$. A configuration
$A\in\mathrm{Adm}_{m,n}(\mathbb{R}^{2})$ is assigned a multilinear map

(100) $c_{[A]}\colon V^{\otimes n}\rightarrow\mathbbm{k}$

of degree $3-m$ (here $[A]$ stands for the configuration chamber of $A$). We
understand that inputs in $V$ are assigned to the edges of the polygon
$\mathrm{Conv}(A)$.

(Partial) invariance under cyclic permutations of inputs. We will say that a
configuration chamber $[A]$, for $A\in\mathrm{Adm}_{m,n}(\mathbb{R}^{2})$, is
“$\mathbb{Z}_{q}$-symmetric,” for $q>1$ a divisor of $n$, if one can (i) move
$A$ staying in the same configuration chamber so that $\mathrm{Conv}(A)$ is a
regular polygon and then (ii) move the configuration of interior points to the
same configuration rotated by the angle $2\pi/q$ while staying in the same
configuration chamber. If $[A]$ is $\mathbb{Z}_{q}$-symmetric, we impose the
condition that the operation $c_{[A]}$ is invariant under the subgroup
$\mathbb{Z}_{q}\subset\mathbb{Z}_{n}$ acting by cyclic permutations of inputs.

######  Example 6.6.

If $A$ is an $n$-tuple of points in $\mathbb{R}^{2}$ in convex position, then
$[A]$ has $\mathbb{Z}_{n}$ symmetry. If $A$ is as in Figure 26, $[A]$ has
$\mathbb{Z}_{3}$ symmetry. If $A$ is as in Figure 29 or Figure 30, then $[A]$
has no $\mathbb{Z}_{q}$ symmetry. The chamber of the configuration of two
floating points in a square in Figure 31 has $\mathbb{Z}_{2}$ symmetry.

Figure 31. Configuration chamber in
$\pi_{0}\mathrm{Adm}_{6,4}(\mathbb{R}^{2})$ with $\mathbb{Z}_{2}$ symmetry.

Differential. The differential $Q$ acts on operations as

(101) $Qc_{[A]}=-c_{[\partial\,\mathrm{sp}(A)]}.$

Here on in the right we take the boundary of the secondary polytope of $A$.
The codimension one faces comprising it correspond to special (“coarse,” in
the terminology of [8]) polygonal subdivisions
$\\{(\mathsf{p}_{i}=\mathrm{Conv}(A_{i}),A_{i})\\}$ of
$(\mathsf{p}=\mathrm{Conv}(A),A)$. For each such subdivision we compute the
contraction of tensors $c_{[A_{i}]}$ assigned to subpolygons, and the inverse
metric $g^{-1}$ assigned to the edges, and then sum over the boundary faces of
$\mathrm{sp}(A)$. I.e., in more detail, (101) is:

(102)
$Qc_{[A]}=-\sum_{\alpha=\\{(\mathsf{p}_{i},A_{i})\\}\;\mathrm{coarse\;subdivision\;of\;}(\mathsf{p},A)}\left\langle\bigotimes_{i}c_{[A_{i}]},\bigotimes_{\mathrm{edges\;of\;}\alpha}g^{-1}\right\rangle.$

######  Remark 6.7.

Specializing the structure to $n$-tuples of points in $\mathbb{R}^{2}$ in
convex position (i.e. vertices of a convex $n$-gon), for $n\geq 3$, we recover
the definition of a cyclic $A_{\infty}$ algebra, with (102) the $A_{\infty}$
relation (cf. Figure 17). We will denote the operation corresponding to a
convex $n$-gon with no floating points by $c_{n}$, to be compatible with
notations in Section 4.1.

Thus, an $\widehat{A}_{\infty}$ algebra is in particular a cyclic $A_{\infty}$
algebra and should be thought of as a refinement of the latter by certain
extra homotopies.

######  Example 6.8.

Let $A$ be a quadruple of points in $\mathbb{R}^{2}$ in non-convex position (a
triangle with a floating point inside). Let us denote the corresponding degree
$-1$ operation $\mu\colon=c_{[A]}\colon V^{\otimes 3}\rightarrow\mathbbm{k}$.
The relation (100) for it takes form

(103)
$(Q\mu)(x,y,z)=\mathrm{Str}_{V}m_{2}(x,m_{2}(y,m_{2}(z,\bullet)))-c_{3}(x,y,z),\qquad
x,y,z\in V,$

where the two terms in the r.h.s. correspond to the two endpoints of the
secondary polytope in Figure 26; $m_{2}$ stands for the operation $c_{3}$ with
one input converted into output by $g^{-1}$.

Note that relation (103) is an up-to-homotopy version of the relation (11)
corresponding to second Pachner move (stellar subdivision/aggregation). The
corresponding homotopy, witnessing the failure of the invariance under the
move to hold strictly, is $\mu$ – the $\widehat{A}_{\infty}$ operation for a
triangle with a floating point.

######  Remark 6.9.

The first few operations in an $\widehat{A}_{\infty}$ algebra, arranged by
degree and arity, are as follows.

degree $\downarrow$; arity $\rightarrow$ | $3\rightarrow 0$ | $4\rightarrow 0$ | $5\rightarrow 0$
---|---|---|---
0 | $c_{3}=c\left(\left[\vbox{\hbox{\includegraphics[scale={0.35}]{Ahat_c3_0.eps}}}\right]\right)$ |  |
-1 | $\mu=c\left(\left[\vbox{\hbox{\includegraphics[scale={0.35}]{Ahat_c3_1.eps}}}\right]\right)$ | $c_{4}=c\left(\left[\vbox{\hbox{\includegraphics[scale={0.35}]{Ahat_c4_0.eps}}}\right]\right)$ |
-2 | $c\left(\left[\vbox{\hbox{\includegraphics[scale={0.35}]{Ahat_c3_2.eps}}}\right]\right)$ | $c\left(\left[\vbox{\hbox{\includegraphics[scale={0.35}]{Ahat_c4_1.eps}}}\right]\right)$ | $c_{5}=c\left(\left[\vbox{\hbox{\includegraphics[scale={0.35}]{Ahat_c5_0.eps}}}\right]\right)$

Note that the left and middle operations in the last row here do not have any
$\mathbb{Z}_{q}$ symmetry, whereas the remaining operations listed in the
table are invariant w.r.t. cyclic permutations of inputs.

######  Remark 6.10.

An $\widehat{A}_{\infty}$ algebra can be formalized as an algebra over a
“cyclic” dg operad $\widehat{A}_{\infty}$ freely generated by the generators
$c_{[A]}$ assigned to elements of $\pi_{0}\mathrm{Adm}_{m,n}(\mathbb{R}^{2})$,
with the differential defined by (101), (102). Here “cyclic” means that
operations have output in the ground field and that there is an extra
structure (contraction with $g^{-1}$) allowing to convert any input into an
output.

Since the operad $\widehat{A}_{\infty}$ is based on the combinatorics of
secondary polytopes for subsets of $\mathbb{R}^{2}$, we will also call this
operad $\mathsf{SP}_{2}$.

###  6.3.  Combinatorial HTQFT on a convex polygon based on secondary
polytopes (“Model 2”)

Let $V$ be an $\widehat{A}_{\infty}$ algebra and
$A\in\mathrm{Adm}_{m,n}(\mathbb{R}^{2})$ an admissible configuration of points
on the plane. Then one has a combinatorial HTQFT on the polygon
$\Sigma=\mathrm{Conv}(A)$ with $A$ the set of admissible vertices of polygonal
decompositions of $\Sigma$. This HTQFT is the cochain on the secondary
polytope

(104) $Z\in C^{\bullet}(\mathrm{sp}(A))\otimes\mathrm{Hom}(V^{\otimes
n},\mathbbm{k})$

defined by

(105)
$Z(e_{\alpha})=\Big{\langle}\bigotimes_{i}c_{[A_{i}]},\bigotimes_{\mathrm{edges\;of\;}\alpha}g^{-1}\Big{\rangle}$

for $\alpha=\\{(\mathsf{p}_{i},A_{i})\\}$ a polygonal subdivision of
$(\Sigma,A)$. Due to the relation (102) in the algebra $V$, the cochain $Z$
satisfies the main equation of HTQFT,

(106) $(\delta+Q)Z=0.$

This construction realizes the “full” 2d combinatorial HTQFT on a convex
polygon, with Pachner flips between triangulations and stellar aggregations
allowed, with a fixed set of admissible vertices.343434In this context one can
think of the set of admissible vertices $A$ as “aether” – a background
geometric structure.

An encouraging property of this construction is that $\mathrm{sp}(A)$ is
contractible (in particular, it is homotopy equivalent to
$E_{2}^{\mathrm{fr}}(0)$), whereas $\Xi_{\mathrm{flip}}$ for a disk with $N$
bulk vertices is expected to have nontrivial homology depending on $N$, cf.
Section 4.2.1.

###  6.4. A toy model: combinatorial 1d HTQM based on secondary polytopes

####  6.4.1. The definition of an $\mathsf{SP}_{1}$ algebra

Similarly to Section 6.2, we can define an algebraic structure corresponding
to secondary polytopes for subsets of $\mathbb{R}^{1}$.

We define a $\mathsf{SP}_{1}$ algebra as a cochain complex $V$ with
differential $Q$ and a collection of endomorphisms $U_{n}\colon V\rightarrow
V$ of degree $-n$ (the operation corresponding to a subset $A$ of $n+2$ points
of $\mathbb{R}$, with the corresponding marked polytope being an interval
$\mathrm{Conv}(A)\subset\mathbb{R}$ with $n$ floating points), for
$n=0,1,2,\ldots$, subject to the relation353535 Understanding that $Q$ acts on
endomorphisms of $V$ via adjoint action, we use notations $QU_{n}$ and
$[Q,U_{n}]$ interchangeably (in the first notation, $Q$ acts on the operation,
in the second – on the output and on the input), cf. footnote 15.

(107) $QU_{n}=\sum_{i=0}^{n-1}(-1)^{i}(U_{i}U_{n-i-1}-U_{n-1}).$

This formula is an explicit form of (102) for $A\subset\mathbb{R}^{1}$; the
$2n$ terms in the r.h.s. correspond to the codimension one faces of the
$n$-cube $\mathrm{sp}(A)$, cf. Example 6.5. In particular, the first relations
are:

(108) $\displaystyle QU$ $\displaystyle=0,$ (109) $\displaystyle QU_{1}$
$\displaystyle=U^{2}-U,$ (110) $\displaystyle QU_{2}$
$\displaystyle=UU_{1}-U_{1}U,$ (111) $\displaystyle QU_{3}$
$\displaystyle=UU_{2}-U_{1}^{2}+U_{2}U-U_{2},$ $\displaystyle\cdots$

where we denoted $U\colon=U_{0}$.

One can introduce the corresponding notion of $\mathsf{SP}_{1}$ operad – a dg
operad freely generated by unary operations $U_{n}$, with differential acting
as in (107).

Note that equation (109) is the up-to-homotopy version of the equation (21) in
strict topological quantum mechanics.

####  6.4.2. Combinatorial HTQM

Fix a set $A\subset\mathbb{R}$ and an $\mathsf{SP}_{1}$ algebra
$(V,Q,\\{U_{n}\\}_{n\geq 0})$. We define a combinatorial HTQM as a cochain $Z$
on the secondary polytope $\mathrm{sp}(A)$ with values in $\mathrm{End}(V)$,
as follows. For $\alpha=\\{(I_{i},A_{i})\\}$ a triangulation of the marked
interval $(I=\mathrm{Conv}(A),A)$ into marked sub-intervals (we assume that
the intervals $I_{i}$ are ordered from left to right in $\mathbb{R}$), the
value of $Z$ on the corresponding face $e_{\alpha}$ of the secondary polytope
is

(112)
$Z(e_{\alpha})=\overleftarrow{\prod_{i}}U_{|A_{i}|-2}\qquad\in\mathrm{End}(V).$

By virtue of the relation (107), we have the main equation of HTQFT:

(113) $(\delta+Q)Z=0.$

######  Example 6.11.

For instance, the value of HTQM on an interval $I$ subdivided into $N$ sub-
intervals with no floating points is $U^{N}$. We think of these sub-intervals
as “elementary” (or “atomic”) intervals. $U_{1}$ is the chain homotopy between
HTQM on an interval consisting of (triangulated into) two elementary ones and
a single elementary interval, cf. (109).

Another example of formula (112), for an interval triangulated into four sub-
intervals, one of them with a floating point:

(114)
$Z\left(\vbox{\hbox{\includegraphics[scale={0.7}]{HTQM_ex.eps}}}\right)=UU_{1}U^{2}.$

######  Example 6.12.

Assume we are given a continuum HTQM as in Example 3.3, with the space of
states being a cochain complex $V,Q$ and with the Hamiltonian $H=[Q,G]$, with
$G\in\mathrm{End}(V)$ a degree $-1$ endomorphism satisfying $G^{2}=0$.

Fix $T>0$ – the length of “elementary/atomic interval.” Then we can construct
an $\mathsf{SP}_{1}$ algebra structure on the complex $(V,Q)$ as follows:

(115) $\displaystyle U$ $\displaystyle=e^{TH},$ (116) $\displaystyle U_{1}$
$\displaystyle=\int_{T}^{2T}e^{tH+dt\,G}=\int_{T}^{2T}dt\,Ge^{tH}=G\frac{e^{2TH}-e^{TH}}{H},$
(117) $\displaystyle U_{\geq 2}$ $\displaystyle=0.$

######  Example 6.13 (“Infinitesimal” $\mathsf{SP}_{1}$ algebra).

Fix a cochain complex $(V,Q)$. Consider the following ansatz for an
$\mathsf{SP}_{1}$ algebra

(118) $U=1+TH+O(T^{2}),\;U_{1}=TG+O(T^{2}),\;U_{2}=T^{2}G_{2}+O(T^{3}),\\\
U_{3}=T^{2}G_{3}+O(T^{3}),\;U_{4}=T^{3}G_{4}+O(T^{4}),\ldots,U_{n}=T^{[n/2]+1}G_{n}+O(T^{[n/2]+2}),\ldots$

with $H=G_{0}$, $G=G_{1}$, $G_{2}$, etc. a sequence of endomorphisms of $V$,
with $G_{n}$ of degree $-n$. Here $T$ is a formal infinitesimal parameter.
Then the $\mathsf{SP}_{1}$ structure equation (107) considered in the leading
order in $T$, yields the sequence of equations

(119) $\displaystyle[Q,H]$ $\displaystyle=0,$ (120) $\displaystyle[Q,G]$
$\displaystyle=H,$ (121) $\displaystyle[Q,G_{2}]$ $\displaystyle=[H,G],$ (122)
$\displaystyle[Q,G_{3}]$ $\displaystyle=G_{2}-G^{2},$ (123)
$\displaystyle[Q,G_{4}]$ $\displaystyle=[H,G_{3}]-[G,G_{2}],$ (124)
$\displaystyle[Q,G_{5}]$ $\displaystyle=G_{4}-[G,G_{3}],$ (125)
$\displaystyle[Q,G_{6}]$ $\displaystyle=[H,G_{5}]-[G,G_{4}]+[G_{2},G_{3}],$
(126) $\displaystyle[Q,G_{7}]$ $\displaystyle=G_{6}-[G,G_{5}]-G_{3}^{2},$
$\displaystyle\vdots$

Generally, the relations are:

(127) $\displaystyle[Q,G_{2k}]$
$\displaystyle=\sum_{i=0}^{k}(-1)^{i}[G_{i},G_{2k-i-1}],$ (128)
$\displaystyle[Q,G_{4k+1}]$
$\displaystyle=G_{4k}-\sum_{i=0}^{k-1}[G_{2i+1},G_{4k-2i-1}],$ (129)
$\displaystyle[Q,G_{4k+3}]$
$\displaystyle=G_{4k+2}-\sum_{i=0}^{k-1}[G_{2i+1},G_{4k-2i+1}]-\frac{1}{2}[G_{2k+1},G_{2k+1}],$

for $k\geq 0$. We call the collection of endomorphisms
$H=G_{0},G=G_{1},G_{2},\ldots$ subject to relations above an “infinitesimal
$\mathsf{SP}_{1}$ algebra.”

Note that in the special case $G_{2}=G_{3}=\cdots=0$, we recover the data of a
continuum HTQM of Example 3.3.

Note also that the relations above involve only supercommutators. Thus, it is
tempting to consider the dg Lie algebra $\mathfrak{sp}_{1}$ with generators
$Q,\\{G_{n}\\}_{n\geq 0}$ and differential $[Q,-]$, subject to relations
(127)–(129). In this terminology, an infinitesimal $\mathsf{SP}_{1}$ algebra
is a representation of the dgLa $\mathfrak{sp}_{1}$ on a complex $(V,Q)$.

###  6.5.  Tentative Model 3 on triangulated surfaces. Conjecture on existence
of the proper Pachner CW complex

In this section we sketch a conjectural combinatorial model of 2d HTQFT –
“Model 3” or “Ideal HTQFT” – combining the features of flip theory of Sections
4–5 (exists on surfaces, contains BV operators) and of the HTQFT on a convex
polygon from secondary polytopes of Section 6.3 (compatible with both Pachner
moves, the number of vertices is not fixed, the homotopy type of $\Xi$ is
under control).

####  6.5.1. Pachner complex

######  Conjecture 6.14.

Let $\Sigma$ be a surface of genus $h$ with $n$ boundary circles. Assume that
the boundary circles are triangulated into $k_{1},\ldots,k_{n}$ intervals. We
conjecture that there exists a CW complex $\Xi(\Sigma,\\{k_{i}\\})$ – “Pachner
complex” – such that:

  1. (i)

Vertices of $\Xi$ correspond to “admissible” triangulations of $\Sigma$,
inducing the given triangulation on the boundary.363636We are blackboxing the
appropriate notion of admissibility. In the setting of secondary polytopes, it
is regularity – existence of a convex piecewise linear function inducing the
given triangulation.

  2. (ii)

Edges of $\Xi$ correspond to Pachner moves between triangulations – flips or
stellar subdivisions/aggregations.

  3. (iii)

$\Xi$ is homotopy equivalent to Zwiebach’s moduli space
$\widetilde{\mathcal{M}}_{h,n}$ (cf. Section 3.3). In particular, for genus
$h=0$, $\Xi$ is homotopy equivalent to the component
$E_{2}^{\mathrm{fr}}(n-1)$ of the framed little disk operad.

We will denote the induced isomorphism on homology

(130) $f_{*}\colon H_{\bullet}(\Xi)\rightarrow
H_{\bullet}(E_{2}^{\mathrm{fr}}(n-1)).$

We assume that this isomorphism is compatible with gluing of
cobordisms/operadic composition in $E_{2}^{\mathrm{fr}}$.

  4. (iv)

Faces of $\Xi$ correspond to “admissible” polygonal decompositions of
$\Sigma$, inducing the given triangulation of the boundary, into polygons with
$\geq 3$ sides equipped with extra geometric data – a linear chart and a
configuration chamber of $\geq 0$ _floating points_ (plus the boundary
vertices of the polygon).

Sub-faces of a face correspond to subdivisions of a given polygonal
decomposition, locally described by the secondary polytope combinatorics.

  5. (v)

For $\Sigma$ a cylinder with $k_{1}=k_{2}=\colon k$, the BV cycle
$\mathsf{c}_{\Delta}^{k}$ (69) is a representative of the generator of
$H_{1}(\Xi,\mathbb{Z})\stackrel{{\scriptstyle
f_{*}}}{{\simeq}}H_{1}(E_{2}^{\mathrm{fr}}(1),\mathbb{Z})=\mathbb{Z}$.

In the rest of this section we will assume the Conjecture above to hold.

######  Remark 6.15.

One can consider a modification of the setup above where the triangulations of
boundary circles are not fixed and there are extra 1-cells of $\Xi$
corresponding to “elementary shelling” moves at the boundary.

######  Remark 6.16.

The Conjecture might require the assumption that $k_{i}$ are large enough.

On a related note, it is possible that the appropriate “admissibility”
property of polygonal decompositions includes in particular that they are
regular as CW complexes, which would imply $k_{i}\geq 2$. It might be
necessary that triangulations are simplicial complexes in a strict sense,
which would imply $k_{i}\geq 3$.

######  Remark 6.17.

If $k_{i}=1$ is allowed, we expect the cycle $\mathsf{c}_{\Delta}$ of Section
4.4 to satisfy (v) above. In particular, we expect it to square to a boundary
in $\Xi$ even though it does not square to a boundary in
$\Xi_{\mathrm{flip}}$.

A similar point applies to $\mathsf{c}_{\Delta}^{k}$ with any odd
$k=3,5,\ldots$: we do not know if it squares to a boundary in
$\Xi_{\mathrm{flip}}$ but (v) above implies that it does in $\Xi$.

####  6.5.2. HTQFT on the Pachner complex

Fix an $\widehat{A}_{\infty}$ algebra $(V,Q,\\{c_{[A]}\\})$. Then one can
define a
$\mathrm{Hom}(\mathcal{H}_{\mathrm{in}},\mathcal{H}_{\mathrm{out}})$-valued
cochain $Z$ (the HTQFT) on $\Xi$ by the formula (105):

(131)
$Z(e_{\alpha})=\Big{\langle}\bigotimes_{\mathrm{polygons\,}(\mathsf{p}_{i},A_{i})\,\mathrm{of}\,\alpha}c_{[A_{i}]},\bigotimes_{\mathrm{edges}\backslash\\{\mathrm{in-
edges}\\}}g^{-1}\Big{\rangle},$

for $\alpha$ a polygonal decomposition of the surface and $e_{\alpha}$ the
corresponding face of $\Xi$. Due to the combinatorics of the face poset of
$\Xi$, the cochain $Z$ automaticaly satisfies the main equation of HTQFT
(106).

####  6.5.3. Chain-level BV algebra structure

Fix $k$ and denote $\mathcal{H}=\mathcal{H}(S^{1}_{(k)})=V^{\otimes k}$. For
all triangulated cobordisms in this section we will assume that the induced
triangulation of each boundary circle is into $k$ intervals.

Product (or “closed string product”). For $\Sigma$ a pair of pants equipped
with any triangulation $T$, the isomorphism $f_{*}$ (130) identifies the
homology class of $T$ in $H_{0}(\Xi)$ with the “product” generator of
$H_{0}(E_{2}^{\mathrm{fr}}(2))$, cf. Section 3.3.1. Evaluating $Z$ on
$(\Sigma,T)$, we obtain a degree $0$ map

(132) $*_{T}=Z(T)\colon\mathcal{H}\otimes\mathcal{H}\rightarrow\mathcal{H}$

– the “closed string product” (as opposed to the $A_{\infty}$ operation
$m_{2}\colon V\otimes V\rightarrow V$ – the “open string product”).

The operation (132) is associative modulo $Q$-exact terms as follows from the
fact that two triangulations $T\circ_{1}T$, $T\circ_{2}T$ of a sphere with
four holes can be transformed one into the other by a sequence of Pachner
moves. A similar argument (comparing $T$ to its rotation by $\pi$,
$T^{\prime}$, which swaps the two in-circles) shows commutativity of (132)
modulo $Q(\cdots)$.

BV operator. Evaluating $Z$ on the chain (69) yields a degree $-1$ BV operator

(133)
$\Delta=Z(\mathsf{c}^{k}_{\Delta})\colon\mathcal{H}\rightarrow\mathcal{H}$

which satisfies $\Delta^{2}=Q(\cdots)$.

BV bracket. Consider a triangulated pair of pants $(\Sigma,T)$ and consider a
1-cycle on $\Xi$ given by composing $T$ with the cycle
$\mathsf{c}_{\Delta}^{k}$ in all possible ways:

(134) $R=\mathsf{c}_{\Delta}^{k}\circ
T-T\circ_{1}\mathsf{c}_{\Delta}^{k}-T\circ_{2}\mathsf{c}_{\Delta}^{k}.$

The isomorphism $f_{*}$ maps the homology class of $R$ to the “BV bracket”
generator of $H_{1}(E_{2}^{\mathrm{fr}}(2))$. Evaluating $Z$ on this 1-cycle,
we get a degree $-1$ map

(135)
$\\{,\\}_{T,\mathsf{c}^{k}_{\Delta}}=Z(R)\colon\mathcal{H}\otimes\mathcal{H}\rightarrow\mathcal{H}$

– the (chain-level) BV bracket. Due to relations in homology of
$E_{2}^{\mathrm{fr}}$, $Z(R)$ satisfies Jacobi and bi-derivation property
modulo $Q$-exact terms. The latter is equivalent to the BV operator $\Delta$
satisfying the 7-term relation (33) modulo $Q$-exact terms.

Unit. For $(D,S)$ a triangulated disk, the class of $S$ in $H_{0}(\Xi)$ maps
by $f_{*}$ to the unit class in $H_{0}(E_{2}^{\mathrm{fr}}(0))$. Thus we have
a (chain-level) “unit”

(136) $1_{S}=Z(S)\in\mathcal{H}.$

The unit properties – (136) being a unit for the product (132) and being
annihilated by $\Delta$ – hold up to $Q(\cdots)$, by the same reason as above.

Figure 32. Chain-level BV algebra operations. $T$ is a triangulation of a pair
of pants, $S$ is a triangulation of a disk.

####  6.5.4. Reduction by passing to $Q$-cohomology and by attaching
triangulated cylinders. The strict BV algebra.

All the objects (132), (133), (135), (136) are $Q$-closed and change by
$Q$-exact terms when triangulations $T,S$ are changed (or the BV cycle is
changed). In particular, passing to cohomology of $Q$, one has a strict BV
algebra structure on the $Q$-cohomology of the space of states,
$H_{Q}(\mathcal{H})$.

Reduction by triangulated cylinders. Evaluating $Z$ on any triangulated
cylinder $(\Sigma,T)$ and passing to $Q$-cohomology, we obtain a map $P\colon
H_{Q}(\mathcal{H})\rightarrow H_{Q}(\mathcal{H})$. Since the composition
square of $(\Sigma,T)$ can be moved to $(\Sigma,T)$ by a series of Pachner
moves, $P$ is a projector: $P^{2}=P$. By construction, the strict BV algebra
on $H_{Q}(\mathcal{H})$ factors through $P$, as one can attach a triangulated
cylinder to any end of any of the triangulated cobordisms giving the BV
algebra operations.

Put another way, one has a splitting

(137)
$H_{Q}(\mathcal{H})=\underbrace{\mathcal{H}^{\mathrm{red}}}_{\mathrm{im}(P)}\oplus\mathrm{ker}(P).$

The operations of the strict BV algebra descend to
$\mathcal{H}^{\mathrm{red}}$ and vanish on $\mathrm{ker}(P)$. Note that
$\mathcal{H}^{\mathrm{red}}$ here plays similar role to the Hochschild
homology appearing in Section 2.4.1.373737Also, our construction of the space
$\mathcal{H}^{\mathrm{red}}$ is similar to the construction of the space of
states as the quotient by the kernel of the partition function for a cylinder
in [23].

Furthermore, we note that the space $\mathcal{H}^{\mathrm{red}}$ and the BV
algebra structure on it are independent of $k$. The isomorphism
$\mathcal{H}^{\mathrm{red}}_{k}\rightarrow\mathcal{H}^{\mathrm{red}}_{k^{\prime}}$
(here we indicate explicitly the triangulation of the circle as a subscript)
is given by any triangulated cylinder with the in- and out-circles
triangulated into $k$ and $k^{\prime}$ intervals.

######  Remark 6.18.

Restricting to $Q$-cohomology and restricting to triangulated cobordisms only
(i.e. with no higher polygons and no floating points) yields the strict TQFT
of Section 2, based on the noncommutative graded Frobenius algebra $H_{Q}(V)$,
with the strictly associative multiplication inherited from the operation
$m_{2}$ on $V$.

In particular, as in Section 2.4.1, one has a model for the reduced space of
states $\mathcal{H}^{\mathrm{red}}$ as the center of $H_{Q}(V)$,

(138) $\mathcal{H}^{\mathrm{red}}=\mathcal{Z}(H_{Q}(V)).$

To summarize the discussion above, we have the following.

######  Corollary 6.19 (of Conjecture 6.14).

For an $\widehat{A}_{\infty}$ algebra $(V,Q,\\{c_{[A]}\\})$, there is a
canonically defined BV algebra structure induced on the center
$\mathcal{Z}(H_{Q}(V))$ of the cohomology of $Q$.

## References

  * [1] M. R. Casali, “A note about bistellar operations on PL-manifolds with boundary.” Geometriae Dedicata 56.3 (1995): 257–262.
  * [2] R. Dijkgraaf, E. Witten, “Topological gauge theories and group cohomology,” Commun. Math. Phys. 129 (1990) 393–429.
  * [3] E. Frenkel, A. Losev, N. Nekrasov, “Instantons beyond topological theory II,” arXiv:0803.3302 [hep-th] (2008).
  * [4] I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky, “Discriminants, Resultants and Multidimensional Determinants,” Birkhäuser, Boston, 1994.
  * [5] E. Getzler, “Batalin-Vilkovisky algebras and two-dimensional topological field theories,” Commun. Math. Phys. 159.2 (1994): 265–285.
  * [6] J. L. Harer, “The virtual cohomological dimension of the mapping class group of an orientable surface,” Invent. Math. 84.1, 157–-176 (1986).
  * [7] T. V. Kadeishvili, “On the theory of homology of fiber spaces,” (Russian) International Topology Conference (Moscow State Univ., Moscow, 1979). Uspekhi Mat. Nauk 35 (1980), no. 3 (213), 183–188. Translated in Russ. Math. Surv. 35 (1980), no. 3 (238) 231–211.
  * [8] M. Kapranov, M. Kontsevich, Y. Soibelman, “Algebra of the infrared and secondary polytopes,” Advances in Mathematics 300 (2016): 616–671.
  * [9] M. Kapranov, Y. Soibelman, L. Soukhanov, “Perverse schobers and the Algebra of the Infrared,” arXiv preprint arXiv:2011.00845 (2020).
  * [10] M. Kontsevich, “Intersection theory on the moduli space of curves and the matrix Airy function.” Communications in Mathematical Physics 147 (1992): 1–23.
  * [11] M. Kontsevich, Yu. Manin, “Gromov-Witten classes, quantum cohomology, and enumerative geometry,” Commun. Math. Phys. 164 (1994): 525–562.
  * [12] J.-L. Loday, B. Vallette, “Algebraic operads.” Vol. 346. Springer Science & Business Media, 2012.
  * [13] A. Losev, P. Mnev, D. R. Youmans, “Two-dimensional abelian BF theory in Lorenz gauge as a twisted N=(2, 2) superconformal field theory,” J. Geom. Phys. 131 (2018): 122–137.
  * [14] A. Losev, S. Shadrin, “From Zwiebach invariants to Getzler relation,” Communications in mathematical physics 271.3 (2007): 649-679.
  * [15] M. Markl, S. Shnider, J. D. Stasheff, “Operads in algebra, topology and physics,” Math. Surveys and Monographs 96, Amer. Math. Soc. 2002.
  * [16] A. A. Migdal, “Recursion equations in gauge theories,” Sov. Phys. JETP 42 (1975) 413.
  * [17] M. Mulase, M. Penkava. “Ribbon Graphs, Quadratic Differentials on Riemann Surfaces, and Algebraic Curves Defined over $\bar{Q}$.” arXiv preprint math-ph/9811024 (1998).
  * [18] U. Pachner, “P.L. homeomorphic manifolds are equivalent by elementary shellings,” European Journal of Combinatorics, 12.2 (1991): 129–145.
  * [19] R. C. Penner,“The decorated Teichmüller space of punctured surfaces.” Commun. Math. Phys. 113, 299–339 (1987).
  * [20] I. Runkel, L. Szegedy, “Area-dependent quantum field theory.” Commun. Math. Phys. 381 (2021): 83-117.
  * [21] T. Tradler, “The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products.” Annales de L’institut Fourier 58.7 (2008) 2351–2379.
  * [22] E. Witten, “On quantum gauge theories in two dimensions,” Commun. Math. Phys. 141 (1991) 153.
  * [23] V. G. Turaev, O. Ya. Viro, “State sum invariants of 3-manifolds and quantum 6j-symbols,” Topology 31 (1992), 865–902.
  * [24] E. Witten, “Chern-Simons gauge theory as a string theory.” The Floer memorial volume. Basel: Birkhäuser Basel, 1995. 637-678.
  * [25] B. Zwiebach “Closed string field theory: Quantum action and the Batalin-Vilkovisky master equation,” Nuclear Physics B 390.1 (1993): 33–152.