A Cellular Topological Field Theory

Abstract

We present a construction of cellular BF theory (in both abelian and non-abelian variants) on cobordisms equipped with cellular decompositions. Partition functions of this theory are invariant under subdivisions, satisfy a version of the quantum master equation, and satisfy Atiyah–Segal-type gluing formula with respect to composition of cobordisms.

Appendices

Appendix A. Determinant Lines, Densities, R-Torsion

1.1 Determinant lines, torsion of a complex of vector spaces

In what follows, line stands for an abstract 1-dimensional real vector space.

Definition A.1

For V a finite-dimensional real vector space, the determinant line is defined as the top exterior power ${\mathrm {Det}}\; V=\wedge ^{\dim V} V$. For $V^\bullet$ a ${\mathbb {Z}}$-graded vector space, one defines the determinant line as

$$\begin{aligned} {\mathrm {Det}}\; V^\bullet =\bigotimes _k ({\mathrm {Det}}\; V^k)^{(-1)^k} \end{aligned}$$

where for L a line, $L^{-1}=L^*$ denotes the dual line.

Here are several useful properties of determinant lines.

• (i) The determinant line of the dual graded vector space is

  $$\begin{aligned} {\mathrm {Det}}\; V^*\cong \left( {\mathrm {Det}}\; V\right) ^{-1} \end{aligned}$$

  (with the grading convention $(V^*)^k=(V^{-k})^*$). In the case of a vector space concentrated in degree 0, the pairing between ${\mathrm {Det}}\; V^*$ and ${\mathrm {Det}}\;V$ is given by

  $$\begin{aligned} \langle v_n^*\wedge \cdots \wedge v_1^*\;,\;v_1\wedge \cdots \wedge v_n \rangle ={\det }\langle v_i^*,v_j\rangle \end{aligned}$$

  with $v_i\in V$, $v_i^*\in V^*$ for $i=1,\ldots ,n=\dim V$. Extension to the graded case is straightforward.

• (ii) Determinant line of the degree-shifted vector space is

  $$\begin{aligned} {\mathrm {Det}}\; V^\bullet [k]\cong \left( {\mathrm {Det}}\; V^\bullet \right) ^{(-1)^k}. \end{aligned}$$

• (iii) Given a short exact sequence of graded vector spaces $0\rightarrow U^\bullet \rightarrow V^\bullet \rightarrow W^\bullet \rightarrow 0$, one has

  $$\begin{aligned} {\mathrm {Det}}\;V^\bullet \cong {\mathrm {Det}}\;U^\bullet \otimes {\mathrm {Det}}\;W^\bullet . \end{aligned}$$

  (180)

  In the case of non-graded vector spaces, the isomorphism sends

  $$\begin{aligned} \underbrace{(u_1\wedge \cdots \wedge u_{\dim U})}_{\in \; {\mathrm {Det}}\;U}\otimes \underbrace{(w_1\wedge \cdots \wedge w_{\dim W})}_{\in \;{\mathrm {Det}}\; W}\quad \mapsto \quad \underbrace{u_1\wedge \cdots \wedge u_{\dim U}\wedge w'_1\wedge \cdots \wedge w'_{\dim W}}_{\in \; {\mathrm {Det}}\;V} \end{aligned}$$

  where on the right, $w'_i$ is some lifting of the element $w_i$ to V. Extension to the graded case is, again, straightforward.

• (iv) If $V^\bullet ,d$ is a cochain complex with cohomology $H^\bullet (V)$, there is a canonical isomorphism of determinant lines

  $$\begin{aligned} {\mathbb {T}}:\; {\mathrm {Det}}\; V^\bullet \xrightarrow {\cong } {\mathrm {Det}}\; H^\bullet (V). \end{aligned}$$

  (181)

  Indeed, one applies property (180) to the two short exact sequences

  $$\begin{aligned} V_\mathrm {closed}^\bullet \hookrightarrow V^\bullet \xrightarrow {d} V^{\bullet +1}_\mathrm {exact},\qquad V^\bullet _\mathrm {exact}\hookrightarrow V^\bullet _\mathrm {closed}\rightarrow H^\bullet (V) \end{aligned}$$

  to obtain isomorphisms.

  $$\begin{aligned} {\mathrm {Det}}\; V^\bullet \cong {\mathrm {Det}}\;V^\bullet _\mathrm {closed} \otimes ({\mathrm {Det}}\;V^\bullet _\mathrm {exact})^{-1},\qquad {\mathrm {Det}}\;V^\bullet _\mathrm {closed}\cong {\mathrm {Det}}\;V^\bullet _\mathrm {exact}\otimes {\mathrm {Det}}\; H^\bullet (V) \end{aligned}$$

  which combine to (181).

All isomorphisms above are canonical (functorial).

It is convenient to work with determinant lines modulo signs, so that one can ignore the question of orientations and Koszul signs. We will use the notation ${\mathrm {Det}}\;V^\bullet /\{\pm 1\}$ for non-zero elements of the determinant line considered modulo sign; so the precise notation should have been $({\mathrm {Det}}\;V^\bullet -\{0\})/\{\pm 1\}$.

Definition A.2

For $V^\bullet ,d$ a cochain complex and $\mu \in {\mathrm {Det}}\;V^\bullet /\{\pm 1\}$ a preferred element of the determinant line, defined up to sign, the torsion is defined as

$$\begin{aligned} \tau (V^\bullet ,d,\mu )={\mathbb {T}}(\mu )\quad \in {\mathrm {Det}}\;H^\bullet (V)/\{\pm 1\} \end{aligned}$$

with ${\mathbb {T}}$ as in (181).

The following Lemma has important consequences in the setting of R-torsion (Sect. A.2).

Lemma A.3

(Multiplicativity of torsions with respect to short exact sequences). Let $0\rightarrow U^\bullet \rightarrow V^\bullet \rightarrow W^\bullet \rightarrow 0$ be a short exact sequence of complexes, equipped with elements $\mu _U,\mu _V,\mu _W$ in respective determinant lines, such that $\mu _V=\mu _U\cdot \mu _W$. Then for the torsions we have

$$\begin{aligned} {\mathbb {T}}_\mathrm {LES}({\mathbb {T}}_U(\mu _U)\cdot {\mathbb {T}}_V(\mu _V)^{-1}\cdot {\mathbb {T}}_W(\mu _W))=1\quad \in \mathbb {R}/\{\pm 1\} \end{aligned}$$

where ${\mathbb {T}}_{U},{\mathbb {T}}_V,{\mathbb {T}}_W$ are the maps (181) for $U^\bullet ,V^\bullet ,W^\bullet$. We denoted $\mathrm {LES}$ the induced long exact sequence in cohomology $\cdots \rightarrow H^k(U)\rightarrow H^k(V)\rightarrow H^k(W)\rightarrow H^{k+1}(U)\rightarrow \cdots$ viewed as an acyclic cochain complex, and

$$\begin{aligned} {\mathbb {T}}_\mathrm {LES}: {\mathrm {Det}}\; H^\bullet (U)\otimes ({\mathrm {Det}}\; H^\bullet (V))^{-1}\otimes {\mathrm {Det}}\; H^\bullet (W)\rightarrow \mathbb {R}\end{aligned}$$

is the corresponding isomorphism (181).

See [25] for details; cf. also [29] for discussion in the language of determinant lines.

1.2 A.2 Densities

Definition A.4

For $\alpha \in \mathbb {R}$ and V a finite-dimensional real vector space, the space ${\mathrm {Dens}}^\alpha (V)$ of $\alpha$-densities on V is defined as the space of maps $\phi :F(V)\rightarrow \mathbb {R}_+$ from the space of bases (frames) in V to positive half-line satisfying the equivariance property: for any automorphism $g\in GL(V)$ and any frame $\underline{v}=(v_1,\ldots ,v_{\dim V})\in F(V)$, one has

$$\begin{aligned} \phi (g\cdot \underline{v})=|\det g|^\alpha \cdot \phi (\underline{v}). \end{aligned}$$

${\mathrm {Dens}}^\alpha (V)$ is a torsor over $\mathbb {R}_+$ (viewed as a multiplicative group), and in the setting of ${\mathbb {Z}}$-graded vector spaces, one defines

$$\begin{aligned} {\mathrm {Dens}}^\alpha (V^\bullet )=\bigotimes _k \left( {\mathrm {Dens}}^\alpha (V^k)\right) ^{(-1)^k} \end{aligned}$$

(tensor product is over $\mathbb {R}_+$); $\alpha$ is called the weight of the density.

By default a “density” has weight $\alpha =1$ (and then we write ${\mathrm {Dens}}$ instead of ${\mathrm {Dens}}^1$), and a “half-density” has, indeed, $\alpha =1/2$.

If $\phi _\alpha$, $\phi _\beta$ are two densities on $V^\bullet$ of weights $\alpha ,\beta$, then the product $\phi _\alpha \cdot \phi _\beta$ is an $(\alpha +\beta )$-density. Also, $\phi _\alpha$ can be raised to any real power $\gamma \in \mathbb {R}$ to yield a density $(\phi _\alpha )^\gamma$ of weight $\alpha \cdot \gamma$. In particular, one has mutually inverse maps

$$\begin{aligned} {\mathrm {Dens}}^{1/2}V^\bullet \xrightarrow {(*)^2}{\mathrm {Dens}}\;V^\bullet ,\quad {\mathrm {Dens}}\;V^\bullet \xrightarrow {\sqrt{*}}{\mathrm {Dens}}^{1/2}V^\bullet . \end{aligned}$$

Evaluation pairing $({\mathrm {Det}}\; V^\bullet /\{\pm 1\})\otimes {\mathrm {Dens}}\;V^\bullet \rightarrow \mathbb {R}_+$ induces a canonical isomorphism of $\mathbb {R}_+$-torsors

$$\begin{aligned} {\mathrm {Det}}\;V^\bullet /\{\pm 1\}\cong {\mathrm {Dens}}\;V^\bullet [1]. \end{aligned}$$

1.3 A.3 R-torsion

Let X be a finite CW-complex and $Y\subset X$ a CW-subcomplex. Let

$$\begin{aligned} h:\pi _1(X)=\pi _1\rightarrow SL_{\pm }(m,\mathbb {R}) \end{aligned}$$

(182)

be some representation of the fundamental group of X by real matrices of determinant $\pm 1$. It extends to a ring homomorphism $h:{\mathbb {Z}}[\pi _1]\rightarrow \mathrm {Mat}(m,\mathbb {R})$ from the group ring of $\pi _1$ to all real matrices of size m. Let $p: \widetilde{X}\rightarrow X$ be the universal cover of X and denote $\widetilde{Y}=p^{-1}(Y)\subset \widetilde{X}$. Consider the cochain complex of vector spaces

$$\begin{aligned} C^\bullet (X,Y;h)=\mathbb {R}^m\otimes _{{\mathbb {Z}}[\pi _1]} C^\bullet (\widetilde{X},\widetilde{Y};{\mathbb {Z}}) \end{aligned}$$

(183)

where on the right we have integral cellular cochains of the pair $(\widetilde{X},\widetilde{Y})$, which is a complex of free ${\mathbb {Z}}[\pi _1]$-modules with elements of $\pi _1$ acting on cells of $\widetilde{X}$ by covering transformations, tensored with $\mathbb {R}^m$ using the representation h. In $C^\bullet (X,Y;h)$ one has a preferred basis of the form

$$\begin{aligned} \{v_i\otimes (\widetilde{e})^* \}_{1\le i\le m,\; e\subset X-Y} \end{aligned}$$

(184)

where $\{v_i\}$ is the standard basis on $\mathbb {R}^m$ (or any unimodular basis, i.e. such that the standard density on $\mathbb {R}^m$ evaluates on it to $\pm 1$) and $\widetilde{e}$ are some liftings of cells e of X not lying in Y to the universal cover; $(\widetilde{e})^*$ is the corresponding basis cochain.

Associated to the basis (184) by construction (29) is an element $\mu \in {\mathrm {Det}}\;C^\bullet (X,Y;h)/\{\pm 1\}$, independent of the choices of liftings of cells $e\mapsto \widetilde{e}$ and independent of the choice of unimodular basis in $\mathbb {R}^m$.

Definition A.5

The R-torsion of the pair (X, Y) of CW-complexes, associated to the representation (182), is defined as the torsion (in the sense of Definition A.2) of the complex $C^\bullet (X,Y;h)$ equipped with element $\mu$:

$$\begin{aligned} \tau (X,Y;h)={\mathbb {T}}(\mu )\quad \in {\mathrm {Det}}\;H^\bullet (X,Y;h)/\{\pm 1\}. \end{aligned}$$

Torsion of a single CW-complex X is defined as $\tau (X;h):=\tau (X,\varnothing ;h)$.

Of particular importance (and historically the most studied) is the acyclic case, when $H^\bullet (X,Y;h)=0$. Then the torsion takes values in the trivial line and thus is a number (modulo sign).

Instead of choosing a representation h of $\pi _1$, one can choose a cellular local $SL_\pm (m)$-system E on X, in the sense of Sect. 3, and define h as the holonomy of E. Cochain complex $C^\bullet (X,Y;E)$ (dual to the chain complex $C_\bullet (X,Y;E^*)$ constructed in Sect. 3) is isomorphic to (183). When we prefer to think in terms of a local system E rather than a representation h of $\pi _1$ (e.g. when we consider restriction to a CW-subcomplex, or gluing of two complexes along a subcomplex), we will write the torsion as $\tau (X,Y;E)$.

The following two properties are consequences of the multiplicativity of the algebraic torsion with respect to short exact sequences of cochain complexes (Lemma A.3).

• (A) For $X\supset Y$ a pair of CW-complexes, one has

  $$\begin{aligned} \tau (X;E)=\tau (X,Y;E)\cdot \tau (Y;E|_Y). \end{aligned}$$

  (185)

  The formula makes sense because ${\mathrm {Det}}\;H^\bullet (X;E)\cong {\mathrm {Det}}\;H^\bullet (X,Y;E)\otimes {\mathrm {Det}}\;H^\bullet (Y;E|_Y)$, since the determinant line of the long exact sequence in homology of the pair (X, Y) (regarded itself as a complex) is ${\mathrm {Det}}\;H^\bullet (X,Y;E)\otimes ({\mathrm {Det}}\;H^\bullet (X;E))^{-1} \otimes {\mathrm {Det}}\;H^\bullet (Y;E|_Y)$ and, on the other hand, is the trivial line, by (181) applied to the long exact sequence.

• (B) For $Z=X\cup Y$ a CW-complex represented as a union of two intersecting subcomplexes, one the gluing (inclusion/exclusion) formula

  $$\begin{aligned} \tau (X\cup Y;E)=\tau (X;E|_X)\cdot \tau (Y;E|_Y)\cdot \tau (X\cap Y;E|_{X\cap Y})^{-1}. \end{aligned}$$

  (186)

  The reason why l.h.s. and r.h.s. can be at all compared is as in (A), but one replaces the long exact sequence of a pair by Mayer–Vietoris sequence.⁷³

In the acyclic case (i.e. when all relevant cohomology spaces vanish), (185, 186) are equalities of numbers.

Theorem A.6

(Combinatorial invariance of R-torsion). If $(X',Y')$ is a cellular subdivision of the pair (X, Y), then

$$\begin{aligned} \tau (X',Y';h)=\tau (X,Y;h). \end{aligned}$$

For the proof, see e.g. [25]. The case $Y=Y'=\varnothing$ is due to Reidemeister, Franz and de Rham.

The combinatorial invariance theorem implies in particular that, for M a compact PL manifold with two different cellular decompositions X and Y, one has $\tau (X;h)=\tau (Y;h)$. Thus in this case it makes sense to talk about the R-torsion of a manifold M, $\tau (M;h)$, forgetting about the cellular subdivision.

Theorem A.7

(Milnor [24]). If M is a piecewise-linear compact oriented n-manifold with boundary $\partial M=\partial _1 M\sqcup \partial _2 M$, one has

$$\begin{aligned} \tau (M,\partial _1 M;h)=(\tau (M,\partial _2 M;h^*))^{(-1)^{n-1}} \end{aligned}$$

where $h^*$ is the dual representation to h.

Note that the l.h.s. belongs to ${\mathrm {Det}}\; H^\bullet (M,\partial _1 M;h)$ while the r.h.s. belongs to $({\mathrm {Det}}\; H^\bullet (M,\partial _2 M;h^*))^{(-1)^{n-1}}$ (modulo signs); these determinant lines are canonically isomorphic due to Poincaré-Lefschetz duality $H^k(M,\partial _1 M;h)\cong (H^{n-k}(M,\partial _2 M;h^*))^*$. Thus it does make sense to compare the two torsions.

Corollary A.8

For M a closed even-dimensional manifold and h such that $H^\bullet (M;h)=0$, the torsion is trivial, $\tau (M;h)=1$.

Appendix B. Two Points of View on “$C_\infty \otimes \mathrm {Lie}=L_\infty$”

In connection with Remark 8.17, we recall two ways to see the $L_\infty$ algebra structure on the tensor product of a $C_\infty$ algebra and a Lie algebra.

Given a $C_\infty$ algebra W with multlinear operations $m_n:W^{\otimes n}\rightarrow W$, and given a Lie algebra $\mathfrak {g}$, one can construct the tensor product $L_\infty$ algebra structure on the graded vector space $W\otimes \mathfrak {g}$ by defining

$$\begin{aligned} l_n(w_1\otimes \alpha _1,\ldots ,w_n\otimes \alpha _n)=\sum _{\sigma \in S_n}\pm m_n(w_{\sigma _1},\ldots ,w_{\sigma _n})\otimes (\alpha _{\sigma _1}\cdots \alpha _{\sigma _n}) \end{aligned}$$

(187)

with $w_1,\ldots ,w_n\in W$ and $\alpha _1,\ldots ,\alpha _n\in \mathfrak {g}$ arbitrary elements. The sum on the r.h.s. is over permuations $\sigma$ of $1,\ldots ,n$. Here the product of $\alpha _i$’s is seen as a product in the universal enveloping algebra $U\mathfrak {g}$. The $C_\infty$ property of the operation $m_n$ (vanishing on shuffle-products) implies that the result lands in $W\otimes \mathfrak {g}\subset W\otimes U\mathfrak {g}$.

Another way to present the same tensor product $L_\infty$ structure on $W\otimes \mathfrak {g}$ is as follows. The $C_\infty$ operations $m_n$ can be written in the form

$$\begin{aligned} m_n(w_1,\ldots ,w_n)=\sum _{T,\pi } m_n^T\circ \pi ^{-1}(w_1\otimes \cdots \otimes w_n) \end{aligned}$$

(188)

where the sum runs over binary rooted trees T with n leaves (viewed up to graph automorphism; for each T we fix arbitrarily a “standard” planar realization) and their planar realizations $\pi$; $m_n^T\in \mathrm {Hom}(W^{\otimes n},W)^{\mathrm {Aut}(T)}$ are some multilinear operations invariant w.r.t. automorphisms of T acting by permutations of factors in $W^{\otimes n}$ (with appropriate signs); $w_1,\ldots ,w_n\in W$ are arbitrary vectors; $\pi ^{-1}(\cdots )$ is understood as a permutation of factors in $W^{\otimes n}$ corresponding to going from the planar representative $\pi$ to the “standard” representative of T. Then the tensor product $L_\infty$ algebra structure on $W\otimes \mathfrak {g}$ is given by

$$\begin{aligned}&l_n(w_1\otimes \alpha _1,\ldots , w_n\otimes \alpha _n)\nonumber \\&\quad =\sum _{\sigma \in S_n}\sum _T \pm \frac{1}{|\mathrm {Aut}(T)|} m_n^T(w_{\sigma _1},\ldots ,w_{\sigma _n}) \otimes \mathrm {Jacobi}_T(\alpha _{\sigma _1},\ldots \alpha _{\sigma _n}) \end{aligned}$$

(189)

with $\mathrm {Jacobi}_T(\cdots )$ the nested commutator determined by the tree T.

Here the first point of view on the tensor product (187) is more direct and does not require splitting $m_n$ into pieces $m_n^T$ possessing different symmetries. However, we wanted to also present the second point of view (189) since it compares directly to the tree part of (115) and explains how to construct the corresponding $C_\infty$ algebra (via (188)).