Abstract
In the present work, we extend our previous work with Gwilliam by realizing \(\hat {A}(X)\) as the projective volume form associated to the BV operator in our quantization of a onedimensional sigma model. We also discuss the associated integration/expectation map. We work in the formalism of L_{ ∞ } spaces, objects of which are computationally convenient presentations for derived stacks. Both smooth and complex geometry embed into L_{ ∞ } spaces and we specialize our results in both of these cases.
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Notes
Note that \(\mathcal {O} (\mathcal {E}) = \widehat {\text {Sym}}(\mathcal {E}^{\vee })\), the subspace \(\mathcal {O}_{loc} (\mathcal {E}) \subset \mathcal {O}(\mathcal {E})\) consists of those functionals determined by Lagrangian densities.
Using densities, the following arguments can be adopted to unoriented manifolds.
This kind of integration notion would work even for infinitedimensional spaces, for which there are no top forms but there is a ring of functions.
Differential operators D_{ X } naturally act on \(C^{\infty }_{X}\) from the left, and they naturally act on top forms from the right. This right action is a consequence of the integration pairing between top forms and functions. In other words, D_{ X } acts on distributions from the right, and hence on top forms as well.
Let p : X →pt denote the map to a point. Then the derived pushforward of \(C^{\infty }_{X,\mu }\) along p is given by (PV (X), div_{ μ }). This construction is sometimes referred to as the Spencer resolution.
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Acknowledgements
The present work grew out of a chapter of the author’s PhD dissertation at the University of Notre Dame. As such, the author is thankful to the university, the math department, and particularly Stephan Stolz and Sam Evens for continued support and discussion. Further, many thanks are due to Kevin Costello and Owen Gwilliam for constant inspiration and collaboration. The anonymous referees greatly enhanced the presentation and readability of the paper. Lastly, thanks to a great mathematical QFT community including Damien Calaque, Si Li, Qin Li, Brian Williams, and Dan BerwickEvans.
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The author was partially supported by the National Science Foundation under Award DMS1309118.
Appendices
Appendix A: BV Theory a la Costello
In this appendix we recall a mathematical take on QFT in the Batalin–Vilkovisky formalism as described by Kevin Costello [10]. Our presentation follows that of [23] and [31].
1.1 A.1 BV Algebras and Observables
Definition 12
A BV algebra is a pair \((\mathcal {A}, {\Delta })\) where

\(\mathcal {A}\) is a \(\mathbb {Z}\)graded commutative associative unital algebra.

\({\Delta }: \mathcal {A} \to \mathcal {A}\) is a secondorder operator of degree 1 such that Δ^{2} = 0.
Here Δ is called the BV operator. Δ being “secondorder” means the following: define the BV bracket{−, −}_{Δ} as the measuring of the failure of Δ being a derivation
In this section we will suppress Δ from the notation, simply writing {−, −}. Then \(\{,\}: \mathcal {A}\otimes \mathcal {A}\to \mathcal {A}\) defines a Poisson bracket of degree 1 satisfying

{a, b}= (− 1)^{ab}{b, a}.

{a, bc}= {a, b}c + (− 1)^{(a+ 1)b}b{a, c}.

Δ{a, b}= −{Δa, b}−(− 1)^{a}{a,Δb}.
Definition 13
A differential BV algebra is a triple \((\mathcal {A}, Q, {\Delta })\) where

\((\mathcal {A}, {\Delta })\) is a BV algebra (see Definition 12).

\(Q: \mathcal {A}\to \mathcal {A}\) is a derivation of degree 1 such that Q^{2} = 0 and [Q, Δ] = 0.
Definition 14
Let \((\mathcal {A}, Q, {\Delta })\) be a differential BV algebra. A degree 0 element \(I_{0}\in \mathcal {A}\) is said to satisfy the classical master equation (CME) if
A degree 0 element \(I\in \mathcal {A}[[\hbar ]]\) is said to satisfy the quantum master equation (QME) if
Here \(\hbar \) is a formal (perturbative) parameter.
The “secondorder” property of Δ implies that QME is equivalent to
If we decompose \(I=\sum \limits _{g\geqslant 0}I_{g}\hbar ^{g}\), then the \(\hbar \to 0\) limit of the QME recovers the CME: \( QI_{0}+\frac {1}{2}\{I_{0}, I_{0}\}= 0. \)
A solution I_{0} of the CME leads to a differential Q + {I_{0}, −}, which is usually called the BRST operator in physics.
Definition 15
Let \((\mathcal {A}, Q, {\Delta })\) be a differential BV algebra and \(I_{0} \in \mathcal {A}\) satisfy the CME. Then the complex of classical observables, Obs^{cl} is given by
Similarly, a solution I of the QME yields a differential and correspondingly a complex of quantum observables.
Definition 16
Let \((\mathcal {A}, Q, {\Delta })\) be a differential BV algebra and \(I \in \mathcal {A}[[\hbar ]]\) satisfy the QME. Then the complex of quantum observables, Obs^{q} is given by
Note that Obs^{cl} has a degree 1 Poisson bracket, so following [13] we call it a P_{0} algebra. Similarly, in ibid. the structure on Obs^{q} is called a BD algebra.
1.2 A.2 Perturbative BV Quantization
The data of a classical field theory over a manifold M consists of a graded vector bundle E (possibly of infinite rank) equipped with a 1 symplectic pairing and a local functional \(S \in \mathcal {O}_{loc} (\mathcal {E})\)^{Footnote 1} expressed as S(e) = 〈e, Q(e)〉+ I_{0}(e), where Q is a square zero differential operator of cohomological degree 1, such that

1.
S satisfies the CME, i.e., {S, S}= 0;

2.
I_{0} is at least cubic; and

3.
\((\mathcal {E}, Q)\) is an elliptic complex.
Definition 17
A classical field theory \((\mathcal {E}, S)\) over M is a cotangent theory if we can write the field content as
We further require that the action S vanishes on tensors where there are at least two sections from the second summand E^{∨}⊗Dens(M).
Quantization of a field theory \((\mathcal {E}, S)\) over M consists of two stages:

1.
Build a BV algebra from the data of the pairing on the bundle E; and

2.
Promote the classical action S to a solution of the QME in this BV algebra.
The first difficulty is that the Poisson kernel K dual to the symplectic pairing is nearly always singular, so the naive definition of the BV operator Δ_{ k } = ∂_{ k } is illdefined. In [10], Costello uses homotopical ideas (built on the heat kernel) to build a family of well defined (smooth) BV operators Δ_{ L } for 0 < L < ∞. Consequently, there is a family of differential BV algebras \(\left \{ (\mathcal {O}(E), Q , {\Delta }_{L})\right \}_{L>0}\). Costello also describes homotopy renormalization group flow (HRG) to relate solutions of the QME between algebras in this family. As we describe in the next section, HRG is expressed in terms of a propagator built from the differential operator Q and a gage fixing operatorQ^{†}; indeed, any parametrix for the generalized Laplacian [Q, Q^{†}] can be used as a propagator.
Definition 18
Let \((\mathcal {E}, S)\) be a classical field theory over M. A perturbative quantization is a family of solutions to the QME, {I[L]}_{L> 0}, linked by the HRG, such that
Remark 2
Flow via the HRG induces a chain homotopy between quantum observables as we vary within the family of BV algebras \(\{(\mathcal {O}(E), Q, {\Delta }_{L})\}_{L>0}\). Thus, we will supress the dependence on L and abusively refer to these chain homotopic complexes as the global quantum observables of our field theory.
1.3 A.3 Homotopy Renormalization Group Flow
The homotopy renormalization group flow equation can be described in terms of Feynman graphs. Note that our description is for an arbitrary functional on a space of fields \(\mathcal {E}\). Further, we will work relative to an arbitrary dg algebra \(\mathcal {A}\) equipped with a nilpotent ideal \(\mathcal {I}\).
Definition 19
A graph \(\mathcal {G}\) consists of the following data:

1.
A finite set of vertices \(V(\mathcal {G})\);

2.
A finite set of halfedges \(H(\mathcal {G})\);

3.
An involution \(\sigma : H(\mathcal {G})\rightarrow H(\mathcal {G})\). The set of fixed points of this map is denoted by \(T(\mathcal {G})\) and is called the set of tails of \(\mathcal {G}\). The set of twoelement orbits is denoted by \(E(\mathcal {G})\) and is called the set of internal edges of \(\mathcal {G}\);

4.
A map \(\pi :H(\mathcal {G})\rightarrow V(\mathcal {G})\) sending a halfedge to the vertex to which it is attached;

5.
A map \(g:V(\mathcal {G})\rightarrow \mathbb {Z}_{\geqslant 0}\) assigning a genus to each vertex.
It is clear how to construct a topological space \(\mathcal {G}\) from the above abstract data. A graph \(\mathcal {G}\) is called connected if \(\mathcal {G}\) is connected. The genus of the graph \(\mathcal {G}\) is defined to be
where \(b_{1}(\mathcal {G})\) denotes the first Betti number of \(\mathcal {G}\). Let
be the subspace consisting of those functionals which are at least cubic modulo \(\hbar \) and the nilpotent ideal \(\mathcal {I}\) in the base ring \(\mathcal {A}\). Let \(F\in \mathcal {O}^{+}(\mathcal {E})\) be a functional, which can be expanded as
We view each \(F_{g}^{(k)}\) as an S_{ k }invariant linear map
With the propagator P_{𝜖→L}, we will describe the (Feynman) graph weights
for any connected graph \(\mathcal {G}\). We label each vertex v in \(\mathcal {G}\) of genus g(v) and valency k by \(F^{(k)}_{g(v)}\). This defines an assignment
where H(v) is the set of halfedges of \(\mathcal {G}\) which are incident to v. Next, we label each internal edge e by the propagator
where \(H(e)\subset H(\mathcal {G})\) is the twoelement set consisting of the halfedges forming e. We can then contract
with
to yield a linear map
Definition 20
We define the (homotopy) RG flow operator with respect to the propagator P_{𝜖→L}
by
where the sum is over all connected graphs.
Equivalently, it is useful to describe the (homotopy) RG flow operator formally via the simple equation
Definition 21
A family of functionals \(F[L] \in \mathcal {O}^{+}(\mathcal {E})\) parametrized by L > 0 is said to satisfy the homotopy renormalization group flow equation (hRGE) if for each 0 < 𝜖 < L
1.4 A.4 BV Theory and Volume Forms: Integration via Homology
We now quickly explain the underlying relationship between BV theory, path integrals, homological methods, and perturbation theory. The story below suggests a relationship between quantizations of a cotangent theory and projective volume forms; the precise relationship is given by Proposition 5 above. More detailed (and eloquent) presentations are given in [1, 15], and [36]. Also, historically much of the mathematical development goes back to Koszul [30].
For simplicity, let X be a connected, orientable, smooth manifold of dimension n.^{Footnote 2} Every top form μ ∈ Ω^{n}(X) then defines a linear functional
which is a natural object from several perspectives. First, from this linear functional — the distribution associated to μ — we can completely reconstruct the top form μ. Second, if μ is a probability measure, then \({\int }_{\mu }\) is precisely the expected value map. Our goal is now to rephrase \({\int }_{\mu }\) in a way that does not explicitly depend on ordinary integration and thus to obtain a version of volume form that can be extended to L_{ ∞ } spaces.
We can understand \({\int }_{\mu }\) in a purely homological way, as follows. We know that integration over X vanishes on total derivatives \(d\omega \in {{\Omega }^{n}_{c}}(X)\), by Stokes’ Theorem, so we have a commutative diagram
where [ω] denotes the cohomology class of the top form ω. (The cohomology group \({H^{n}_{c}}(X)\) is 1dimensional by Poincaré duality.) In consequence, we can identify \({\int }_{\mu }\) with the composition
where ι_{ μ } denotes “multiplication by μ”(or “contraction with μ”). We thus have a purely homological version of integration againstμ.
It is natural to extend the map “contract with μ” to the whole de Rham complex, and not just the top forms:
where \(P{V^{k}_{c}}(X) := {\Gamma }_{c}(X, {\Lambda }^{k} T_{X})\) denotes the compactlysupported polyvector fields and div_{ μ } denotes “divergence with respect to μ.” We require now that μ is nowherevanishing, so that the divergence is welldefined. This map of cochain complexes ι_{ μ } is then an isomorphism.
The significance of the bottom row is that it fully encodes integration against μ but the relevant data of μ is contained in the differential div_{ μ }. We would like to characterize such differentials on the polyvector fields PV (X), in order to describe a version of integration that applies to spaces more general than manifolds (at least spaces that possess a good notion of polyvector fields).^{Footnote 3}
Note that a choice of top form μ induces a \(C^{\infty }_{X}\)linear map from \(C^{\infty }_{X}\) to \({{\Omega }^{n}_{X}}\), and so we can pullback the natural rightD_{ X }module structure on \({{\Omega }^{n}_{X}}\).^{Footnote 4} Let \(C^{\infty }_{X,\mu }\) denote \(C^{\infty }_{X}\) equipped with this right D_{ X }module structure.^{Footnote 5} Although a volume form makes \(C^{\infty }_{X}\) into a right D_{ X }module, the converse need not hold. Observe that for any nonzero constant c, the operators div_{ μ } and div_{ c μ } are the same. Hence, a right D_{ X }module structure on \(C^{\infty }_{X}\) locally gives a volume form only up to scale.
The complex (PV (X), div_{ μ }) is a fundamental example of a BV algebra (see Appendix A.1), the associated bracket {−, −}is the Schouten bracket. Further, note that we have an equivalence
Hence, we witness the intimate relationship between BV algebras, shifted cotangent bundles, integration via homological algebra, and (projective) volume forms.
Appendix B: The \(\hat {A}\) Genus
Recall that (following Hirzebruch [26]) the Todd class can be defined in terms of Chern classes by the power series Q(x) and the \(\hat {A}\) class is given in Pontryagin classes via P(x) where
We define a new power series by log(Q(x)) −x/2 and denote the corresponding characteristic class by \(\log (e^{c_{1}/2} \text {Td})\). We have an equivalence of power series (see [39] and [27])
where ζ is the Riemann zeta function.
We now use standard arguments about characteristic classes. For a sum of complex line bundles E = L_{1} ⊕ ⋯ ⊕L_{ n }, the Todd class is
Thus, (2) tells us
As ch_{2k}(E) = (c_{1}(L_{1})^{2k} + ⋯ + c_{1}(L_{ n })^{2k})/(2k!), we obtain a general formula for an arbitrary bundle E,
Putting together the above discussion we make the following definition for an L_{ ∞ } space \(B \mathfrak {g}\).
Definition 22
Let V be a vector bundle over \((X, \mathfrak {g})\) (e.g. the tangent bundle as given by the module \(\mathfrak {g}[1]\)) then the we define
Similarly, we will need an equivariant class, so we define for any smooth manifold X the class \(\log (\hat {A}_{u} (X))\) to be
This is the usual logarithm of the \(\hat {A}\) class weighted by powers of u.
1.1 B.1 The Many Faces of the \(\hat {A}\) Genus
The \(\hat {A}\) genus/class makes many appearances in math and physics, as we now briefly sketch.
1.1.1 B.1.1 Index of the Dirac Operator
In the late 1950s Borel and Hirzebruch [7] proved that \(\hat {A} (X)\) was an integer provided that M was a spin manifold. The question of why the spin condition implied integrality was quickly sorted by Hirzebruch and Atiyah [4]. In extended work with Singer (and Bott and Hirzebruch) [5], Atiyah fleshed out this picture through the formulation (and proof) of the index theorem. In particular, for a spin manifold X, the integer \(\hat {A} (X)\) is the Fredholm index of an elliptic operator: the Dirac operator.
1.1.2 B.1.2 An Obstruction to Positive Scalar Curvature
Using a Bochner/Weitzenböck formula, Lichnerowicz [32] showed that \(\hat {A} (X)\) is an obstruction to a spin manifold X admitting a metric of positive scalar curvature. Hitchin [28] refined this result and introduced a Ktheoretic invariant, the α invariant. Significant progress on the converse statement was made by Gromov and Lawson [25], before the problem was solved–in the simply connected case–by Stolz [37].
1.1.3 B.1.3 Twisted Todd Class
In the holomorphic setting we use the characteristic class \(e^{c_{1}(E)/2} Td(E)\). Note that for real bundles, this agrees with \(\hat {A} (E)\). Following [27] or [18] there is an index theoretic explanation. Indeed, suppose that a manifold X is both complex and spin. Then there are two canonical spin^{c} structures and corresponding Dirac operators. Let be the Dirac operator corresponding to the spin^{c} structure determined by the complex structure and let denote the Dirac operator determined by the spin structure. One can show that where K^{1/2} denotes the square root of the canonical bundle. Then via the AtiyahSinger index theorem we have
1.1.4 B.1.4 As a Partition Function
The relationship between the \(\hat {A}\) class/genus and (supersymmetric) physics has an extended history, for highlights see [3, 16], or [17]. Indeed, in our previous work, [19], we demonstrated \(\hat {A}(M)\) as the partition function of a one dimensional perturbative BV theory.
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Grady, R. The Âgenus as a Projective Volume form on the Derived Loop Space. Math Phys Anal Geom 21, 13 (2018). https://doi.org/10.1007/s1104001892691
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DOI: https://doi.org/10.1007/s1104001892691