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The Â-genus as a Projective Volume form on the Derived Loop Space

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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 one-dimensional 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

  1. 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.

  2. Using densities, the following arguments can be adopted to unoriented manifolds.

  3. This kind of integration notion would work even for infinite-dimensional spaces, for which there are no top forms but there is a ring of functions.

  4. 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.

  5. 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 Berwick-Evans.

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Correspondence to Ryan Grady.

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The author was partially supported by the National Science Foundation under Award DMS-1309118.

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 second-order operator of degree 1 such that Δ2 = 0.

Here Δ is called the BV operator. Δ being “second-order” means the following: define the BV bracket{−, −}Δ as the measuring of the failure of Δ being a derivation

$$\{a,b\}_{{\Delta}}:={\Delta}(ab)-({\Delta} a)b- (-1)^{\lvert a \rvert}a {\Delta} b. $$

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)|a||b|{b, a}.

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

  • Δ{a, b}= −{Δa, b}−(− 1)|a|{ab}.

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 Q2 = 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

$$QI_{0}+\frac{1}{2} \{I_{0},I_{0}\}= 0. $$

A degree 0 element \(I\in \mathcal {A}[[\hbar ]]\) is said to satisfy the quantum master equation (QME) if

$$QI+\hbar {\Delta} I+\frac{1}{2}\{I,I\}= 0. $$

Here \(\hbar \) is a formal (perturbative) parameter.

The “second-order” property of Δ implies that QME is equivalent to

$$(Q+\hbar {\Delta})e^{I/\hbar}= 0. $$

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 I0 of the CME leads to a differential Q + {I0, −}, 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, Obscl is given by

$$Obs^{cl} \overset{\text{def}}{=}(\mathcal{A}, Q+\{I_{0},-\}). $$

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, Obsq is given by

$$Obs^{q} \overset{\text{def}}{=}(\mathcal{A}[[\hbar]], Q+\hbar {\Delta} + \{I,-\}). $$

Note that Obscl has a degree 1 Poisson bracket, so following [13] we call it a P0 algebra. Similarly, in ibid. the structure on Obsq 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)〉+ I0(e), where Q is a square zero differential operator of cohomological degree 1, such that

  1. 1.

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

  2. 2.

    I0 is at least cubic; and

  3. 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

$$\mathcal{E} = {\Gamma} \left( M ; E[1] \oplus \left( E^{\vee} \otimes \text{Dens}(M) [-2] \right) \right). $$

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. 1.

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

  2. 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 ill-defined. 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

$$\lim\limits_{L \to 0} I[L] \equiv I_{0} \quad \quad (\text{modulo\ } \hbar). $$

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. 1.

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

  2. 2.

    A finite set of half-edges \(H(\mathcal {G})\);

  3. 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 two-element orbits is denoted by \(E(\mathcal {G})\) and is called the set of internal edges of \(\mathcal {G}\);

  4. 4.

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

  5. 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

$$g(\mathcal{G}):=b_{1}(|\mathcal{G}|)+\sum\limits_{v\in V(\mathcal{G})}g(v), $$

where \(b_{1}(|\mathcal {G}|)\) denotes the first Betti number of \(|\mathcal {G}|\). Let

$$\mathcal{O}^{+}(\mathcal{E})\subset \mathcal{O}(\mathcal{E})[[\hbar]] $$

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

$$F= \sum\limits_{g,k\geqslant 0}\hbar^{g} F_{g}^{(k)}, \quad F_{g}^{(k)}\in \mathcal{O}^{(k)}(\mathcal{E}). $$

We view each \(F_{g}^{(k)}\) as an S k -invariant linear map

$$F_{g}^{(k)}: \mathcal{E}^{\otimes k}\rightarrow\mathcal{A}. $$

With the propagator P𝜖L, we will describe the (Feynman) graph weights

$$W_{\mathcal{G}}(P_{\epsilon \to L},F)\in \mathcal{O}^{+}(\mathcal{E}) $$

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

$$F(v):\mathcal{E}^{\otimes H(v)}\rightarrow \mathcal{A}, $$

where H(v) is the set of half-edges of \(\mathcal {G}\) which are incident to v. Next, we label each internal edge e by the propagator

$$P_{e}=P_{\epsilon \to L}\in\mathcal{E}^{\otimes H(e)}, $$

where \(H(e)\subset H(\mathcal {G})\) is the two-element set consisting of the half-edges forming e. We can then contract

$$\otimes_{v\in V(\mathcal{G})}F(v): \mathcal{E}^{H(\mathcal{G})}\rightarrow \mathcal{A} $$

with

$$\otimes_{e\in E(\mathcal{G})} P_{e}\in\mathcal{E}^{H(\mathcal{G})\setminus T(\mathcal{G})} $$

to yield a linear map

$$W_{\mathcal{G}}(P_{\epsilon \to L},F) : \mathcal{E}^{\otimes T(\mathcal{G})}\rightarrow \mathcal{A}. $$

Definition 20

We define the (homotopy) RG flow operator with respect to the propagator P𝜖L

$$W(P_{\epsilon \to L}, -): \mathcal{O}^{+}(\mathcal{E})\to \mathcal{O}^{+}(\mathcal{E}), $$

by

$$ W(P_{\epsilon \to L}, F):=\sum\limits_{\mathcal{G}}\frac{\hbar^{g(\mathcal{G})}}{\lvert \text{Aut}(\mathcal{G})\rvert}W_{\mathcal{G}}(P_{\epsilon \to L}, F) $$
(1)

where the sum is over all connected graphs.

Equivalently, it is useful to describe the (homotopy) RG flow operator formally via the simple equation

$$e^{W(P_{\epsilon \to L}, F)/\hbar}=e^{\hbar \partial_{P_{\epsilon \to L}}} e^{F/\hbar}. $$

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

$$F[L]=W(P_{\epsilon \to L}, F[\epsilon]). $$

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 ∈ Ωn(X) then defines a linear functional

$$\begin{array}{cccc} {\int}_{\mu}:& C^{\infty}_{c}(X)& \to &\mathbb{R} \\ & f & \mapsto & {\int}_{X} f \mu \end{array}, $$

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 1-dimensional 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 compactly-supported polyvector fields and div μ denotes “divergence with respect to μ.” We require now that μ is nowhere-vanishing, so that the divergence is well-defined. 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

$$\mathcal{O} (T^{\ast} [-1]X) \cong PV(X). $$

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

$$Q(x) = \frac{x}{1- e^{-x}} \text{ \; \; \; and \; \; \; } P(x) = \frac{x/2}{\sinh x/2} . $$

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])

$$ \log \left( \frac{x}{1-e^{-x}} \right) - \frac{x}{2} = \sum\limits_{k \geqslant 1} 2 \zeta (2k) \frac{x^{2k}}{2k (2 \pi i)^{2k}} , $$
(2)

where ζ is the Riemann zeta function.

We now use standard arguments about characteristic classes. For a sum of complex line bundles E = L1 ⊕ ⋯ ⊕L n , the Todd class is

$$Td(E) = Q(c_{1}(L_{1})) {\cdots} Q(c_{1}(L_{n})). $$

Thus, (2) tells us

$$\log (e^{-c_{1}(E)/2} Td(E) ) = \sum\limits_{k \geqslant 1} \frac{2 \zeta (2k)}{2k (2 \pi i)^{2k}} (c_{1}(L_{1})^{2k} + {\cdots} + c_{1}(L_{n})^{2k}). $$

As ch2k(E) = (c1(L1)2k + ⋯ + c1(L n )2k)/(2k!), we obtain a general formula for an arbitrary bundle E,

$$\log (e^{-c_{1}(E)/2} Td(E) ) = \sum\limits_{k \geqslant 1} \frac{2 \zeta (2k)}{2k (2 \pi i)^{2k}} (2k)! ch_{2k}(E). $$

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

$$\log(\hat{A} (V)) \overset{\text{def}}{=} \sum\limits_{k \geqslant 1} \frac{2 \zeta (2k)}{2k (2 \pi i)^{2k}} (2k)! ch_{2k}(V) \in {\Omega}^{-\ast}_{B \mathfrak{g}} . $$

Similarly, we will need an equivariant class, so we define for any smooth manifold X the class \(\log (\hat {A}_{u} (X))\) to be

$$\log (\hat{A}_{u} (X)) \overset{def}{=} \sum\limits_{k \geqslant 1} \frac{2 (2k-1)!}{(2 \pi i)^{2k}} u^{2k} \zeta(2k) ch_{2k} (X) \in ({\Omega}^{-\ast}_{X} [[u]], ud ). $$

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 K-theoretic 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 spinc structures and corresponding Dirac operators. Let be the Dirac operator corresponding to the spinc structure determined by the complex structure and let denote the Dirac operator determined by the spin structure. One can show that where K1/2 denotes the square root of the canonical bundle. Then via the Atiyah-Singer 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 Â-genus as a Projective Volume form on the Derived Loop Space. Math Phys Anal Geom 21, 13 (2018). https://doi.org/10.1007/s11040-018-9269-1

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