The factorization property and the partition function

We have seen in Proposition 3.4.4 that the quantum observables on a fiberwise non-compact, connected family of supermanifolds are given by various completions of the

vector space of quantum point observables. This situation changes dramatically for families of closed manifolds, where the solution theory of elliptic operators gives us finite-dimensional complexes computing them.

We fix a quantized sigma modelΣ=_{ΣX,g¯h,η¯h} determined by a Riemannian manifold throughout. By equivariance with respect to Euclidean isometries, the quantum observ-ablesOBS^∗_q(_T²_M^|1→ M;ΣX,g_¯h,η_¯h)form a vector bundle overMwhich is pulled back from Eucl^nb_2|1. For any open subsupermanifold of the torus, we obtain an induced map, which can only be nonzero on the zeroth cohomology group. By density, it suffices to consider point observables.

Definition 4.1.1 Thetorus expectation valueEM : Obsq(pt)→_Γ(M; Obsq(_T²_M^|1→ M)) is the map which sends a pair of a parameter and a quantum point observableOto the realization map of quantum point observablesρ_(0,0,0)(O)(cf. Definition 3.5.9) defined by the origin in the torus fiber with the Euclidean structure specified by the parameter.

The choice of name is motivated by the fact that for a closed, oriented Riemannian target we will define a further mapObs_q(_T²_M^|1 → M)→ O(M)((¯h)), and the composite is a rigorous version of the expectation value with respect to the (suitably normalized) path integral as a function of the moduli parameters.

A general family of non-bounding tori is glued together from pinned families using Euclidean isometries. There is another (non-smooth) moduli space classifying two pinned families with a choice of isometry between them, whose description can be found in Proposition A.2.3: it is the productM ×(U(1)×SL(2,Z)×_T^2|1), where the three factors correspond to a change of trivialization of theU(1)_-torsor,SL(2,Z)-torsor, and section defining the pinning, respectively.

Proposition 4.1.2 The mapEMisU(1)×SL(2,Z)-equivariant, whereSL(2,Z)acts trivially on point observables. Furthermore, it intertwines the map D⁺on point observables with the action of the odd Euclidean isometryvol^−1/2D⁺_M.

Proof Recall that the realization map of quantum point observables is defined in a centered Euclidean chart, with the choice of chart fixed by a trivialization of theU(₁) -torsor at this point. For the universal family of pinned tori, we calculated the chart centered at the chosen section in Proposition A.2.2; it is given by

z= w₁x+w₂y

z= w₁x+w₂y−2ζ v(w₁x+w₂y) +v(w₁x+w₂y) ζe=vol^1/2ζ−vol^−1/2 v(w₁x+w₂y) +v(w₁x+w₂y)

TheU(1)×SL(2,Z)-equivariance then follows from the equivariance of quantum ob-servables and the realization map with respect to Euclidean symmetries.

In this centered Euclidean chart, the odd isometryD⁺_Mis given by the usual Euclidean symmetryvol^1/2·D⁺, as both Euclidean vector fields agree at the origin. Note that

it is not tangent to the fibers ofT²_M^|1 → M, but rather covers the C^0|1-action on the base determined by the cohomological vector field2∑i(w_iv+w_iv)∂w_i. This shows the

second equivariance.

Corollary 4.1.3 IfO ∈ Obs_q(pt)is antiholomorphic, i.e., satisfies D⁺O = 0, its torus ex-pectation valueEM(O)_{defines a} SL(2,Z)-invariant section ofObs_q(_T²_M^|1 → M)_{which is} annihilated by∇_D⁺

M. Furthermore, ifOhas approximate conformal bidegree(r,r), the resulting section transforms under theU(1)_{-action as}u·_EM(O) =u^r−rEM(O)_.

From the computations in Corollary 3.5.8 and Proposition 3.5.12, we see that the as-sumptionD⁺O=0is quite strong: on the associated graded of the weight filtration, the observableOmust be represented by a flat tensor (paired with holomorphic distributions at the origin). There is then an infinite tower of potential higher obstructions depending on the choice of quantization. However, we can always takeO=1.

Definition 4.1.4 Thepartition functionisEM(1)∈Obs_q(_T²_M^|1→ M)_. We now come to a crucial property of the quantum observables: They form a factoriza-tion algebra, in the sense that the value on any manifold is determined by the values on arbitrarily small submanifolds, together with the factorization products between them.

The mathematical statement of this property isWeiss codescent: The augmented Čech complex for the precosheaf of quantum observables on a Weiss cover{U_i}_i∈I →U, i.e., a cover such that for each finite subsetS ⊂Uthere isi∈ IwithS⊂U_i, is a homotopy colimit cone. This is shown in [CG21, Theorem 8.6.0.1] by a spectral sequence argument, which reduces the statement first to classical observables via the¯h-adic filtration and then to classical observables for free theories via the weight filtration. We note that their argu-ments do not apply verbatim in our setting, as we work with chain complexes infiltered convenient vector spaces throughout. From our perspective, it is more natural to reverse the direction of the argument by using the weight filtration to reduce to a free quantum field theory and then the¯h-adic filtration or the arguments in [Gwi12, Theorem 5.3.10] to reduce again to a classical free theory. This algebraic argument works in the category of filtered convenient vector spaces since there is a contraction of the quantum observables on an open supermanifold to their cohomology, which is concentrated in degree0. By truncation, this zeroth cohomology group then defines a (non-derived) factorization algebra in convenient vector spaces.

Note that these methods only show that the precosheafU 7→ OBS^∗_q(U ⊂ E)_{is a} factorization algebra, whereas we would like to produce a factorization algebra on the site of (families of) Euclidean supermanifolds. This was investigated in [Mur20], where given a groupGaction on a manifoldM, it is shown that the restriction from factorization algebras on the site of (families of) manifolds equipped with a rigid(G,M)-structure in the sense of [ST11] toG-equivariant factorization algebras onMis an equivalence, with inverse given by left Kan extension.

It remains to see what this factorization property can tell us about the quantum observables. On an open supermanifold, it is essentially the same information as the existence of the weight filtration spectral sequence, together with the fact that super-harmonic distributions form a cosheaf, so that the (completed) symmetric algebra on them forms a Weiss cosheaf. This leaves only tori, for which the factorization property should characterize the quantum observables in terms of the multicategory of subsets ofC^1|1 and pairwise disjoint embeddings. However, since this multicategory does not have an algebraic description, it is hard to turn this into a non-tautologous statement. We therefore close this section by two remarks explaining how the calculations of the preceding chapter give, at least in principle, a characterization of the zeroth cohomology group of the quantum observables on a torus via a trace-like property. The toy models are the Feigin–

Felder–Shoikhet formula for the normalized trace on a deformation quantization [FFS05]

and the identification of the partition function of the holomorphic sigma model on an elliptic curve [GGW20, Proposition 10.3].

Remark 4.1.5 Recall from Remark 3.4.16 that quantum observables on cylinders of a fixed radiuslgive a non-locally constant, translation-invariant factorization algebra on R, which one can think of as differential operators on the spinor bundle of the free loop space onX. The factorization algebra structure is encoded in multiplication mapsµt;t₁,t₂, which takes values in a certain completion A(t+t₁+t₂)of a naive definition of this algebra. The mapET,tfrom quantum observables on a cylinder(_0,t) +iR/lZto those on a torusT²_l,T,S^|1 =_C/(_ilZ+ (T+iS)_Z)then satisfies the identity

ET,t(µ_t;t1,t2(O₁,O₂)) =_ET,t µ_{T−t1−t2;t2;t1}(O₂,R_S(O₁))_.

(The astute reader will notice that the left side seems to demandT≥ t+t₁+t₂, rendering the equation senseless. In fact, one can define the torus expectation value also fort>_T:

Quantum observables are functorial with respect to local Euclidean embeddings by reducing to covers of Euclidean charts, for which one can sum over the fibers.) Using Propositions 3.4.13 and 3.4.14 and induction over the weight filtration, one can check that the quotient ofA(t)[h¯⁻¹]by this relation is a quotient of volume forms on the target Xby a subset containing the total derivatives. In particular, forXclosed, connected, and oriented, this space is at most one-dimensional. TheBV integration mapin Definition 4.2.5 defines an explicit trivialization, and the computation of the partition function can be seen as a generalization of the algebraic index theorem to supersymmetric equivariant

quantum mechanics on the loop space.

Remark 4.1.6 We can similarly try to characterize the kernel of the torus expectation valueEM. The idea is to take a point observableO₁and a family of observablesQε ∈ Obs(B^˚_ε(0))with support in an increasingly smaller punctured disk such thatι

Bε(0),→T˚²_M^|1Q_ε vanishes in the punctured torus, and evaluateO₁(0)∗Q_ε(0)in the limitε→0using the operator product expansion. The result is a point observable whose image under the

torus expectation value is the factorization product ofO₁and the zero observable on the punctured torus and therefore vanishes.

For holomorphic theories, one can takeQ_ε = R

|z|=εO₂(z)αdz, whereαdzis a mero-morphic one-form on the elliptic curveEwith poles only at the origin. This observable is visibly supported inB_2ε(0), and its cohomology class is independent ofεand vanishes in observables of the punctured torus by Stokes’s theorem. We have

O₁(0)∗Qε =

Z

|z|=ε

(O₁(0)∗O₂(z))αdz−−→^ε→0 Res_z=0 (O₁(0)∗^OPEO₂(z))αdz which can be evaluated as a formal Laurent series using the operator-state correspon-dence. The resulting elements are exactly those that map to0in the quotient to conformal blocks onE[Zhu96; FB04; EH21].

In our case, this ansatz does not work as in general there are no nontrivial holomorphic or antiholomorphic cohomology classes of observables. One can attempt to salvage it by building the observablesQ(ε)by induction over the weight filtration, but it seems un-avoidable to use smearings of observables at multiple points, so thatlimε→0O₁(₀)∗Q_ε involves the non-algebraic n-point OPE with n > 2. The computations in Proposi-tions 3.5.10 and 3.5.12 should again be sufficient to see that the quotient by the resulting relations is at most one-dimensional for a closed, connected, oriented target: In the clas-sical limit, it kills the ideal of super-harmonic point distributions generated by those that map to zero on the torus, leaving only the exterior algebra generated by the observables ω_D. One can then run a variant of the above calculation withα=1, picking up the residue of the OPEω_D(0)∗ω_D⁰ (Z)∼z⁻¹g⁻¹(ω,ω⁰)to eliminate all products of less thandimX of these generators. Finally, the factorization productω_∂(0)∗ω⁰_D(Z)∼ ¹_z(∇_ω]ω⁰)_D+. . . can be used to show that total derivatives are among the relations. Each of these argu-ments only works modulo higher terms in the¯h-adic or weight filtration, however, and canceling these terms requires the use ofn-point observables.