The chain complex of classical observables

boundary terms and fermionic signs are handled by the functor of points formalism.

dI_A(v) =−¹ 2

Z

C^1|1

(∇^VA)(v; dv,Dv,∂v) +A(v;Ddv,∂v) +A(v;Dv,∂dv)µ_E

= −¹ 2 Z

C^1|1 D(A(v; dv,∂v)) +∂(A;Dv, dv)µ_E

| {z }

=0

+ ¹ 2

Z

C^1|1 A(v;D∂v, dv) +A(v; dv,D∂v)

+ (∇^VA)(v;Dv, dv,∂v) + (∇^VA)(v;∂v,Dv, dv)

−(∇^VA)(v; dv,Dv,∂v)

! µ_E

The last expression is given by evaluating dvon a smooth section and therefore extends to distributional sections.

Let g(v;X,Y) = g₀+ ¹₂(A(v;X,Y) +A(v;Y,X)) be the symmetrization of g₀+_A, which is a non-degenerate symmetric bilinear form (non-degeneracy can be checked forv =0, where it is equivalent to non-degeneracy ofg₀). We can then define a tensor Γ∈VB((V^∨)^⊗2⊗V)_by

g(_Γ(v;X,Y),Z) = ¹ 2

(∇^VA)(v;X,Y,Z) + (∇^VA)(v;Z,X,Y)−(∇^VA)(v;Y,X,Z) IfAis symmetric, this defines the connection1-form of the Levi-Civita connection ofg in the trivialization of the tangent bundle determined by the affine structure onV. As expected from the classification of deformations of the classical action, the antisymmetric component of Aonly contributes through its de Rham derivative, and it is given by raising one index of the three-formηwith the metric. Together, this defines a connection

∇^η =∇^V+Γ. Note that∇^ηg=0, whereas the torsion of∇^η is given by raising an index ofηusingg.

The BV Poisson bracket then becomes the Koszul differential computing the derived zero locus of the Euler–Lagrange equations∇^η_∂Dv =∇⁻_D^η∂v=0. Concretely, applying it to a linear functionalX7→ hω,Xi(0)of the antifields shows thathω,∇^η

∂Dvi(0)_{is exact} under the BV bracket with the classical action, which is a differential by the curved

Classical Master Equation.

measurements is encapsulated in thefactorization product, which allows us to multiply measurements in different regions. In classical field theories, this is possible without such a restriction, and classical observables form a presheaf of cdgas. After recalling the definition of the classical observables from [CG21, Definition 5.1.0.2], we calculate their values on open and closed (families of) Euclidean supermanifolds and identify the leading terms in the action of Euclidean symmetries under a filtration which corresponds to the order of vanishing at the constant maps.

Recall from Definition A.3.1 that forE → Ba family of Euclidean supermanifolds, the subspaceO_D∂=0(U)_ofsuper-harmonic functionsis the kernel of the differential op-eratorD∂=0(observe that this is independent of the trivialization of theU(1)_-torsor parametrizing different choices of the vector fields Dand∂). It is a convenient vector space, with dual thesuper-harmonic distributions.

Definition 2.3.1 LetΣ= (X,A,E_Σ,θ)be a classical2|1-sigma model, and letp:E→ B be a family of2|1-Euclidean supermanifolds. Theclassical observablesOBS^∗_cl(E→B;Σ) are the filtered cdgaO_/(X×B,A⊗πO(B)),cs,sm(E×X;F_Σ), with the differential given by the sum of the background differential d∇−ι_Rand the bracket with the classical actionS_θ and the filtration given by the polynomial degree.

To simplify notation, we denote its degree0cohomology group byObs_cl(E→B;Σ)_. It assigns a commutative algebra in complete filtered vector spaces to any family of Eu-clidean supermanifolds. We also drop the sigma model from the notation if no confusion

is possible.

By the Jacobi identity, the differential squares to zero iffS_θsatisfies the curved Classical Master Equation.

The classical observables are local both on the target and the source. In fact, we have functoriality with respect to change of the ground supermanifold B and (fiberwise) embeddings:

Proposition 2.3.2 LetΣ = (X,A,E_Σ,θ)be a classical2|1-sigma model, and let(p⁰ : E⁰ → B⁰)→(p:E→ B)be a morphism of families of Euclidean supermanifolds. Then pulling back and extending a compactly supported functional by0defines a chain mapOBS^∗_cl(E⁰ →B⁰;Σ)→ OBS_cl^∗(E×_BB⁰ →B⁰;Σ).

Proof Every morphism of families of supermanifolds can be written as the composition of a cocartesian morphism, i.e., a pullback diagram, and the inclusion of an open subsu-permanifold. For the first, compatibility with the differential follows since the classical action is invariant underIsom(_C^1|1), while for the second, it follows since the differential

{S_θ,−}preserves the support of observables.

Proposition 2.3.3 LetΣ=_{ΣX,g,η be the}2|1-sigma model defined by a Riemannian manifold (X,g)and a closed 3-formη, and letE → Bbe a family of supermanifolds whose fiber has no compact connected component. Then the cohomology of the classical observables onE is

concentrated in degree0, and the associated graded of the weight filtration has degreencomponent given by the quotient of the projective tensor product

(O_D∂=0,0(E^×Bn)^∨⊗^π_Γ(X;(T^∨X)^⊗n))_Sn

of super-harmonic distributions onE^×Bn→ Bwhich vanish on functions which are constant in one variable and (covariant) tensors on the targetX.

For a morphism of Euclidean supermanifolds(E⁰ → B⁰)→(E→ B)whose source and target have no compact connected component and which induces a fiberwise surjection on connected components, the induced mapObs_cl(E⁰ → B⁰)→Obs_cl(E×_BB⁰ →B⁰)is surjective.

Proof The classical observables are the symmetric algebra in convenient vector spaces generated by the linear dual of the fields, which as a convenient vector space is equipped with a bigrading by the differential form and polynomial degree. The differential d∇+ {S,−} does not preserve either one of these degrees, but only the (weight) filtration defined by their sum. Expanding the Maurer–Cartan elementθ= _∑d≥−1τ^(d)+_∑d>0A^(d), the components{I_θ(d),−}_and{I_A(d),−}increase the weight filtration degree byd+1 andd, respectively; the component d_∇ increases it by1, whereas the free component Q={S_A⁽0),−}preserves it. Consequently, the cohomology of the associated graded is calculated by the same space equipped with the differential{I_θ(−1),−}+_Q.

The first component is essentially the formal de Rham differential of Example 1.4.16.

There is an explicit contraction for it, given by combining the contraction of the Poincaré lemma on the algebraic factor with multiplication by a section of the Berezinian which integrates to1, showing that the inclusion of the subcomplex of distributional sections which pair trivially with the constant function is a quasi-isomorphism. The differentialQ is the continuous extension of the Laplace–Dolbeault operator f 7→ D∂(f)µ_E, tensored with the isomorphism[: TX∼=T^∨Xdefined by the Riemannian metric, to a derivation of the completed symmetric algebra. By elliptic regularity, the inclusion of smooth sections is a quasi-isomorphism, and the cohomology of the corresponding complex is concentrated in degree0by Proposition A.3.11: Essentially, this is a statement about solving the equations∆if = gand∂_if = gon powers of the fiber. The first equation can be reduced to solving the equation∂_ief = g_etwice, which is possible since the compactly supported Dolbeault cohomology of a holomorphic vector bundle over a power of the fiber, which is a Stein complex manifold, has a contraction onto its cohomology in top degree. Combining these results, one obtains that theE²-page is concentrated in cohomological degree0, with the indicated convenient vector space in the filtration stage n. The statement then follows by passing to theE^∞-page, noting that the weight filtration is complete and that the analytic differentialQhas a continuous contraction, so that degeneration of the spectral sequence on theE²-page and splitting of the filtration can be proved using Lemma 0.2.

Equivariance with respect to Euclidean symmetries is clear from the definition of the weight filtration. The induced map on the associated graded is given by the induced map on super-harmonic distributions. By uniqueness of analytic continuation, this

restriction is injective if the inclusion is surjective on fiberwise connected components, which dualizes to surjectivity of extension of super-harmonic distributions.

Remark 2.3.4 From the perspective of classical field theory, it is perhaps surprising to obtain the classical observables in this implicit form, as the general Lagrangian formalism suggests that they should be given by functions on the space of solutions of the classical equations of motion∇^η

∂(Dx) = 0. We consider only the formal neighborhood of the constant solutions, i.e., the Taylor expansions of observables along this subspace. The weight filtration corresponds to the filtration by order of vanishing on this subspace on these Taylor expansions, and the fact that we only obtain an explicit description of the associated graded is the familiar statement that then-th derivative of a function only transforms tensorially if all previous derivatives vanish. The splitting of the filtration is then defined by the Levi-Civita connection, which does give the possibility to obtain diffeomorphism-invariant higher derivatives at the price of picking up polynomials in the covariant derivatives of the Riemann curvature tensor.

Higher cohomology groups of the classical observables capture a non-transverse intersection of the Euler–Lagrange equations with the zero section of the cotangent bundle of the space of fields. The fact that these do not show up on an open Euclidean supermanifold means that the equations of motion cut out a smooth subspace of the

formal neighborhood of constant maps.

On a closed Euclidean supermanifold, i.e., a super torus, nonzero cohomology groups do appear, since the constant classical solutions are invariant under even translations.

For the free field theory, this is reflected in a nonzero first cohomology group of the space of fields. The Hodge parametrix constructed in Proposition A.3.12 allows us to describe the space of classical observables explicitly.

Proposition 2.3.5 LetE→ Bbe a family of non-bounding super tori. Then there is an explicit contraction of the chain complex of classical observablesOBS_cl^∗(E→ B)onto a subalgebra with trivial differential and cohomology isomorphic to

Ob B×T^∨[−1](LΠTX)=

∏

n

Γ

B×X; Symⁿ TX[1]⊕LΠTX[1]⊕L^∨ΠT^∨X whereL →Bis a line bundle constructed from the familyE.

Proof First suppose thatE→ Bis pulled back from the universal familyT²_M^|1 → M_of pinned bounding super tori from Proposition A.2.2 (unwrapping this statement, one may suppose thatE→B∼= B×_C^1|1_/Λis the quotient of the model geometryC^1|1by a lattice determined by two even morphisms(w_i,w_i) : B → _Cand odd morphisms v,v : B → _C^0|1_satisfying vv = 0). The usual weight filtration argument identifies the E²-page of the weight filtration spectral sequence with the symmetric algebra of the subspace of the cohomology of the Laplace–Dolbeault operator, tensored with the cotangent and tangent bundle ofXin degrees0and1, respectively, where the Berezinian

component must pair trivially with the constant function 1. The cohomology of the Laplace–Dolbeault operator is spanned byT²-invariant sections, and we construct an explicit contraction onto them, theHodge parametrixΦ∞, in Proposition A.3.12. Since this is not concentrated in degree0, a priori there might be higher differentials. To see that these are impossible, we use instead the filtration by the augmentation ideal of the background fieldsΩ^∗(X), i.e., by differential form degree.

On the associated graded, only the interaction componentQ+{I_A,−}survives, which is given by plugging a distribution on the complex of fields into the Euler–Lagrange equation defined by the (Taylor expansion of a) metric and3-form A. It follows that this differential vanishes on products ofT²-invariant distributions, so that theE¹-page is given by

∏

Γ

B×X; Symⁿ T^∨X[−₁]⊕T^∨X⊕TX[₁]⊕_ΠTX[₁]⊕_ΠT^∨X .

The differential is defined by the bracket with the symmetry component{I_τ,−}_{of the} classical action, which is the tensor product of the algebraic de Rham differential with the identity of the (filtered and degreewise finite dimensional, graded) vector bundle Symd^∗ TX[₁]⊕_ΠTX[₁]⊕_ΠT^∨X

. In particular this differential is a derivation, and the usual filtration argument shows that there is an explicit contraction onto a subalgebra such that setting the formal derivatives along X given by T^∨X and the background differential formsT^∨X[−₁]_to0defines a quasi-isomorphism; these are exactly the jets of sections of this infinite-dimensional bundle.

In general, the familyEis obtained by choosing a coverB=^S_i∈IU_iand gluing together families over theU_ipulled back fromM. By Proposition A.2.3 the gluing data is defined by maps toU(1)×SL(2,Z)×_T^2|1on intersectionsU_i∩U_j, and the projection toU(1) defines the cocycle used to construct L. A more careful analysis shows that the odd symmetry does not preserve the splitting of the fields into even and odd components, so that the isomorphism must be constructed by induction over the filtration by nilpotents.

Remark 2.3.6 For non-bounding super tori, a straightforward modification of this argu-ment identifies the classical observables withΓ(B×X; Sym^∗T^∨[1]X), since the Dolbeault operator of the non-bounding spin structure is invertible.

The map induced from the inclusion of an open subsupermanifold into a bounding super torus can in principle be described explicitly as a sum over trees whose edges are labeled by the Hodge parametrix and whose vertices are labeled by the classical action, with one special vertex labeled by the observable on the subsupermanifold. We revisit this idea in Section 4.2, where this procedure is generalized to the quantum observables.