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A construction of observables for AKSZ sigma models

A construction of gauge-invariant observables is suggested for a class of topological field theories, the AKSZ sigma-models. The observables are associated to extensions of the target Q-manifold of the sigma model to a Q-bundle over it with additional Hamiltonian structure in fibers.


Introduction
The interest in observables in topological field theories is largely due to the applications in algebraic topology, where expectation values of observables are known to yield (at least in some examples) invariants of knots and links under ambient isotopy or, more generally, cohomology classes of spaces of embeddings. The seminal example here was given in the work of Witten [16], where the expectation value of the Wilson loop observable in Chern-Simons theory with gauge group SU (2), associated to a knot in the 3-sphere, was found out to yield the Jones polynomial of the knot.
Given a diffeomorphism-invariant action S Σ of a topological field theory on a manifold Σ with space of fields F Σ , one is particularly interested in observables O ∈ C ∞ (F Σ ) associated to embedded submanifolds i : γ ֒→ Σ which depend on fields only via their pull-back from Σ to γ: where X is the field. Such observables are automatically invariant with respect to diffeomorphisms φ : Σ → Σ in the sense that where (φ −1 ) * X is the pull-back of the field by φ −1 . On a formal level, the expectation value of the observable due to (2), diffeomorphism-invariance of the action S Σ and of the path integral measure DX. This means in particular that the expectation value is an invariant of the embedding i : γ ֒→ Σ under ambient isotopy. Since topological field theories possess gauge symmetry, one also requires that the observable is gauge-invariant, so that in the path integral (3) one could pass to integration over gauge equivalence classes of fields.
In this paper we employ Batalin-Vilkovisky (BV) formalism to treat systems with gauge symmetry (cf. e.g. [15] for details). In particular the space of fields F Σ becomes extended to an odd-symlectic Q-manifold F Σ with the action S Σ extended to a function on F Σ , satisfying the master equation 1 {S Σ , S Σ } = 0 and generating the cohomological vector field Q Σ as its Hamiltonian vector field. In this formalism the gauge-invariance of the observable is expressed as (5) Q Σ O = 0 and the path integral (3) for the expectation value is traded for an integral over a Lagrangian submanifold L in F Σ , with L playing the role of the gauge-fixing condition.
One idea how to construct an observable for a gauge theory in BV formalism is to consider an extension of the BV theory (F Σ , S Σ ) to a BV theory on a larger space (F Σ × F aux , S Σ + S aux ) with F aux the space of "auxiliary fields" and S aux 1 In this outline we do not make explicit distinction between classical and quantum master equation. We write the classical one which also coincides with the quantum one if ∆ Σ S = 0 for ∆ Σ the BV Laplacian. Likewise, we write the classical gauge-invariance condition on the observable Q Σ O = 0 which coincides with the quantum one if ∆ Σ O = 0, cf. section 4 for details.
the action for auxiliary fields Y which is also allowed to depend on the "ambient" fields X ∈ F Σ . Then one uses the BV push-forward construction (6) O(X) = L aux ⊂F aux DY e iS aux (X,Y ) (propositions 1, 3 in section 4, cf. also [13]) to integrate out the auxiliary fields and produce an observable for the ambient theory (F Σ , S Σ ). In this paper we call such extensions by auxiliary fields "pre-observables" (to be more precise, we use a slightly better definition with equation (40) on S aux instead of the master equation on S Σ + S aux , which still suffices to produce an observable, cf. definition 6 and remark 5).
In this paper we consider observables obtained by the construction outlined above for topological sigma models coming from AKSZ construction [1] (cf. a brief reminder in section 2), where F Σ is the mapping space from the tangent bundle of Σ with shifted parity of fibers to a target Q-manifold M with additional Hamiltonian structure (cf. definition 1). In this setting we give a construction of pre-observables (section 5), which associates a pre-observable for the AKSZ sigma model to an embedded submanifold i : γ ֒→ Σ and an extension of the target M to a Qbundle over M with additional Hamiltonian structure in fibers (the "Hamiltonian Q-bundle", cf. section 3). For the ambient and auxiliary actions to have degree zero, degrees of Hamiltonian structures in the base and the fiber of the target Q-bundle have to match 2 dimensions of Σ and γ respectively.
Having a pre-observable for an AKSZ sigma model, we construct the corresponding observable by the BV push-forward (6). By construction, this observable is gauge-invariant in the sense of (5) and satisfies (1), and so formally produces expectation values which are invariant under ambient isotopy. Of course, the problem with this definition of the observable is that (6) is generally a path integral. In section 6 we consider several simple cases when this integral can be made sense of and its expected properties can be rigorously checked: • case when the fiber in the target is a point (which gives rise to observables given by an exponential of an integral of a local expression), • case of one-dimensional observables (when the path integral becomes a path integral of quantum mechanical type and can be regularized via geometric quantization, which gives rise to certain generalization of Wilson loops), • case when the integral (6) is Gaussian which gives rise to a class of observables which we call "torsion-like" for their similarity to Ray-Singer torsion.
In section 7 we give explicit examples of observables falling within one of the three classes above. In particular, we recover the usual Wilson loop observable in Chern-Simons theory, with the corresponding path integral expression being the Alekseev-Faddeev-Shatashvili path integral formula for the Wilson loop [2]. We also recover the Cattaneo-Rossi "Wilson surface" observable for BF theory [9].
In this paper we concentrate only on the construction of observables, we are not calculating their expectation values.

Logic of the construction.
(i) For an AKSZ sigma model on manifold Σ with target M, find a Hamiltonian Q-bundle E over M. Then for every embedded submanifold i : γ ֒→ Σ of fixed dimension matching the degree of Hamiltonian structure in fibers of E, we construct a pre-observable.
(ii) Take the fiber BV integral (6) over the auxiliary fields of the pre-observable constructed in step (i) to obtain an observable for the AKSZ sigma model.
Unlike step (i), step (ii) is not canonical, in the sense that it is a quantization problem: the path integral has to be made sense of (and the gauge-invariance of the result has to be checked) which can be done for certain classes of target Qbundles, but it is not clear whether it is possible to do in greater generality 3 .

1.2.
Plan of the paper. The logical organization of the exposition is as follows.
• Section 3 -auxiliary construction for step (i) of the main construction.
• Section 4 -motivation for step (ii) of the main construction. Section 2 is a short reminder of the AKSZ construction of topological sigma models in Batalin-Vilkovisky formalism. We also recall how some well-known topological field theories fit into the construction: Chern-Simons theory, BF theory, Poisson sigma model.
In section 3 we first briefly recall the standard notion of a Q-bundle and then define a "Hamiltonian" Q-bundle, preparing the grounds for the construction of pre-observables in section 5. We start with the definition 2 of a trivial Hamiltonian Q-bundle ("trivial" here means trivial as a fiber bundle; the cohomological vector field is not required to be a sum of a cohomological vector field on the base and another one on the fiber). All our examples are of this type, but for the completeness of the exposition, we also give a slightly more general definition 3, which does not explicitly rely on the total space being a direct product. However in a local trivialization definition 3 boils down to definition 2.
In section 4 we recall the standard notions of classical and quantum observables in BV formalism and introduce the notion of a pre-observable, which comes in three modifications: (i) Classical pre-observable, definition 6: essentially, an extension of the action of the ambient classical BV theory to a solution of classical master equation on the space of fields extended by auxiliary fields. We give a technically more convenient definition with equation (40) required instead of the classical master equation on extended space of fields (cf. remark 5 for the relation between the two). (ii) Semi-quantum pre-observable, definition 7, suited for integrating out auxiliary fields to obtain an observable for the ambient theory (proposition 1). (iii) Quantum pre-observable for a quantum ambient BV theory (i.e. one with the action satisfying the quantum master equation), definition 9: this is also an extension of the ambient theory by auxiliary fields, plus an extension of the action to a solution of quantum master equation on the extended space. From a quantum pre-observable one can induce a quantum observable for the ambient theory, by integrating out auxiliary fields (proposition 3). 3 One can indeed try to define the path integral perturbatively, as sum of Feynman diagrams represented by integrals over compactified configuration spaces of tuples of points on γ. However, for the formal argument of proposition 1 that the result is gauge-invariant to become rigorous, one has to prove that the hidden boundary strata of the configuration spaces appearing in the calculation of Q Σ O (cf. the proof of proposition 6) do not contribute, which is not true in general case.
In this section in the discussion of quantization we always work with spaces of fields as with finite dimensional spaces and the integrals are over finite-dimensional super-manifolds. In the context of local field theory, the BV push-forward becomes a path integral, so the proofs given in section 4 stop to work and the propositions become conjectures that have to be proven by more delicate means (cf. the proofs of propositions 5, 6 in section 6). Section 5 is the logical core of the paper. Here we give the construction of a classical pre-observable for an AKSZ theory out of an extension of the target to a Hamiltonian Q-bundle.
In section 6 we treat several classes of situations when the BV push-forward yielding an observable out of a pre-observable constructed in section 5 can be performed rigorously: case of target fiber being a point, case of 1-dimensional observables (via geometric quantization), case when the BV push-forward is given by a Gaussian integral.
In section 7 we specialize the constructions of section 6 to present several explicit examples, including the Wilson loop together with its path integral representation known from [2], Cattaneo-Rossi codimension 2 observable for BF theory, torsion observables in Chern-Simons and BF theories, etc.
1.3. Acknowledgements. I wish to thank Anton Alekseev, Alberto Cattaneo, Andrei Losev and Nicolai Reshetikhin for inspiring discussions. This work was partially supported by RFBR grant 11-01-00570-a and by SNF grant 200021 137595.
• Q-manifold (or bundle) = differential graded manifold (or bundle). • Ghost number = internal degree (to distinguish from de Rham degree of a differential form) = Z-grading on functions on Z-graded manifolds. • Observable = gauge-invariant functional on the space of fields (cf. section 4 for definitions). • Expectation value = correlator.
Conventions. We set the Planck's constant = 1 (cf. remark 6 on how to re-introduce ).
Notations. We use M, N , E, F etc. for Z-graded manifolds, M, N etc. for ordinary (non-graded) manifolds; Σ always denotes the source (spacetime) manifold of the sigma model, γ is typically a submanifold of Σ on which the observable is supported. We use L for a Lagrangian submanifold of a degree -1 symplectic graded manifold.
We denote by X(M) the Lie algebra of vector fields on M. For a fiber bundle π : E → M, we denote by X vert (E) the Lie algebra of vertical vector fields on the total space E, i.e. the space of sections Γ(E, T vert E) of the vertical distribution on We denote the degree (the ghost number) of functions/differential forms/vector fields on a graded manifold by | · · · |.

AKSZ reminder
What follows is a very short reminder of the AKSZ construction of topological sigma models in Batalin-Vilkovisky formalism, to fix the terminology and notation. We refer the reader to the original paper [1] and the later expositions in [7], [14] for details.
2.1. Target data. Let M be a degree n sympectic Q-manifold, i.e. a Z-graded manifold endowed with a degree 1 vector field Q satisfying Q 2 = 0 (the cohomological vector field) and with a degree 4 n symplectic form ω ∈ Ω 2 (M) which is compatible with Q, i.e. L Q ω = 0.
Definition 1. We call the set of data (M, Q, ω = δα, Θ) a Hamiltonian Q-manifold of degree n.
2.2. The AKSZ sigma model. Fix a Hamiltonian Q-manifold M of degree n ≥ −1 and let Σ be an oriented closed manifold, dim Σ = n + 1. Then one constructs the space of fields as the space of graded maps between graded manifolds from the degree-shifted tangent bundle T [1]Σ to M: It is a Q-manifold with the cohomological vector field coming from the lifting of Q on the target and of the de Rham operator d Σ on Σ (viewed as a cohomological vector field on T [1]Σ) to the mapping space: Transgression map. The following natural maps (10) where p is the projection to the first factor) allow us to define the transgression map Here p * is the fiber integration over T [1]Σ (with the canonical integration measure). Map τ Σ preserves the de Rham degree of a form, but changes the internal grading (the ghost number) by − dim Σ. If u ∈ X(T [1]Σ), v ∈ X(M) are any vector fields on the source and the target and ψ ∈ Ω • (M) is a form on the target, then for the Lie derivatives of the transgressed form along the lifted vector fields we have The finite version of (12) is that for Φ : where Φ * : F Σ → F Σ is the lifting of Φ to the mapping space and (Φ * ) * : is the pull-back by Φ * . 4 By "degree" here we mean the internal Z-grading coming from the grading on M. 5 In fact (cf. [14]), for n = −1 the symplectic property of the cohomological vector field (L Q ω = 0) implies existence and uniqueness of the Hamiltonian Θ = 1 n+1 ι E ι Q ω where E is the Euler vector field. Maurer-Cartan equation (7) follows from Q 2 = 0 for n = −2. Also, for n = 0, a closed form ω is automatically exact.
The BV 2-form and the master action. One obtains the degree −1 symplectic form (the "BV 2-form") on F Σ from the target by transgression 6 : The Hamiltonian function for Q Σ (the master action) is constructed as It automatically satisfies the classical master equation One can summarize the construction above by saying that we have a degree −1 Hamiltonian Q-manifold structure on the mapping space (8): (F Σ , Q Σ , Ω Σ , S Σ ). The primitive 1-form for the BV 2-form can also be constructed by transgression as τ Σ (α).
Note that exact shifts of the target 1-form, Why AKSZ theory is topological. (14) and because d Σ ∈ X(T [1]Σ) commutes withφ, since the latter is a tangent lift. In coordinates. Let x a be local homogeneous coordinates on the target M, let u µ be local coordinates on Σ and θ µ = du µ be the associated degree 1 fiber coordinates on T [1]Σ. Then locally an element of F Σ is parameterized by Coefficient functions X a µ1···µ k (u) are local coordinates of degree |x a | − k on the mapping space F Σ . Expression (18) is known as (the component of) the superfield, and it can be regarded as a generating function for the coordinates on the mapping space F Σ .
For any function f ∈ C ∞ (M), we have where we denoted X a ≥1 = k≥1 X a (k) the part of the superfield of positive de Rham degree with respect to Σ; ev is the horizontal arrow in (10).
Let α and ω be locally given as α = α a (x) δx a and ω = 1 2 ω ab (x) δx a ∧ δx b . Then the BV 2-form and its primitive are: We introduce the sign (−1) dim Σ in this definition to avoid signs in the formula for the action below. The reader may encounter different sign conventions for the AKSZ construction in the literature.
(Note that we use δ to denote the de Rham differential on the target M and on the mapping space F Σ . We reserve symbol d for the de Rham differential on the source Σ.) The master action is: If the cohomological vector field on the target is locally written as Q = Q a (x) ∂ ∂x a then the cohomological vector field (9) is determined by its action on the components of the superfield: Q Σ X a = dX a + Q a (X) Critical points of S Σ are (with our sign conventions) Q-anti-morphisms between T

Examples.
Here we recall some of the standard examples of the AKSZ construction.
Chern-Simons theory [1]. Let g be a quadratic Lie algebra, i.e. a Lie algebra with a non-degenerate invariant pairing (, ). Denote by ψ : g[1] → g the degree 1 g-valued coordinate on g [1]. We choose the target Hamiltonian Q-manifold of degree 2 as where , is the canonical pairing 7 between g and g * . The associated AKSZ sigma model on a closed oriented 3-manifold Σ has the space of fields The superfield is with A (k) a coordinate on F Σ with values in g-valued k-forms on Σ, with internal degree (ghost number) 1 − k, for k = 0, 1, 2, 3. The BV 2-form is and the action is This is the action of Chern-Simons theory in Batalin-Vilkovisky formalism.
In case g = R with abelian Lie algebra structure, we have Q = Θ = 0 on the target and This is the abelian Chern-Simons theory in BV formalism. BF theory. For g a Lie algebra (not necessarily quadratic 8 ) and D a non-negative integer, we define the target Hamiltonian Q-manifold of degree D − 1 as 7 We will generally be using notation , for the canonical pairing V ⊗ V * → R between a vector space V and its dual; sometimes we will indicate the respective V as a subscript: , V . 8 However, for the consistency of quantization, in particular for the quantum master equation (48), one has to require that g is unimodular.
Here ψ : g [1] → g is as before and ξ : g * [D − 2] → g * is the g * -valued coordinate on g * [D − 2] of degree D − 2; ad * is the coadjoint action of g on g * .
The associated AKSZ sigma model on a closed oriented D-manifold Σ has the space of fields The superfields associated to ψ and ξ are respectively The BV 2-form and the action are: This is the BF theory in BV formalism.
In abelian case, g = R, we have Q = Θ = 0 on the target and , Poisson sigma model [7]. Let M be a manifold endowed with a Poisson bivector π ∈ Γ(M, ∧ 2 T M ). We construct the target Hamiltonian Q-manifold of degree 1 as Here x and p stand for the local base and fiber coordinates on T * [1]M respectively. Note that all objects in (27) are globally well defined.
The corresponding AKSZ sigma model on an oriented closed surface Σ has the space of fields with superfields associated to local coordinates x and p on the target, respectively. Here X (k) and η (k) are k-forms on Σ with internal degrees −k and 1 − k respectively. The BV 2-form and the action are:

Hamiltonian Q-bundles
In this section we briefly recall the standard notion of a Q-bundle and then introduce "Hamiltonian Q-bundles", an auxiliary notion necessary for our construction of observables.
3.1. Q-bundles reminder. Recall that a "Q-bundle" [11] is a fiber bundle in the category of Q-manifolds.
In particular, a trivial Q-bundle is trivial bundle of graded manifolds where the base M is endowed with a cohomological vector field Q and the total space is endowed with a cohomological vector field Q tot in such a way that π is a Q-morphism, i.e. dπ(Q tot ) = Q. This implies the following ansatz for Q tot : plus the cohomological property for Q. Note that in the first term on the l.h.s. of (29) we are thinking of Q as acting on C ∞ (M) part of A. Equivalently, we can lift Q to a horizontal vector field on E and write the first term as [Q, A].

Trivial Hamiltonian Q-bundles.
Definition 2. We define a trivial Hamiltonian Q-bundle of degree n ′ ∈ Z as the following collection of data.
The fiber N is endowed with a degree n ′ exact symplectic form ω ′ = δα ′ and a degree (n ′ + 1) where E is the Euler vector field on N . Remark 2. If n ′ = 0, then ω ′ is not automatically exact. In this situation, exactness condition may be relaxed 9 to the Bohr-Sommerfeld condition that ω ′ /2π defines an integral cohomology class in H 2 (N ). Then the role of primitive 1-form α ′ is taken by a Hermitian line bundle Observe that this argument fails for the fiber N : applying the same reasoning to derive the equation (30) from (29), we obtain that l.h.s. of (30) is a pull-back of some function on the base of degree n ′ + 2, which does not imply that it is zero. 9 Cf. remark 8 for the motivation: primitive α ′ (or connection ∇ ′ in the relaxed version) will be needed in the construction of section 5 to define the kinetic part of the auxiliary action.

General Hamiltonian Q-bundles.
The assumption of triviality of the bundle E, used in the definition 2, can be relaxed as follows.
Definition 3. A Hamiltonian Q-bundle is the following set of data: We also require that where the splitting Q tot = Q hor + Q vert of the cohomological vector field on E into the horizontal and the vertical part is defined by connection ∇ ω ′ .
Remark 4. Note that one can introduce a local trivialization of E consistent with the flat connection ∇ ω ′ (i.e. where the horizontal distribution ker ω ′ ⊂ T E is defined by the direct product structure on E| U coming from the local trivialization, where U ⊂ M is a trivializing neighborhood). In such a trivialization, we recover back the definition 2. In this sense, definition 3 does not bring in a drastic increase of generality.

Examples.
(i) Let g be a Lie algebra and let ρ : g → so(R) ⊂ End(R) be a representation of g on by anti-symmetric matrices on a Euclidean vector space R, (, ). Then for any k ∈ Z we can define a degree 4k + 2 trivial Hamiltonian Q-bundle with base M = g [1] and fiber N = R[2k + 1]. Let ψ be the degree 1 g-valued coordinate on g [1] and let x be the degree 2k + 1 R-valued coordinate on R[2k + 1]. Then we set -the Chevalley-Eilenberg differential, and where , denotes the canonical pairing between a vector space and its dual. It is a straightforward check that this comprises the data of a trivial Hamiltonian Q-bundle of degree 4k + 2, according to definition 2.
As a variation of this example, we may require that the pairing (, ) is antisymmetric (so that R is a symplectic space instead of Euclidean) and that ρ : g → sp(R) ⊂ End(R) is a symplectic representation. Then formulae (31,32) define a Hamiltonian Q-bundle structure of degree 4k on g[1] × R[2k] (note that here we choose an even degree shift for the fiber).
(ii) Example above generalizes straightforwardly to the setting of L ∞ algebras and L ∞ modules. Namely, let g be an L ∞ algebra with operations {l j : ∧ j g → g} j≥1 . Also, let R be graded vector space with graded symmetric pairing (, ) of degree q and with the structure of an L ∞ module over g with the module operations {ρ j : ∧ j g ⊗ R → R} j≥0 , where we additionally require that operations (•, ρ j (· · · ; •)) are graded anti-symmetric w.r.t. the two entries in the module. For ψ a g-valued degree 1 coordinate on g [1] and x a R-valued degree 2k + 1 coordinate on R[2k + 1], we construct a trivial Hamiltonian Q-bundle with M = g [1], N = R[2k + 1] and the data are: for the base and This comprises the data of a trivial Hamiltonian Q-bundle of degree 4k +2+q. (iii) Example (ii) admits the following variation. If h is an L ∞ algebra with I ⊂ h an L ∞ ideal 10 , then there is a natural Q-bundle with bundle projection being the quotient map and with the fiber I. If in addition we fix a section of the quotient map h → h/I, i.e. a splitting of the short exact sequence and if I carries a degree q cyclic 11 inner product, then (35) becomes a Hamiltonian Q-bundle of degree 2 + q. Explicitly, the structure on the base M = (h/I) [1] is again given by (33) with {l j } the quotient L ∞ operations on h/I, ψ the degree 1 (h/I)-valued coordinate on (h/I) [1]. The structure on the fiber N = I[1] is: where

BV pre-observables and observables
In this section we recall some basic structures of BV formalism: classical and quantum BV theory (cf. [15]), observables in classical and quantum setting . We also introduce the notion of a pre-observable 12 : an extension of a BV theory by auxiliary fields endowed with an odd-symplectic structure and an action (which is also dependent on the fields of the ambient theory). Given a pre-observable, one can construct an observable by integrating out the auxiliary fields (propositions 1, 3). Pre-observables come in three versions: purely classical, semi-quantumsuited for integrating out auxiliary fields to produce a classical observable for the ambient theory, and quantum -suited for integrating out auxiliary fields to produce a quantum observable for the ambient theory.
Throughout this section in our treatment of quantum objects we are assuming that the spaces of fields are finite-dimensional. This is almost never the case in the interesting examples. In the setting of local quantum field theory, when the spaces of fields are infinite-dimensional, measures on fields are problematic to define; however the BV Laplacians and the integrals over auxiliary fields (which are perturbative path integrals now) can be made sense of with an appropriate regularization/renormalization procedure (one systematic approach is via Wilson's renormalization flow, cf. [10]). Proofs of propositions 1, 3 cannot be taken for granted in the infinite-dimensional setting, and should be redone within the framework of perturbative path integrals (cf. e.g. the proof of proposition 6 in section 6.3).
In view of this, our treatment of quantum objects in this section should be viewed as a simplified discussion, to motivate the interest in pre-observables and argue that given a pre-observable one can expect a path integral expression for an observable. 4.1. Pre-observables and observables for a classical ambient BV theory. Definition 4. We will call a (classical) BV theory a Hamiltonian Q-manifold of degree −1, i.e. a quadruple (F , Q, Ω, S) consisting of a Q-manifold (F , Q) (the space of fields with the BRST operator ), the degree −1 symplectic structure (the BV 2-form) Ω and the degree 0 action S satisfying {S, •} Ω = Q and the classical master equation: 12 This is not a part of the standard BV lore; the term is sometimes used in a totally different sense in the literature. 13 We will be assuming degree 0 condition for observables by default, but sometimes it is interesting to relax it. In such cases we will indicate the degree explicitly.
i.e. O ′ and O give the same class in Q-cohomology.
Definition 6. For a classical BV theory (F , Q, Ω, S) (which we will call the ambient theory), a pre-observable is a degree −1 Hamiltonian Q-bundle over F . We will call the fiber the space of auxiliary fields F aux , which comes with its own degree −1 symplectic structure Ω aux and the action for auxiliary fields S aux ∈ C ∞ (F × F aux ) satisfying Remark 5. Note that if in addition to (40) we have the property then the pre-observable gives a new classical BV theory Note also that this cohomological vector field on F × F aux differs from the one arising from the Q-bundle structure by the term {S aux , •} Ω and is in general not projectable to F .
In the case when F aux is finite-dimensional, it makes sense to introduce the following notion.
Definition 7. For a classical BV-theory (F , Q, Ω, S), a semi-quantum pre-observable is a quadruple (F aux , Ω aux , µ aux , S aux ) consisting of a Z-graded manifold F aux and a degree −1 symplectic form Ω aux on it. Further, • F aux is endowed with a volume element µ aux compatible with Ω aux in the sense that the associated BV Laplacian on C ∞ (F aux ), Infinitesimally, this equivalence condition can be written as In particular, a classical pre-observable can be promoted to a semi-quantum preobservable if there exists a volume element µ aux on F aux compatible with Ω aux , such that in addition to (40) we have ∆ aux S aux = 0.
Given a semi-quantum pre-observable, one can construct an observable for the ambient classical BV theory using fiber BV integrals [13] as follows.
for L ⊂ F aux a Lagrangian submanifold. Then: (ii) If Lagrangian submanifolds L and L ′ can be connected by a Lagrangian homotopy, then the difference Given two equivalent semi-quantum pre-observables S aux andS aux , the associated observables O L andÕ L are equivalent.
Proof. Applying Q to the integral (45), we obtain where we used (42) and the Stokes' theorem for BV integrals [15], that the BV integral of a ∆-coboundary vanishes. We suppress the measure in the short-hand notation for integrals here. This proves (i). Now let {L t } t∈[0,1] be a smooth family of Lagrangian submanifolds of (F aux , Ω aux ) connecting L = L 0 and L ′ = L 1 . An infinitesimal deformation of a Lagrangian submanifold L t is given by a graph of a Hamiltonian vector field 14 viewed as a section of the normal bundle of L t , with Ψ t ∈ C ∞ (L t ), |Ψ t | = −1 the generator. Hence, The term ∆ aux (Ψ t ) in the second line comes from the transformation of measure √ µ aux | Lt by the infinitesimal shift of Lagrangian (47). This calculation implies that and thus proves (ii). Item (iii) follows immediately from the construction (45), our definition of equivalence (43) and the Stokes' theorem for BV integrals: 14 A general infinitesimal Lagrangian deformation is given by graph(ι (Ω aux ) −1 χt) for any degree −1 closed 1-form χt ∈ Ω 1 (Lt). Since in non-zero degree a closed form is automatically exact (cf. remark 1), we have (47) with the generator Ψt = ι E χt.
Here we are implicitly assuming convergence of the integral (45), and for the proof of (ii) the integral over L t should converge for every t ∈ [0, 1].
The following construction allows one to produce new semi-quantum pre-observables out of old ones by partially integrating out the auxiliary fields.
Proposition 2. Given a semi-quantum pre-observable (F aux , Ω aux , µ aux , S aux ), assume that F aux is given as a product 15 , F aux = F aux z ×F aux , so that Ω aux and µ aux split: withL a Lagrangian submanifold of (F aux ,Ω aux ). Then , Ω aux z , µ aux z , S aux z ) is a semi-quantum pre-observable for the same ambient classical BV theory. (ii) The observable for the ambient theory induced from S aux z using (45) with Lagrangian L z ⊂ F aux z is equivalent to the one induced directly from S aux using Lagrangian L ⊂ F aux , provided that L can be connected with L z ×L by a Lagrangian homotopy in F aux .
Proof. Similarly to (46), we have ) e iS aux = 0 which proves (i). Item (ii) is an immediate corollary of (ii) of proposition 1.

4.2.
Pre-observables and observables for a quantum ambient BV theory.
In the case when the space of fields of the ambient theory F is finite-dimensional, it makes sense to introduce the following notions.
Definition 8 (cf. [15]). A quantum BV theory is a quadruple (F , Ω, µ, S) where (F , Ω) is a Z-graded manifold with a degree −1 symplectic form Ω; µ is a volume element on F , compatible with Ω as in definition 7, i.e. the BV Laplacian ∆ : f → 1 2 div µ {f, •} Ω satisfies ∆ 2 = 0; the degree 0 action S ∈ C ∞ (F ) is required to satisfy the quantum master equation For a quantum BV theory it is convenient to introduce a degree 1 second order differential operator which satisfies (δ BV ) 2 = 0. Given a quantum observable, one can define its correlator as where L is a Lagrangian submanifold of (F , Ω) and we are assuming convergence. By the Stokes' theorem for BV integrals, this expression does not change with Lagrangian homotopy of L: O L = O L ′ , and also correlators of equivalent observables coincide: O L = O ′ L , cf. [15]. Finally, if both F and F aux are finite-dimensional, the following definition makes sense.

Definition 10.
A quantum pre-observable for a quantum BV theory is a semiquantum pre-observable (F aux , Ω aux , µ aux , S aux ) where instead of equation (41) we require that S + S aux satisfies the quantum master equation on F × F aux : The analog of proposition 1 in the setting of quantum BV theory is as follows.  16 Here we define a quantum observable in the framework of Lagrangian field theory in BV formalism, as something that can be averaged over the space of fields to yield a correlator (49). In other contexts one has quite different definitions of quantum observables: e.g. in the framework of Atiyah's topological quantum field theory [12], an observable is a pair consisting of a submanifold γ ⊂ Σ of the spacetime manifold and a vectorÔγ in the space of states associated to the boundary of a tubular neighborhood of γ in Σ. The passage from the first picture to the second one goes through path integral quantization of the ambient topological theory on the tubular neighborhood (as on a manifold with boundary) with the insertion of observable.
BV Laplacian rescales as ∆ → ∆. For the structure on auxiliary fields, one does the same: With these redefinitions the equation (41) on semi-quantum pre-observables becomes and can be solved by obstruction theory order by order in for S aux = S aux (0) + S aux (1) + 2 S aux (2) + · · · starting from S aux (0) a solution of (40). Likewise, the BV push-forward formula (45) becomes and can be evaluated by the stationary phase formula. Note that one can also introduce two independent Planck's constants , aux for the ambient (50) and auxiliary (51) theory respectively.

AKSZ pre-observable associated to a Hamiltonian Q-bundle
Let (M, Q, ω = δα, Θ) be a Hamiltonian Q-manifold and Σ a closed oriented spacetime manifold, dim Σ = |ω| + 1. Then we have the AKSZ theory with as in section 2. Given a trivial Hamiltonian Q-bundle over M with fiber data (N , A, ω ′ = δα ′ , Θ ′ ), a closed oriented manifold γ of dimension dim γ = |ω ′ | + 1 and a smooth map 17 i : γ → Σ, we can construct a pre-observable for the AKSZ theory (52) with target M as follows. Set Here the notations are as follows.
• τ γ : Ω • (N ) → Ω • (F γ ) and τ tot is the lifting of the de Rham differential on γ to a vector field on the mapping space Map(T [1]γ, N ). • A lifted is the lifting of the vertical vector field A ∈ X vert (E) to a vertical vector field on the mapping space Map(T [1]γ, E), so that p * A lifted becomes a vertical vector field on F Σ × F γ .
Proposition 4. The data (53) defines a classical pre-observable for the AKSZ theory (52), in particular Proof. First let us check that A γ as defined in (53) is the Hamiltonian vector field for S γ . Introduce the notations for the source (kinetic) and target parts of A γ and S γ : Here δ vert stands for the de Rham differential in the fiber direction, as in section 3.3. Collecting (55,56), we get Next, let us prove (54). Note that Q Σ S kin γ = 0 since S kin γ does not depend on the ambient fields (i.e. is a function on F Σ × F γ , constant in the direction of F Σ ); also Q kin Σ S target γ = p * (Q kin Σ | γ τ tot γ (Θ ′ )) = 0 as in (12) Here the first term on the r.h.s. has ((d γ ) lifted ) 2 = 0 as its Hamiltonian vector field on F γ , thus it is a degree 1 constant function on F γ , and hence vanishes. Therefore, continuing the calculation (57) and using (13), we have since the target is a Hamiltonian Q-bundle and equation (30) holds there. This finishes the proof.
In coordinates. Alongside with the superfield (18) for the ambient theory on Σ, we now have a superfield for the auxiliary theory on γ: associated to local homogeneous coordinates y i on N .
Here v 1 , . . . , v dim γ are local coordinates on γ and ξ ν stands for dv ν , ν = 1, . . . , dim γ. Let α ′ and ω ′ be locally given on N as α ′ = α ′ i (y) δy i , ω ′ = 1 2 ω ′ ij (y) δy i ∧ δy j . Then the BV 2-form for auxiliary fields is: Its primitive is: The action for auxiliary fields is: Vertical part of the cohomological vector field A γ acts on the auxiliary superfield by if on the target we have A = A i (x, y) ∂ ∂y i . Remark 7. In the case when i(γ) ⊂ Σ is an embedded submanifold of positive codimension, the expression (60) {S γ , S γ } ΩΣ is ill-defined, because it formally contains Σ (δ γ ) 2 · · · where δ γ is the delta function supported on on i(γ) ⊂ Σ. On the other hand, if we use adapted local coordinates u 1 , . . . , u dim Σ on Σ in which i(γ) is given by u dim γ+1 = · · · = u dim Σ = 0, the auxiliary action S γ only depends on the components of the ambient superfield X a µ1···µ k with µ 1 , . . . , µ k ≤ dim γ, whereas the BV 2-form Ω Σ couples the components X a µ1···µ k and X b ν1···ν l if and only if {µ 1 , . . . , µ k } ⊔{ν 1 , . . . , ν l } = {1, . . . , dim Σ} (for the sake of this argument we choose local Darboux coordinates on M in which ω ab (x) does not depend on x). So, formally (60) is ∞ · 0, and we naively regularize it to zero. Thus, by remark 5, we have (in the sense of our regularization) the classical master equation on F Σ × F γ : In the case when i : γ → Σ is a diffeomorphism, we have Remark 8. If dim γ = 1, ω ′ is not automatically exact. The interesting case is when ω ′ is not exact but satisfies Bohr-Sommerfeld integrality condition, see remark 2. We assume that γ is a circle; the discussion extends to a disjoint union of circles trivially. The kinetic part of the action for auxiliary fields S kin γ cannot be defined as in (53), because the primitive α ′ does not exist globally on N . Instead one can define -the holonomy around γ of the pull-back of the connection ∇ ′ (cf. remark 2) by Y (0) : γ → N . Here the l.h.s. should be viewed as one symbol. Target part of the action is defined as before, so we have Otherwise, using Stokes' theorem one can rewrite the kinetic part of the auxiliary action as for D a disc bounded by γ andỸ (0) an arbitrary extension of Y (0) from γ to D. Due to integrality of ω ′ /2π, arbitrariness ofỸ (0) results in (64) being defined modulo multiples of 2π: S kin γ (Y ) ∈ R/2πZ From this point of view, the l.h.s. of (63) is well-defined.
It is easy to see that the proof of proposition 4 still works in this case (e.g. because one can find a primitive of ω ′ in the tubular neighborhood of the loop Y (0) (γ) ⊂ N ).

From pre-observables to observables
The general idea of passage from pre-observables constructed in section 5 to observables for the underlying AKSZ theory is by means of integrating out the auxiliary fields as in proposition 1, i.e.
where L is a Lagrangian submanifold of (F γ , Ω γ ). When we want to emphasize the dependence of O γ on the map i : γ → Σ, we will write O γ,i . Assuming that we can make sense of the path integral (66), the observable O γ is expected to have the following properties: (v) Invariance under ambient diffeomorphisms: for Φ ∈ Diff(Σ), we have Here (i) follows from our construction of the pre-observable (53) and (v) follows from (i). Properties (ii) and (iii) are expected to hold in view of the proposition 1 for finite-dimensional fiber BV integrals. Property (iv) formally follows from (65), (iii) and Diff(γ)-invariance of the measure on F γ : Here L DY · · · should be understood as L √ µ| L · · · where µ is the path integral measure on F γ . Note that we do use the fact that φ is an isotopy rather than a general diffeomorphism, since otherwise Lagrangians L and (φ −1 ) * L are not guaranteed to be homotopic. Property (vi) formally follows from (v), Diff(Σ)-invariance of the AKSZ action (17) and of the path integral measure on F Σ , and from the BV-Stokes' theorem: Remark 10. For our construction of observable O γ it is not a priori necessary to impose any restrictions on the map i : γ → Σ. However, if we want to make sense of the correlator O γ via perturbation theory for the path integral (which is outside of the scope of this paper), we have to require that i is an embedding. The technical reason is that if i : γ → Σ is not an embedding, it does not lift to a map between compactified configuration spaces of pairs of points on γ and Σ, Conf 2 (γ) → Conf 2 (Σ), which prevents one from defining the pull-back of the propagator of the ambient theory to γ, which in turn leads to Feynman diagrams for the correlator being ill-defined.
We will not try to give meaning to the path integral (66) in the most general situation here, but rather will discuss several special cases. 6.1. Case N = point. In the case when N is a point the target Hamiltonian Qbundle is described in example (v) in section 3.4: A = 0, ω ′ = 0, Θ ′ = θ ∈ C ∞ (M) -a Q-cocycle of degree p for the target M. The associated pre-observable (53) for a p-dimensional closed manifold γ and a map i : γ → Σ is: The passage to the observable (66) is trivial in this setting, since there are no auxiliary fields: This observable obviously satisfies properties (i-v) above.

6.2.
Case of 1-dimensional observables. Let i : γ = S 1 → Σ be a circle mapped into Σ and suppose we have a Hamiltonian Q-bundle of degree 0 over the AKSZ target (M, Q, ω = δα, Θ) with fiber data (N , A, ω ′ = δα ′ , Θ ′ ). We will assume for simplicity that N is concentrated in degree 0, i.e. (N , ω ′ ) is an ordinary (nongraded) symplectic manifold. In local coordinates, the auxiliary superfield (59) is: where u is the coordinate on γ = S 1 and index i corresponds to local coordinates on N . For the pre-observable (53) we have , One natural choice of Lagrangian in (66) Note that with this gauge fixing, one formally has a strict version of property (iv), namely that O γ is invariant under isotopy of γ (instead of only the class of O γ in Q Σ -cohomology being invariant). This follows from invariance of L under Diff(γ), together with Diff(γ)-invariance of the path integral measure DY (0) . The quantum mechanical path integral (72) can be understood in the Hamiltonian formalism, using the fiber geometric quantization of the target Hamiltonian Q-bundle, as follows.
Proposition 5. Assume that the symplectic manifold (N , ω ′ ) can be geometrically quantized to a complex vector space of states H and the Hamiltonian Θ ′ ∈ C ∞ (M× N ) can be quantized (viewing coordinates on M as external parameters) to an operator-valued function on M,Θ ′ ∈ C ∞ (M) ⊗ End(H) satisfying (the quantum counterpart of the equation (30)). Then define O γ as the trace of a path ordered exponential 18 18 To define the path ordered exponential, we need to choose a starting point p ∈ γ, but the dependence on p is cancelled by taking the trace.
Proof. Let j : [0, 1] → Σ be a path in Σ parameterized by t ∈ [0, 1]. Let us denote ψ =Θ ′ (j * X) ∈ Ω • ([0, 1])⊗C ∞ (F Σ ) ⊗ End(H), also let ψ (0) (t) and ψ (1) (t, dt) be the 0-and 1-form components of ψ. Consider the path ordered exponential where we used that due to (73), we have For j = i : S 1 → Σ, we have j(0) = j(1) and applying (76), we get Note that the construction of proposition 5 does not require ω ′ to be exact, it is enough to require that ω ′ satisfies Bohr-Sommerfeld integrality condition. 6.3. Torsion-like observables. Now let again i : γ → Σ be an oriented closed manifold of arbitrary dimension mapped into Σ and suppose that N is a finitedimensional graded vector space, ω ′ is a graded-antisymmetric pairing on N of degree dim Σ − 1 and Θ ′ is quadratic in N directions: We also assume that where Str N stands for the super-trace over N . One way to make sense of the expression (66) is to choose a Riemannian metric on γ. Then we have the Hodge decomposition for differential forms on γ with values in N F γ which can be used to first take the BV integral over the complement of harmonic forms, as in proposition 2, an then take the BV integral over the zero-modes (harmonic forms): where Y = Y z +Ỹ is the splitting of the auxiliary superfield into the harmonic part and the part in the complement; F z γ comes with the degree −1 symplectic structure Ω z γ induced from ω ′ and the Poincaré pairing on cohomology of γ; L z ⊂ F z γ is a Lagrangian vector subspace w.r.t. Ω z γ . The internal integral in (80) is a path integral and can be made sense of using perturbation theory 19 : The notations here are: • Conf k (γ) is the Fulton-Macpherson-Axelrod-Singer compactification of the configuration space of k points on γ, cf. [5].
its integral kernel extends to a smooth (dim γ − 1)-form η on Conf 2 (γ) (the Hodge propagator ), cf. [5]. By P harm we denote the orthogonal projection to harmonic forms, using the Hodge inner product. • π j+1,j : Conf k → Conf 2 is the map associated to forgetting all points except points j and j + 1; also π j : Conf k → Conf 1 = γ is forgetting all points except the point j. By convention, π k+1,k = π 1,k . • In the first term in the exponential in (81), θ(i * X) is understood as a multiplication operator on Ω • (γ) and an endomorphism of N , depending on ambient fields. Expression (id + G θ(i * X)) −1 is understood as ∞ k=0 (−1) k (G θ(i * X)) k . The BV integral over zero-modes in (80) then is: where µ z is some fixed translation-invariant volume element on F z .
Remark 11. The requirement that the integral over zero-modes in (82) should converge imposes the following restriction on L z . If {Y I Lz } are coordinates on L z and if we write then we require the even-even block (i.e. the block pairing coordinates Y I Lz of even degree with coordinates Y J Lz of even degree) of the matrix (H IJ ) to be invertible. One universal way to satisfy this condition is to choose L z ⊂ F z γ by setting all even 19 We are working modulo constants here. We are also implicitly assuming convergence of the sum over k in (81).
coordinates on F z γ to zero. With this choice, the integral over L z in (82) is (up to normalization) the Pfaffian of (H IJ ). (82) is an observable, i.e.

Proposition 6. (i) Expression
(ii) Dependence of O γ on deformations of Riemannian metric on γ and on defor- (81) and O z γ = e iS z γ : where θ actually means θ(i * X). First let us check that S z γ is a semi-quantum pre-observable, i.e. satisfies the equation (41): Using Stokes' theorem, (87) can be written as Two last terms here cancel due to the contribution of principal boundary strata of Conf k (γ), where points j and j + 1 collapse (and we use the property of the propagator that the fiber integral of η over fibers of ∂Conf 2 (γ) → γ is the constant function −(−1) dim γ 1 ∈ C ∞ (γ)). Hidden boundary strata (i.e. those corresponding to the collapse of ≥ 3 points) do not contribute due to the fact that the collapsed subgraph of the "wheel" graph either has a vertex of valence 2, and then the integral vanishes by (anti-) symmetry of η on ∂Conf 2 (γ) w.r.t. the antipodal involution, or the collapsed subgraph has an isolated vertex, and then the integral vanishes for degree reasons, cf. [6]. Since [d, G] = id − P harm and since dη = −(−1) dim γ P harm where P harm ∈ Ω dim γ (γ × γ) is the integral kernel of P harm , collecting (85,86,88) we get the semiquantum master equation (84).
To prove (ii) we first note that deformations of L z induce Q Σ -exact shifts in O γ by proposition 1.
Next we check the dependence of S z γ on metric on γ. Let us now be more careful with the space of zero-modes and define it as F z γ = N ⊗ H • (γ) -the Nvalued de Rham cohomology of γ; it comes with a metric-dependent embedding of cohomology into differential forms i : H • (γ) ֒→ Ω • (γ) which realizes cohomology classes by harmonic forms. In view of this redefinition we should now write i(Y z ) instead of Y z in (83). Now assume that we have a path in the space of Riemannian metrics on γ, g t ∈ Met(γ) with t ∈ [0, 1] the parameter. Then we have the t-dependent Hodgetheoretic inverse G t for de Rham operator, the t-dependent propagator η t and the t-dependent embedding of cohomology into differential forms i t with exact timederivatives: ; H t , ζ t and j t depend on g t and ∂ ∂t g t . Then one can introduce the "extended" versions of G, η and i: They satisfy respectively (83). Repeating our proof of (84) in the extended case we find that which, when we decomposeS z γ into the 0-form and 1-form parts w.r.t. t,S z γ = S z γ (t) + dt R z γ (t), is equivalent to semi-quantum master equation (84) for S z γ (t) plus the equation Therefore (cf. (44)) semi-quantum pre-observables S z γ (t) are equivalent for all values of the parameter t and so by item (iii) of proposition 1, dependence of the induced This finishes the proof of (ii).

Examples of observables
Here we will illustrate the construction of AKSZ pre-observables/observables of sections 5, 6 by several explicit examples. 7.1. Observables with N a point. First we give some examples of the situation described in section 6.1.
• For abelian Chern-Simons theory (22) we may take θ = cψ ∈ C ∞ (R [1]) with c ∈ R any constant. This yields by (69) a 1-dimensional observable for i : γ = S 1 ֒→ Σ O γ = e ic γ i * A (1) -the abelian Wilson loop; parameter c can be viewed as the weight of a one-dimensional representation of the gauge group R. Note that this is also an observable for the abelian BF theory (26).
• For D-dimensional abelian BF theory, take θ = cξ with c ∈ R a parameter.
The associated observable for a closed oriented submanifold i : γ ֒→ Σ of codimension 2 is • For 2-dimensional BF theory assume that g is quadratic so that we can identify g * with g and let ρ : g → End(R) be a representation. Set θ = −i log tr R f (ρ(ξ)) for f ∈ C ∞ (R) a function. The corresponding 0dimensional observable for γ ∈ Σ a point is • For 2-dimensional abelian BF theory take θ = f (ξ)ψ for f ∈ C ∞ (R). The corresponding 1-dimensional observable for i : γ = S 1 ֒→ Σ is Note that for f non-constant, O γ depends non-trivially on components of the superfields of ghost number = 0, namely on A (0) and B (1) : Thus (94) defines a map from Poisson cohomology of M in degree one to Q Σ -cohomology in degree zero. Note that the observable (91) is a special case of (93) when we take M to be g * with Kirillov-Kostant Poisson structure. Also, (92) is a special case of (94) for M = R with π = 0. 7.2. Wilson loop observable in Chern-Simons theory. Here we will combine the discussion of section 6.2 with the example (iv) of section 3.4.
Let g be a quadratic Lie algebra and let (M, ω M ) be a symplectic manifold with a Hamiltonian action of g with equivariant moment map µ : M → g * . Further, we assume that ω M satisfies Bohr-Sommerfeld integrality condition and is the curvature of a U (1)-connection ∇ on a line bundle L over M .
Then we have a degree 0 Hamiltonian Q-bundle structure on g [1] × M → g [1], with base structure given by (20) and fiber structure (38), with the correction that now we relaxed the exactness condition on ω M . Applying the construction of section 5 to this target data we get the following 1-dimensional pre-observable for Chern-Simons theory: is the superfield of the ambient Chern-Simons theory, Y = Y (0) + Y (1) is the auxiliary superfield (21), with components Y (0) , Y (1) having internal degrees 0 and −1 respectively; indices i, j, k refer to local coordinates on M .
To construct an observable out of the pre-observable (95), we impose the gauge fixing (71) which consists in setting Y (1) = 0. The path integral (72) reads Assume that there exists a Lagrangian polarization P of (M, ω M ) such that the geometric quantization of (M, ω, L, ∇, P ) yields a space of states H and components µ a ∈ C ∞ (M ) of the moment map are quantized into operatorsμ a ∈ End(H) such that the mapμ : g → End(H) sending generator T a ∈ g intoμ a is a Lie algebra homomorphism (i.e.μ is a representation of g on H). Then The corresponding 1-dimensional pre-observable for Poisson sigma model associated to (M, π) is given by F γ , Ω γ as in (95), changing M to N , and the auxiliary action is The corresponding observable is formally given (in the gauge (71)) by the path integral (72): Assume that (N, ω N ) can be geometrically quantized to a space of states H and F is quantized to an operator-valued vector fieldF ∈ End(H)⊗X(M ) satisfying (101) [π,F ] + iF ∧F = 0 Then (74) gives which can be regarded as the evaluation of the path integral (100)

7.4.
Torsion observables in Chern-Simons theory. Fix n ′ ∈ {−1, 0, 1, 2} and m ∈ Z (only the parity of m will matter for our discussion). Let again g be a quadratic Lie algebra and let ρ : g → End(R) be a representation on a vector space R with values in traceless matrices (cf. (78)). Then we have a degree n ′ Hamiltonian Q-bundle structure on with base structure (20) and fiber structure where ρ * : g → End(R * ) is the representation dual to ρ; q is the R-valued coordinate of degree m on R[m] and p is the R * -valued coordinate of degree n ′ −m on R * [n ′ −m].
The corresponding pre-observable for Chern-Simons theory on a closed oriented 3-manifold Σ, for i : γ ֒→ Σ an embedded closed oriented (n ′ + 1)-manifold, reads: k=0 q (k) and p = n ′ +1 k=0 p (k) are the auxiliary superfields corresponding to q and p; component q (k) is a R-valued k-form on γ with internal degree m− k and p (k) is a R * -valued k-form on γ of internal degree n ′ − m − k.
Pre-observable (104) can be pushed forward to zero-modes (in the sense of proposition 2) as in section 6.3: where we introduced the notation Notations G, η, π are the same as in section 6.3. We use harmonic representatives for zero-modes q z , p z in F γ . Taking the BV integral over zero-modes in (105) we obtain the observable for Chern-Simons theory on Σ: with L z ⊂ F z γ a Lagrangian subspace and µ z a translation invariant volume element on F z γ . Proposition 6 implies that O γ is Q Σ -closed and that deformations of gaugefixing induce Q Σ -exact shifts in O γ .
In case n ′ = 0, m = 1, with γ = S 1 , (107) can be evaluated explicitly 21 : One trick to simplify the calculation is to use gauge-invariance of Oγ to pick the constant representative of the gauge equivalence class of connection i * A.
where W = P exp γ i * A (1) is the holonomy of the ambient connection around γ (since this requires the choice of a starting point on γ, one can think of W as a group element defined modulo conjugation). We also denote A = log W . In (108) we used the gauge-fixing Lagrangian (71). In particular, the gauge-fixing for the integral over zero-modes is Remark 13. Observable (108) is only interesting (not identically zero) for representations ρ such that det ρ ≡ 0, i.e. ρ(x) does not have zero eigenvalue for x in an open dense subset of g. In particular, (108) is identically zero for ρ = ad the adjoint representation.
Remark 14. Observable (108) has the property that it only depends on i * A (1) and not on the other components of the ambient field. Also, it is the only representative of its class in Q Σ -cohomology which depends only on i * A, for degree reasons (since C ∞ (F Σ | γ ) is non-negatively graded; we denote F Σ | γ = Map(T [1]γ, M) the space of pull-backs of ambient fields to γ).
In case n ′ > 0, O γ defined by (107) generally depends on i * A (k) for 0 ≤ k ≤ n ′ +1. Also, there is no canonical representative of the class of O γ in Q Σ -cohomology and no canonical choice for the gauge-fixing Lagrangian L ⊂ F γ .
For n ′ = −1, convergence requirement for the integral over zero-modes (107) forces (cf. remark 11) For n ′ = 2, for γ a connected closed oriented 3-manifold with non-zero first Betti number, we may take m = 1 and choose the gauge-fixing for zero-modes as where λ is any line in H 1 (γ) and λ ⊥ is its orthogonal complement in H 2 (γ) w.r.t. the Poincaré duality. If the condition det ρ ≡ 0 as in remark 13 holds, O γ defined with zero-mode gauge-fixing (109) is well-defined on an open subset of F Σ | γ and is not identically zero.
In case of γ a rational homology sphere, H 1 (γ) = 0, construction (109) is not applicable. However, one can take Remark 15. In case n ′ = 2 and γ = Σ, i = id, property (62) does not generally hold, so (104) is a pre-observable in the sense of definition 6, but S Σ + S γ does not satisfy the classical master equation (61) on the space F Σ × F γ . On the level of observable O γ this means that Q Σ O γ = 0, but S Σ − i log O γ does not satisfy the classical master equation on F Σ .
In case n ′ = 1, for γ a connected closed oriented surface of genus g > 0, we may take m = 0 and set where λ is again any line H 1 (γ) and λ ⊥ is its orthogonal in H 1 (γ) w.r.t. the Poincaré duality. Assuming det ρ ≡ 0, with this choice of L z we produce an observable O γ of degree (2g − 2) dim R which is well-defined and non-zero on an open subset of F Σ | γ .
Expression (107) also is an observable for BF theory in dimension D. In this case n ′ should satisfy −1 ≤ n ′ ≤ D − 1. For n ′ = 2k and γ a closed oriented submanifold of Σ of dimension 2k + 1, we may take m = 2k + 1 and choose the gauge-fixing L z as For L ′ to exist, we require that for γ the combinations of Betti numbers on the r.h.s. of (111) are non-negative. Assuming det ρ ≡ 0, property (110) ensures that the integral over zero-modes in (107) exists and is non-zero in an open subset of F Σ | γ . 7.5. A codimension 2 (pre-)observable in BF theory. The following example is due to Cattaneo-Rossi [9]. Let M = g Here q is the g-valued coordinate of degree 0 on g and p is the g * -valued coordinate of degree D − 3 on g * [D − 3]; ψ and ξ are coordinates on M as in section 2.3. Given a closed oriented D-manifold Σ and a closed oriented submanifold i : γ ֒→ Σ of codimension 2, by the construction of section 5 we get a pre-observable for the BF theory on Σ associated to the Hamiltonian Q-bundle (23,112): where q = D−2 k=0 q (k) and p = D−2 k=0 p (k) are the auxiliary superfields corresponding to q and p; q (k) and p (k) are k-forms on γ with values in g and g * respectively, having internal degrees −k and D − 3 − k respectively.
Push-forward of the pre-observable (113-115) to zero-modes (in the sense of proposition 2) is evaluated as in section 6.3: where tor is as in (106), for ρ = ad the adjoint representation of g.
In the case of abelian BF theory, g = R, (116) yields (117) e iS z γ = e i γ i * B,qz which is an observable if we understand q z ∈ H • (γ) as an external parameter (the crucial point here is that by construction (117) satisfies the semi-quantum master equation (42), but the last term in (42) vanishes since S z γ depends only on q z ). Taking q z = c · 1 ∈ H 0 (γ), we obtain the observable (90).
In [9] it is shown that in the case of "long knots" R D−2 ∼ γ ⊂ Σ = R D , embeddings of R D−2 into R D with prescribed linear asymptotics, the push-forward of pre-observable (113-115) to zero-modes yields, upon fixing the values of zeromodes to q z = c ⊗ 1 ∈ g * ⊗ Ω 0 (γ) and p z = 0, an observable, generalizing (90) in non-abelian case.

Final remarks
In this paper we presented a two-step construction of classical observables in AKSZ sigma models, which consists of: (i) constructing an extension of the AKSZ theory to a BV theory on a larger space of fields using an extension of the target to a Hamiltonian Q-bundle, (ii) using the BV push-forward to the original space of fields to produce an observable. We also provided some examples, and in particular recovered the well-known Wilson loop observable in Chern-Simons theory together with its path integral representation [2] and the Cattaneo-Rossi "Wilson surface" observable in BF theory.
There are natural questions to this construction not answered here, to which we hope to return in a future publication: • Extend the zoo of explicit examples. In particular, -give more explicit examples of "Wilson loops" in Poisson sigma model, section 7.3 (which requires a solution of (98) as the input for step (i) of the construction and its quantization satisfying (101) for step (ii)) other than those mentioned in remark 12; -give an example corresponding to a non-trivial Hamiltonian Q-bundle over the target of the AKSZ sigma model. • Calculate the isotopy invariants of embeddings given by expectation values of our observables.
8.1. Extension to source manifolds with boundary. Our construction of observables possesses an extension to the case when the source manifold Σ has a boundary ∂Σ, and the submanifold γ ⊂ Σ on which the observable is supported is also allowed to have boundary, ∂γ ⊂ ∂Σ. We will outline this extension here, treating gauge theories with boundary in "BV-BFV formalism" [8]. For a more detailed example, Chern-Simons theory on a manifold with boundary with Wilson lines ending on the boundary, the reader is referred to [4].
We can also define a pre-observable in the BV-BFV context as the following data.