Factorization Algebras for Classical Bulk-Boundary Systems
aa r X i v : . [ m a t h . QA ] A ug FACTORIZATION ALGEBRAS FOR CLASSICAL BULK-BOUNDARYSYSTEMS
EUGENE RABINOVICHA
BSTRACT . We study a certain class of bulk-boundary systems in the Batalin-Vilkovisky(BV) formalism. We construct factorization algebras of observables for such bulk-boundary systems, and show that these factorization algebras have a natural Poissonbracket of cohomological degree 1. C ONTENTS
1. Introduction 11.1. Comparison to Related Work 31.2. Conventions 41.3. Outline 41.4. Acknowledgements 42. Classical Bulk-Boundary Systems 42.1. Classical Field Theories on Manifolds with Boundary 52.2. Boundary conditions and homotopy pullbacks 83. The Factorization Algebras of Observables 134. Topological Mechanics 18Appendix A. Going under the hood: functional analysis 18A.1. The topological tensor product of function spaces with boundaryconditions 18A.2. The strong topological duals of function spaces with boundary conditions 20References 221. I
NTRODUCTION
The main goal of this work is to construct, along the lines of [CG], factorizationalgebras of observables for a general class of interacting perturbative classical fieldtheories on manifolds with boundary. In a companion paper [Rab], we study the anal-ogous constructions for quantum field theories. Factorization algebras are local-to-global objects which are meant to encode the structure of observables in perturbativequantum field theory. They bear a strong relationship to the theory of E n algebrasand vertex algebras, which are objects used in the study of topological and chiral confor-mal field theories, respectively. In [CG], Costello and Gwilliam construct factorizationalgebras of observables for classical and quantum field theories on manifolds with-out boundary. They also exhibit a P ( + + free theories on manifolds with boundary. Inthe present work, we restrict to classical field theories but allow interactions. In futurework [Rab], we carry out the quantization of such interacting theories.In the version of the BV formalism of [Cos11], [CG17], and [CG], a classical per-turbative BV theory on the manifold without boundary ˚ M : = M \ ∂ M is specified bya Z -graded vector bundle E → ˚ M (whose space of sections E furnishes the space offields for the theory) together with a ( − ) -shifted fiberwise symplectic structure h· , ·i E and a cohomological degree 0 local action functional S satisfying the classical masterequation { S , S } =
0. One may easily extend E , h· , ·i E , and S to M ; however, thepresence of the boundary ∂ M makes the consideration of the classical master equa-tion more subtle, since, in almost all cases, one uses integration by parts to verify that { S , S } = M . Put in a different way, the space of fields ceases to be (–1)-shiftedsymplectic once we pass from ˚ M to M .Our solution is to impose boundary conditions on the fields (the space E ) so thatthe boundary terms obstructing the classical master equation vanish when restrictedto the fields with this boundary condition. We note that the imposition of boundaryterms is not a novel idea. What is notable about our approach, however, is that weimpose boundary conditions in a way that is consistent with the interpretation of theBV formalism in terms of the geometry of shifted symplectic spaces. Namely, we arecareful to ensure that the imposition of the boundary conditions happens in a homo-topically coherent way. The payoff is two-fold. First, we find that the space of fieldswith the boundary condition imposed is indeed (–1)-shifted symplectic, using generalfacts about shifted symplectic geometry. Second, we guarantee that our constructionsnaturally take into account the gauge symmetry of the problem—there is no need toseparately impose boundary conditions on the fields, ghosts, etc.Furthermore, we ask for our boundary conditions to be suitably local (see Defini-tion 2.19). The locality condition ensures that even after the boundary condition isimposed, the fields remain the space of global sections of a sheaf on M : this allows usto carry through the constructions of [CG17] almost without change.Let us say a bit more about the class of field theories we consider. These are thetheories which are so-called “topological normal to the boundary” (first defined in[BY16]; see also Definition 2.4). We will often use the acronym TNBFT to stand for“field theory which is topological normal to the boundary.” For a TNBFT we requirethat, roughly speaking, in a tubular neighborhood T ∼ = ∂ M × [ ε ) of the boundary ∂ M , the space of fields E (the sheaf of sections of the bundle E introduced above)admits a decomposition E ∂ | T ⊗ Ω • [ ε ) , LASSICAL BULK-BOUNDARY FACTORIZATION ALGEBRAS 3 where E ∂ is a 0-shifted symplectic space living on the boundary ∂ M , and arising, like E , from a Z -graded bundle E ∂ → ∂ M . We require all relevant structures on E todecompose in a natural way with respect to this product structure. A pair consistingof a TNBFT E and a boundary condition for E will be termed a bulk-boundary system .Though this term properly applies to a much more general class of objects, we use ithere to avoid bulky terminology. The reason for restricting our attention to TNBFTsis that such theories have a prescribed simple behavior near the boundary ∂ M . Thisbehavior makes it easier to verify the Weiss cosheaf condition for the factorizationalgebra of observables of a bulk-boundary system. This behavior also enables us tostudy heat kernel renormalization of bulk-boundary systems in the companion paper[Rab].Important examples of TNBFTs are of course the topological theories: BF theory,Chern-Simons theory, the Poisson sigma model, and topological mechanics. How-ever, even for topological theories, we may consider boundary conditions which arenot topological in nature. For example, one may study in Chern-Simons theory thechiral Wess-Zumino-Witten boundary condition, whose definition requires the choiceof a complex structure on the boundary surface. One may also study theories whichhave a dependence on arbitrary geometric structures on the boundary ∂ M , as longas their dependence in the direction normal to ∂ M is topological. Mixed BF theory,Example 2.11 is one such example; it depends on the choice of a complex structure onthe boundary of a three manifold of the form N × R ≥ , where N is a surface.Butson and Yoo provide another large source of examples of TNBFTs: the so-called“universal bulk theories.” These are TNBFTs associated to “degenerate field theo-ries” (degenerate in the sense that they are specified by shifted Poisson geometry asopposed to shifted symplectic geometry). The universal bulk theory has a canoni-cal boundary condition encoding the boundary degenerate field theory. For example,the Poisson sigma model on H (the upper half plane) is the universal bulk theory forPoisson topological mechanics on R . A general degenerate field theory F may begeometric in nature, i.e. it may depend on a metric or other geometric data on N .Hence, the factorization algebras we construct in this paper may be used to probe ar-bitrary degenerate field theories. For the sake of brevity, we do not discuss universalbulk theories at any length; however, the techniques we use here for bulk-boundarysystems apply equally well to universal bulk theories (with their canonical boundarycondition).1.1. Comparison to Related Work.
We are aware of a number of other works dis-cussing the quantization of gauge theories on manifolds with boundary using thetechniques of homological algebra. Among them are [CMR18], [MSW19], [BCQZ19],[MSTW19]. Our approach has a number of features in common with these references.For example, all of the mentioned references use the Batalin-Vilkovisky formalismand study what happens when the classical master equation fails to be satisfied. In[MSW19], the possibility of operators which are not simply integrals over all of space-time is introduced. In [MSTW19], the authors impose boundary conditions via ho-motopy pullbacks. These ideas are present also in this work; however, by restricting
EUGENE RABINOVICH to the class of TNBFTs, we enable a simple construction of factorization algebras ofclassical observables.1.2.
Conventions.
Throughout, the spacetime manifold will be denoted M and itsboundary ∂ M . The inclusion ∂ M → M will be denoted ι . If N is a tubular neighbor-hood of ∂ M in M with a specified diffeomorphism N ∼ = ∂ M × [ ε ) , then we will use t and s exclusively to denote the normal coordinate in N . • Given a manifold M , Dens M is the bundle of densities on M and Ω nM , tw is itsheaf of sections. • In the sequel, whenever we use a normal-font letter (e.g. E ) for a bundle on M ,we use the script version of that latter (e.g. E ) for the corresponding sheaf ofsections. This contrasts slightly with our usage in the introduction, where weused the script letter for the space of global sections of the bundle.1.3. Outline.
In Section 2, we introduce TNBFTs, as well as the class of local boundaryconditions we consider. We show that the imposition of such boundary conditions onTNBFTs is consistent with locality and gauge symmetry (Lemma 2.26). In Section 3,we construct the P factorization algebra of classical observables for a bulk-boundarysystem. In the Appendix, we discuss some functional analytic details necessary in themain body of the paper.1.4. Acknowledgements.
This work is the author’s own interpretation of researchconducted with Benjamin Albert. The author would like to thank Benjamin for manydiscussions on the subject.The author would also like to thank Dylan Butson, Kevin Costello, Owen Gwilliam,Si Li, Peter Teichner, Brian Williams, and Philsang Yoo for the many discussions re-lated to the work presented here.This material is based upon work supported by the National Science FoundationGraduate Research Fellowship Program under Grant No. DGE 1752814.2. C
LASSICAL B ULK -B OUNDARY S YSTEMS
In this section, we introduce the basic classical field-theoretic objects with whichwe work. We use the Batalin-Vilkovisky (BV) formalism as the natural language forquantum field theory, which encodes both the equations of motion and the symmetriesof a field theory in an intrinsically homotopically invariant fashion. The theories weconsider are called “topological normal to the boundary” (TNBFT). The definition weuse here is due to Butson and Yoo [BY16]. For a manifold with boundary M , a TNBFTis a field theory on ˚ M (the interior of M ) which has a specified behavior of the fieldtheory near the boundary. As the name suggests, the solutions to the equations ofmotion of a TNBFT are constant along the direction normal to the boundary ∂ M insome tubular neighborhood T ∼ = ∂ M × [ ε ) of the boundary.In Section 2.2, we introduce boundary conditions for TNBFTs and discuss the ho-motopical interpretation of the resulting bulk-boundary systems. LASSICAL BULK-BOUNDARY FACTORIZATION ALGEBRAS 5
Classical Field Theories on Manifolds with Boundary.
Before elaborating onthe definition of a TNBFT, let us first recall what is meant by a perturbative classicalfield theory on ˚ M . The definition here follows [CG]. Definition 2.1.
A perturbative classical BV theory on ˚ M is the information of(1) A Z -graded vector bundle E → ˚ M ,(2) Sheaf maps ℓ k : ( E [ − ]) ⊗ k → E [ − ] of degree 2 − k ;(3) a degree − h· , ·i loc : E ⊗ E → Dens M ;This information is required to satisfy the following properties: • The maps ℓ k turn E [ − ] into a sheaf of L ∞ algebras, • The maps ℓ k are polydifferential operators (we will call the pair ( E [ − ] , ℓ k ) a local L ∞ algebra on ˚ M ; • The complex ( E , ℓ ) is elliptic, • The pairing h· , ·i loc is fiberwise non-degenerate, • Let us denote by h· , ·i the induced map of precosheaves E c ⊗ E c → C given by using the fiberwise pairing h· , ·i loc , and then integrating the re-sult. (Here, C is the constant pre-cosheaf assigning C to each open subset.)We use the same notation for the pairing induced on E c [ − ] . We requirethat, endowed with this pairing, E [ − ] becomes a precosheaf of cyclic L ∞ algebras (on E c [ − ] , the pairing has degree –3). Remark : One can extract a BV-BFV theory ([CMR18]) from a classical TNBFT, using E as the bulk space of fields and E ∂ as the boundary fields. The assumptions in thedefinition guarantee that the general procedure of symplectic reduction in [CMR18]outputs E ∂ as the phase space for E . ♦ Remark : The ellipticity of the complex ( E , ℓ ) is not a necessary requirement fromthe standpoint of physics, and in fact, there is no need for this requirement in thedefinition of a classical BV theory. We include it here because the theory of ellipticPDE furnishes a wealth of tools which make it possible to develop a framework forrenormalization. This latter point will be clearer in the companion work [Rab]. ♦ We now specify the precise definition of a TNBFT. The following definition is adaptedfrom Definitions 3.8 and 3.9 of [BY16].
Definition 2.4. A field theory on M which is topological normal to the boundary is specified, as in Definition 2.1, by a Z -graded bundle E → M , a collection of sheafmaps ℓ k , and a bundle map h· , ·i loc . We also specify the following data:(1) A Z -graded bundle E ∂ → ∂ M , EUGENE RABINOVICH (2) A collection of sheaf maps ℓ k , ∂ : ( E ∂ [ − ]) ⊗ k → E ∂ [ − ] (3) a degree 0 bundle map h· , ·i loc , ∂ : E ∂ ⊗ E ∂ → Dens ∂ M .(4) In some tubular neighborhood T ∼ = ∂ M × [ ε ) of ∂ M , an isomorphism φ : E | T ∼ = E ∂ ⊠ Λ • T ∗ [ ε ) .We require these data to satisfy • The k -th cohomological degree part E k of E is zero for | k | >> • When E , ℓ k , and h· , ·i loc are restricted to ˚ M , the resulting data satisfy the condi-tions to be a classical BV theory on ˚ M , • the data ( E ∂ , ℓ k , ∂ , h· , ·i loc , ∂ ) satisfy all the requirements of Definition 2.1 (withthe degree − h· , ·i loc replaced by the degree 0 pairing h· , ·i loc , ∂ ), • The isomorphism φ respects all relevant structures. More precisely, we require – Over T , the induced isomorphism ϕ : E [ − ] | T ∼ = E ∂ [ − ] ⊗ Ω • [ ε ) is an isomorphism of (sheaves of) L ∞ algebras. Here, the target of ϕ hasthe L ∞ -algebra structure induced from its decomposition as a tensor prod-uct of an L ∞ algebra with a commutative differential graded algebra. – Over T , the fiberwise pairing h· , ·i loc is identified with the tensor productof the pairing h· , ·i loc , ∂ and the wedge product pairing V on Λ • T ∗ [ • ) .Before listing examples of TNBFTs, let us mention a few of their properties whichfollow directly from the definitions. First, however, we need to establish a definition: Definition 2.5.
Let ι : ∂ M → M denote the inclusion. Given a TNBFT ( E , E ∂ , · · · ) , the canonical submersion is the composite sheaf map E → ι ∗ E ∂ which arises as the composite of(1) The restriction E ( U ) → E ( U ∩ T ) ,(2) the isomorphism E ( U ∩ T ) ∼ = ( E ∂ ⊗ Ω • [ ε ) )( U ∩ T ) ,(3) and the “pullback to t =
0” map ( E ∂ ⊗ Ω • [ ε ) )( U ∩ T ) → E ∂ ( U ∩ ∂ M ) . Remark : Occasionally, when we wish to emphasize the interpretation of E (respec-tively, E ∂ ) as formal moduli spaces (as in [CG]), we will refer to E (respectively, E ∂ ) asa local, (–1)-shifted (respectively, 0-shifted) formal moduli problem . ♦ The following facts follow in a straightforward manner from the definitions.
LASSICAL BULK-BOUNDARY FACTORIZATION ALGEBRAS 7
Proposition 2.7. (1) The canonical surjection ρ , when thought of as a map E [ − ] → ι ∗ E ∂ [ − ] ,is a strict map of sheaves of L ∞ algebras on M (i.e. it strictly intertwines theoperations ℓ k and ℓ k , ∂ ).(2) The only failure of ( E c [ − ] , ℓ k , h· , ·i ) to be a precosheaf of cyclic L ∞ algebras on M is encoded in the equation(2.1) h ℓ e , e i + ( − ) | e | h e , ℓ e i = h ρ e , ρ e i ∂ ,where e and e are compactly-supported sections of E → M (the support ofsuch sections may intersect ∂ M ). Remark : We avoid interpreting TNBFTs in terms of local action functionals solv-ing the classical master equation because of the subtleties which arise in defining thePoisson bracket of local functionals on manifolds with boundary. These subtleties willdisappear once we impose boundary conditions. ♦ Let us move on to discuss some examples of TNBFTs.
Example : Let M = [ a , b ] , and V a symplectic vector space. For E we take Ω • M ⊗ V with the de Rham differential and the “wedge-and-integrate” pairing. It is straightfor-ward to verify that E is a TNBFT, with E ∂ = V ⊕ V . This theory is called topologicalmechanics . ♦ Example : Let M be an oriented n -manifold with boundary ∂ M ; let g be a finite-rankLie algebra. (Topological) BF theory on M has space of fields E = Ω • M ⊗ g [ ] ⊕ Ω • M ⊗ g ∨ [ n − ] .The L ∞ structure on E [ − ] is the natural such structure obtained from considering E [ − ] as the tensor product of the differential graded commutative algebra Ω • M withthe graded Lie algebra g ⊕ g [ n − ] . The pairing h· , ·i loc is given by wedge product offorms, followed by projection onto the top form degree.The boundary data are given similarly by E ∂ = Ω • ∂ M ⊗ g [ ] ⊕ Ω • ∂ M ⊗ g ∨ [ n − ] with analogous L ∞ structure and pairing h· , ·i loc , ∂ .In general, if g is an L ∞ algebra, the same definitions can be made. Furthermore,if M is not orientable, one can make the same definitions by replacing the g ∨ -valuedforms with g ∨ -valued twisted forms. ♦ Example : Let Σ be a Riemann surface and g a Lie algebra. Mixed BF theory is atheory on Σ × R ≥ whose space of fields is E = Ω • Σ ⊗ Ω • R ≥ ⊗ g [ ] ⊕ Ω • Σ ⊗ Ω • R ≥ ⊗ g ∨ ;the differential on the space of fields is ¯ ∂ + d dR , t (where t denotes the coordinate on R ≥ ); the only non-zero bracket of arity greater than 1 is the two-bracket, which arisesfrom the wedge product of forms and the Lie algebra structure of g (and its action on EUGENE RABINOVICH g ∨ ). One defines h· , ·i loc in an analogous way to the corresponding object in topologicalBF theory.The boundary data are given by the space of boundary fields E ∂ = Ω • Σ ⊗ g [ ] ⊕ Ω • Σ ⊗ g ,with similar definitions for the brackets and pairing. ♦ Example : Let M be an oriented 3-manifold with boundary, and g a Lie algebra witha symmetric, non-degenerate pairing κ . Chern-Simons theory on M is given by spaceof fields E = Ω • M ⊗ g [ ] ; E [ − ] has the L ∞ structure induced from considering it as the tensor product of thecommutative differential graded algebra Ω • M with the Lie algebra g . The pairing h· , ·i loc uses the wedge product of forms and the invariant pairing g .The boundary data are encoded in the boundary fields E ∂ = Ω • ∂ M ⊗ g [ ] ,with brackets and pairing h· , ·i loc , ∂ defined similarly to the analogous structures on thebulk fields E .We note that, on manifolds of the form Σ × R ≥ , Chern-Simons theory is a defor-mation of mixed BF theory (see [ACMV17] or [GW19] for details). ♦ Remark : Most of the theories we consider are fully topological in nature (topo-logical BF theory, Chern-Simons theory, the Poisson sigma model, and topologicalmechanics are all of this form). However, we will see that even for topological the-ories, we can choose a non-topological boundary condition. This is notably true forChern-Simons theory. ♦ Remark : As noted in the introduction, one of the main sources of examples ofTNBFTs is the class of so-called “degenerate” BV theories ([BY16]). Every degeneratetheory on a manifold N gives rise to a (“non-degenerate”) TNBFT on N × R ≥ , whichButson and Yoo call the “universal bulk theory”. We refer the reader to [BY16], Defi-nitions 2.34 and 3.18 for the definitions of these notions, since we do not use them atany length. We note only that BF theory, mixed BF theory, the Poisson sigma model,and Chern-Simons theory (on spaces of the form N × R ≥ ) arise in this way. ♦ Boundary conditions and homotopy pullbacks.
Boundary conditions.
Equation (2.1) tells us that the space of fields E of a TNBFTon M is not ( − ) -shifted symplectic. However, we do have a map ρ : E ( M ) → E ∂ ( ∂ M ) and E ∂ ( ∂ M ) has a 0-shifted symplectic structure. We will construct a Lagrangianstructure on the map ρ , using Equation (2.1). If we are given another Lagrangian L ( M ) → E ∂ ( M ) , then by Example 3.2 of [Cal14], we expect that the homotopy fiberproduct E ( M ) × h E ∂ ( ∂ M ) L ( M ) have a ( − ) -shifted symplectic structure. We will callthis second Lagrangian L ( M ) a boundary condition . The purpose of this section is tocarry out this general philosophy more precisely, and in a sheaf-theoretic manner on M . LASSICAL BULK-BOUNDARY FACTORIZATION ALGEBRAS 9
Definition 2.15.
Let F be the sheaf of sections of a bundle F → M and supposethat F [ − ] is a local L ∞ -algebra on M with brackets ℓ i . Suppose that G is a 0-shiftedsymplectic local formal moduli problem on ∂ M (i.e. G satisfies the axioms that E ∂ does in Definition 2.4), and ρ : F [ − ] → ι ∗ G [ − ] a map of sheaves with the followingproperties: • ρ intertwines the L ∞ (strictly), and • ρ is given by the action of a differential operator acting on F , followed byevaluation at the boundary, followed by a differential operator C ∞ ( ∂ M , F | ∂ M ) → G .Suppose h F : F ⊗ F → Dens M .is a bundle map. We write h loc for the induced map h loc : F ⊗ F → Ω nM , tw and h for the induced pairing on compactly-supported sections of F .(1) The pairing h is a constant local isotropic structure on ρ precisely if(2.2) h ( ℓ f , f ) + ( − ) | f | h ( f , ℓ f ) = h ρ f , ρ f i G ,where h· , ·i G is the symplectic pairing on G , and(2.3) h ( ℓ k ( f , · · · , f k ) , f k + ) ± h ( f , ℓ k ( f , · · · , f k + )) = k > Ψ : Cone ( ρ ) → F ∨ c , Ψ ( f , g )( f ′ ) = h ( f , f ′ ) − ω ( g , ρ ( f ′ )) .We say that the isotropic structure is Lagrangian (i.e. that there is a constantlocal Lagrangian structure on ρ ) if this map of complexes of sheaves is a quasi-isomorphism. Here, the symbol ∨ denotes the sheaf which, to an open U ⊂ M ,assigns the strong continuous dual to F c ( U ) . In other words, F ∨ c is the sheafof distributional sections of F ! . Remark : Let us give a more geometric interpretation of the definition of isotropicstructure. To this end, let g and h be L ∞ algebras and ρ : g → h a strict L ∞ map. Thealgebras g and h define formal moduli spaces B g and B h . Let us suppose that B h is0-shifted symplectic with symplectic form h· , ·i h . Then, an isotropic structure on themap ρ is an element h ∈ Ω cl B h such that(2.4) Q TOT h = ρ ∗ (cid:16) h· , ·i h (cid:17) ,where Q TOT is the total differential on Ω cl B h , which includes a Chevalley-Eilenbergterm and a de Rham term. If one requires that both h· , ·i h and h be constant on B h (i.e.are specified by elements of Λ ( h [ ]) ∨ ), then one obtains precisely Equations (2.2) and(2.3) from Equation (2.4). ♦ Remark : Definition 2.15 is an adaptation to the setting of formal derived algebraicgeometry of the corresponding definitions for derived Artin stacks given in Section2.2 of[PTVV13]. ♦ Lemma 2.18.
The pairing h· , ·i E endows the map E → E ∂ with a constant local La-grangian structure. Proof.
Equation (2.1) implies Equation (2.2), while the invariance of h· , ·i E under thehigher ( k >
1) brackets implies Equation (2.3). Hence h· , ·i E gives a constant localisotropic structure on ρ . We need only to check that the induced map of sheaves ( E [ ] ⊕ ι ∗ E ∂ , ℓ + ℓ ∂ ± ρ ) → E ∨ c is a quasi-isomorphism of sheaves. We prove this as follows: first, we note that for anopen U ⊂ M which does not intersect ∂ M , we are studying the map E [ ]( U ) → E ! ( U ) → E c ( U ) ∨ ,where the first map is induced from the isomorphism E [ ] → E ! arising from h· , ·i E .The composite map is a quasi-isomorphism because the first map is an isomorphismand the second map is the quasi-isomorphism of the Atiyah-Bott lemma (see, e.g.,Appendix D of [CG17]).Now, let U ∼ = U ′ × [ δ ) , where U ′ is an open subset of ∂ M . We will show that both ( E ( U )[ ] ⊕ E ∂ ( U ′ ) , Q cone ) and E ∨ c are acyclic. Let’s start with the latter complex. Write e = e + e dt for e ∈ E c ( U ) , and consider the map K : E c ( U )[ − ] → E c ( U ) K ( e )( t ) = ( − ) | e | + Z δ t e ( s ) ds ;one verifies that K is a contracting homotopy for E c ( U ) : ℓ K ( e ) = e ∧ dt + ( − ) | e | + Z δ t ℓ ∂ e ( s ) dsK ℓ ( e )( t ) = − e ( ε ) + e ( t ) + ( − ) | e | Z δ t ℓ ∂ e ( s ) ds ,so that ℓ K + K ℓ = id . It follows that E c ( U ) , and hence E ∨ c ( U ) , is acyclic. Next,consider ( E ( U )[ ] ⊕ E ∂ ( U ′ ) , Q cone ) . There is a natural map φ : E ∂ ( U ′ ) → E ( U ) ∼ = E ∂ ( U ′ ) ⊗ Ω • [ ε ) ([ δ )) induced from the map C → Ω • [ ε ) ([ δ )) . Let C denote themapping cone for the identity map on E ∂ ( U ′ ) , and define the map Φ : C → cone ( ρ )( U ) Φ ( α , α ) = ( φ ( α ) , α ) .It is straightforward to check that Φ is a quasi-isomorphism. Hence, Cone ( ρ )( U ) isacyclic. As a consequence, we have shown that, for every point x ∈ M , and anyneighborhood V of x , we have a neighborhood U ⊂ V of x on which the sheaf map LASSICAL BULK-BOUNDARY FACTORIZATION ALGEBRAS 11 under study is a quasi-isomorphism. This map is therefore a quasi-isomorphism ofcomplexes of sheaves. (cid:3)
We would now like to choose another Lagrangian L → E ∂ , so that the homotopypullback L × h E ∂ E is ( − ) -shifted. To make this precise, we need to choose a modelcategory in which to take the homotopy pullback, and we need to introduce an ap-propriate class of Lagrangians L which we will study. The following definition isintended to fulfill the latter aim. Definition 2.19.
Let E be a TNBFT. A local boundary condition for E is a subbundle L ⊂ E ∂ endowed with brackets ℓ L , i making L [ − ] into a local L ∞ algebra satisfyingthe following properties: • The induced map L → E ∂ of sheaves of sections on ∂ M intertwines (strictly)the brackets, • h· , ·i E ∂ is identically zero on L ⊗ L , and • there exists a vector bundle complement L ′ ⊂ E ∂ on which h· , ·i E ∂ is also zero.Such data are considered a local boundary condition because the map L → E ∂ arisesfrom a bundle map L ⊂ E ∂ . Since we have no need for boundary conditions whichare not of this form, we will use the term “boundary condition” when we mean “localboundary condition.” Example : Recall from Example 2.9 that a symplectic vector space V provides aTNBFT on R ≥ . It is straightforward to check that a Lagrangian subspace L ⊂ V = E ∂ gives a local boundary condition for this theory. ♦ Example : One can define two boundary conditions for BF theory (Example 2.10):the space of boundary fields is E ∂ = Ω • ∂ M ⊗ g [ ] ⊕ Ω • ∂ M , tw ⊗ g ∨ [ n − ] ; we may takeeither of these two summands as a boundary condition. We will call the former bound-ary condition the A condition and the latter the B condition. ♦ Example : For Chern-Simons theory on an oriented 3-manifold M , the sheaf ofboundary fields is Ω • ∂ M ⊗ g [ ] ;let us choose a complex structure on ∂ M . Then, it is straightforward to verify that Ω • ∂ M ⊗ g gives a local boundary condition, the chiral Wess-Zumino-Witten boundarycondition . The Chern-Simons/chiral Wess-Zumino-Witten system for abelian g is thecentral example of [GRW20]. ♦ The L -conditioned fields. Definition 2.23.
Given a TNBFT ( E , E ∂ , h· , ·i ) and a boundary condition L ⊂ E ∂ , the space of L -conditioned fields E L is the complex of sheaves E L ( U ) : = { e ∈ E ( U ) | ρ ( e ) ∈ ( ι ∗ L )( U ) } of fields in E satisfying the boundary condition specified by L . In other words, E L is the (strict) pullback E × ι ∗ E ∂ ( ι ∗ L ) taken in the category of presheaves of complexeson M . A bulk-boundary system is a TNBFT together with a boundary condition. Remark : The term “bulk-boundary system” is appropriate for a much more gen-eral class of field theoretic information. However, since we study only this specifictype of bulk-boundary system, we omit qualifying adjectives from the terminology. ♦ Lemma 2.25.
The brackets on E [ − ] descend to brackets on E L [ − ] , and ( E L , ℓ i , h· , ·i E ) forms a classical BV theory in the sense of [Cos11], except that E L is not the sheaf ofsections of a vector bundle on M . Proof.
The first statement follows from the fact that E L [ − ] is a pullback of sheaves of L ∞ -algebras on M . The only thing that remains to be verified is that the the pairing h· , ·i E is cyclic with respect to the brackets ℓ i , once we restrict to E L . By our assump-tions, the only failure of the brackets to be cyclic for E is captured in Equation (2.1).Upon restriction to E L , however, the boundary term in that equation vanishes. (cid:3) The previous lemma shows that the pullback E × ι ∗ E ∂ ( ι ∗ L ) in the category of presheavesof shifted L ∞ algebras on M has a ( − ) -shifted symplectic structure. In the rest of thissection, we explain why this is not an accident.As we have noted in Lemma 2.18, the map E → ι ∗ E ∂ is a Lagrangian map. More-over, the map L → E ∂ is also Lagrangian by assumption. We should therefore expectthe homotopy pullback E × h ι ∗ E ∂ ( ι ∗ L ) (in an appropriate model category) to have a ( − ) -shifted symplectic structure. The space of L -conditioned fields E L is a priori only the strict pullback. However, on Lemma 2.26 below, we will show that E L isindeed a model for this homotopy pullback.Let us first, however, describe the model category in which we would take the ho-motopy fiber product. The sheaves E [ − ] , ι ∗ E ∂ [ − ] , ι ∗ L [ − ] are presheaves of L ∞ -algebras on M . In [Hin05] (Theorem 2.2.1), a model structure on the category of suchobjects is given. The weak equivalences in this model category are those which inducequasi-isomorphisms of complexes of sheaves after sheafification, and the fibrations f : M → N are the maps such that f ( U ) : M ( U ) → N ( U ) is surjective (degree bydegree) for each open U and such that for any hypercover V • → U , the correspondingdiagram M ( U ) ˇ C ( V • , M ) N ( U ) ˇ C ( V • , N ) is a homotopy pullback. This information about the model category is enough to showthe following lemma: Lemma 2.26.
In the model category briefly described in the preceding paragraph, thespace of L -conditioned fields E L [ − ] is a model for the homotopy pullback ( E [ − ]) × h ι ∗ E ∂ [ − ] ( ι ∗ L [ − ]) of presheaves of L ∞ -algebras on M . LASSICAL BULK-BOUNDARY FACTORIZATION ALGEBRAS 13
Proof.
We first note the following: E satisfies ˇCech descent for arbitrary covers in M ,and E ∂ , L do the same on ∂ M , by Lemma B.7.6 of [CG17]. Because the ˇCech com-plex for ι ∗ L (resp. ι ∗ E ∂ ) with cover { U α } is identically the ˇCech complex for L (resp. E ∂ ) with cover { U α ∩ ∂ M } , L and E ∂ satisfy ˇCech descent as presheaves on M . ByTheorem 7.2.3.6 and Proposition 7.2.1.10 of [Lur09], these presheaves satisfy descentfor arbitrary hypercovers. (Strictly speaking, the cited results are only proved forpresheaves of simplicial sets on M ; however, using the boundedness of E stipulatedin Definition 2.4, we can shift all objects involved to be concentrated in non-positivedegree and then use the Dold-Kan correspondence to show that hyperdescent anddescent coincide for presheaves of (globally) bounded complexes.)We claim that the map E → ι ∗ E ∂ is a fibration, whence the lemma follows immedi-ately. It is manifest that the maps E ( U ) → E ∂ ( U ∩ ∂ M ) are surjective for every open U ⊂ M . So, it remains to check that the square E ( U ) ˇ C ( V • , E ) E ∂ ( U ) ˇ C ( V • , E ∂ ) is a homotopy pullback square of complexes for any hypercover V • . This is true be-cause the above square is the outer square of the diagram E ( U ) ˇ C ( V • , E ) ˇ C ( V • , E ) ˇ C ( V • , E ) ˇ C ( V • , E ∂ ) ˇ C ( V • , E ∂ ) E ∂ ( U ) ˇ C ( V • , E ∂ ) ∼ ∼ idid ∼ ∼ ;all the diagonal maps in the diagram are quasi-isomorphisms, and the inner square isclearly a homotopy pullback square. (cid:3)
3. T HE F ACTORIZATION A LGEBRAS OF O BSERVABLES
In this section, given a bulk-boundary system ( E , L ) , we construct a factorizationalgebra Obs cl of classical observables for the bulk-boundary system on M . Obs cl will be constructed as “functions” on E L . Obs cl has the advantage of being easyto define; however, it does not manifestly carry the P (shifted Poisson) structurethat one expects to find on the space of functions on a (–1)-shifted symplectic space. Hence, we construct also a P factorization algebra ^ Obs cl and a quasi-isomorphism ^ Obs cl → Obs cl . We closely follow [CG].For the definition of a factorization algebra, we refer the reader to Definitions 3.1.2and 6.1.4 of [CG17]. Remark : The implicit functional analytic context in which we work is that of differ-entiable vector spaces (Appendix B.2 of [CG17]). The category of differentiable vectorspaces is abelian, by contrast with the category of locally convex topological vectorspaces. There is, however, a functor di f : LCTV S → DV S from the category of locally convex topological vector spaces to the category of dif-ferentiable vector spaces. All of the differentiable vector spaces we consider will bein the image of this functor, so we will present all of the relevant differentiable vec-tor spaces as topological vector spaces. There is one important subtlety, however: di f does not preserve colimits, and hence it does not preserve quasi-isomorphismsof chain complexes in general. However, di f does preserve homotopy equivalences,so that if f : V → W is part of a homotopy equivalence of chain complexes inLCTVS, di f ( f ) will also be part of a homotopy equivalence in DVS. We will there-fore “pretend” that we are working in the category LCTVS, making sure to only ob-tain quasi-isomorphisms via homotopy equivalences and general properties of abeliancategories. ♦ Definition 3.2.
Let ( E , L ) be a bulk-boundary system. DefineObs cl ( U ) : = (cid:16) [ Sym ( E ∨ L ( U )) , d CE (cid:17) = ∏ k ≥ Sym k ( E ∨ L ( U )) , d CE ! ,where the symmetric algebra is taken with respect to the completed projective ten-sor product of nuclear topological vector spaces. Here, d CE denotes the Chevalley-Eilenberg differential constructed from the structure of L ∞ algebra on E L ( U )[ − ] .Given disjoint open subsets U , . . . , U k ⊂ M all contained in an open subset V ⊂ M ,define maps m VU , ··· , U k : Obs cl ( U ) ⊗ · · · ⊗ Obs cl ( U k ) → Obs cl ( V ) as the composite mapsObs cl ( U ) ⊗ · · · ⊗ Obs cl ( U k ) → [ Sym ( ⊕ i E ∨ L ( U i )) ∼ = Obs cl ( U ⊔ · · · ⊔ U k ) → Obs cl ( V ) ,where the first map is a version of the natural isomorphism Sym ( A ⊕ B ) ∼ = Sym ( A ) ⊗ Sym ( B ) , and the last map is induced from the extension of compactly-supported dis-tributions E ∨ L ( ⊔ i U i ) → E ∨ L ( V ) . Remark : The construction here is almost identical to that of [CG]. We simply imposeboundary conditions on the fields of interest. ♦ Theorem 3.4.
The differentiable vector spaces Obs cl ( U ) , together with the structuremaps m VU ,..., U k , form a factorization algebra. LASSICAL BULK-BOUNDARY FACTORIZATION ALGEBRAS 15
Proof.
The proof is very similar to the analogous proof in [CG]. The proof that theclassical observables form a prefactorization algebra is identical to that in [CG]. Forthe Weiss cosheaf condition, it suffices, as in [CG], to check the condition for freetheories. A similar situation arises in [GRW20], though there one uses Sym ( E L , c [ ]) for the classical observables. By the same arguments as in the proof of Theorem 3.2 of[GRW20], we need only to show that, given any Weiss cover U = { U i } i ∈ I of an opensubset U ⊂ M , the map(3.1) ∞ M n = M i , ··· , i n Sym m (cid:0) E ∨ L ( U i ∩ · · · ∩ U i n ) (cid:1) [ n − ] → Sym m ( E ∨ L ( U )) is a quasi-isomorphism, where the left-hand side is endowed with the ˇCech differen-tial.To this end, consider the following commuting diagram (recall that we are assuming E is a free theory, i.e. can be described by an elliptic complex):(3.2) L ∞ n = L i , ··· , i n Sym m ( E L , c [ ]( U i ∩ · · · ∩ U i n )) [ n − ] Sym m ( E L , c [ ]( U )) L ∞ n = L i , ··· , i n Sym m ( E ∨ L ( U i ∩ · · · ∩ U i n )) [ n − ] Sym m ( E ∨ L ( U )) The vertical maps are quasi-isomorphisms, by Proposition A.1 of [GRW20]. In theproof of Theorem 3.2 of [GRW20], it is shown that the top horizontal map is a quasi-isomorphism. Hence, the bottom horizontal map is also a quasi-isomorphism. (cid:3)
In [CG], Theorem 6.4.0.1, it is also shown that Obs cl possesses a sub-factorizationalgebra ^ Obs cl which has a P structure. We have the same situation here. Let us firstmake a definition: Definition 3.5.
Let I ∈ Sym k ( E ∨ L ( U )) with k ≥ I induces a mapSym k − ( E ∨ L ( U )) → E ∨ L ( U ) ;we say that I has smooth first derivative if this map has image in E L , c [ ]( U ) ⊂ E ∨ L ( U ) . We consider this condition to be vacuously satisfied when k =
0. Given J ∈ ∏ k ≥ Sym k ( E ∨ L ( U )) , we say that J has smooth first derivative if each of its Taylorcomponents does. Theorem 3.6.
Let ^ Obs cl ( U ) denote the subspace of Obs cl ( U ) consisting of functionalswith smooth first derivative.(1) ^ Obs cl is a sub-factorization algebra of Obs cl .(2) ^ Obs cl posseses a Poisson bracket of degree + ^ Obs cl → Obs cl is a quasi-isomorphism. Proof.
Again, the proof follows [CG]. The proof that ^ Obs cl is a sub-prefactorizationalgebra of Obs cl is identical to the one in [CG]. One point requires comment, however: ] Obcl is closed under the Chevalley-Eilenberg differential on Obs cl because ( E L , c [ − ] , ℓ l , h· , ·i ) is a precosheaf of cyclic L ∞ algebras. Indeed, any functional with smooth first deriva-tive is of a symmetrized sum of functionals of the form I ( e , . . . , e k ) = h D ( e , · · · , e k − ) , e k i ,where D : E ⊗ ( k − ) L → E L , c [ ] is a continuous map. One can check directly that apply-ing the Chevalley-Eilenberg differential to such a functional gives another such func-tional. It is important that ( E L , c [ − ] , ℓ , h· , ·i ) is a precosheaf of cyclic L ∞ algebrasbecause this allows one to “integrate by parts,” i.e. use the equality h D ( e , . . . , e k − ) , ℓ e k i = ± h ℓ D ( e , . . . , e k − ) , e k i and its analogues for the higher brackets ℓ , ℓ , . . ..The construction of the shifted Poisson bracket is also identical to the constructionin [CG]. The Weiss cosheaf condition will be satisfied once we prove that ^ Obs cl → Obs cl is a quasi-isomorphism. Hence, statement (3) of the theorem is the only one thatremains to be proved.The essential ingredient in the proof of statement (3) is the fact that the inclusion E L ( U )[ ] → E ∨ L ( U ) is a quasi-isomorphism with a homotopy inverse for certain (called “somewhat nice”in [GRW20]) open subsets U , as shown in [GRW20], Proposition A.1.Just as in [CG], we may assume that the theory is free. We let Σ n : ( E ∨ L ( U )) ⊗ n → Sym n ( E ∨ L ( U )) denote the symmetrization map. We let Γ n denote ( Σ n ) − ^ Obs cl ( U ) . Just as in [CG], wecan identify Γ n = ∩ n − k = E ∨ L ( U ) ⊗ k ⊗ E L , c ( U )[ ] ⊗ E ∨ L ( U ) ⊗ ( n − k − ) .It suffices to show that the inclusion(3.3) Γ n ֒ → E ∨ L ( U ) ⊗ n is an equivalence, since symmetrization is an exact functor. More generally, let { U i } ni = be open subsets of M , and define Γ n , { U i } : = ∩ nk = k O i = E ∨ L ( U i ) ⊗ E L , c ( U k + )[ ] ⊗ n O i = k + E ∨ L ( U i ) ! We will show that the natural inclusion(3.4) Γ n , { U i } ֒ → n O i = E ∨ L ( U i ) is a quasi-isomorphism possessing a homotopy inverse when each U i is either con-tained entirely in M \ ∂ M or is of the form U ′ i × [ δ i ) for U ′ i open in ∂ M (and usingour fixed tubular neighborhood of ∂ M ). We will call such U i “somewhat nice.” Let us LASSICAL BULK-BOUNDARY FACTORIZATION ALGEBRAS 17 explain why this proves that the inclusion in Equation (3.3) is a quasi-isomorphism. Itfollows from Theorem A.1 and Proposition A.2 that k O i = E ∨ L ( U i ) ⊗ E L , c ( U k + )[ ] ⊗ n O i = k + E ∨ L ( U i ) is the space of compactly-supported distributional sections of ( E ! ) ⊠ n over × i U i whichare smooth in the ( k + ) -st coordinate and lie in L ⊕ E ∂ dt when the ( k + ) -st coor-dinate lies on ∂ M , modulo the space of those distributions which depend on the L ′ component of ρ applied to one of their inputs. Now, let V = U n . We may cover V by products of somewhat nice sets in M . Let V = { U i } i ∈ I be such an open cover.By taking the dual statement to that of Lemma A.5.7 of [CG17], we find a contractinghomotopy for the mapping cone of the mapˇ C ( V , ( E ∨ ) ⊗ n ) → ( E ∨ ( V )) ⊗ n .This contracting homotopy involves only multiplication by smooth, compactly-supportedfunctions on M n = M × · · · × M . We have seen that this cochain homotopy descendsto one for the map ˇ C ( V , ( E ∨ L ) ⊗ n ) → ( E ∨ L ( V )) ⊗ n .For similar reasons, the cochain homotopy also descends to one for Γ n , { U i } .Hence, we have a commuting diagramˇ C ( V , Γ n , · ) Γ n , V ˇ C ( V , ⊗ i E ∨ L ) ( ⊗ i E ∨ L )( V ) , ∼∼ where the top and bottom maps are quasi-isomorphisms. Here, we abuse notationslightly and let Γ n , V ′ denote Γ n , { U i } when V ′ = U × · · · U n . We are interested in show-ing that the right-hand arrow in the diagram is a quasi-isomorphism. The finite in-tersection of any number of products of somewhat nice subsets is again a product ofsomewhat nice sets. Therefore, if the map of Equation (3.4) is a quasi-isomorphismfor U somewhat nice, the left-hand map in the above commuting diagram will be aquasi-isomorphism. It follows that the map of (3.3) will be a quasi-isomorphism, sincethat map is also the right-hand map in the commuting diagram.Let us now proceed to show that the map of Equation (3.3) is a quasi-isomorphism.To prove this, it suffices—just as in the corresponding proof in [CG]—to show theexistence of a continuous homotopy inverse to the map E L , c ( U )[ ] → E ∨ L ( U ) when U ∩ ∂ M = ∅ or when U ∼ = U ′ × [ δ ) . For U ∩ ∂ M = ∅ , this is the Atiyah-Bottlemma (Appendix D of [CG17]). For U ∼ = U ′ × [ δ ) , this is shown in Proposition A.1of [GRW20]. (cid:3)
4. T
OPOLOGICAL M ECHANICS
The goal of this section is to study the factorization algebra of topological mechan-ics. Recall from Examples 2.9 and 2.20 that a symplectic vector space V and a La-grangian L ⊂ V define a free bulk-boundary system on R ≥ known as topologicalmechanics. The procedure of the previous section constructs a factorization algebraObs qV , L on R ≥ for these choices. Our goal is to study this factorization algebra.Given an associative algebra A and a right A -module M , there is a factorizationalgebra F A , M on R ≥ which assigns A to any open interval, and M to any intervalcontaining 0 (see § A and the right-module action of A on M . We will see that the co-homology factorization algebra of topological mechanics is isomorphic to one of theform F A , M , for appropriate A and M .Let O ( V ) = Sym ( V ∨ ) denote the symmetric algebra of polynomial functions on V ,and similarly for O ( L ) . The inclusion L → V induces a restriction of functions map O ( V ) → O ( L ) which defines a right O ( V ) -module structure on O ( L ) .We would like to say that Obs clV , L is equivalent to F O ( V ) , O ( L ) ; however, Obs cl is de-fined in terms of a space of power series on E L , while O ( V ) and O ( L ) are polynomialalgebras. To remedy this, one may also consider, for each U , the space Obs clV , L , poly ( U ) consisting only of polynomial functions on E L ( U ) . It is easy to verify that Obs clV , L , poly forms a sub-factorization algebra of Obs clV , L . Lemma 4.1.
The cohomology of the factorization algebra Obs clV , L , poly is isomorphic tothe factorization algebra F O ( V ) , O ( L ) . Proof.
In Proposition 5.1 of [GRW20], the same statement is shown for a slightly dif-ferent model of the classical observables, namely the one that assignsObs clmoll ( U ) : = Sym ( E L , c [ ]( U )) to each open subset U ⊂ M . There is a natural map Obs clmoll → Obs cl . It follows fromProposition A.1 (ibid.) that this map is a quasi-isomorphism. The statement of thelemma follows. (cid:3) A PPENDIX
A. G
OING UNDER THE HOOD : FUNCTIONAL ANALYSIS
In the body of the text, we have taken a number of tensor products and duals of(locally convex) topological vector spaces; the goal of this appendix is to describe whatis meant by these tensor products and duals.A.1.
The topological tensor product of function spaces with boundary conditions.
Given a pair of bundles V → M and V → M , let V and V denote the nuclearFr´echet spaces of smooth global sections of V and V , respectively. This notation dif-fers from that of the body of the text, where V i denotes the sheaf on M i of sections of V i . The completed projective tensor product of these two spaces satisfies the incrediblyuseful property: V b ⊗ π V ∼ = C ∞ ( M × M , V ⊠ V ) . LASSICAL BULK-BOUNDARY FACTORIZATION ALGEBRAS 19
We would like a version of this isomorphism to be true when the M i have bound-aries and we impose boundary conditions on sections in the spaces V i . To this end, let M , · · · , M k be manifolds with boundary, V → M , · · · , V k → M k be vector bundleson the M i , and L ⊂ V | ∂ M , · · · , L k ⊂ V k | ∂ M k subbundles of the indicated bundles.We introduce the following notations: • ( V i ) L i denotes the (locally convex topological vector) space of sections of V i lying in L i on ∂ M i , endowed with the topology which it inherits as a subspaceof ( V i ) L i ; ( V i ) L i is nuclear Fr´echet, since it is a closed subpsace of V i , which isnuclear Fr´echet. • V ··· , k denotes the space of global sections of the bundle V ⊠ · · · ⊠ V k over M × · · · M k . • ( V ··· , k ) L , ··· , L k denotes the following space { σ ∈ V ··· , k | σ ( x , · · · , x k ) ∈ ( V ) x ⊗ · · · ⊗ ( L i ) x i ⊗ · · · ⊗ ( V k ) x k , x i ∈ ∂ M i } ;we put on ( V ··· , k ) L , ··· , L k the topology which it inherits as a (closed) subspaceof V ··· , k . Just as ( V i ) L i was nuclear Fr´echet, so too is ( V ··· , k ) L , ··· , L k .Note that the continuous map V × · · · × V k → V ··· , k ,when restricted to ( V ) L × · · · × ( V k ) L k , has image in ( V ··· , k ) L , ··· , L k , so there is a nat-ural map S : ( V ) L b ⊗ π ( V ) L b ⊗ π · · · b ⊗ π ( V k ) L k → ( V ··· , k ) L , ··· , L k .We introduce similar notations for sections with support on a specified compact set.Let K , · · · , K k be compact subsets of M , · · · , M k , respectively. We introduce the fol-lowing notations: • ( V i ) K i denotes the subspace of ( V i ) consisting of sections with compact supporton K i . • ( V i ) K i , L i denotes the intersection ( V i ) K i ∩ ( V i ) L i . • ( V ··· , k ) K ×···× K k denotes the subspace of V ··· , k consisting of sections with com-pact support on K × · · · × K k . • ( V ··· , k ) L , ··· , L k , K ×···× K k denotes the intersection ( V ··· , k ) K ×···× K k ∩ ( V ··· , k ) L , ··· , L k .All four spaces are nuclear Fr´echet. As before, there is a map S c . s . : ( V ) K , L b ⊗ π ( V ) K , L b ⊗ π · · · b ⊗ π ( V k ) K k , L k → ( V ··· , k ) L , ··· , L k , K ×···× K k .In Appendix A of [GRW20], the following result is shown: Theorem A.1.
The maps S and S c . s . are isomorphisms (for the topological vectorspace structures).This gives an explicit characterization of the completed projective tensor product ofvector spaces of the form ( V i ) L i or ( V i ) K i , L i . The proof of Theorem A.1 also shows that one can omit boundary conditions onany number of tensor factors on both the domain and codomain of S and S c . s . . Forexample, we have an isomorphism ( V ) L b ⊗ π ( V ) ∼ = ( V ) L ,where ( V ) L denotes the space of sections of V ⊠ V over M × M which lie in L when the first coordinate lies on ∂ M .A.2. The strong topological duals of function spaces with boundary conditions.
In the previous section, we studied the tensor product ( V ) L b ⊗ π ( V ) L . We foundthat the tensor product was a subspace of V b ⊗ π V . In this section, we study the dual (( V ) L ) ∨ . We will find that this dual is a quotient of ( V ) ∨ .Let V be a bundle on a manifold with boundary ( M , ∂ M ) , and let L be a subbundleof V | ∂ M . In this section, we would like to study the duals V ∨ L and V ∨ L , c . We will assumethat there is a tubular neighborhood T ∼ = ∂ M × [ ε ) of ∂ M over which V ∼ = V | ∂ M ⊠ R ,where R is the trivial bundle over [ ε ) . (This assumption is satisfied for the space offields of a TNBFT.)Let C denote the bundle ( V | ∂ M ) / L on ∂ M . There are natural surjective maps P : V → C P c : V c → C c ;hence, there are injective maps P ∨ : C ∨ → V ∨ P ∨ c : C ∨ c → V ∨ c (which may or may not be embeddings). Similarly, we have surjective maps i ∨ : V ∨ → V ∨ L i ∨ c : V ∨ c → V ∨ L , c induced from the appropriate inclusions i , i c of spaces of smooth sections. Note thatim P ∨ ( c ) ⊂ ker i ∨ ( c ) ; hence there are induced surjective maps Φ : coker P ∨ → V ∨ L Φ c : coker P ∨ c → V ∨ L , c The main result of this subsection is that these are isomorphisms of topological vectorspaces.
Proposition A.2.
The maps P ∨ , P ∨ c embed C ∨ and C ∨ c as closed subspaces of V ∨ and V ∨ c , respectively. Moreover, the maps Φ and Φ c are isomorphisms of topological vec-tor spaces. Hence, we have V ∨ L ∼ = V ∨ / C ∨ and V ∨ L , c ∼ = V ∨ c / C ∨ c . In fact, choosing asplitting Ψ : C → ( V | ∂ M ) LASSICAL BULK-BOUNDARY FACTORIZATION ALGEBRAS 21 gives isomorphisms V ∼ = V L ⊕ CV c ∼ = V L , c ⊕ C c Proof.
The first and second statements will follow from the third. Towards the end ofproving the third statement, let us construct a splitting I of P with the property thatim ( − IP ) ⊂ V L . To see this latter fact, first note that V L = ker P , and consider themap T : V → V L ⊕ C T ( v ) = (( − IP ) v , Pv ) . T is a continuous linear isomorphism, by standard arguments. T has an inverse S given by i ⊕ I . In other words, we will have(A.1) V ∼ = V L ⊕ C .Equation (A.1) induces an isomorphism(A.2) V ∨ L ⊕ C ∨ → V ∨ .Now, the direct summands of a direct sum of Haussdorff spaces are always closedin the direct sum, and the induced map C ∨ → V ∨ L ⊕ C ∨ → V ∨ is P ∨ . This proves that P ∨ embeds C ∨ as a closed subspace of V ∨ . It follows that Φ is an isomorphism.Let us now construct the splitting C → V . To this end, choose a tubular neighbor-hood T ∼ = ∂ M × [ ε ) of ∂ M in M satisfying the assumptions laid out in the introduc-tion to this section. Let χ be a compactly-supported function on [ ε ) which is 1 in aneighborhood of 0 and with support a subset of [ ε / ] . Let c ∈ C be a section of C .Then, we set I ( c ) = χΨ ( c ) . It is straightforward to verify that I satisfies the property PI = id and im ( − IP ) ⊂ V L .Let us now cover M by a countable collection K ⊂ K ⊂ · · · of compact subsets.Let us assume, by replacing K i with K i ∪ ( K i ∩ ∂ M ) × [ ε / ] if necessary, that K i contains ( K i ∩ ∂ M ) × [ ε / ] . Then, the formulas for I and P send C K i ∩ ∂ M → V K i and V K i → C K i ∩ ∂ M . We therefore obtain an isomorphism V K i ∼ = V K i , L ⊕ C K i ∩ ∂ M .The isomorphism respects the maps induced from the inclusions K i ⊂ K i + and K i ∩ ∂ M ⊂ K i + ∩ ∂ M , and so we have also isomorphisms V c = colim i V K i ∼ = colim i ( V L , K i ) ⊕ colim i C K i ∩ ∂ M = V L , c ⊕ C c .The rest of the proof follows along the same lines as for the sections without compactsupport. (cid:3) Corollary A.
There are isomorphisms ( V ∨ L ) ⊗ k ∼ = ( V ∨ ) ⊗ k / (cid:16) C ∨ ⊗ ( V ∨ ) ⊗ ( k − ) + ( V ∨ ) ⊗ C ∨ ⊗ ( V ∨ ) ⊗ ( k − ) + · · · + ( V ∨ ) ⊗ ( k − ) ⊗ C ∨ (cid:17) ,Sym ( V ∨ L ) ∼ = Sym ( V ∨ ) / (cid:0) C ∨ ⊗ Sym ( V ∨ ) (cid:1) ,where all tensor products are completed projective tensor products. R EFERENCES [ACMV17] Mina Aganagic, Kevin Costello, Jacob McNamara, and Cumrun Vafa. Topological chern-simons/matter theories. 2017.[ASZK97] M. Alexandrov, A. Schwarz, O. Zaboronsky, and M. Kontsevich. The geometry of the masterequation and topological quantum field theory.
Internat. J. Modern Phys. A , 12(7):1405–1429,1997.[BCQZ19] Francesco Bonechi, Alberto S. Cattaneo, Jian Qiu, and Maxim Zabzine. Equivariant Batalin-Vilkovisky formalism. 2019.[BV81] I. A. Batalin and G. A. Vilkovisky. Gauge algebra and quantization.
Phys. Lett. B , 102(1):27–31,1981.[BY16] Dylan Butson and Philsang Yoo. Degenerate Classical Field Theories and Boundary Theories. arXiv e-prints , page arXiv:1611.00311, Nov 2016.[Cal14] Damien Calaque. Three lectures on derived symplectic geometry and topological field the-ories.
Indagationes Mathematicae , 25(5):926 – 947, 2014. Poisson 2012: Poisson Geometry inMathematics and Physics.[CG] Kevin Costello and Owen Gwilliam.
Factorization algebras in quantum field theory. Vol. 2 . Avail-able at http://people.mpim-bonn.mpg.de/gwilliam/vol2may8.pdf .[CG17] Kevin Costello and Owen Gwilliam.
Factorization algebras in quantum field theory. Vol. 1 , vol-ume 31 of
New Mathematical Monographs . Cambridge University Press, Cambridge, 2017.[CMR18] Alberto S. Cattaneo, Pavel Mnev, and Nicolai Reshetikhin. Perturbative quantum gauge the-ories on manifolds with boundary.
Comm. Math. Phys. , 357(2):631–730, 2018.[Cos11] Kevin Costello.
Renormalization and effective field theory , volume 170 of
Mathematical Surveysand Monographs . American Mathematical Society, Providence, RI, 2011.[GRW20] Owen Gwilliam, Eugene Rabinovich, and Brian R. Williams. Factorization algebras andabelian cs/wzw-type correspondences, 2020.[GW19] Owen Gwilliam and Brian R. Williams. A one-loop exact quantization of chern-simons the-ory, 2019.[Hin05] Vladimir Hinich. Deformations of sheaves of algebras.
Adv. Math. , 195(1):102–164, 2005.[Lur09] Jacob Lurie.
Higher topos theory , volume 170 of
Annals of Mathematics Studies . Princeton Uni-versity Press, Princeton, NJ, 2009.[MSTW19] Philippe Mathieu, Alexander Schenkel, Nicholas J. Teh, and Laura Wells. Homological per-spective on edge modes in linear Yang-Mills theory. 2019.[MSW19] Pavel Mnev, Michele Schiavina, and Konstantin Wernli. Towards holography in the BV-BFVsetting. 2019.[PTVV13] Tony Pantev, Bertrand To¨en, Michel Vaqui´e, and Gabriele Vezzosi. Shifted symplectic struc-tures.
Publications math´ematiques de lIH ´ES , 117(1):271328, May 2013.[Rab] Eugene Rabinovich. Factorization algebras for quantum field theories on manifolds withboundary.[Sch93] Albert Schwarz. Geometry of Batalin-Vilkovisky quantization.