Introduction BV-BFV formalism, outline Examples Motivation I: perturbative Chern-Simons theory Motivation I: Chern-Simons theory – the perturbative answer (Witten-Axelrod-Singer): Z pert ( M, G, A 0 , � , ϕ ) = �� � � − χ (Γ) 4 ψ A 0 ,g · exp � S ( A 0 ) · τ ( M, A 0 ) | Aut(Γ) | i E + V · Φ A 0 ,g i 1 πi 2 · e = e · Γ Γ · e ic ( � ) S grav ( g,φ ) M is closed , A 0 is an acyclic flat connection. Γ ∈ { , , , · · · } – connected 3-valent, � � Φ Γ = η ( x e in , x e out ) Conf V ( M ) edges e Here η ∈ Ω 2 (Conf 2 ( M ) , E ⊠ E ) is the propagator – the integral kernel of d ∗ E / ∆ E .
Introduction BV-BFV formalism, outline Examples Motivation I: perturbative Chern-Simons theory Motivation I: Chern-Simons theory – the perturbative answer (Witten-Axelrod-Singer): Z pert ( M, G, A 0 , � , ϕ ) = �� � � − χ (Γ) 4 ψ A 0 ,g · exp � S ( A 0 ) · τ ( M, A 0 ) | Aut(Γ) | i E + V · Φ A 0 ,g i 1 πi 2 · e = e · Γ Γ · e ic ( � ) S grav ( g,φ ) M is closed , A 0 is an acyclic flat connection. Γ ∈ { , , , · · · } – connected 3-valent, � � Φ Γ = η ( x e in , x e out ) Conf V ( M ) edges e Here η ∈ Ω 2 (Conf 2 ( M ) , E ⊠ E ) is the propagator – the integral kernel of d ∗ E / ∆ E . g – an arbitrary Riemannian metric, ϕ – framing of M , c ( � ) ∈ C [[ � ]] a universal element.
Introduction BV-BFV formalism, outline Examples Motivation II: cut/paste approach in field theory Motivation II: calculating partition functions by cut/paste. Idea: � � � � �� � � Z = Z , Z
Introduction BV-BFV formalism, outline Examples Motivation II: cut/paste approach in field theory Motivation II: calculating partition functions by cut/paste. Idea: � � � � �� � � Z = Z , Z Functorial description (Atiyah-Segal): Closed ( n − 1) -manifold Σ H Σ n -cobordism M Partition function Z M : H Σ in → H Σ out Gluing Composition Z M I ∪ M II = Z M II ◦ Z M I
Introduction BV-BFV formalism, outline Examples Motivation II: cut/paste approach in field theory Motivation II: calculating partition functions by cut/paste. Idea: � � � � �� � � Z = Z , Z Functorial description (Atiyah-Segal): Closed ( n − 1) -manifold Σ H Σ n -cobordism M Partition function Z M : H Σ in → H Σ out Gluing Composition Z M I ∪ M II = Z M II ◦ Z M I Atiyah: TQFT is a functor of monoidal categories (Cob n , ⊔ ) → (Vect C , ⊗ ) .
Introduction BV-BFV formalism, outline Examples Motivation II: cut/paste approach in field theory Example: 2D TQFT Z can be expressed in terms of building blocks: � � Z : C → H S 1 1 � � Z : H S 1 → C 2 Z : H S 1 ⊗ H S 1 → H S 1 3 Z : H S 1 → H S 1 ⊗ H S 1 4 – Universal local building blocks for 2D TQFT!
Introduction BV-BFV formalism, outline Examples Corners For n > 2 we want to glue along pieces of boundary/ glue-cut with corners. Building blocks: balls with stratified boundary (cells)
Introduction BV-BFV formalism, outline Examples Corners For n > 2 we want to glue along pieces of boundary/ glue-cut with corners. Building blocks: balls with stratified boundary (cells) Extension of Atiyah’s axioms to gluing with corners: extended TQFT (Baez-Dolan-Lurie).
Introduction BV-BFV formalism, outline Examples Corners For n > 2 we want to glue along pieces of boundary/ glue-cut with corners. Building blocks: balls with stratified boundary (cells) Extension of Atiyah’s axioms to gluing with corners: extended TQFT (Baez-Dolan-Lurie). Example: Turaev-Viro 3D state-sum model. building block - 3-simplex q6j-symbol gluing sum over spins on edges
Introduction BV-BFV formalism, outline Examples Goal Problems: Very few examples! Some natural examples do not fit into Atiyah axiomatics. Goal: Construct TQFT with corners and gluing out of perturbative path integrals for diffeomorphism-invariant action functionals. Investigate interesting examples.
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories Reminder. A classical BV theory on a closed spacetime manifold M :
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories Reminder. A classical BV theory on a closed spacetime manifold M : F
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories Reminder. A classical BV theory on a closed spacetime manifold M : F ω ∈ Ω 2 ( F ) odd-symplectic, gh = − 1
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories Reminder. A classical BV theory on a closed spacetime manifold M : F ω ∈ Ω 2 ( F ) odd-symplectic, gh = − 1 Q ∈ X ( F ) , odd, gh = 1 , Q 2 = 0
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories Reminder. A classical BV theory on a closed spacetime manifold M : F ω ∈ Ω 2 ( F ) odd-symplectic, gh = − 1 Q ∈ X ( F ) , odd, gh = 1 , Q 2 = 0 S ∈ C ∞ ( F ) , gh = 0 , ι Q ω = δS
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories Reminder. A classical BV theory on a closed spacetime manifold M : F ω ∈ Ω 2 ( F ) odd-symplectic, gh = − 1 Q ∈ X ( F ) , odd, gh = 1 , Q 2 = 0 S ∈ C ∞ ( F ) , gh = 0 , ι Q ω = δS Note: { S, S } ω = 0 .
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories BV-BFV formalism for gauge theories on manifolds with boundary Reference: A. S. Cattaneo, P. Mnev, N. Reshetikhin, Classical BV theories on manifolds with boundary, Comm. Math. Phys. 332 2 (2014) 535–603. For M with boundary: M − − − − → ( F , ω, Q, S ) – space of fields � π � π ∗ ∂M − − − − → ( F ∂ , ω ∂ = δα ∂ , Q ∂ , S ∂ ) – phase space
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories BV-BFV formalism for gauge theories on manifolds with boundary Reference: A. S. Cattaneo, P. Mnev, N. Reshetikhin, Classical BV theories on manifolds with boundary, Comm. Math. Phys. 332 2 (2014) 535–603. For M with boundary: M − − − − → ( F , − 1 , ω Q , S 0 ) – space of fields 1 � π � π ∗ ∂M − − − − → ( F ∂ , ω ∂ = δα ∂ , Q ∂ , S ∂ 1 ) – phase space 0 1 Subscripts =“ghost numbers”.
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories BV-BFV formalism for gauge theories on manifolds with boundary Reference: A. S. Cattaneo, P. Mnev, N. Reshetikhin, Classical BV theories on manifolds with boundary, Comm. Math. Phys. 332 2 (2014) 535–603. For M with boundary: M − − − − → ( F , ω, Q, S ) – space of fields � π � π ∗ ∂M − − − − → ( F ∂ , ω ∂ = δα ∂ , Q ∂ , S ∂ ) – phase space Q 2 = 0 , ι Q ω = δS + π ∗ α ∂ . Relations: Q 2 ∂ = 0 , ι Q ∂ ω ∂ = δS ∂ ; ⇒ CME: 1 2 ι Q ι Q ω = π ∗ S ∂
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories BV-BFV formalism for gauge theories on manifolds with boundary Reference: A. S. Cattaneo, P. Mnev, N. Reshetikhin, Classical BV theories on manifolds with boundary, Comm. Math. Phys. 332 2 (2014) 535–603. For M with boundary: M − − − − → ( F , ω, Q, S ) – space of fields � π � π ∗ ∂M − − − − → ( F ∂ , ω ∂ = δα ∂ , Q ∂ , S ∂ ) – phase space Q 2 = 0 , ι Q ω = δS + π ∗ α ∂ . Relations: Q 2 ∂ = 0 , ι Q ∂ ω ∂ = δS ∂ ; ⇒ CME: 1 2 ι Q ι Q ω = π ∗ S ∂ Gluing: M I ∪ Σ M II → F M I × F Σ F M II
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories BV-BFV formalism for gauge theories on manifolds with boundary Reference: A. S. Cattaneo, P. Mnev, N. Reshetikhin, Classical BV theories on manifolds with boundary, Comm. Math. Phys. 332 2 (2014) 535–603. For M with boundary: M − − − − → ( F , ω, Q, S ) – space of fields � π � π ∗ ∂M − − − − → ( F ∂ , ω ∂ = δα ∂ , Q ∂ , S ∂ ) – phase space Q 2 = 0 , ι Q ω = δS + π ∗ α ∂ . Relations: Q 2 ∂ = 0 , ι Q ∂ ω ∂ = δS ∂ ; ⇒ CME: 1 2 ι Q ι Q ω = π ∗ S ∂ Gluing: M I ∪ Σ M II → F M I × F Σ F M II This picture extends to higher-codimension strata!
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories Example: abelian Chern-Simons theory, dim M = 3 . M − − − − → ( F , ω, Q, S ) � π � π ∗ ∂M − − − − → ( F ∂ , ω ∂ = δα ∂ , Q ∂ , S ∂ )
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories Example: abelian Chern-Simons theory, dim M = 3 . (Ω • ( M )[1] , M − − − − → ω, Q, S ) � π : A�→A| ∂ � π ∗ → (Ω • ( ∂M )[1] , ω ∂ = δα ∂ , Q ∂ , S ∂ ) ∂M − − − − + A + − 1 + c + Superfield A = c + A ���� ���� − 2 � �� � ghost , 1 classical field , 0 antifields
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories Example: abelian Chern-Simons theory, dim M = 3 . � (Ω • ( M )[1] , 1 M − − − − → M δ A ∧ δ A , Q, S ) 2 � π : A�→A| ∂ � π ∗ � → (Ω • ( ∂M )[1] , 1 ∂M − − − − ∂ δ A ∧ δ A , Q ∂ , S ∂ ) 2 + A + − 1 + c + Superfield A = c + A ���� ���� − 2 � �� � ghost , 1 classical field , 0 antifields
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories Example: abelian Chern-Simons theory, dim M = 3 . � � (Ω • ( M )[1] , 1 M d A δ M − − − − → M δ A ∧ δ A , δ A , S ) 2 � π : A�→A| ∂ � π ∗ � � → (Ω • ( ∂M )[1] , 1 ∂ d A δ ∂M − − − − ∂ δ A ∧ δ A , δ A , S ∂ ) 2 + A + − 1 + c + Superfield A = c + A ���� ���� − 2 � �� � ghost , 1 classical field , 0 antifields
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories Example: abelian Chern-Simons theory, dim M = 3 . � � � (Ω • ( M )[1] , 1 M d A δ δ A , 1 M − − − − → M δ A ∧ δ A , M A ∧ d A ) 2 2 � π : A�→A| ∂ � π ∗ � � � → (Ω • ( ∂M )[1] , 1 ∂ d A δ 1 ∂M − − − − ∂ δ A ∧ δ A , δ A , ∂ A ∧ d A ) 2 2 + A + − 1 + c + Superfield A = c + A ���� ���� − 2 � �� � ghost , 1 classical field , 0 antifields
Introduction BV-BFV formalism, outline Examples Classical BV-BFV theories Example: abelian Chern-Simons theory, dim M = 3 . � � � (Ω • ( M )[1] , 1 M d A δ δ A , 1 M − − − − → M δ A ∧ δ A , M A ∧ d A ) 2 2 � π : A�→A| ∂ � π ∗ � � � → (Ω • ( ∂M )[1] , 1 ∂ d A δ 1 ∂M − − − − ∂ δ A ∧ δ A , δ A , ∂ A ∧ d A ) 2 2 + A + − 1 + c + Superfield A = c + A ���� ���� − 2 � �� � ghost , 1 classical field , 0 antifields Euler-Lagrange moduli spaces: → H • ( M )[1] M − − − − � ι ∗ → H • ( ∂M )[1] ∂M − − − −
Introduction BV-BFV formalism, outline Examples Quantum BV-BFV theories Quantum BV-BFV formalism. ( H • Σ closed, dim Σ = n − 1 �→ Σ , Ω Σ )
Introduction BV-BFV formalism, outline Examples Quantum BV-BFV theories Quantum BV-BFV formalism. ( H • Σ closed, dim Σ = n − 1 �→ Σ , Ω Σ ) M , dim M = n �→ ( F res , ω res )
Introduction BV-BFV formalism, outline Examples Quantum BV-BFV theories Quantum BV-BFV formalism. ( H • Σ closed, dim Σ = n − 1 �→ Σ , Ω Σ ) M , dim M = n �→ ( F res , ω res ) 1 2 ( F res ) ⊗ H ∂M Z M ∈ Dens
Introduction BV-BFV formalism, outline Examples Quantum BV-BFV theories Quantum BV-BFV formalism. ( H • Σ closed, dim Σ = n − 1 �→ Σ , Ω Σ ) M , dim M = n �→ ( F res , ω res ) 1 2 ( F res ) ⊗ H ∂M satisfying mQME: Z M ∈ Dens � i � � Ω ∂M − i � ∆ res Z M = 0
Introduction BV-BFV formalism, outline Examples Quantum BV-BFV theories Quantum BV-BFV formalism. ( H • Σ closed, dim Σ = n − 1 �→ Σ , Ω Σ ) M , dim M = n �→ ( F res , ω res ) 1 2 ( F res ) ⊗ H ∂M satisfying mQME: Z M ∈ Dens � i � � Ω ∂M − i � ∆ res Z M = 0 Reminder: In Darboux coordinates ( x i , ξ i ) on F res , ∂ ∂ ∆ res = ∂x i ∂ξ i
Introduction BV-BFV formalism, outline Examples Quantum BV-BFV theories Quantum BV-BFV formalism. ( H • Σ closed, dim Σ = n − 1 �→ Σ , Ω Σ ) M , dim M = n �→ ( F res , ω res ) 1 2 ( F res ) ⊗ H ∂M satisfying mQME: Z M ∈ Dens � i � � Ω ∂M − i � ∆ res Z M = 0 � i � Gauge-fixing ambiguity ⇒ Z M ∼ Z M + � Ω ∂M − i � ∆ res ( · · · ) .
Introduction BV-BFV formalism, outline Examples Quantum BV-BFV theories Quantum BV-BFV formalism. ( H • Σ closed, dim Σ = n − 1 �→ Σ , Ω Σ ) M , dim M = n �→ ( F res , ω res ) 1 2 ( F res ) ⊗ H ∂M satisfying mQME: Z M ∈ Dens � i � � Ω ∂M − i � ∆ res Z M = 0 � i � Gauge-fixing ambiguity ⇒ Z M ∼ Z M + � Ω ∂M − i � ∆ res ( · · · ) . Gluing: Z M I ∪ Σ M II = P ∗ ( Z M I ∗ Σ Z M II )
Introduction BV-BFV formalism, outline Examples Quantum BV-BFV theories Quantum BV-BFV formalism. ( H • Σ closed, dim Σ = n − 1 �→ Σ , Ω Σ ) M , dim M = n �→ ( F res , ω res ) 1 2 ( F res ) ⊗ H ∂M satisfying mQME: Z M ∈ Dens � i � � Ω ∂M − i � ∆ res Z M = 0 � i � Gauge-fixing ambiguity ⇒ Z M ∼ Z M + � Ω ∂M − i � ∆ res ( · · · ) . Gluing: Z M I ∪ Σ M II = P ∗ ( Z M I ∗ Σ Z M II ) ∗ Σ — pairing of states in H Σ ,
Introduction BV-BFV formalism, outline Examples Quantum BV-BFV theories Quantum BV-BFV formalism. ( H • Σ closed, dim Σ = n − 1 �→ Σ , Ω Σ ) M , dim M = n �→ ( F res , ω res ) 1 2 ( F res ) ⊗ H ∂M satisfying mQME: Z M ∈ Dens � i � � Ω ∂M − i � ∆ res Z M = 0 � i � Gauge-fixing ambiguity ⇒ Z M ∼ Z M + � Ω ∂M − i � ∆ res ( · · · ) . Gluing: Z M I ∪ Σ M II = P ∗ ( Z M I ∗ Σ Z M II ) ∗ Σ — pairing of states in H Σ , P ∗ — BV pushforward (fiber BV integral) for P F M I res × F M II → F M I ∪ Σ M II − res res
Introduction BV-BFV formalism, outline Examples Aside: BV pushforward Aside: BV pushforward. V = V ′ × � V — splitting of odd-symplectic manifolds, P : V → V ′
Introduction BV-BFV formalism, outline Examples Aside: BV pushforward Aside: BV pushforward. V = V ′ × � V — splitting of odd-symplectic manifolds, P : V → V ′ L ⊂ � V Lagrangian
Introduction BV-BFV formalism, outline Examples Aside: BV pushforward Aside: BV pushforward. V = V ′ × � V — splitting of odd-symplectic manifolds, P : V → V ′ L ⊂ � V Lagrangian BV pushforward: 1 1 2 ( V ) 2 ( V ′ ) P ∗ : Dens → Dens
Introduction BV-BFV formalism, outline Examples Aside: BV pushforward Aside: BV pushforward. V = V ′ × � V — splitting of odd-symplectic manifolds, P : V → V ′ L ⊂ � V Lagrangian BV pushforward: 1 1 2 ( V ) 2 ( V ′ ) P ∗ : Dens → Dens � ψ �→ V ψ L⊂ �
Introduction BV-BFV formalism, outline Examples Aside: BV pushforward Aside: BV pushforward. V = V ′ × � V — splitting of odd-symplectic manifolds, P : V → V ′ L ⊂ � V Lagrangian BV pushforward: 1 1 2 ( V ) 2 ( V ′ ) P ∗ : Dens → Dens � ψ �→ V ψ L⊂ � Theorem P ∗ is a chain map: P ∗ (∆ V ψ ) = ∆ V ′ P ∗ ψ 1
Introduction BV-BFV formalism, outline Examples Aside: BV pushforward Aside: BV pushforward. V = V ′ × � V — splitting of odd-symplectic manifolds, P : V → V ′ L ⊂ � V Lagrangian BV pushforward: 1 1 2 ( V ) 2 ( V ′ ) P ∗ : Dens → Dens � ψ �→ V ψ L⊂ � Theorem P ∗ is a chain map: P ∗ (∆ V ψ ) = ∆ V ′ P ∗ ψ 1 For L 0 ∼ L 1 , P ( L 1 ) ψ = P ( L 0 ) ψ + ∆ V ′ ( · · · ) 2 ∗ ∗
Introduction BV-BFV formalism, outline Examples Aside: BV pushforward Aside: BV pushforward. V = V ′ × � V — splitting of odd-symplectic manifolds, P : V → V ′ L ⊂ � V Lagrangian BV pushforward: 1 1 2 ( V ) 2 ( V ′ ) P ∗ : Dens → Dens � ψ �→ V ψ L⊂ � Theorem P ∗ is a chain map: P ∗ (∆ V ψ ) = ∆ V ′ P ∗ ψ 1 For L 0 ∼ L 1 , P ( L 1 ) ψ = P ( L 0 ) ψ + ∆ V ′ ( · · · ) 2 ∗ ∗ Reference: P. Mnev, Discrete BF theory, arXiv:0809.1160
Introduction BV-BFV formalism, outline Examples Quantization Quantization Choose p : F ∂ → B Lagrangian fibration, α ∂ | p − 1 ( b ) = 0 . 1 2 ( B ) , Ω ∂ = � H ∂ = Dens S ∂ ∈ End( H ∂ ) 1 .
Introduction BV-BFV formalism, outline Examples Quantization Quantization Choose p : F ∂ → B Lagrangian fibration, α ∂ | p − 1 ( b ) = 0 . 1 2 ( B ) , Ω ∂ = � H ∂ = Dens S ∂ ∈ End( H ∂ ) 1 . F � π F ∂ � p B
Introduction BV-BFV formalism, outline Examples Quantization Quantization Choose p : F ∂ → B Lagrangian fibration, α ∂ | p − 1 ( b ) = 0 . 1 2 ( B ) , Ω ∂ = � H ∂ = Dens S ∂ ∈ End( H ∂ ) 1 . = π − 1 p − 1 { b } F ⊃ F b � π F ∂ p � B ∋ b boundary condition
Introduction BV-BFV formalism, outline Examples Quantization Quantization Choose p : F ∂ → B Lagrangian fibration, α ∂ | p − 1 ( b ) = 0 . 2 ( B ) , Ω ∂ = � 1 H ∂ = Dens S ∂ ∈ End( H ∂ ) 1 . = π − 1 p − 1 { b } F ⊃ F b � π F ∂ p � B ∋ b boundary condition Partition function: � i 1 � S , 2 ( B ) Z M ( b ) = e Z M ∈ Dens L⊂F b L ⊂ F b gauge-fixing Lagrangian. Problem: Z M may be ill-defined due to zero-modes.
Introduction BV-BFV formalism, outline Examples Quantization Quantization Choose p : F ∂ → B Lagrangian fibration, α ∂ | p − 1 ( b ) = 0 . 1 2 ( B ) , Ω ∂ = � H ∂ = Dens S ∂ ∈ End( H ∂ ) 1 . = π − 1 p − 1 { b } F ⊃ F b � π F ∂ � p B ∋ b boundary condition Solution: Split F b = F res × � F ∋ ( φ res , � φ ) . Partition function: � � S ( b,φ res , � i 1 1 φ ) , 2 ( B ) ⊗ Dens 2 ( F res ) Z M ( b, φ res ) = e Z M ∈ Dens L⊂ � F L ⊂ � F gauge-fixing Lagrangian.
Introduction BV-BFV formalism, outline Examples Quantization Quantization Choose p : F ∂ → B Lagrangian fibration, α ∂ | p − 1 ( b ) = 0 . 1 2 ( B ) , Ω ∂ = � H ∂ = Dens S ∂ ∈ End( H ∂ ) 1 . = π − 1 p − 1 { b } F ⊃ F b � π F ∂ � p B ∋ b boundary condition Solution: Split F b = F res × � F ∋ ( φ res , � φ ) . Partition function: � � S ( b,φ res , � i 1 1 φ ) , 2 ( B ) ⊗ Dens 2 ( F res ) Z M ( b, φ res ) = e Z M ∈ Dens L⊂ � F L ⊂ � F gauge-fixing Lagrangian. P → F ′ Z ′ F res − ⇒ M = P ∗ Z M res
Introduction BV-BFV formalism, outline Examples Abelian BF theory Abelian BF theory: the continuum model. Input: M a closed oriented n -manifold M . E an SL ( m ) -local system.
Introduction BV-BFV formalism, outline Examples Abelian BF theory Abelian BF theory: the continuum model. Input: M a closed oriented n -manifold M . E an SL ( m ) -local system. Space of BV fields: F = Ω • ( M, E )[1] ⊕ Ω • ( M, E ∗ )[ n − 2] ∋ ( A, B ) � Action: S = M � B, d E A � . Reference: A. S. Schwarz, The partition function of degenerate quadratic functional and Ray-Singer invariants, Lett. Math. Phys. 2, 3 (1978) 247–252. A. S. Schwarz: For M closed and E acyclic, the partition function is the R -torsion τ ( M, E ) ∈ R .
Introduction BV-BFV formalism, outline Examples Abelian BF theory Result, C-M-R arXiv:1507.01221 For M closed, E possibly non-acyclic, F res = H • ( M, E )[1] ⊕ H • ( M, E ∗ )[ n − 2] and Z M = ξ · τ ( M, E )
Introduction BV-BFV formalism, outline Examples Abelian BF theory Result, C-M-R arXiv:1507.01221 For M closed, E possibly non-acyclic, F res = H • ( M, E )[1] ⊕ H • ( M, E ∗ )[ n − 2] and Z M = ξ · τ ( M, E ) 1 2 ( F res ) is the R-torsion where τ ( M, E ) ∈ Det H • ( M, E ) = Dens
Introduction BV-BFV formalism, outline Examples Abelian BF theory Result, C-M-R arXiv:1507.01221 For M closed, E possibly non-acyclic, F res = H • ( M, E )[1] ⊕ H • ( M, E ∗ )[ n − 2] and Z M = ξ · τ ( M, E ) 1 2 ( F res ) is the R-torsion and where τ ( M, E ) ∈ Det H • ( M, E ) = Dens � n � n 2 k ( − 1) k ) · dim H k ( M,E ) · ( e − πi 2 k ( − 1) k ) · dim H k ( M,E ) k =0 ( − 1 4 − 1 k =0 ( 1 4 − 1 2 � ) ξ = (2 π � )
Introduction BV-BFV formalism, outline Examples Abelian BF theory Result, C-M-R arXiv:1507.01221 For M closed, E possibly non-acyclic, F res = H • ( M, E )[1] ⊕ H • ( M, E ∗ )[ n − 2] and Z M = ξ · τ ( M, E ) 1 2 ( F res ) is the R-torsion and where τ ( M, E ) ∈ Det H • ( M, E ) = Dens � n � n 2 k ( − 1) k ) · dim H k ( M,E ) · ( e − πi 2 k ( − 1) k ) · dim H k ( M,E ) k =0 ( − 1 4 − 1 k =0 ( 1 4 − 1 2 � ) ξ = (2 π � ) 16 s with 2 πi In particular Z M contains a mod16 phase e s = � n k =0 ( − 1 + 2 k ( − 1) k ) · dim H k ( M, E ) .
Introduction BV-BFV formalism, outline Examples Abelian BF theory Result, C-M-R arXiv:1507.01221 For M with boundary, E possibly non-acyclic,
Introduction BV-BFV formalism, outline Examples Abelian BF theory Result, C-M-R arXiv:1507.01221 For M with boundary, E possibly non-acyclic, Z M = ξ · τ ( M, Σ in ; E ) · �� � � � · exp i B a + b A − B ( x ) η ( x, y ) A ( y ) � Σ out Σ in Σ out × Σ in ∋ ( x,y )
Introduction BV-BFV formalism, outline Examples Abelian BF theory Result, C-M-R arXiv:1507.01221 For M with boundary, E possibly non-acyclic, Z M = ξ · τ ( M, Σ in ; E ) · �� � · exp i � � B a + b A − B ( x ) η ( x, y ) A ( y ) � Σ out Σ in Σ out × Σ in ∋ ( x,y ) Where: F res = H • ( M, Σ in ; E )[1] ⊕ H • ( M, Σ out ; E ∗ )[ n − 2] ∋ ( a , b )
Introduction BV-BFV formalism, outline Examples Abelian BF theory Result, C-M-R arXiv:1507.01221 For M with boundary, E possibly non-acyclic, Z M = ξ · τ ( M, Σ in ; E ) · �� � · exp i � � B a + b A − B ( x ) η ( x, y ) A ( y ) � Σ out Σ in Σ out × Σ in ∋ ( x,y ) B = Ω • (Σ in )[1] ⊕ Ω • (Σ out )[ n − 2] ∋ ( A , B ) Where: � � 1 2 ( B ) H Σ = Dens ∋ Conf k (Σ in ) × Conf l (Σ out ) k,l ≥ 0 Ψ( x 1 , . . . , x k ; y 1 , . . . , y l ) A ( x 1 ) · · · A ( x k ) B ( y 1 ) · · · B ( y l )
Introduction BV-BFV formalism, outline Examples Abelian BF theory Result, C-M-R arXiv:1507.01221 For M with boundary, E possibly non-acyclic, Z M = ξ · τ ( M, Σ in ; E ) · �� � · exp i � � B a + b A − B ( x ) η ( x, y ) A ( y ) � Σ out Σ in Σ out × Σ in ∋ ( x,y ) Where: ξ as before (but for relative cohomology),
Introduction BV-BFV formalism, outline Examples Abelian BF theory Result, C-M-R arXiv:1507.01221 For M with boundary, E possibly non-acyclic, Z M = ξ · τ ( M, Σ in ; E ) · �� � · exp i � � B a + b A − B ( x ) η ( x, y ) A ( y ) � Σ out Σ in Σ out × Σ in ∋ ( x,y ) Where: τ - relative R-torsion,
Introduction BV-BFV formalism, outline Examples Abelian BF theory Result, C-M-R arXiv:1507.01221 For M with boundary, E possibly non-acyclic, Z M = ξ · τ ( M, Σ in ; E ) · �� � · exp i � � B a + b A − B ( x ) η ( x, y ) A ( y ) � Σ out Σ in Σ out × Σ in ∋ ( x,y ) η ∈ Ω n − 1 (Conf 2 ( M ) , E ⊠ E ∗ ) – propagator, i.e. Where: � M ∋ y η ( x, y ) α ( y ) is a chain contraction from Ω • ( M, Σ in ; E ) to α �→ H • ( M, Σ in ; E ) .
Introduction BV-BFV formalism, outline Examples Abelian BF theory Result, C-M-R arXiv:1507.01221 For M with boundary, E possibly non-acyclic, Z M = ξ · τ ( M, Σ in ; E ) · �� � · exp i � � B a + b A − B ( x ) η ( x, y ) A ( y ) � Σ out Σ in Σ out × Σ in ∋ ( x,y ) This result satisfies: gluing
Introduction BV-BFV formalism, outline Examples Abelian BF theory Result, C-M-R arXiv:1507.01221 For M with boundary, E possibly non-acyclic, Z M = ξ · τ ( M, Σ in ; E ) · �� � · exp i � � B a + b A − B ( x ) η ( x, y ) A ( y ) � Σ out Σ in Σ out × Σ in ∋ ( x,y ) This result satisfies: gluing mQME
Introduction BV-BFV formalism, outline Examples Abelian BF theory Result, C-M-R arXiv:1507.01221 For M with boundary, E possibly non-acyclic, Z M = ξ · τ ( M, Σ in ; E ) · �� � · exp i � � B a + b A − B ( x ) η ( x, y ) A ( y ) � Σ out Σ in Σ out × Σ in ∋ ( x,y ) This result satisfies: gluing mQME � i � change of η shifts Z M by � Ω ∂ − i � ∆ res -exact term.
Introduction BV-BFV formalism, outline Examples Abelian BF theory Result, C-M-R arXiv:1507.01221 For M with boundary, E possibly non-acyclic, Z M = ξ · τ ( M, Σ in ; E ) · �� � · exp i � � B a + b A − B ( x ) η ( x, y ) A ( y ) � Σ out Σ in Σ out × Σ in ∋ ( x,y ) This result satisfies: gluing mQME � i � change of η shifts Z M by � Ω ∂ − i � ∆ res -exact term. �� � � Σ out d E B δ Σ in d E A δ BFV operator: Ω ∂ = − i � δ B + δ A
Introduction BV-BFV formalism, outline Examples Abelian BF theory Gluing in two steps: � � i Σ2 B 2 A 2 · Z M I ( B 2 , A 1 ; a I , b I ) . � Z M = A 2 , B 2 Z M II ( B 3 , A 2 ; a II , b II ) · e � 1 Z M = P ∗ � Z M , for P : F I res × F II res → F res . 2
Introduction BV-BFV formalism, outline Examples Gluing of propagators Result, C-M-R arXiv:1507.01221 η I , η II – propagators on M I , M II . Assume H • ( M, Σ 1 ) = H • ( M I , Σ 1 ) ⊕ H • ( M II , Σ 2 ) . Then the glued propagator on M is: η I ( x, y ) if x, y ∈ M I η II ( x, y ) if x, y ∈ M II η ( x, y ) = 0 if x ∈ M I , y ∈ M II � η II ( x, z ) η I ( z, y ) if x ∈ M II , y ∈ M I z ∈ Σ 2
Introduction BV-BFV formalism, outline Examples Poisson sigma model Example: Poisson sigma model, n = 2 . � M � B, dA � + 1 Action: S = 2 � π ( B ) , A ⊗ A � π = � ij π ij ( u ) ∂ ∂ ∂u j Poisson bivector on R m . ∂u i ∧ Result, C-M-R arXiv:1507.01221 � Z M = ξ · τ · exp i � graphs
Introduction BV-BFV formalism, outline Examples Poisson sigma model Example: Poisson sigma model, n = 2 . � M � B, dA � + 1 Action: S = 2 � π ( B ) , A ⊗ A � π = � ij π ij ( u ) ∂ ∂ ∂u j Poisson bivector on R m . ∂u i ∧ Result, C-M-R arXiv:1507.01221 � Z M = ξ · τ · exp i � graphs Z M satisfies: gluing mQME � i � change of η shifts Z M by � Ω ∂ − i � ∆ res -exact term.
Introduction BV-BFV formalism, outline Examples Poisson sigma model Example: Poisson sigma model, n = 2 . � M � B, dA � + 1 Action: S = 2 � π ( B ) , A ⊗ A � π = � ij π ij ( u ) ∂ ∂ ∂u j Poisson bivector on R m . ∂u i ∧ Result, C-M-R arXiv:1507.01221 � Z M = ξ · τ · exp i � graphs Z M satisfies: gluing mQME � i � change of η shifts Z M by � Ω ∂ − i � ∆ res -exact term. Ω ∂ = standard-ordering quantization ( B �→ − i � δ δ A on Σ in , A �→ − i � δ δ B � B i d A i + 1 2Π ij ( B ) A i A j where Π ij ( u ) = u i ∗ u j − u j ∗ u i on Σ out ) of is i � ∂ Kontsevich’s deformation of π .
Introduction BV-BFV formalism, outline Examples Poisson sigma model Rules for calculating Φ Γ (“Feynman rules”). Decorate half-edges by i ∈ { 1 , . . . , m } , put internal vertices to z 1 . . . , z p ∈ M , boundary in-vertices to x 1 , . . . , x k ∈ Σ in , boundary out-vertices to y 1 , . . . , y l ∈ Σ out . Assign: Sum over i -labels, integrate over positions of vertices.
Introduction BV-BFV formalism, outline Examples Exact discretizations Reference. Abelian and non-abelian BF : P. Mnev, Discrete BF theory, arXiv:0809.1160 (– for M closed), A. S. Cattaneo, P. Mnev, N. Reshetikhin, Cellular BV-BFV- BF theory. (– with gluing). 1D Chern-Simons: A. Alekseev, P. Mnev, One-dimensional Chern-Simons theory, Comm. Math. Phys. 307 1 (2011) 185–227.
Introduction BV-BFV formalism, outline Examples Exact discretizations Example: abelian BF theory on a cobordism with a cell decomposition. Reference. A. S. Cattaneo, P. Mnev, N. Reshetikhin, Cellular BV-BFV- BF theory. M an n -cobordism, T a cell decomposition. T ∨ – dual decomposition.
Introduction BV-BFV formalism, outline Examples Exact discretizations Example: abelian BF theory on a cobordism with a cell decomposition. Reference. A. S. Cattaneo, P. Mnev, N. Reshetikhin, Cellular BV-BFV- BF theory. M an n -cobordism, T a cell decomposition. T ∨ – dual decomposition. F T = C • ( T )[1] ⊕ C • ( T ∨ )[ n − 2] ∋ ( A, B ) .
Introduction BV-BFV formalism, outline Examples Exact discretizations Example: abelian BF theory on a cobordism with a cell decomposition. Reference. A. S. Cattaneo, P. Mnev, N. Reshetikhin, Cellular BV-BFV- BF theory. M an n -cobordism, T a cell decomposition. T ∨ – dual decomposition. F T = C • ( T )[1] ⊕ C • ( T ∨ )[ n − 2] ∋ ( A, B ) . BV 2-form ω comes from the Lefschetz pairing C k ( T, T in ) ⊗ C n − k ( T ∨ , T ∨ out ) → R , extended by zero to T in , T ∨ out .
Introduction BV-BFV formalism, outline Examples Exact discretizations Example: abelian BF theory on a cobordism with a cell decomposition. Reference. A. S. Cattaneo, P. Mnev, N. Reshetikhin, Cellular BV-BFV- BF theory. M an n -cobordism, T a cell decomposition. T ∨ – dual decomposition. F T = C • ( T )[1] ⊕ C • ( T ∨ )[ n − 2] ∋ ( A, B ) . BV 2-form ω comes from the Lefschetz pairing C k ( T, T in ) ⊗ C n − k ( T ∨ , T ∨ out ) → R , extended by zero to T in , T ∨ out . S = � B, dA � T − � B, A � T out .
Introduction BV-BFV formalism, outline Examples Exact discretizations Example: abelian BF theory on a cobordism with a cell decomposition – continued. Quantization – as in continuum case, but replacing differential forms by cellular cochains. R -torsion appears as a measure-theoretic integral rather than regularized ∞ -dimensional integral.
Introduction BV-BFV formalism, outline Examples Exact discretizations Example: abelian BF theory on a cobordism with a cell decomposition – continued. Quantization – as in continuum case, but replacing differential forms by cellular cochains. R -torsion appears as a measure-theoretic integral rather than regularized ∞ -dimensional integral. Data on T can itself be viewed as quantum BV-BFV theory: � S · µ T, � satisfies mQME ( i i Z = e � Ω − i � ∆ T ) Z = 0 with Ω = − i � � d A , ∂ ∂ A � T in − i � � d B , ∂ ∂ B � T out .
Introduction BV-BFV formalism, outline Examples Exact discretizations Example: abelian BF theory on a cobordism with a cell decomposition – continued. Quantization – as in continuum case, but replacing differential forms by cellular cochains. R -torsion appears as a measure-theoretic integral rather than regularized ∞ -dimensional integral. Data on T can itself be viewed as quantum BV-BFV theory: � S · µ T, � satisfies mQME ( i i Z = e � Ω − i � ∆ T ) Z = 0 with Ω = − i � � d A , ∂ ∂ A � T in − i � � d B , ∂ ∂ B � T out . Consistent with BV pushforwards along cellular aggregations T ′ → T .
Introduction BV-BFV formalism, outline Examples Conclusion Further program → Corners. 1 Partition function for a “building block” (cell) in interesting 2 examples. Compute cohomology of Ω ∂ , e.g. in PSM. 3 More general polarizations, generalized Hitchin’s connection. 4 Chern-Simons theory in BV-BFV formalism: extension of 5 Axelrod-Singer’s treatment to 3-manifolds with boundary/corners. Comparison with Witten-Reshetikhin-Turaev non-perturbative answers. Prove the conjecture that k → ∞ asymptotics of the RT invariant on a closed 3-manifold is given by Axelrod-Singer expansion. Observables supported on submanifolds. 6
Recommend
More recommend
