Perturbative topological field theory with Segal-like gluing Pavel Mnev Max Planck Institute for Mathematics, Bonn ICMP, Santiago de Chile, July 27, 2015 Joint work with Alberto S. Cattaneo and Nikolai Reshetikhin
Introduction BV-BFV formalism, outline Examples Plan Introduction: calculating partition functions by cut/paste. 1
Introduction BV-BFV formalism, outline Examples Plan Introduction: calculating partition functions by cut/paste. 1 BV-BFV formalism for gauge theories on manifolds with boundary: 2 an outline.
Introduction BV-BFV formalism, outline Examples Plan Introduction: calculating partition functions by cut/paste. 1 BV-BFV formalism for gauge theories on manifolds with boundary: 2 an outline. Abelian BF theory in BV-BFV formalism. 3
Introduction BV-BFV formalism, outline Examples Plan Introduction: calculating partition functions by cut/paste. 1 BV-BFV formalism for gauge theories on manifolds with boundary: 2 an outline. Abelian BF theory in BV-BFV formalism. 3 Further examples: Poisson sigma model, cellular models. 4
Introduction BV-BFV formalism, outline Examples Cut/paste philosophy Introduction: calculating partition functions by cut/paste. Idea: � � � � �� � � Z = Z , Z
Introduction BV-BFV formalism, outline Examples Cut/paste philosophy Introduction: 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 Cut/paste philosophy Introduction: 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 Cut/paste philosophy Example: 2D TQFT Z can be expressed in terms of building blocks: � � Z : C → H S 1 1 � � : H S 1 → C Z 2 Z : H S 1 ⊗ H S 1 → H S 1 3 : H S 1 → H S 1 ⊗ H S 1 Z 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
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 Q ∈ X ( F ) , odd, 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 Q ∈ X ( F ) , odd, Q 2 = 0 S ∈ C ∞ ( F ) , 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 Q ∈ X ( F ) , odd, Q 2 = 0 S ∈ C ∞ ( F ) , 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: − − − − → ( F , Q, S ) – space of fields M ω, � 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: − − − − → ( F , 0 ) – space of fields M − 1 , ω Q , S 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: − − − − → ( F , Q, S ) – space of fields M ω, � 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: − − − − → ( F , Q, S ) – space of fields M ω, � 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: − − − − → ( F , Q, S ) – space of fields M ω, � 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 δ A ∧ δ A , Q, S ) M 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
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