BF theory on cobordisms endowed with cellular decomposition Pavel - PowerPoint PPT Presentation

BF theory on cobordisms endowed with cellular decomposition Pavel Mnev Max Planck Institute for Mathematics, Bonn Poisson 2016, ETH Z¨ urich, July 4, 2016 Joint work with Alberto S. Cattaneo and Nikolai Reshetikhin

Introduction BV-BFV formalism, outline Abelian cellular BF theory Non-abelian cellular BF theory Plan BV-BFV formalism for gauge theories on manifolds with boundary: 1 an outline.

Introduction BV-BFV formalism, outline Abelian cellular BF theory Non-abelian cellular BF theory Plan BV-BFV formalism for gauge theories on manifolds with boundary: 1 an outline. Cellular abelian BF theory. 2

Introduction BV-BFV formalism, outline Abelian cellular BF theory Non-abelian cellular BF theory Plan BV-BFV formalism for gauge theories on manifolds with boundary: 1 an outline. Cellular abelian BF theory. 2 Cellular non-abelian BF theory 3

Introduction BV-BFV formalism, outline Abelian cellular BF theory Non-abelian cellular BF theory Classical BV-BFV theories Reminder. A classical BV theory on a closed spacetime manifold M :

Introduction BV-BFV formalism, outline Abelian cellular BF theory Non-abelian cellular BF theory Classical BV-BFV theories Reminder. A classical BV theory on a closed spacetime manifold M : F

Introduction BV-BFV formalism, outline Abelian cellular BF theory Non-abelian cellular BF theory 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 Abelian cellular BF theory Non-abelian cellular BF theory 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 Abelian cellular BF theory Non-abelian cellular BF theory 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 Abelian cellular BF theory Non-abelian cellular BF theory 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 Abelian cellular BF theory Non-abelian cellular BF theory 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 − − − − → (Φ ∂ , ω ∂ = δα ∂ , Q ∂ , S ∂ ) – phase space

Introduction BV-BFV formalism, outline Abelian cellular BF theory Non-abelian cellular BF theory 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 − − − − → (Φ ∂ , ω ∂ = δα ∂ , Q ∂ , S ∂ 1 ) – phase space 0 1 Subscripts =“ghost numbers”.

Introduction BV-BFV formalism, outline Abelian cellular BF theory Non-abelian cellular BF theory 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 − − − − → (Φ ∂ , 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 Abelian cellular BF theory Non-abelian cellular BF theory 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 − − − − → (Φ ∂ , 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 × Φ Σ F M II

Introduction BV-BFV formalism, outline Abelian cellular BF theory Non-abelian cellular BF theory 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 − − − − → (Φ ∂ , 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 × Φ Σ F M II This picture extends to higher-codimension strata!

Introduction BV-BFV formalism, outline Abelian cellular BF theory Non-abelian cellular BF theory Quantum BV-BFV theories Quantum BV-BFV formalism. Reference: A. S. Cattaneo, P. Mnev, N. Reshetikhin, Perturbative quantum gauge theories on manifolds with boundary, arXiv:1507.01221. ( H • Σ closed, dim Σ = n − 1 �→ Σ , Ω Σ )

Introduction BV-BFV formalism, outline Abelian cellular BF theory Non-abelian cellular BF theory Quantum BV-BFV theories Quantum BV-BFV formalism. Reference: A. S. Cattaneo, P. Mnev, N. Reshetikhin, Perturbative quantum gauge theories on manifolds with boundary, arXiv:1507.01221. ( H • Σ closed, dim Σ = n − 1 �→ Σ , Ω Σ ) M , dim M = n �→ ( F res , ω res )

Introduction BV-BFV formalism, outline Abelian cellular BF theory Non-abelian cellular BF theory Quantum BV-BFV theories Quantum BV-BFV formalism. Reference: A. S. Cattaneo, P. Mnev, N. Reshetikhin, Perturbative quantum gauge theories on manifolds with boundary, arXiv:1507.01221. ( 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 Abelian cellular BF theory Non-abelian cellular BF theory Quantum BV-BFV theories Quantum BV-BFV formalism. Reference: A. S. Cattaneo, P. Mnev, N. Reshetikhin, Perturbative quantum gauge theories on manifolds with boundary, arXiv:1507.01221. ( 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 Abelian cellular BF theory Non-abelian cellular BF theory Quantum BV-BFV theories Quantum BV-BFV formalism. Reference: A. S. Cattaneo, P. Mnev, N. Reshetikhin, Perturbative quantum gauge theories on manifolds with boundary, arXiv:1507.01221. ( 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 Abelian cellular BF theory Non-abelian cellular BF theory Quantum BV-BFV theories Quantum BV-BFV formalism. Reference: A. S. Cattaneo, P. Mnev, N. Reshetikhin, Perturbative quantum gauge theories on manifolds with boundary, arXiv:1507.01221. ( 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 Abelian cellular BF theory Non-abelian cellular BF theory Quantum BV-BFV theories Quantum BV-BFV formalism. Reference: A. S. Cattaneo, P. Mnev, N. Reshetikhin, Perturbative quantum gauge theories on manifolds with boundary, arXiv:1507.01221. ( 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 )