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Higher Structures in Algebraic Quantum Field Theory Alexander - PowerPoint PPT Presentation

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Higher Structures in Algebraic Quantum Field Theory Alexander Schenkel School of Mathematical Sciences, University of Nottingham Mathematics of Interacting QFT Models, 15 July 2019, York. Based on joint works with M. Benini and different

  1. � AQFTs and colored operads ⋄ The definition of AQFT allows for the following generalizations: • Vec K � cocomplete closed symmetric monoidal category T • Loc , causally disjoint and Cauchy � orthogonal category C = ( C , ⊥ ) Def: AQFT ( C , T ) denotes the category of T -valued AQFTs on C Thm: (i) There exists a colored operad O C such that AQFT ( C , T ) ≃ Alg O C ( T ) (ii) Every orthogonal functor F : C → D defines an adjunction � AQFT ( D , T ) : F ∗ F ! : AQFT ( C , T ) Alexander Schenkel ∞ AQFT York 19 4 / 13

  2. � AQFTs and colored operads ⋄ The definition of AQFT allows for the following generalizations: • Vec K � cocomplete closed symmetric monoidal category T • Loc , causally disjoint and Cauchy � orthogonal category C = ( C , ⊥ ) Def: AQFT ( C , T ) denotes the category of T -valued AQFTs on C Thm: (i) There exists a colored operad O C such that AQFT ( C , T ) ≃ Alg O C ( T ) (ii) Every orthogonal functor F : C → D defines an adjunction � AQFT ( D , T ) : F ∗ F ! : AQFT ( C , T ) ⋄ This is extremely useful for local-to-global constructions: Alexander Schenkel ∞ AQFT York 19 4 / 13

  3. � AQFTs and colored operads ⋄ The definition of AQFT allows for the following generalizations: • Vec K � cocomplete closed symmetric monoidal category T • Loc , causally disjoint and Cauchy � orthogonal category C = ( C , ⊥ ) Def: AQFT ( C , T ) denotes the category of T -valued AQFTs on C Thm: (i) There exists a colored operad O C such that AQFT ( C , T ) ≃ Alg O C ( T ) (ii) Every orthogonal functor F : C → D defines an adjunction � AQFT ( D , T ) : F ∗ F ! : AQFT ( C , T ) ⋄ This is extremely useful for local-to-global constructions: • Let j : Loc ⋄ → Loc be full orthogonal subcat of “diamonds” ( M ∼ = R m ) Alexander Schenkel ∞ AQFT York 19 4 / 13

  4. � � AQFTs and colored operads ⋄ The definition of AQFT allows for the following generalizations: • Vec K � cocomplete closed symmetric monoidal category T • Loc , causally disjoint and Cauchy � orthogonal category C = ( C , ⊥ ) Def: AQFT ( C , T ) denotes the category of T -valued AQFTs on C Thm: (i) There exists a colored operad O C such that AQFT ( C , T ) ≃ Alg O C ( T ) (ii) Every orthogonal functor F : C → D defines an adjunction � AQFT ( D , T ) : F ∗ F ! : AQFT ( C , T ) ⋄ This is extremely useful for local-to-global constructions: • Let j : Loc ⋄ → Loc be full orthogonal subcat of “diamonds” ( M ∼ = R m ) • We get extension-restriction adjunction � AQFT ( Loc , T ) : j ∗ = res ext = j ! : AQFT ( Loc ⋄ , T ) Alexander Schenkel ∞ AQFT York 19 4 / 13

  5. � � AQFTs and colored operads ⋄ The definition of AQFT allows for the following generalizations: • Vec K � cocomplete closed symmetric monoidal category T • Loc , causally disjoint and Cauchy � orthogonal category C = ( C , ⊥ ) Def: AQFT ( C , T ) denotes the category of T -valued AQFTs on C Thm: (i) There exists a colored operad O C such that AQFT ( C , T ) ≃ Alg O C ( T ) (ii) Every orthogonal functor F : C → D defines an adjunction � AQFT ( D , T ) : F ∗ F ! : AQFT ( C , T ) ⋄ This is extremely useful for local-to-global constructions: • Let j : Loc ⋄ → Loc be full orthogonal subcat of “diamonds” ( M ∼ = R m ) • We get extension-restriction adjunction � AQFT ( Loc , T ) : j ∗ = res ext = j ! : AQFT ( Loc ⋄ , T ) ∼ = • Descent condition: A ∈ AQFT ( Loc , T ) is j -local iff ǫ A : ext res A − → A Alexander Schenkel ∞ AQFT York 19 4 / 13

  6. � � AQFTs and colored operads ⋄ The definition of AQFT allows for the following generalizations: • Vec K � cocomplete closed symmetric monoidal category T • Loc , causally disjoint and Cauchy � orthogonal category C = ( C , ⊥ ) Def: AQFT ( C , T ) denotes the category of T -valued AQFTs on C Thm: (i) There exists a colored operad O C such that AQFT ( C , T ) ≃ Alg O C ( T ) (ii) Every orthogonal functor F : C → D defines an adjunction � AQFT ( D , T ) : F ∗ F ! : AQFT ( C , T ) ⋄ This is extremely useful for local-to-global constructions: • Let j : Loc ⋄ → Loc be full orthogonal subcat of “diamonds” ( M ∼ = R m ) • We get extension-restriction adjunction � AQFT ( Loc , T ) : j ∗ = res ext = j ! : AQFT ( Loc ⋄ , T ) ∼ = • Descent condition: A ∈ AQFT ( Loc , T ) is j -local iff ǫ A : ext res A − → A Ex: Linear Klein-Gordon theory is j -local (uses also results by [Lang] ) Alexander Schenkel ∞ AQFT York 19 4 / 13

  7. Higher structures in gauge theory Alexander Schenkel ∞ AQFT York 19 5 / 13

  8. What is a gauge theory? “Ordinary” field theory: Set/Space of fields Φ ′ Φ ′′ Φ Alexander Schenkel ∞ AQFT York 19 5 / 13

  9. What is a gauge theory? “Ordinary” field theory: Gauge theory: Set/Space of fields Grpd/Stack of fields Φ ′ A ′ g ′′ g ′ g Φ ′′ A ′′ Φ A Alexander Schenkel ∞ AQFT York 19 5 / 13

  10. What is a gauge theory? “Ordinary” field theory: Gauge theory: Set/Space of fields Grpd/Stack of fields Φ ′ A ′ g ′′ g ′ g Φ ′′ A ′′ Φ A Ex: Groupoid of principal G -bundles with connection on U ∼ = R m   A ∈ Ω 1 ( U, g ) Obj: BG con ( U ) = g ∈ C ∞ ( U,G ) � A ⊳ g = g − 1 Ag + g − 1 d g  Mor: A Alexander Schenkel ∞ AQFT York 19 5 / 13

  11. What is a gauge theory? “Ordinary” field theory: Gauge theory: Set/Space of fields Grpd/Stack of fields Φ ′ A ′ g ′′ g ′ g Φ ′′ A ′′ Φ A Ex: Groupoid of principal G -bundles with connection on U ∼ = R m   A ∈ Ω 1 ( U, g ) Obj: BG con ( U ) = g ∈ C ∞ ( U,G ) � A ⊳ g = g − 1 Ag + g − 1 d g  Mor: A Invariant information: = Ω 1 ( U, g ) /C ∞ ( U, G ) (“gauge orbit space”) � � • π 0 BG con ( U ) Alexander Schenkel ∞ AQFT York 19 5 / 13

  12. What is a gauge theory? “Ordinary” field theory: Gauge theory: Set/Space of fields Grpd/Stack of fields Φ ′ A ′ g ′′ g ′ g Φ ′′ A ′′ Φ A Ex: Groupoid of principal G -bundles with connection on U ∼ = R m   A ∈ Ω 1 ( U, g ) Obj: BG con ( U ) = g ∈ C ∞ ( U,G ) � A ⊳ g = g − 1 Ag + g − 1 d g  Mor: A Invariant information: = Ω 1 ( U, g ) /C ∞ ( U, G ) (“gauge orbit space”) � � • π 0 BG con ( U ) = { g ∈ C ∞ ( U, G ) : A = A ⊳ g } (stabilizers/“loops”) � � • π 1 BG con ( U ) , A Alexander Schenkel ∞ AQFT York 19 5 / 13

  13. What is a gauge theory? “Ordinary” field theory: Gauge theory: Set/Space of fields Grpd/Stack of fields Φ ′ A ′ g ′′ g ′ g Φ ′′ A ′′ Φ A Ex: Groupoid of principal G -bundles with connection on U ∼ = R m   A ∈ Ω 1 ( U, g ) Obj: BG con ( U ) = g ∈ C ∞ ( U,G ) � A ⊳ g = g − 1 Ag + g − 1 d g  Mor: A Invariant information: = Ω 1 ( U, g ) /C ∞ ( U, G ) (“gauge orbit space”) � � • π 0 BG con ( U ) = { g ∈ C ∞ ( U, G ) : A = A ⊳ g } (stabilizers/“loops”) � � • π 1 BG con ( U ) , A ! Grpd is a 2 -category (or model category) with weak equivalences the categorical equivalences ⇒ need for higher (or derived) functors! Alexander Schenkel ∞ AQFT York 19 5 / 13

  14. Why are these higher structures important? 1. π 1 encodes essential information of the gauge theory: Alexander Schenkel ∞ AQFT York 19 6 / 13

  15. Why are these higher structures important? 1. π 1 encodes essential information of the gauge theory: Consider structure group G = U (1) or R . Then � � ∼ � � ∼ = Ω 1 ( U ) / dΩ 0 ( U ) π 0 BG con ( U ) , π 1 BG con ( U ) , A = G � �� � � �� � sees G doesn’t see G Alexander Schenkel ∞ AQFT York 19 6 / 13

  16. Why are these higher structures important? 1. π 1 encodes essential information of the gauge theory: Consider structure group G = U (1) or R . Then � � ∼ � � ∼ = Ω 1 ( U ) / dΩ 0 ( U ) π 0 BG con ( U ) , π 1 BG con ( U ) , A = G � �� � � �� � sees G doesn’t see G 2. Higher structures are crucial for descent: Alexander Schenkel ∞ AQFT York 19 6 / 13

  17. � � �� ��� Why are these higher structures important? 1. π 1 encodes essential information of the gauge theory: Consider structure group G = U (1) or R . Then � � ∼ � � ∼ = Ω 1 ( U ) / dΩ 0 ( U ) π 0 BG con ( U ) , π 1 BG con ( U ) , A = G � �� � � �� � sees G doesn’t see G 2. Higher structures are crucial for descent: Let M be manifold with good open cover { U i ⊆ M } . Then � � � �� � � holim i BG con ( U i ) ij BG con ( U ij ) ijk BG con ( U ijk ) � · · · is the groupoid of all principal G -bundles with connection on M Alexander Schenkel ∞ AQFT York 19 6 / 13

  18. � � �� ��� Why are these higher structures important? 1. π 1 encodes essential information of the gauge theory: Consider structure group G = U (1) or R . Then � � ∼ � � ∼ = Ω 1 ( U ) / dΩ 0 ( U ) π 0 BG con ( U ) , π 1 BG con ( U ) , A = G � �� � � �� � sees G doesn’t see G 2. Higher structures are crucial for descent: Let M be manifold with good open cover { U i ⊆ M } . Then � � � �� � � holim i BG con ( U i ) ij BG con ( U ij ) ijk BG con ( U ijk ) � · · · is the groupoid of all principal G -bundles with connection on M 3. π 1 detects “fake” gauge symmetries: Alexander Schenkel ∞ AQFT York 19 6 / 13

  19. � �� � ��� Why are these higher structures important? 1. π 1 encodes essential information of the gauge theory: Consider structure group G = U (1) or R . Then � � ∼ � � ∼ = Ω 1 ( U ) / dΩ 0 ( U ) π 0 BG con ( U ) , π 1 BG con ( U ) , A = G � �� � � �� � sees G doesn’t see G 2. Higher structures are crucial for descent: Let M be manifold with good open cover { U i ⊆ M } . Then � � � �� � � holim i BG con ( U i ) ij BG con ( U ij ) ijk BG con ( U ijk ) � · · · is the groupoid of all principal G -bundles with connection on M 3. π 1 detects “fake” gauge symmetries: Let M be manifold and consider groupoid   (Φ 1 , Φ 2 ) ∈ C ∞ ( M ) × C ∞ ( M ) Obj: P ( M ) = ǫ ∈ C ∞ ( M ) � (Φ 1 + ǫ, Φ 2 + ǫ )  Mor: (Φ 1 , Φ 2 ) Alexander Schenkel ∞ AQFT York 19 6 / 13

  20. ��� � �� � Why are these higher structures important? 1. π 1 encodes essential information of the gauge theory: Consider structure group G = U (1) or R . Then � � ∼ � � ∼ = Ω 1 ( U ) / dΩ 0 ( U ) π 0 BG con ( U ) , π 1 BG con ( U ) , A = G � �� � � �� � sees G doesn’t see G 2. Higher structures are crucial for descent: Let M be manifold with good open cover { U i ⊆ M } . Then � � � �� � � holim i BG con ( U i ) ij BG con ( U ij ) ijk BG con ( U ijk ) � · · · is the groupoid of all principal G -bundles with connection on M 3. π 1 detects “fake” gauge symmetries: Let M be manifold and consider groupoid   (Φ 1 , Φ 2 ) ∈ C ∞ ( M ) × C ∞ ( M ) Obj: P ( M ) = ǫ ∈ C ∞ ( M ) � (Φ 1 + ǫ, Φ 2 + ǫ )  Mor: (Φ 1 , Φ 2 ) Because all π 1 ’s are trivial, this is a “fake” gauge symmetry. Alexander Schenkel ∞ AQFT York 19 6 / 13

  21. ��� � �� � Why are these higher structures important? 1. π 1 encodes essential information of the gauge theory: Consider structure group G = U (1) or R . Then � � ∼ � � ∼ = Ω 1 ( U ) / dΩ 0 ( U ) π 0 BG con ( U ) , π 1 BG con ( U ) , A = G � �� � � �� � sees G doesn’t see G 2. Higher structures are crucial for descent: Let M be manifold with good open cover { U i ⊆ M } . Then � � � �� � � holim i BG con ( U i ) ij BG con ( U ij ) ijk BG con ( U ijk ) � · · · is the groupoid of all principal G -bundles with connection on M 3. π 1 detects “fake” gauge symmetries: Let M be manifold and consider groupoid   (Φ 1 , Φ 2 ) ∈ C ∞ ( M ) × C ∞ ( M ) Obj: P ( M ) = ǫ ∈ C ∞ ( M ) � (Φ 1 + ǫ, Φ 2 + ǫ )  Mor: (Φ 1 , Φ 2 ) Because all π 1 ’s are trivial, this is a “fake” gauge symmetry. Indeed, there is an equivalence P ( M ) → C ∞ ( M ) , (Φ 1 , Φ 2 ) �→ Φ 1 − Φ 2 to the scalar field. Alexander Schenkel ∞ AQFT York 19 6 / 13

  22. Smooth cochain algebras on stacks ⋄ To study geometry of gauge fields, one needs “smooth groupoids” a.k.a. stacks Alexander Schenkel ∞ AQFT York 19 7 / 13

  23. Smooth cochain algebras on stacks ⋄ To study geometry of gauge fields, one needs “smooth groupoids” a.k.a. stacks ⋄ A stack is a presheaf X : Cart op → Grpd satisfying homotopical descent Alexander Schenkel ∞ AQFT York 19 7 / 13

  24. Smooth cochain algebras on stacks X ⋄ To study geometry of gauge fields, one needs “smooth groupoids” a.k.a. stacks ⋄ A stack is a presheaf X : Cart op → Grpd satisfying homotopical descent Alexander Schenkel ∞ AQFT York 19 7 / 13

  25. Smooth cochain algebras on stacks R 0 X ⋄ To study geometry of gauge fields, one needs “smooth groupoids” a.k.a. stacks ⋄ A stack is a presheaf X : Cart op → Grpd satisfying homotopical descent Alexander Schenkel ∞ AQFT York 19 7 / 13

  26. Smooth cochain algebras on stacks R 0 X ⋄ To study geometry of gauge fields, one needs “smooth groupoids” a.k.a. stacks R 1 ⋄ A stack is a presheaf X : Cart op → Grpd satisfying homotopical descent Alexander Schenkel ∞ AQFT York 19 7 / 13

  27. Smooth cochain algebras on stacks R 0 X ⋄ To study geometry of gauge fields, one needs “smooth groupoids” a.k.a. stacks R 1 ⋄ A stack is a presheaf X : Cart op → Grpd satisfying homotopical descent R 2 Alexander Schenkel ∞ AQFT York 19 7 / 13

  28. Smooth cochain algebras on stacks R 0 X ⋄ To study geometry of gauge fields, one needs “smooth groupoids” a.k.a. stacks R 1 ⋄ A stack is a presheaf X : Cart op → Grpd satisfying homotopical descent R 2 ⋄ (A)QFT needs “observable algebras”, but what are functions on stacks? Alexander Schenkel ∞ AQFT York 19 7 / 13

  29. � Smooth cochain algebras on stacks R 0 X ⋄ To study geometry of gauge fields, one needs “smooth groupoids” a.k.a. stacks R 1 ⋄ A stack is a presheaf X : Cart op → Grpd satisfying homotopical descent R 2 ⋄ (A)QFT needs “observable algebras”, but what are functions on stacks? ⋄ Smooth normalized cochains: N ∗∞ ( − , K ) � PSh ( Cart , Ch K ) � Ch op Stacks K Map ∞ ( − , K ) N ∗ ( − , K ) Alexander Schenkel ∞ AQFT York 19 7 / 13

  30. � Smooth cochain algebras on stacks R 0 X ⋄ To study geometry of gauge fields, one needs “smooth groupoids” a.k.a. stacks R 1 ⋄ A stack is a presheaf X : Cart op → Grpd satisfying homotopical descent R 2 ⋄ (A)QFT needs “observable algebras”, but what are functions on stacks? ⋄ Smooth normalized cochains: N ∗∞ ( − , K ) � PSh ( Cart , Ch K ) � Ch op Stacks K Map ∞ ( − , K ) N ∗ ( − , K ) (i) N ∗∞ ( − , K ) is left Quillen functor, i.e. left derived functor L N ∗∞ ( − , K ) exists Prop: Alexander Schenkel ∞ AQFT York 19 7 / 13

  31. � Smooth cochain algebras on stacks R 0 X ⋄ To study geometry of gauge fields, one needs “smooth groupoids” a.k.a. stacks R 1 ⋄ A stack is a presheaf X : Cart op → Grpd satisfying homotopical descent R 2 ⋄ (A)QFT needs “observable algebras”, but what are functions on stacks? ⋄ Smooth normalized cochains: N ∗∞ ( − , K ) � PSh ( Cart , Ch K ) � Ch op Stacks K Map ∞ ( − , K ) N ∗ ( − , K ) (i) N ∗∞ ( − , K ) is left Quillen functor, i.e. left derived functor L N ∗∞ ( − , K ) exists Prop: (ii) L N ∗∞ ( − , K ) : Stacks → Alg E ∞ ( Ch K ) op takes values in E ∞ -algebras Alexander Schenkel ∞ AQFT York 19 7 / 13

  32. � Smooth cochain algebras on stacks R 0 X ⋄ To study geometry of gauge fields, one needs “smooth groupoids” a.k.a. stacks R 1 ⋄ A stack is a presheaf X : Cart op → Grpd satisfying homotopical descent R 2 ⋄ (A)QFT needs “observable algebras”, but what are functions on stacks? ⋄ Smooth normalized cochains: N ∗∞ ( − , K ) � PSh ( Cart , Ch K ) � Ch op Stacks K Map ∞ ( − , K ) N ∗ ( − , K ) (i) N ∗∞ ( − , K ) is left Quillen functor, i.e. left derived functor L N ∗∞ ( − , K ) exists Prop: (ii) L N ∗∞ ( − , K ) : Stacks → Alg E ∞ ( Ch K ) op takes values in E ∞ -algebras ! Main observation: Classical observables in a gauge theory are described by dg-algebras that are only homotopy-coherently commutative! Alexander Schenkel ∞ AQFT York 19 7 / 13

  33. Higher structures in AQFT Alexander Schenkel ∞ AQFT York 19 8 / 13

  34. Homotopy AQFTs: Definition and basic properties ⋄ Recall AQFT ( C , T ) = Alg O C ( T ) requires choice of target category T Alexander Schenkel ∞ AQFT York 19 8 / 13

  35. Homotopy AQFTs: Definition and basic properties ⋄ Recall AQFT ( C , T ) = Alg O C ( T ) requires choice of target category T ⋄ Take T = Ch K to include dg-algebras of observables of a gauge theory! Alexander Schenkel ∞ AQFT York 19 8 / 13

  36. Homotopy AQFTs: Definition and basic properties ⋄ Recall AQFT ( C , T ) = Alg O C ( T ) requires choice of target category T ⋄ Take T = Ch K to include dg-algebras of observables of a gauge theory! ⋄ But what about homotopy coherent algebraic structures (e.g. E ∞ -algebras)? Alexander Schenkel ∞ AQFT York 19 8 / 13

  37. Homotopy AQFTs: Definition and basic properties ⋄ Recall AQFT ( C , T ) = Alg O C ( T ) requires choice of target category T ⋄ Take T = Ch K to include dg-algebras of observables of a gauge theory! ⋄ But what about homotopy coherent algebraic structures (e.g. E ∞ -algebras)? Def: A Ch K -valued homotopy AQFT on C is an algebra over any Σ -cofibrant ∼ resolution O C , ∞ ։ O C of the AQFT dg-operad O C ∈ Op ( Ch K ) . Alexander Schenkel ∞ AQFT York 19 8 / 13

  38. Homotopy AQFTs: Definition and basic properties ⋄ Recall AQFT ( C , T ) = Alg O C ( T ) requires choice of target category T ⋄ Take T = Ch K to include dg-algebras of observables of a gauge theory! ⋄ But what about homotopy coherent algebraic structures (e.g. E ∞ -algebras)? Def: A Ch K -valued homotopy AQFT on C is an algebra over any Σ -cofibrant ∼ resolution O C , ∞ ։ O C of the AQFT dg-operad O C ∈ Op ( Ch K ) . AQFT ∞ ( C ) := Alg O C , ∞ ( Ch K ) denotes category of homotopy AQFTs. Alexander Schenkel ∞ AQFT York 19 8 / 13

  39. Homotopy AQFTs: Definition and basic properties ⋄ Recall AQFT ( C , T ) = Alg O C ( T ) requires choice of target category T ⋄ Take T = Ch K to include dg-algebras of observables of a gauge theory! ⋄ But what about homotopy coherent algebraic structures (e.g. E ∞ -algebras)? Def: A Ch K -valued homotopy AQFT on C is an algebra over any Σ -cofibrant ∼ resolution O C , ∞ ։ O C of the AQFT dg-operad O C ∈ Op ( Ch K ) . AQFT ∞ ( C ) := Alg O C , ∞ ( Ch K ) denotes category of homotopy AQFTs. Prop: (i) AQFT ∞ ( C ) is a model category with weak equivalences the natural quasi-isomorphisms Alexander Schenkel ∞ AQFT York 19 8 / 13

  40. Homotopy AQFTs: Definition and basic properties ⋄ Recall AQFT ( C , T ) = Alg O C ( T ) requires choice of target category T ⋄ Take T = Ch K to include dg-algebras of observables of a gauge theory! ⋄ But what about homotopy coherent algebraic structures (e.g. E ∞ -algebras)? Def: A Ch K -valued homotopy AQFT on C is an algebra over any Σ -cofibrant ∼ resolution O C , ∞ ։ O C of the AQFT dg-operad O C ∈ Op ( Ch K ) . AQFT ∞ ( C ) := Alg O C , ∞ ( Ch K ) denotes category of homotopy AQFTs. Prop: (i) AQFT ∞ ( C ) is a model category with weak equivalences the natural quasi-isomorphisms (ii) AQFT ∞ ( C ) does not depend on the choice of resolution (up to zig-zags of Quillen equivalences) Alexander Schenkel ∞ AQFT York 19 8 / 13

  41. Homotopy AQFTs: Definition and basic properties ⋄ Recall AQFT ( C , T ) = Alg O C ( T ) requires choice of target category T ⋄ Take T = Ch K to include dg-algebras of observables of a gauge theory! ⋄ But what about homotopy coherent algebraic structures (e.g. E ∞ -algebras)? Def: A Ch K -valued homotopy AQFT on C is an algebra over any Σ -cofibrant ∼ resolution O C , ∞ ։ O C of the AQFT dg-operad O C ∈ Op ( Ch K ) . AQFT ∞ ( C ) := Alg O C , ∞ ( Ch K ) denotes category of homotopy AQFTs. Prop: (i) AQFT ∞ ( C ) is a model category with weak equivalences the natural quasi-isomorphisms (ii) AQFT ∞ ( C ) does not depend on the choice of resolution (up to zig-zags of Quillen equivalences) ∼ (iii) If char K = 0 , then id : O C ։ O C is Σ -cofibrant resolution ⇒ all homotopy AQFTs in char K = 0 (i.e. in physics) can be strictified Alexander Schenkel ∞ AQFT York 19 8 / 13

  42. Homotopy AQFTs: Definition and basic properties ⋄ Recall AQFT ( C , T ) = Alg O C ( T ) requires choice of target category T ⋄ Take T = Ch K to include dg-algebras of observables of a gauge theory! ⋄ But what about homotopy coherent algebraic structures (e.g. E ∞ -algebras)? Def: A Ch K -valued homotopy AQFT on C is an algebra over any Σ -cofibrant ∼ resolution O C , ∞ ։ O C of the AQFT dg-operad O C ∈ Op ( Ch K ) . AQFT ∞ ( C ) := Alg O C , ∞ ( Ch K ) denotes category of homotopy AQFTs. Prop: (i) AQFT ∞ ( C ) is a model category with weak equivalences the natural quasi-isomorphisms (ii) AQFT ∞ ( C ) does not depend on the choice of resolution (up to zig-zags of Quillen equivalences) ∼ (iii) If char K = 0 , then id : O C ։ O C is Σ -cofibrant resolution ⇒ all homotopy AQFTs in char K = 0 (i.e. in physics) can be strictified ∼ ։ O C ⊗ Com ∼ Ex: Component-wise tensor product O C ⊗ E ∞ = O C with the Barratt-Eccles operad defines a Σ -cofibrant resolution. Alexander Schenkel ∞ AQFT York 19 8 / 13

  43. Homotopy AQFTs: Definition and basic properties ⋄ Recall AQFT ( C , T ) = Alg O C ( T ) requires choice of target category T ⋄ Take T = Ch K to include dg-algebras of observables of a gauge theory! ⋄ But what about homotopy coherent algebraic structures (e.g. E ∞ -algebras)? Def: A Ch K -valued homotopy AQFT on C is an algebra over any Σ -cofibrant ∼ resolution O C , ∞ ։ O C of the AQFT dg-operad O C ∈ Op ( Ch K ) . AQFT ∞ ( C ) := Alg O C , ∞ ( Ch K ) denotes category of homotopy AQFTs. Prop: (i) AQFT ∞ ( C ) is a model category with weak equivalences the natural quasi-isomorphisms (ii) AQFT ∞ ( C ) does not depend on the choice of resolution (up to zig-zags of Quillen equivalences) ∼ (iii) If char K = 0 , then id : O C ։ O C is Σ -cofibrant resolution ⇒ all homotopy AQFTs in char K = 0 (i.e. in physics) can be strictified ∼ ։ O C ⊗ Com ∼ Ex: Component-wise tensor product O C ⊗ E ∞ = O C with the Barratt-Eccles operad defines a Σ -cofibrant resolution. Smooth normalized cochain algebras on (a diagram X : C op → Stacks of) stacks leads to homotopy AQFTs of this type. Alexander Schenkel ∞ AQFT York 19 8 / 13

  44. Example: Linear Yang-Mills as a homotopy AQFT ⋄ Linear gauge theory: (derived) stacks � chain complexes Ch K Alexander Schenkel ∞ AQFT York 19 9 / 13

  45. Example: Linear Yang-Mills as a homotopy AQFT ⋄ Linear gauge theory: (derived) stacks � chain complexes Ch K ⋄ Yang-Mills with structure group R is defined on M ∈ Loc by Alexander Schenkel ∞ AQFT York 19 9 / 13

  46. � Example: Linear Yang-Mills as a homotopy AQFT ⋄ Linear gauge theory: (derived) stacks � chain complexes Ch K ⋄ Yang-Mills with structure group R is defined on M ∈ Loc by (0) (1) � d � Ω 1 ( M ) Ω 0 ( M ) (1) Field complex F ( M ) = Alexander Schenkel ∞ AQFT York 19 9 / 13

  47. � Example: Linear Yang-Mills as a homotopy AQFT ⋄ Linear gauge theory: (derived) stacks � chain complexes Ch K ⋄ Yang-Mills with structure group R is defined on M ∈ Loc by (0) (1) � d � Ω 1 ( M ) Ω 0 ( M ) (1) Field complex F ( M ) = (2) Action functional S ( A ) = 1 � M d A ∧ ∗ d A 2 Alexander Schenkel ∞ AQFT York 19 9 / 13

  48. � � � � � Example: Linear Yang-Mills as a homotopy AQFT ⋄ Linear gauge theory: (derived) stacks � chain complexes Ch K ⋄ Yang-Mills with structure group R is defined on M ∈ Loc by (0) (1) � d � Ω 1 ( M ) Ω 0 ( M ) (1) Field complex F ( M ) = (2) Action functional S ( A ) = 1 � M d A ∧ ∗ d A 2 ⋄ Variation of action defines section of cotangent bundle   0 d Ω 1 ( M ) Ω 0 ( M ) 0 F ( M )     =  0 � (id ,δ d) �  δ v S � id �   T ∗ F ( M ) Ω 0 ( M ) Ω 1 ( M ) × Ω 1 ( M ) Ω 0 ( M ) − δπ 2 ι 1 d Alexander Schenkel ∞ AQFT York 19 9 / 13

  49. � � � � � � � Example: Linear Yang-Mills as a homotopy AQFT ⋄ Linear gauge theory: (derived) stacks � chain complexes Ch K ⋄ Yang-Mills with structure group R is defined on M ∈ Loc by (0) (1) � d � Ω 1 ( M ) Ω 0 ( M ) (1) Field complex F ( M ) = (2) Action functional S ( A ) = 1 � M d A ∧ ∗ d A 2 ⋄ Variation of action defines section of cotangent bundle   0 d Ω 1 ( M ) Ω 0 ( M ) 0 F ( M )     =  0 � (id ,δ d) �  δ v S � id �   T ∗ F ( M ) Ω 0 ( M ) Ω 1 ( M ) × Ω 1 ( M ) Ω 0 ( M ) − δπ 2 ι 1 d � F ( M ) Def: The solution complex is defined as the (linear) Sol ( M ) h derived critical locus of the action S , i.e. the δ v S � T ∗ F ( M ) following homotopy pullback in Ch K F ( M ) 0 Alexander Schenkel ∞ AQFT York 19 9 / 13

  50. � � � � � � � � � � Example: Linear Yang-Mills as a homotopy AQFT ⋄ Linear gauge theory: (derived) stacks � chain complexes Ch K ⋄ Yang-Mills with structure group R is defined on M ∈ Loc by (0) (1) � d � Ω 1 ( M ) Ω 0 ( M ) (1) Field complex F ( M ) = (2) Action functional S ( A ) = 1 � M d A ∧ ∗ d A 2 ⋄ Variation of action defines section of cotangent bundle   0 d Ω 1 ( M ) Ω 0 ( M ) 0 F ( M )     =  0 � (id ,δ d) �  δ v S � id �   T ∗ F ( M ) Ω 0 ( M ) Ω 1 ( M ) × Ω 1 ( M ) Ω 0 ( M ) − δπ 2 ι 1 d � F ( M ) Def: The solution complex is defined as the (linear) Sol ( M ) h derived critical locus of the action S , i.e. the δ v S � T ∗ F ( M ) following homotopy pullback in Ch K F ( M ) 0 � � ( − 2) ( − 1) (0) (1) δ δ d d Ω 0 ( M ) Ω 1 ( M ) Ω 1 ( M ) Ω 0 ( M ) Prop: Sol ( M ) = Alexander Schenkel ∞ AQFT York 19 9 / 13

  51. � � � � � � � � � � Example: Linear Yang-Mills as a homotopy AQFT ⋄ Linear gauge theory: (derived) stacks � chain complexes Ch K ⋄ Yang-Mills with structure group R is defined on M ∈ Loc by (0) (1) � d � Ω 1 ( M ) Ω 0 ( M ) (1) Field complex F ( M ) = (2) Action functional S ( A ) = 1 � M d A ∧ ∗ d A 2 ⋄ Variation of action defines section of cotangent bundle   0 d Ω 1 ( M ) Ω 0 ( M ) F ( M ) 0     =  0 � (id ,δ d) �  δ v S � id �   T ∗ F ( M ) Ω 0 ( M ) Ω 1 ( M ) × Ω 1 ( M ) Ω 0 ( M ) − δπ 2 ι 1 d � F ( M ) Def: The solution complex is defined as the (linear) Sol ( M ) h derived critical locus of the action S , i.e. the δ v S � T ∗ F ( M ) following homotopy pullback in Ch K F ( M ) 0 � � C ‡ A ‡ A C δ δ d d Ω 0 ( M ) Ω 1 ( M ) Ω 1 ( M ) Ω 0 ( M ) Prop: Sol ( M ) = Rem: Interpretation of Sol ( M ) in terms of BRST/BV formalism from physics Alexander Schenkel ∞ AQFT York 19 9 / 13

  52. Example: Linear Yang-Mills as a homotopy AQFT II ⋄ Every derived critical locus carries a [1] -shifted Poisson structure, explicitly: Alexander Schenkel ∞ AQFT York 19 10 / 13

  53. � � � Example: Linear Yang-Mills as a homotopy AQFT II ⋄ Every derived critical locus carries a [1] -shifted Poisson structure, explicitly: ( − 1) (0) (1) (2) − δ − d � δ d � Ω 0 Ω 1 Ω 1 Ω 0 • Smooth dual L ( M ) = c ( M ) c ( M ) c ( M ) c ( M ) Alexander Schenkel ∞ AQFT York 19 10 / 13

  54. � � � Example: Linear Yang-Mills as a homotopy AQFT II ⋄ Every derived critical locus carries a [1] -shifted Poisson structure, explicitly: ( − 1) (0) (1) (2) − δ − d � δ d � Ω 0 Ω 1 Ω 1 Ω 0 • Smooth dual L ( M ) = c ( M ) c ( M ) c ( M ) c ( M ) ⊆ • Canonical inclusion j : L ( M ) − → L pc / fc ( M ) − → Sol ( M )[1] Alexander Schenkel ∞ AQFT York 19 10 / 13

  55. � � � Example: Linear Yang-Mills as a homotopy AQFT II ⋄ Every derived critical locus carries a [1] -shifted Poisson structure, explicitly: ( − 1) (0) (1) (2) − δ − d � δ d � Ω 0 Ω 1 Ω 1 Ω 0 • Smooth dual L ( M ) = c ( M ) c ( M ) c ( M ) c ( M ) ⊆ • Canonical inclusion j : L ( M ) − → L pc / fc ( M ) − → Sol ( M )[1] id ⊗ j ev • Shifted Poisson structure Υ : L ( M ) ⊗ L ( M ) − → L ( M ) ⊗ Sol ( M )[1] − → R [1] Alexander Schenkel ∞ AQFT York 19 10 / 13

  56. � � � � � � � � � � � � � � � � � � � � � � � � Example: Linear Yang-Mills as a homotopy AQFT II ⋄ Every derived critical locus carries a [1] -shifted Poisson structure, explicitly: ( − 1) (0) (1) (2) − δ − d � δ d � Ω 0 Ω 1 Ω 1 Ω 0 • Smooth dual L ( M ) = c ( M ) c ( M ) c ( M ) c ( M ) ⊆ • Canonical inclusion j : L ( M ) − → L pc / fc ( M ) − → Sol ( M )[1] id ⊗ j ev • Shifted Poisson structure Υ : L ( M ) ⊗ L ( M ) − → L ( M ) ⊗ Sol ( M )[1] − → R [1] (i) ∃ (unique up to homotopy) contracting homotopy G ± for L pc / fc ( M ) , e.g. Thm: − δ − d 0 δ d 0 Ω 0 Ω 1 Ω 1 Ω 0 0 pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) 0 − G ± G ± − δ G ± 0 � d � � 0 0 id id id id 0 Ω 0 Ω 1 Ω 1 Ω 0 0 pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) 0 0 − δ δ d − d 0 Alexander Schenkel ∞ AQFT York 19 10 / 13

  57. � � � � � � � � � � � � � � � � � � � � � � � � Example: Linear Yang-Mills as a homotopy AQFT II ⋄ Every derived critical locus carries a [1] -shifted Poisson structure, explicitly: ( − 1) (0) (1) (2) − δ − d � δ d � Ω 0 Ω 1 Ω 1 Ω 0 • Smooth dual L ( M ) = c ( M ) c ( M ) c ( M ) c ( M ) ⊆ • Canonical inclusion j : L ( M ) − → L pc / fc ( M ) − → Sol ( M )[1] id ⊗ j ev • Shifted Poisson structure Υ : L ( M ) ⊗ L ( M ) − → L ( M ) ⊗ Sol ( M )[1] − → R [1] (i) ∃ (unique up to homotopy) contracting homotopy G ± for L pc / fc ( M ) , e.g. Thm: − δ − d 0 δ d 0 Ω 0 Ω 1 Ω 1 Ω 0 0 pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) 0 − G ± G ± − δ G ± 0 � d � � 0 0 id id id id 0 Ω 0 Ω 1 Ω 1 Ω 0 0 pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) 0 0 − δ δ d − d 0 (ii) j = ∂ G ± and Υ = ∂ ( something ) are exact Alexander Schenkel ∞ AQFT York 19 10 / 13

  58. � � � � � � � � � � � � � � � � � � � � � � � � Example: Linear Yang-Mills as a homotopy AQFT II ⋄ Every derived critical locus carries a [1] -shifted Poisson structure, explicitly: ( − 1) (0) (1) (2) − δ − d � δ d � Ω 0 Ω 1 Ω 1 Ω 0 • Smooth dual L ( M ) = c ( M ) c ( M ) c ( M ) c ( M ) ⊆ • Canonical inclusion j : L ( M ) − → L pc / fc ( M ) − → Sol ( M )[1] id ⊗ j ev • Shifted Poisson structure Υ : L ( M ) ⊗ L ( M ) − → L ( M ) ⊗ Sol ( M )[1] − → R [1] (i) ∃ (unique up to homotopy) contracting homotopy G ± for L pc / fc ( M ) , e.g. Thm: − δ − d 0 δ d 0 Ω 0 Ω 1 Ω 1 Ω 0 0 pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) 0 − G ± G ± − δ G ± 0 � d � � 0 0 id id id id 0 Ω 0 Ω 1 Ω 1 Ω 0 0 pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) 0 0 − δ δ d − d 0 (ii) j = ∂ G ± and Υ = ∂ ( something ) are exact (iii) Difference G := G + − G − defines unshifted Poisson structure id ⊗G ev τ : L ( M ) ⊗ L ( M ) − → L ( M ) ⊗ Sol ( M ) − → R (unique up to homotopy τ + ∂ρ ) Alexander Schenkel ∞ AQFT York 19 10 / 13

  59. � � � � � � � � � � � � � � � � � � � � � � � � Example: Linear Yang-Mills as a homotopy AQFT II ⋄ Every derived critical locus carries a [1] -shifted Poisson structure, explicitly: ( − 1) (0) (1) (2) − δ − d � δ d � Ω 0 Ω 1 Ω 1 Ω 0 • Smooth dual L ( M ) = c ( M ) c ( M ) c ( M ) c ( M ) ⊆ • Canonical inclusion j : L ( M ) − → L pc / fc ( M ) − → Sol ( M )[1] id ⊗ j ev • Shifted Poisson structure Υ : L ( M ) ⊗ L ( M ) − → L ( M ) ⊗ Sol ( M )[1] − → R [1] (i) ∃ (unique up to homotopy) contracting homotopy G ± for L pc / fc ( M ) , e.g. Thm: − δ − d 0 δ d 0 Ω 0 Ω 1 Ω 1 Ω 0 0 pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) 0 − G ± G ± − δ G ± 0 � d � � 0 0 id id id id 0 Ω 0 Ω 1 Ω 1 Ω 0 0 pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) 0 0 − δ δ d − d 0 (ii) j = ∂ G ± and Υ = ∂ ( something ) are exact (iii) Difference G := G + − G − defines unshifted Poisson structure id ⊗G ev τ : L ( M ) ⊗ L ( M ) − → L ( M ) ⊗ Sol ( M ) − → R (unique up to homotopy τ + ∂ρ ) (iv) Quantization CCR : PoCh R → Alg As ( Ch C ) preserves quasi-isomorphisms and homotopic Poisson structures, i.e. CCR ( V, τ + ∂ρ ) ≃ CCR ( V, τ ) Alexander Schenkel ∞ AQFT York 19 10 / 13

  60. � � � � � � � � � � � � � � � � � � � � � � � � Example: Linear Yang-Mills as a homotopy AQFT II ⋄ Every derived critical locus carries a [1] -shifted Poisson structure, explicitly: ( − 1) (0) (1) (2) − δ − d � δ d � Ω 0 Ω 1 Ω 1 Ω 0 • Smooth dual L ( M ) = c ( M ) c ( M ) c ( M ) c ( M ) ⊆ • Canonical inclusion j : L ( M ) − → L pc / fc ( M ) − → Sol ( M )[1] id ⊗ j ev • Shifted Poisson structure Υ : L ( M ) ⊗ L ( M ) − → L ( M ) ⊗ Sol ( M )[1] − → R [1] (i) ∃ (unique up to homotopy) contracting homotopy G ± for L pc / fc ( M ) , e.g. Thm: − δ − d 0 δ d 0 Ω 0 Ω 1 Ω 1 Ω 0 0 pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) 0 − G ± G ± − δ G ± 0 � d � � 0 0 id id id id 0 Ω 0 Ω 1 Ω 1 Ω 0 0 pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) pc / fc ( M ) 0 0 − δ δ d − d 0 (ii) j = ∂ G ± and Υ = ∂ ( something ) are exact (iii) Difference G := G + − G − defines unshifted Poisson structure id ⊗G ev τ : L ( M ) ⊗ L ( M ) − → L ( M ) ⊗ Sol ( M ) − → R (unique up to homotopy τ + ∂ρ ) (iv) Quantization CCR : PoCh R → Alg As ( Ch C ) preserves quasi-isomorphisms and homotopic Poisson structures, i.e. CCR ( V, τ + ∂ρ ) ≃ CCR ( V, τ ) (v) Loc ∋ M �→ A YM ( M ) := CCR ( L ( M ) , τ ) defines a homotopy AQFT Alexander Schenkel ∞ AQFT York 19 10 / 13

  61. � Towards descent in homotopy AQFT ⋄ The inclusion Loc ⋄ ֒ → Loc of “diamonds” defines Quillen adjunction � AQFT ∞ ( Loc ) : res ext : AQFT ∞ ( Loc ⋄ ) Alexander Schenkel ∞ AQFT York 19 11 / 13

  62. � Towards descent in homotopy AQFT ⋄ The inclusion Loc ⋄ ֒ → Loc of “diamonds” defines Quillen adjunction � AQFT ∞ ( Loc ) : res ext : AQFT ∞ ( Loc ⋄ ) ⋄ Homotopical descent condition: A ∈ AQFT ∞ ( Loc ) is homotopy j -local iff ∼ derived counit � ǫ A : L ext res A − → A is weak equivalence Alexander Schenkel ∞ AQFT York 19 11 / 13

  63. � Towards descent in homotopy AQFT ⋄ The inclusion Loc ⋄ ֒ → Loc of “diamonds” defines Quillen adjunction � AQFT ∞ ( Loc ) : res ext : AQFT ∞ ( Loc ⋄ ) ⋄ Homotopical descent condition: A ∈ AQFT ∞ ( Loc ) is homotopy j -local iff ∼ derived counit � ǫ A : L ext res A − → A is weak equivalence ⋄ Main question: Is this condition fulfilled in examples, e.g. linear Yang-Mills? Alexander Schenkel ∞ AQFT York 19 11 / 13

  64. � Towards descent in homotopy AQFT ⋄ The inclusion Loc ⋄ ֒ → Loc of “diamonds” defines Quillen adjunction � AQFT ∞ ( Loc ) : res ext : AQFT ∞ ( Loc ⋄ ) ⋄ Homotopical descent condition: A ∈ AQFT ∞ ( Loc ) is homotopy j -local iff ∼ derived counit � ǫ A : L ext res A − → A is weak equivalence ⋄ Main question: Is this condition fulfilled in examples, e.g. linear Yang-Mills? � Technically super (really, super!) complicated! Needs better technology for dealing with derived adjunctions, maybe ∞ -categories ` a la Joyal/Lurie? Alexander Schenkel ∞ AQFT York 19 11 / 13

  65. � Towards descent in homotopy AQFT ⋄ The inclusion Loc ⋄ ֒ → Loc of “diamonds” defines Quillen adjunction � AQFT ∞ ( Loc ) : res ext : AQFT ∞ ( Loc ⋄ ) ⋄ Homotopical descent condition: A ∈ AQFT ∞ ( Loc ) is homotopy j -local iff ∼ derived counit � ǫ A : L ext res A − → A is weak equivalence ⋄ Main question: Is this condition fulfilled in examples, e.g. linear Yang-Mills? � Technically super (really, super!) complicated! Needs better technology for dealing with derived adjunctions, maybe ∞ -categories ` a la Joyal/Lurie? � We can prove that simple (topological) toy-models of non-quantized gauge theories have this property! Alexander Schenkel ∞ AQFT York 19 11 / 13

  66. � Towards descent in homotopy AQFT ⋄ The inclusion Loc ⋄ ֒ → Loc of “diamonds” defines Quillen adjunction � AQFT ∞ ( Loc ) : res ext : AQFT ∞ ( Loc ⋄ ) ⋄ Homotopical descent condition: A ∈ AQFT ∞ ( Loc ) is homotopy j -local iff ∼ derived counit � ǫ A : L ext res A − → A is weak equivalence ⋄ Main question: Is this condition fulfilled in examples, e.g. linear Yang-Mills? � Technically super (really, super!) complicated! Needs better technology for dealing with derived adjunctions, maybe ∞ -categories ` a la Joyal/Lurie? � We can prove that simple (topological) toy-models of non-quantized gauge theories have this property! But already this requires heavy machinery, given by the following theorem based on Lurie’s Seifert-van Kampen Theorem Alexander Schenkel ∞ AQFT York 19 11 / 13

  67. � Towards descent in homotopy AQFT ⋄ The inclusion Loc ⋄ ֒ → Loc of “diamonds” defines Quillen adjunction � AQFT ∞ ( Loc ) : res ext : AQFT ∞ ( Loc ⋄ ) ⋄ Homotopical descent condition: A ∈ AQFT ∞ ( Loc ) is homotopy j -local iff ∼ derived counit � ǫ A : L ext res A − → A is weak equivalence ⋄ Main question: Is this condition fulfilled in examples, e.g. linear Yang-Mills? � Technically super (really, super!) complicated! Needs better technology for dealing with derived adjunctions, maybe ∞ -categories ` a la Joyal/Lurie? � We can prove that simple (topological) toy-models of non-quantized gauge theories have this property! But already this requires heavy machinery, given by the following theorem based on Lurie’s Seifert-van Kampen Theorem Thm: Let Man m be category of oriented m -manifolds and j : Disk m → Man m the full subcategory of m -disks. Alexander Schenkel ∞ AQFT York 19 11 / 13

  68. � Towards descent in homotopy AQFT ⋄ The inclusion Loc ⋄ ֒ → Loc of “diamonds” defines Quillen adjunction � AQFT ∞ ( Loc ) : res ext : AQFT ∞ ( Loc ⋄ ) ⋄ Homotopical descent condition: A ∈ AQFT ∞ ( Loc ) is homotopy j -local iff ∼ derived counit � ǫ A : L ext res A − → A is weak equivalence ⋄ Main question: Is this condition fulfilled in examples, e.g. linear Yang-Mills? � Technically super (really, super!) complicated! Needs better technology for dealing with derived adjunctions, maybe ∞ -categories ` a la Joyal/Lurie? � We can prove that simple (topological) toy-models of non-quantized gauge theories have this property! But already this requires heavy machinery, given by the following theorem based on Lurie’s Seifert-van Kampen Theorem Thm: Let Man m be category of oriented m -manifolds and j : Disk m → Man m the full subcategory of m -disks. Let A : Disk m → Alg E ∞ ( Ch K ) be functor that is weakly equivalent to a constant functor with value A ∈ Alg E ∞ ( Ch K ) . Alexander Schenkel ∞ AQFT York 19 11 / 13

  69. � Towards descent in homotopy AQFT ⋄ The inclusion Loc ⋄ ֒ → Loc of “diamonds” defines Quillen adjunction � AQFT ∞ ( Loc ) : res ext : AQFT ∞ ( Loc ⋄ ) ⋄ Homotopical descent condition: A ∈ AQFT ∞ ( Loc ) is homotopy j -local iff ∼ derived counit � ǫ A : L ext res A − → A is weak equivalence ⋄ Main question: Is this condition fulfilled in examples, e.g. linear Yang-Mills? � Technically super (really, super!) complicated! Needs better technology for dealing with derived adjunctions, maybe ∞ -categories ` a la Joyal/Lurie? � We can prove that simple (topological) toy-models of non-quantized gauge theories have this property! But already this requires heavy machinery, given by the following theorem based on Lurie’s Seifert-van Kampen Theorem Thm: Let Man m be category of oriented m -manifolds and j : Disk m → Man m the full subcategory of m -disks. Let A : Disk m → Alg E ∞ ( Ch K ) be functor that is weakly equivalent to a constant functor with value A ∈ Alg E ∞ ( Ch K ) . L Then the derived extension L j ! A ( M ) = Sing( M ) ⊗ A at M ∈ Man m is given by derived higher Hochschild chains on Sing( M ) with values in A . Alexander Schenkel ∞ AQFT York 19 11 / 13

  70. � � Toy-model: R -Chern-Simons theory on 2 -surfaces ⋄ Linear observables for flat R -connections on oriented 2 -manifold M are � � ( − 1) (0) (1) d d L ( M ) = Ω • Ω 2 Ω 1 Ω 0 c ( M )[1] = c ( M ) c ( M ) c ( M ) Alexander Schenkel ∞ AQFT York 19 12 / 13

  71. � � Toy-model: R -Chern-Simons theory on 2 -surfaces ⋄ Linear observables for flat R -connections on oriented 2 -manifold M are � � ( − 1) (0) (1) d d L ( M ) = Ω • Ω 2 Ω 1 Ω 0 c ( M )[1] = c ( M ) c ( M ) c ( M ) � � max ) ∼ ⋄ Define A ∈ AQFT ∞ ( Man 2 = Fun Man 2 , Alg E ∞ ( Ch K ) by � � A ( M ) = E ∞ L ( M ) [No quantization here!] Alexander Schenkel ∞ AQFT York 19 12 / 13

  72. � � Toy-model: R -Chern-Simons theory on 2 -surfaces ⋄ Linear observables for flat R -connections on oriented 2 -manifold M are � � ( − 1) (0) (1) d d L ( M ) = Ω • Ω 2 Ω 1 Ω 0 c ( M )[1] = c ( M ) c ( M ) c ( M ) � � max ) ∼ ⋄ Define A ∈ AQFT ∞ ( Man 2 = Fun Man 2 , Alg E ∞ ( Ch K ) by � � A ( M ) = E ∞ L ( M ) [No quantization here!] ⋄ Restriction j ∗ A to 2 -disks is weakly equivalent to constant functor with value � U : Ω • E ∞ ( R [ − 1]) because c ( U )[1] → R [ − 1] is quasi-iso, for all U ∈ Disk 2 Alexander Schenkel ∞ AQFT York 19 12 / 13

  73. � � Toy-model: R -Chern-Simons theory on 2 -surfaces ⋄ Linear observables for flat R -connections on oriented 2 -manifold M are � � ( − 1) (0) (1) d d L ( M ) = Ω • Ω 2 Ω 1 Ω 0 c ( M )[1] = c ( M ) c ( M ) c ( M ) � � max ) ∼ ⋄ Define A ∈ AQFT ∞ ( Man 2 = Fun Man 2 , Alg E ∞ ( Ch K ) by � � A ( M ) = E ∞ L ( M ) [No quantization here!] ⋄ Restriction j ∗ A to 2 -disks is weakly equivalent to constant functor with value � U : Ω • E ∞ ( R [ − 1]) because c ( U )[1] → R [ − 1] is quasi-iso, for all U ∈ Disk 2 ⋄ Compute derived extension at M ∈ Man 2 : � � � � L ( L j ! j ∗ A )( M ) ≃ Sing( M ) ⊗ E ∞ R [ − 1] ≃ Sing( M ) ⊗ E ∞ R [ − 1] Alexander Schenkel ∞ AQFT York 19 12 / 13

  74. � � Toy-model: R -Chern-Simons theory on 2 -surfaces ⋄ Linear observables for flat R -connections on oriented 2 -manifold M are � � ( − 1) (0) (1) d d L ( M ) = Ω • Ω 2 Ω 1 Ω 0 c ( M )[1] = c ( M ) c ( M ) c ( M ) � � max ) ∼ ⋄ Define A ∈ AQFT ∞ ( Man 2 = Fun Man 2 , Alg E ∞ ( Ch K ) by � � A ( M ) = E ∞ L ( M ) [No quantization here!] ⋄ Restriction j ∗ A to 2 -disks is weakly equivalent to constant functor with value � U : Ω • E ∞ ( R [ − 1]) because c ( U )[1] → R [ − 1] is quasi-iso, for all U ∈ Disk 2 ⋄ Compute derived extension at M ∈ Man 2 : � � � � L ( L j ! j ∗ A )( M ) ≃ Sing( M ) ⊗ E ∞ R [ − 1] ≃ Sing( M ) ⊗ E ∞ R [ − 1] � � [Fresse] ≃ N ∗ (Sing( M ) , R ) ⊗ R [ − 1] E ∞ Alexander Schenkel ∞ AQFT York 19 12 / 13

  75. � � Toy-model: R -Chern-Simons theory on 2 -surfaces ⋄ Linear observables for flat R -connections on oriented 2 -manifold M are � � ( − 1) (0) (1) d d L ( M ) = Ω • Ω 2 Ω 1 Ω 0 c ( M )[1] = c ( M ) c ( M ) c ( M ) � � max ) ∼ ⋄ Define A ∈ AQFT ∞ ( Man 2 = Fun Man 2 , Alg E ∞ ( Ch K ) by � � A ( M ) = E ∞ L ( M ) [No quantization here!] ⋄ Restriction j ∗ A to 2 -disks is weakly equivalent to constant functor with value � U : Ω • E ∞ ( R [ − 1]) because c ( U )[1] → R [ − 1] is quasi-iso, for all U ∈ Disk 2 ⋄ Compute derived extension at M ∈ Man 2 : � � � � L ( L j ! j ∗ A )( M ) ≃ Sing( M ) ⊗ E ∞ R [ − 1] ≃ Sing( M ) ⊗ E ∞ R [ − 1] � � [Fresse] ≃ N ∗ (Sing( M ) , R ) ⊗ R [ − 1] E ∞ � � [de Rham] Ω • ≃ c ( M )[1] = A ( M ) E ∞ Alexander Schenkel ∞ AQFT York 19 12 / 13

  76. � � Toy-model: R -Chern-Simons theory on 2 -surfaces ⋄ Linear observables for flat R -connections on oriented 2 -manifold M are � � ( − 1) (0) (1) d d L ( M ) = Ω • Ω 2 Ω 1 Ω 0 c ( M )[1] = c ( M ) c ( M ) c ( M ) � � max ) ∼ ⋄ Define A ∈ AQFT ∞ ( Man 2 = Fun Man 2 , Alg E ∞ ( Ch K ) by � � A ( M ) = E ∞ L ( M ) [No quantization here!] ⋄ Restriction j ∗ A to 2 -disks is weakly equivalent to constant functor with value � U : Ω • E ∞ ( R [ − 1]) because c ( U )[1] → R [ − 1] is quasi-iso, for all U ∈ Disk 2 ⋄ Compute derived extension at M ∈ Man 2 : � � � � L ( L j ! j ∗ A )( M ) ≃ Sing( M ) ⊗ E ∞ R [ − 1] ≃ Sing( M ) ⊗ E ∞ R [ − 1] � � [Fresse] ≃ N ∗ (Sing( M ) , R ) ⊗ R [ − 1] E ∞ � � [de Rham] Ω • ≃ c ( M )[1] = A ( M ) E ∞ max → Man 2 max ⇒ This toy-model is homotopy j -local for j : Disk 2 Alexander Schenkel ∞ AQFT York 19 12 / 13

  77. Conclusions and outlook ⋄ Describing gauge theories requires higher categorical structures, in particular stacks and derived stacks Alexander Schenkel ∞ AQFT York 19 13 / 13

  78. Conclusions and outlook ⋄ Describing gauge theories requires higher categorical structures, in particular stacks and derived stacks ⋄ Shadows of such structures are well-known in physics under the name BRST/BV formalism Alexander Schenkel ∞ AQFT York 19 13 / 13

  79. Conclusions and outlook ⋄ Describing gauge theories requires higher categorical structures, in particular stacks and derived stacks ⋄ Shadows of such structures are well-known in physics under the name BRST/BV formalism ⋄ The homotopical AQFT program that I initiated with M. Benini aims to introduce the relevant higher algebraic structures into AQFT Alexander Schenkel ∞ AQFT York 19 13 / 13

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