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

SLIDE 1

Higher Structures in Algebraic Quantum Field Theory

Alexander Schenkel

School of Mathematical Sciences, University of Nottingham

Mathematics of Interacting QFT Models, 1–5 July 2019, York. Based on joint works with M. Benini and different subsets of

S. Bruinsma, M. Perin, U. Schreiber, R. J. Szabo, L. Woike
Alexander Schenkel

∞AQFT York 19 1 / 13

SLIDE 2

Introduction and motivation

⋄ Common feature of (all?) approaches to QFT: Geometry Algebra

QFT

Alexander Schenkel ∞AQFT York 19 2 / 13

SLIDE 3

Introduction and motivation

⋄ Common feature of (all?) approaches to QFT: Geometry Algebra

QFT

⋄ Why interesting?

(1) QFT is a tool to learn something about geometry, e.g. invariants of manifolds via TQFT or Factorization Homology

Alexander Schenkel ∞AQFT York 19 2 / 13

SLIDE 4

Introduction and motivation

⋄ Common feature of (all?) approaches to QFT: Geometry Algebra

QFT

⋄ Why interesting?

(1) QFT is a tool to learn something about geometry, e.g. invariants of manifolds via TQFT or Factorization Homology (2) QFT leads to interesting algebraic structures that are “parametrized” by geometries, e.g. En-algebras

Alexander Schenkel ∞AQFT York 19 2 / 13

SLIDE 5

Introduction and motivation

⋄ Common feature of (all?) approaches to QFT: Geometry Algebra

QFT

⋄ Why interesting?

(1) QFT is a tool to learn something about geometry, e.g. invariants of manifolds via TQFT or Factorization Homology (2) QFT leads to interesting algebraic structures that are “parametrized” by geometries, e.g. En-algebras

⋄ AQFT is obtained from physically relevant choices for geometry and algebra:

Lorentzian manifolds = Einstein’s general relativity

Alexander Schenkel ∞AQFT York 19 2 / 13

SLIDE 6

Introduction and motivation

⋄ Common feature of (all?) approaches to QFT: Geometry Algebra

QFT

⋄ Why interesting?

(1) QFT is a tool to learn something about geometry, e.g. invariants of manifolds via TQFT or Factorization Homology (2) QFT leads to interesting algebraic structures that are “parametrized” by geometries, e.g. En-algebras

⋄ AQFT is obtained from physically relevant choices for geometry and algebra:

Lorentzian manifolds = Einstein’s general relativity
associative and unital algebras = quantum mechanics

Alexander Schenkel ∞AQFT York 19 2 / 13

SLIDE 7

Introduction and motivation

⋄ Common feature of (all?) approaches to QFT: Geometry Algebra

QFT

⋄ Why interesting?

(1) QFT is a tool to learn something about geometry, e.g. invariants of manifolds via TQFT or Factorization Homology (2) QFT leads to interesting algebraic structures that are “parametrized” by geometries, e.g. En-algebras

⋄ AQFT is obtained from physically relevant choices for geometry and algebra:

Lorentzian manifolds = Einstein’s general relativity
associative and unital algebras = quantum mechanics

⋄ Goals of this talk:

1. Brief introduction to (the algebraic structure of) AQFT

Alexander Schenkel ∞AQFT York 19 2 / 13

SLIDE 8

Introduction and motivation

⋄ Common feature of (all?) approaches to QFT: Geometry Algebra

QFT

⋄ Why interesting?

(1) QFT is a tool to learn something about geometry, e.g. invariants of manifolds via TQFT or Factorization Homology (2) QFT leads to interesting algebraic structures that are “parametrized” by geometries, e.g. En-algebras

⋄ AQFT is obtained from physically relevant choices for geometry and algebra:

Lorentzian manifolds = Einstein’s general relativity
associative and unital algebras = quantum mechanics

⋄ Goals of this talk:

1. Brief introduction to (the algebraic structure of) AQFT
2. Explain why gauge theory requires higher categorical structures

Alexander Schenkel ∞AQFT York 19 2 / 13

SLIDE 9

Introduction and motivation

⋄ Common feature of (all?) approaches to QFT: Geometry Algebra

QFT

⋄ Why interesting?

(1) QFT is a tool to learn something about geometry, e.g. invariants of manifolds via TQFT or Factorization Homology (2) QFT leads to interesting algebraic structures that are “parametrized” by geometries, e.g. En-algebras

⋄ AQFT is obtained from physically relevant choices for geometry and algebra:

Lorentzian manifolds = Einstein’s general relativity
associative and unital algebras = quantum mechanics

⋄ Goals of this talk:

1. Brief introduction to (the algebraic structure of) AQFT
2. Explain why gauge theory requires higher categorical structures
3. Survey of our homotopical AQFT program for quantum gauge theories

Alexander Schenkel ∞AQFT York 19 2 / 13

SLIDE 10

Algebraic quantum field theory

Alexander Schenkel ∞AQFT York 19 3 / 13

SLIDE 11

What is an AQFT?

⋄ Spacetime := oriented and time-oriented globally hyperbolic Lorentzian manifold N

time N Σ

Alexander Schenkel ∞AQFT York 19 3 / 13

SLIDE 12

What is an AQFT?

⋄ Spacetime := oriented and time-oriented globally hyperbolic Lorentzian manifold N ⋄ Spacetime embedding := (time-)orientation preserving isometric embedding f : M → N s.t. f(M) ⊆ N open and causally convex

time N M

Alexander Schenkel ∞AQFT York 19 3 / 13

SLIDE 13

What is an AQFT?

⋄ Spacetime := oriented and time-oriented globally hyperbolic Lorentzian manifold N ⋄ Spacetime embedding := (time-)orientation preserving isometric embedding f : M → N s.t. f(M) ⊆ N open and causally convex ⋄ Denote by Loc the category of spacetimes and spacetime embeddings

time N M

Alexander Schenkel ∞AQFT York 19 3 / 13

SLIDE 14

What is an AQFT?

⋄ Spacetime := oriented and time-oriented globally hyperbolic Lorentzian manifold N ⋄ Spacetime embedding := (time-)orientation preserving isometric embedding f : M → N s.t. f(M) ⊆ N open and causally convex ⋄ Denote by Loc the category of spacetimes and spacetime embeddings

time N M

Def: An AQFT is a functor A : Loc − → AlgAs(VecK) to the category of associative and unital K-algebras that satisfies:

Alexander Schenkel ∞AQFT York 19 3 / 13

SLIDE 15

What is an AQFT?

⋄ Spacetime := oriented and time-oriented globally hyperbolic Lorentzian manifold N ⋄ Spacetime embedding := (time-)orientation preserving isometric embedding f : M → N s.t. f(M) ⊆ N open and causally convex ⋄ Denote by Loc the category of spacetimes and spacetime embeddings

time N M1 M2

Def: An AQFT is a functor A : Loc − → AlgAs(VecK) to the category of associative and unital K-algebras that satisfies:

(i) Einstein causality: If M1

f1

− → N

f2

← − M2 causally disjoint, then µN ◦

A(f1) ⊗ A(f2)
= µop

N ◦

A(f1) ⊗ A(f2)
: A(M1) ⊗ A(M2) −

→ A(N)

Alexander Schenkel ∞AQFT York 19 3 / 13

SLIDE 16

What is an AQFT?

⋄ Spacetime := oriented and time-oriented globally hyperbolic Lorentzian manifold N ⋄ Spacetime embedding := (time-)orientation preserving isometric embedding f : M → N s.t. f(M) ⊆ N open and causally convex ⋄ Denote by Loc the category of spacetimes and spacetime embeddings

time N M Σ

Def: An AQFT is a functor A : Loc − → AlgAs(VecK) to the category of associative and unital K-algebras that satisfies:

(i) Einstein causality: If M1

f1

− → N

f2

← − M2 causally disjoint, then µN ◦

A(f1) ⊗ A(f2)
= µop

N ◦

A(f1) ⊗ A(f2)
: A(M1) ⊗ A(M2) −

→ A(N) (ii) Time-slice: If f : M → N is Cauchy morphism, then A(f) : A(M)

∼ =

− → A(N)

Alexander Schenkel ∞AQFT York 19 3 / 13

SLIDE 17

What is an AQFT?

⋄ Spacetime := oriented and time-oriented globally hyperbolic Lorentzian manifold N ⋄ Spacetime embedding := (time-)orientation preserving isometric embedding f : M → N s.t. f(M) ⊆ N open and causally convex ⋄ Denote by Loc the category of spacetimes and spacetime embeddings

time N M Σ

Def: An AQFT is a functor A : Loc − → AlgAs(VecK) to the category of associative and unital K-algebras that satisfies:

(i) Einstein causality: If M1

f1

− → N

f2

← − M2 causally disjoint, then µN ◦

A(f1) ⊗ A(f2)
= µop

N ◦

A(f1) ⊗ A(f2)
: A(M1) ⊗ A(M2) −

→ A(N) (ii) Time-slice: If f : M → N is Cauchy morphism, then A(f) : A(M)

∼ =

− → A(N)

AQFT links Lorentzian geometry to algebraic structure

Alexander Schenkel ∞AQFT York 19 3 / 13

SLIDE 18

AQFTs and colored operads

⋄ The definition of AQFT allows for the following generalizations:

VecK cocomplete closed symmetric monoidal category T

Alexander Schenkel ∞AQFT York 19 4 / 13

SLIDE 19

AQFTs and colored operads

⋄ The definition of AQFT allows for the following generalizations:

VecK cocomplete closed symmetric monoidal category T
Loc, causally disjoint and Cauchy orthogonal category C = (C, ⊥)

Alexander Schenkel ∞AQFT York 19 4 / 13

SLIDE 20

AQFTs and colored operads

⋄ The definition of AQFT allows for the following generalizations:

VecK 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

Alexander Schenkel ∞AQFT York 19 4 / 13

SLIDE 21

AQFTs and colored operads

⋄ The definition of AQFT allows for the following generalizations:

VecK 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 OC such that AQFT(C, T) ≃ AlgOC(T)

Alexander Schenkel ∞AQFT York 19 4 / 13

SLIDE 22

AQFTs and colored operads

⋄ The definition of AQFT allows for the following generalizations:

VecK 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 OC such that AQFT(C, T) ≃ AlgOC(T) (ii) Every orthogonal functor F : C → D defines an adjunction F! : AQFT(C, T)

AQFT(D, T) : F ∗

Alexander Schenkel

∞AQFT York 19 4 / 13

SLIDE 23

AQFTs and colored operads

⋄ The definition of AQFT allows for the following generalizations:

VecK 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 OC such that AQFT(C, T) ≃ AlgOC(T) (ii) Every orthogonal functor F : C → D defines an adjunction F! : AQFT(C, T)

AQFT(D, T) : F ∗

⋄ This is extremely useful for local-to-global constructions:

Alexander Schenkel ∞AQFT York 19 4 / 13

SLIDE 24

AQFTs and colored operads

⋄ The definition of AQFT allows for the following generalizations:

VecK 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 OC such that AQFT(C, T) ≃ AlgOC(T) (ii) Every orthogonal functor F : C → D defines an adjunction F! : AQFT(C, T)

AQFT(D, T) : F ∗

⋄ This is extremely useful for local-to-global constructions:
Let j : Loc⋄ → Loc be full orthogonal subcat of “diamonds” (M ∼

= Rm)

Alexander Schenkel ∞AQFT York 19 4 / 13

SLIDE 25

AQFTs and colored operads

⋄ The definition of AQFT allows for the following generalizations:

VecK 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 OC such that AQFT(C, T) ≃ AlgOC(T) (ii) Every orthogonal functor F : C → D defines an adjunction F! : AQFT(C, T)

AQFT(D, T) : F ∗

⋄ This is extremely useful for local-to-global constructions:
Let j : Loc⋄ → Loc be full orthogonal subcat of “diamonds” (M ∼

= Rm)

We get extension-restriction adjunction

ext = j! : AQFT(Loc⋄, T)

AQFT(Loc, T) : j∗ = res

Alexander Schenkel

∞AQFT York 19 4 / 13

SLIDE 26

AQFTs and colored operads

⋄ The definition of AQFT allows for the following generalizations:

VecK 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 OC such that AQFT(C, T) ≃ AlgOC(T) (ii) Every orthogonal functor F : C → D defines an adjunction F! : AQFT(C, T)

AQFT(D, T) : F ∗

⋄ This is extremely useful for local-to-global constructions:
Let j : Loc⋄ → Loc be full orthogonal subcat of “diamonds” (M ∼

= Rm)

We get extension-restriction adjunction

ext = j! : AQFT(Loc⋄, T)

AQFT(Loc, T) : j∗ = res

Descent condition: A ∈ AQFT(Loc, T) is j-local iff ǫA : ext res A

∼ =

− → A

Alexander Schenkel ∞AQFT York 19 4 / 13

SLIDE 27

AQFTs and colored operads

⋄ The definition of AQFT allows for the following generalizations:

VecK 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 OC such that AQFT(C, T) ≃ AlgOC(T) (ii) Every orthogonal functor F : C → D defines an adjunction F! : AQFT(C, T)

AQFT(D, T) : F ∗

⋄ This is extremely useful for local-to-global constructions:
Let j : Loc⋄ → Loc be full orthogonal subcat of “diamonds” (M ∼

= Rm)

We get extension-restriction adjunction

ext = j! : AQFT(Loc⋄, T)

AQFT(Loc, T) : j∗ = res

Descent condition: A ∈ AQFT(Loc, T) is j-local iff ǫA : ext res A

∼ =

− → A Ex: Linear Klein-Gordon theory is j-local (uses also results by [Lang])

Alexander Schenkel ∞AQFT York 19 4 / 13

SLIDE 28

Higher structures in gauge theory

Alexander Schenkel ∞AQFT York 19 5 / 13

SLIDE 29

What is a gauge theory?

“Ordinary” field theory: Set/Space of fields

Φ Φ′ Φ′′

Alexander Schenkel ∞AQFT York 19 5 / 13

SLIDE 30

What is a gauge theory?

“Ordinary” field theory: Gauge theory: Set/Space of fields Grpd/Stack of fields

Φ Φ′ Φ′′ A A′ A′′

g g

′

g′′

Alexander Schenkel ∞AQFT York 19 5 / 13

SLIDE 31

What is a gauge theory?

“Ordinary” field theory: Gauge theory: Set/Space of fields Grpd/Stack of fields

Φ Φ′ Φ′′ A A′ A′′

g g

′

g′′ Ex: Groupoid of principal G-bundles with connection on U ∼ = Rm BGcon(U) =    Obj: A ∈ Ω1(U, g) Mor: A

g∈C∞(U,G) A ⊳ g = g−1Ag + g−1dg

Alexander Schenkel ∞AQFT York 19 5 / 13

SLIDE 32

What is a gauge theory?

“Ordinary” field theory: Gauge theory: Set/Space of fields Grpd/Stack of fields

Φ Φ′ Φ′′ A A′ A′′

g g

′

g′′ Ex: Groupoid of principal G-bundles with connection on U ∼ = Rm BGcon(U) =    Obj: A ∈ Ω1(U, g) Mor: A

g∈C∞(U,G) A ⊳ g = g−1Ag + g−1dg

Invariant information:

π0
BGcon(U)
= Ω1(U, g)/C∞(U, G) (“gauge orbit space”)

Alexander Schenkel ∞AQFT York 19 5 / 13

SLIDE 33

What is a gauge theory?

“Ordinary” field theory: Gauge theory: Set/Space of fields Grpd/Stack of fields

Φ Φ′ Φ′′ A A′ A′′

g g

′

g′′ Ex: Groupoid of principal G-bundles with connection on U ∼ = Rm BGcon(U) =    Obj: A ∈ Ω1(U, g) Mor: A

g∈C∞(U,G) A ⊳ g = g−1Ag + g−1dg

Invariant information:

π0
BGcon(U)
= Ω1(U, g)/C∞(U, G) (“gauge orbit space”)
π1
BGcon(U), A
= {g ∈ C∞(U, G) : A = A ⊳ g} (stabilizers/“loops”)

Alexander Schenkel ∞AQFT York 19 5 / 13

SLIDE 34

What is a gauge theory?

“Ordinary” field theory: Gauge theory: Set/Space of fields Grpd/Stack of fields

Φ Φ′ Φ′′ A A′ A′′

g g

′

g′′ Ex: Groupoid of principal G-bundles with connection on U ∼ = Rm BGcon(U) =    Obj: A ∈ Ω1(U, g) Mor: A

g∈C∞(U,G) A ⊳ g = g−1Ag + g−1dg

Invariant information:

π0
BGcon(U)
= Ω1(U, g)/C∞(U, G) (“gauge orbit space”)
π1
BGcon(U), A
= {g ∈ C∞(U, G) : A = A ⊳ g} (stabilizers/“loops”)

! 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

SLIDE 35

Why are these higher structures important?

1. π1 encodes essential information of the gauge theory:

Alexander Schenkel ∞AQFT York 19 6 / 13

SLIDE 36

Why are these higher structures important?

1. π1 encodes essential information of the gauge theory:

Consider structure group G = U(1) or R. Then π0

BGcon(U)

∼ = Ω1(U)/dΩ0(U)

doesn’t see G

, π1

BGcon(U), A

∼ = G

sees G

Alexander Schenkel ∞AQFT York 19 6 / 13

SLIDE 37

Why are these higher structures important?

1. π1 encodes essential information of the gauge theory:

Consider structure group G = U(1) or R. Then π0

BGcon(U)

∼ = Ω1(U)/dΩ0(U)

doesn’t see G

, π1

BGcon(U), A

∼ = G

sees G
2. Higher structures are crucial for descent:

Alexander Schenkel ∞AQFT York 19 6 / 13

SLIDE 38

Why are these higher structures important?

1. π1 encodes essential information of the gauge theory:

Consider structure group G = U(1) or R. Then π0

BGcon(U)

∼ = Ω1(U)/dΩ0(U)

doesn’t see G

, π1

BGcon(U), A

∼ = G

sees G
2. Higher structures are crucial for descent:

Let M be manifold with good open cover {Ui ⊆ M}. Then holim

i BGcon(Ui) ij BGcon(Uij)

ijk BGcon(Uijk)
· · ·
is the groupoid of all principal G-bundles with connection on M

Alexander Schenkel ∞AQFT York 19 6 / 13

SLIDE 39

Why are these higher structures important?

1. π1 encodes essential information of the gauge theory:

Consider structure group G = U(1) or R. Then π0

BGcon(U)

∼ = Ω1(U)/dΩ0(U)

doesn’t see G

, π1

BGcon(U), A

∼ = G

sees G
2. Higher structures are crucial for descent:

Let M be manifold with good open cover {Ui ⊆ M}. Then holim

i BGcon(Ui) ij BGcon(Uij)

ijk BGcon(Uijk)
· · ·
is the groupoid of all principal G-bundles with connection on M
3. π1 detects “fake” gauge symmetries:

Alexander Schenkel ∞AQFT York 19 6 / 13

SLIDE 40

Why are these higher structures important?

1. π1 encodes essential information of the gauge theory:

Consider structure group G = U(1) or R. Then π0

BGcon(U)

∼ = Ω1(U)/dΩ0(U)

doesn’t see G

, π1

BGcon(U), A

∼ = G

sees G
2. Higher structures are crucial for descent:

Let M be manifold with good open cover {Ui ⊆ M}. Then holim

i BGcon(Ui) ij BGcon(Uij)

ijk BGcon(Uijk)
· · ·
is the groupoid of all principal G-bundles with connection on M
3. π1 detects “fake” gauge symmetries:

Let M be manifold and consider groupoid P(M) =    Obj: (Φ1, Φ2) ∈ C∞(M) × C∞(M) Mor: (Φ1, Φ2)

ǫ∈C∞(M) (Φ1 + ǫ, Φ2 + ǫ)

Alexander Schenkel ∞AQFT York 19 6 / 13

SLIDE 41

Why are these higher structures important?

1. π1 encodes essential information of the gauge theory:

Consider structure group G = U(1) or R. Then π0

BGcon(U)

∼ = Ω1(U)/dΩ0(U)

doesn’t see G

, π1

BGcon(U), A

∼ = G

sees G
2. Higher structures are crucial for descent:

Let M be manifold with good open cover {Ui ⊆ M}. Then holim

i BGcon(Ui) ij BGcon(Uij)

ijk BGcon(Uijk)
· · ·
is the groupoid of all principal G-bundles with connection on M
3. π1 detects “fake” gauge symmetries:

Let M be manifold and consider groupoid P(M) =    Obj: (Φ1, Φ2) ∈ C∞(M) × C∞(M) Mor: (Φ1, Φ2)

ǫ∈C∞(M) (Φ1 + ǫ, Φ2 + ǫ)

Because all π1’s are trivial, this is a “fake” gauge symmetry.

Alexander Schenkel ∞AQFT York 19 6 / 13

SLIDE 42

Why are these higher structures important?

1. π1 encodes essential information of the gauge theory:

Consider structure group G = U(1) or R. Then π0

BGcon(U)

∼ = Ω1(U)/dΩ0(U)

doesn’t see G

, π1

BGcon(U), A

∼ = G

sees G
2. Higher structures are crucial for descent:

Let M be manifold with good open cover {Ui ⊆ M}. Then holim

i BGcon(Ui) ij BGcon(Uij)

ijk BGcon(Uijk)
· · ·
is the groupoid of all principal G-bundles with connection on M
3. π1 detects “fake” gauge symmetries:

Let M be manifold and consider groupoid P(M) =    Obj: (Φ1, Φ2) ∈ C∞(M) × C∞(M) Mor: (Φ1, Φ2)

ǫ∈C∞(M) (Φ1 + ǫ, Φ2 + ǫ)

Because all π1’s are trivial, this is a “fake” gauge symmetry. Indeed, there is an equivalence P(M) → C∞(M) , (Φ1, Φ2) → Φ1 − Φ2 to the scalar field.

Alexander Schenkel ∞AQFT York 19 6 / 13

SLIDE 43

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

SLIDE 44

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 : Cartop → Grpd satisfying homotopical descent

Alexander Schenkel ∞AQFT York 19 7 / 13

SLIDE 45

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 : Cartop → Grpd satisfying homotopical descent

X

Alexander Schenkel ∞AQFT York 19 7 / 13

SLIDE 46

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 : Cartop → Grpd satisfying homotopical descent

X R0

Alexander Schenkel ∞AQFT York 19 7 / 13

SLIDE 47

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 : Cartop → Grpd satisfying homotopical descent

X R0 R1

Alexander Schenkel ∞AQFT York 19 7 / 13

SLIDE 48

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 : Cartop → Grpd satisfying homotopical descent

X R0 R1 R2

Alexander Schenkel ∞AQFT York 19 7 / 13

SLIDE 49

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 : Cartop → Grpd satisfying homotopical descent

X R0 R1 R2

⋄ (A)QFT needs “observable algebras”, but what are functions on stacks?

Alexander Schenkel ∞AQFT York 19 7 / 13

SLIDE 50

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 : Cartop → Grpd satisfying homotopical descent

X R0 R1 R2

⋄ (A)QFT needs “observable algebras”, but what are functions on stacks? ⋄ Smooth normalized cochains: Stacks

N ∗∞(−,K)

N∗(−,K)

PSh(Cart, ChK)

Map∞(−,K)

Chop

K

Alexander Schenkel ∞AQFT York 19 7 / 13

SLIDE 51

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 : Cartop → Grpd satisfying homotopical descent

X R0 R1 R2

⋄ (A)QFT needs “observable algebras”, but what are functions on stacks? ⋄ Smooth normalized cochains: Stacks

N ∗∞(−,K)

N∗(−,K)

PSh(Cart, ChK)

Map∞(−,K)

Chop

K

Prop:

(i) N ∗∞(−, K) is left Quillen functor, i.e. left derived functor LN ∗∞(−, K) exists

Alexander Schenkel ∞AQFT York 19 7 / 13

SLIDE 52

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 : Cartop → Grpd satisfying homotopical descent

X R0 R1 R2

⋄ (A)QFT needs “observable algebras”, but what are functions on stacks? ⋄ Smooth normalized cochains: Stacks

N ∗∞(−,K)

N∗(−,K)

PSh(Cart, ChK)

Map∞(−,K)

Chop

K

Prop:

(i) N ∗∞(−, K) is left Quillen functor, i.e. left derived functor LN ∗∞(−, K) exists (ii) LN ∗∞(−, K) : Stacks → AlgE∞(ChK)op takes values in E∞-algebras

Alexander Schenkel ∞AQFT York 19 7 / 13

SLIDE 53

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 : Cartop → Grpd satisfying homotopical descent

X R0 R1 R2

⋄ (A)QFT needs “observable algebras”, but what are functions on stacks? ⋄ Smooth normalized cochains: Stacks

N ∗∞(−,K)

N∗(−,K)

PSh(Cart, ChK)

Map∞(−,K)

Chop

K

Prop:

(i) N ∗∞(−, K) is left Quillen functor, i.e. left derived functor LN ∗∞(−, K) exists (ii) LN ∗∞(−, K) : Stacks → AlgE∞(ChK)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

SLIDE 54

Higher structures in AQFT

Alexander Schenkel ∞AQFT York 19 8 / 13

SLIDE 55

Homotopy AQFTs: Definition and basic properties

⋄ Recall AQFT(C, T) = AlgOC(T) requires choice of target category T

Alexander Schenkel ∞AQFT York 19 8 / 13

SLIDE 56

Homotopy AQFTs: Definition and basic properties

⋄ Recall AQFT(C, T) = AlgOC(T) requires choice of target category T ⋄ Take T = ChK to include dg-algebras of observables of a gauge theory!

Alexander Schenkel ∞AQFT York 19 8 / 13

SLIDE 57

Homotopy AQFTs: Definition and basic properties

⋄ Recall AQFT(C, T) = AlgOC(T) requires choice of target category T ⋄ Take T = ChK 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

SLIDE 58

Homotopy AQFTs: Definition and basic properties

⋄ Recall AQFT(C, T) = AlgOC(T) requires choice of target category T ⋄ Take T = ChK to include dg-algebras of observables of a gauge theory! ⋄ But what about homotopy coherent algebraic structures (e.g. E∞-algebras)? Def: A ChK-valued homotopy AQFT on C is an algebra over any Σ-cofibrant resolution OC,∞

∼

։ OC of the AQFT dg-operad OC ∈ Op(ChK).

Alexander Schenkel ∞AQFT York 19 8 / 13

SLIDE 59

Homotopy AQFTs: Definition and basic properties

⋄ Recall AQFT(C, T) = AlgOC(T) requires choice of target category T ⋄ Take T = ChK to include dg-algebras of observables of a gauge theory! ⋄ But what about homotopy coherent algebraic structures (e.g. E∞-algebras)? Def: A ChK-valued homotopy AQFT on C is an algebra over any Σ-cofibrant resolution OC,∞

∼

։ OC of the AQFT dg-operad OC ∈ Op(ChK). AQFT∞(C) := AlgOC,∞(ChK) denotes category of homotopy AQFTs.

Alexander Schenkel ∞AQFT York 19 8 / 13

SLIDE 60

Homotopy AQFTs: Definition and basic properties

⋄ Recall AQFT(C, T) = AlgOC(T) requires choice of target category T ⋄ Take T = ChK to include dg-algebras of observables of a gauge theory! ⋄ But what about homotopy coherent algebraic structures (e.g. E∞-algebras)? Def: A ChK-valued homotopy AQFT on C is an algebra over any Σ-cofibrant resolution OC,∞

∼

։ OC of the AQFT dg-operad OC ∈ Op(ChK). AQFT∞(C) := AlgOC,∞(ChK) 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

SLIDE 61

Homotopy AQFTs: Definition and basic properties

⋄ Recall AQFT(C, T) = AlgOC(T) requires choice of target category T ⋄ Take T = ChK to include dg-algebras of observables of a gauge theory! ⋄ But what about homotopy coherent algebraic structures (e.g. E∞-algebras)? Def: A ChK-valued homotopy AQFT on C is an algebra over any Σ-cofibrant resolution OC,∞

∼

։ OC of the AQFT dg-operad OC ∈ Op(ChK). AQFT∞(C) := AlgOC,∞(ChK) 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

SLIDE 62

Homotopy AQFTs: Definition and basic properties

⋄ Recall AQFT(C, T) = AlgOC(T) requires choice of target category T ⋄ Take T = ChK to include dg-algebras of observables of a gauge theory! ⋄ But what about homotopy coherent algebraic structures (e.g. E∞-algebras)? Def: A ChK-valued homotopy AQFT on C is an algebra over any Σ-cofibrant resolution OC,∞

∼

։ OC of the AQFT dg-operad OC ∈ Op(ChK). AQFT∞(C) := AlgOC,∞(ChK) 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 : OC

∼

։ OC 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

SLIDE 63

Homotopy AQFTs: Definition and basic properties

⋄ Recall AQFT(C, T) = AlgOC(T) requires choice of target category T ⋄ Take T = ChK to include dg-algebras of observables of a gauge theory! ⋄ But what about homotopy coherent algebraic structures (e.g. E∞-algebras)? Def: A ChK-valued homotopy AQFT on C is an algebra over any Σ-cofibrant resolution OC,∞

∼

։ OC of the AQFT dg-operad OC ∈ Op(ChK). AQFT∞(C) := AlgOC,∞(ChK) 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 : OC

∼

։ OC is Σ-cofibrant resolution ⇒ all homotopy AQFTs in char K = 0 (i.e. in physics) can be strictified

Ex: Component-wise tensor product OC ⊗ E∞

∼

։ OC ⊗ Com ∼ = OC with the Barratt-Eccles operad defines a Σ-cofibrant resolution.

Alexander Schenkel ∞AQFT York 19 8 / 13

SLIDE 64

Homotopy AQFTs: Definition and basic properties

⋄ Recall AQFT(C, T) = AlgOC(T) requires choice of target category T ⋄ Take T = ChK to include dg-algebras of observables of a gauge theory! ⋄ But what about homotopy coherent algebraic structures (e.g. E∞-algebras)? Def: A ChK-valued homotopy AQFT on C is an algebra over any Σ-cofibrant resolution OC,∞

∼

։ OC of the AQFT dg-operad OC ∈ Op(ChK). AQFT∞(C) := AlgOC,∞(ChK) 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 : OC

∼

։ OC is Σ-cofibrant resolution ⇒ all homotopy AQFTs in char K = 0 (i.e. in physics) can be strictified

Ex: Component-wise tensor product OC ⊗ E∞

∼

։ OC ⊗ Com ∼ = OC with the Barratt-Eccles operad defines a Σ-cofibrant resolution. Smooth normalized cochain algebras on (a diagram X : Cop → Stacks of) stacks leads to homotopy AQFTs of this type.

Alexander Schenkel ∞AQFT York 19 8 / 13

SLIDE 65

Example: Linear Yang-Mills as a homotopy AQFT

⋄ Linear gauge theory: (derived) stacks chain complexes ChK

Alexander Schenkel ∞AQFT York 19 9 / 13

SLIDE 66

Example: Linear Yang-Mills as a homotopy AQFT

⋄ Linear gauge theory: (derived) stacks chain complexes ChK ⋄ Yang-Mills with structure group R is defined on M ∈ Loc by

Alexander Schenkel ∞AQFT York 19 9 / 13

SLIDE 67

Example: Linear Yang-Mills as a homotopy AQFT

⋄ Linear gauge theory: (derived) stacks chain complexes ChK ⋄ Yang-Mills with structure group R is defined on M ∈ Loc by

(1) Field complex F(M) =

(0)

Ω1(M)

(1)

Ω0(M)

d

Alexander Schenkel

∞AQFT York 19 9 / 13

SLIDE 68

Example: Linear Yang-Mills as a homotopy AQFT

⋄ Linear gauge theory: (derived) stacks chain complexes ChK ⋄ Yang-Mills with structure group R is defined on M ∈ Loc by

(1) Field complex F(M) =

(0)

Ω1(M)

(1)

Ω0(M)

d

(2) Action functional S(A) = 1

2

M dA ∧ ∗dA

Alexander Schenkel ∞AQFT York 19 9 / 13

SLIDE 69

Example: Linear Yang-Mills as a homotopy AQFT

⋄ Linear gauge theory: (derived) stacks chain complexes ChK ⋄ Yang-Mills with structure group R is defined on M ∈ Loc by

(1) Field complex F(M) =

(0)

Ω1(M)

(1)

Ω0(M)

d

(2) Action functional S(A) = 1

2

M dA ∧ ∗dA

⋄ Variation of action defines section of cotangent bundle

F(M)

δvS

T ∗F(M) =      Ω1(M)

(id,δd)

Ω0(M)

d

id

Ω0(M) Ω1(M) × Ω1(M)

−δπ2

Ω0(M)

ι1d



   

Alexander Schenkel ∞AQFT York 19 9 / 13

SLIDE 70

Example: Linear Yang-Mills as a homotopy AQFT

⋄ Linear gauge theory: (derived) stacks chain complexes ChK ⋄ Yang-Mills with structure group R is defined on M ∈ Loc by

(1) Field complex F(M) =

(0)

Ω1(M)

(1)

Ω0(M)

d

(2) Action functional S(A) = 1

2

M dA ∧ ∗dA

⋄ Variation of action defines section of cotangent bundle

F(M)

δvS

T ∗F(M) =      Ω1(M)

(id,δd)

Ω0(M)

d

id

Ω0(M) Ω1(M) × Ω1(M)

−δπ2

Ω0(M)

ι1d



   

Def: The solution complex is defined as the (linear) derived critical locus of the action S, i.e. the following homotopy pullback in ChK

Sol(M)

F(M)

h δvS

F(M)

T ∗F(M)

Alexander Schenkel ∞AQFT York 19 9 / 13

SLIDE 71

Example: Linear Yang-Mills as a homotopy AQFT

⋄ Linear gauge theory: (derived) stacks chain complexes ChK ⋄ Yang-Mills with structure group R is defined on M ∈ Loc by

(1) Field complex F(M) =

(0)

Ω1(M)

(1)

Ω0(M)

d

(2) Action functional S(A) = 1

2

M dA ∧ ∗dA

⋄ Variation of action defines section of cotangent bundle

F(M)

δvS

T ∗F(M) =      Ω1(M)

(id,δd)

Ω0(M)

d

id

Ω0(M) Ω1(M) × Ω1(M)

−δπ2

Ω0(M)

ι1d



   

Def: The solution complex is defined as the (linear) derived critical locus of the action S, i.e. the following homotopy pullback in ChK

Sol(M)

F(M)

h δvS

F(M)

T ∗F(M)

Prop: Sol(M) =

(−2)

Ω0(M)

(−1)

Ω1(M)

δ

(0)

Ω1(M)

δd

(1)

Ω0(M)

d

Alexander Schenkel

∞AQFT York 19 9 / 13

SLIDE 72

Example: Linear Yang-Mills as a homotopy AQFT

⋄ Linear gauge theory: (derived) stacks chain complexes ChK ⋄ Yang-Mills with structure group R is defined on M ∈ Loc by

(1) Field complex F(M) =

(0)

Ω1(M)

(1)

Ω0(M)

d

(2) Action functional S(A) = 1

2

M dA ∧ ∗dA

⋄ Variation of action defines section of cotangent bundle

F(M)

δvS

T ∗F(M) =      Ω1(M)

(id,δd)

Ω0(M)

d

id

Ω0(M) Ω1(M) × Ω1(M)

−δπ2

Ω0(M)

ι1d



   

Def: The solution complex is defined as the (linear) derived critical locus of the action S, i.e. the following homotopy pullback in ChK

Sol(M)

F(M)

h δvS

F(M)

T ∗F(M)

Prop: Sol(M) =

C‡

Ω0(M)

A‡

Ω1(M)

δ

A

Ω1(M)

δd

C

Ω0(M)

d

Rem: Interpretation of Sol(M) in terms of BRST/BV formalism from physics

Alexander Schenkel ∞AQFT York 19 9 / 13

SLIDE 73

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

SLIDE 74

Example: Linear Yang-Mills as a homotopy AQFT II

⋄ Every derived critical locus carries a [1]-shifted Poisson structure, explicitly:

Smooth dual L(M) =
(−1)

Ω0

c(M) (0)

Ω1

c(M) −δ

(1)

Ω1

c(M) δd

(2)

Ω0

c(M) −d

Alexander Schenkel

∞AQFT York 19 10 / 13

SLIDE 75

Example: Linear Yang-Mills as a homotopy AQFT II

⋄ Every derived critical locus carries a [1]-shifted Poisson structure, explicitly:

Smooth dual L(M) =
(−1)

Ω0

c(M) (0)

Ω1

c(M) −δ

(1)

Ω1

c(M) δd

(2)

Ω0

c(M) −d

Canonical inclusion j : L(M)

⊆

− → Lpc/fc(M) − → Sol(M)[1]

Alexander Schenkel ∞AQFT York 19 10 / 13

SLIDE 76

Example: Linear Yang-Mills as a homotopy AQFT II

⋄ Every derived critical locus carries a [1]-shifted Poisson structure, explicitly:

Smooth dual L(M) =
(−1)

Ω0

c(M) (0)

Ω1

c(M) −δ

(1)

Ω1

c(M) δd

(2)

Ω0

c(M) −d

Canonical inclusion j : L(M)

⊆

− → Lpc/fc(M) − → Sol(M)[1]

Shifted Poisson structure Υ : L(M) ⊗ L(M)

id⊗j

− → L(M) ⊗ Sol(M)[1]

ev

− → R[1]

Alexander Schenkel ∞AQFT York 19 10 / 13

SLIDE 77

Example: Linear Yang-Mills as a homotopy AQFT II

⋄ Every derived critical locus carries a [1]-shifted Poisson structure, explicitly:

Smooth dual L(M) =
(−1)

Ω0

c(M) (0)

Ω1

c(M) −δ

(1)

Ω1

c(M) δd

(2)

Ω0

c(M) −d

Canonical inclusion j : L(M)

⊆

− → Lpc/fc(M) − → Sol(M)[1]

Shifted Poisson structure Υ : L(M) ⊗ L(M)

id⊗j

− → L(M) ⊗ Sol(M)[1]

ev

− → R[1]

Thm:

(i) ∃ (unique up to homotopy) contracting homotopy G± for Lpc/fc(M), e.g.

Ω0

pc/fc(M)

id
−G±

d

Ω1

pc/fc(M) −δ

id
G±
Ω1

pc/fc(M) δd

id
−δ G±
Ω0

pc/fc(M) −d

id
Ω0

pc/fc(M)

Ω1

pc/fc(M) −δ

Ω1

pc/fc(M) δd

Ω0

pc/fc(M) −d

Alexander Schenkel

∞AQFT York 19 10 / 13

SLIDE 78

Example: Linear Yang-Mills as a homotopy AQFT II

⋄ Every derived critical locus carries a [1]-shifted Poisson structure, explicitly:

Smooth dual L(M) =
(−1)

Ω0

c(M) (0)

Ω1

c(M) −δ

(1)

Ω1

c(M) δd

(2)

Ω0

c(M) −d

Canonical inclusion j : L(M)

⊆

− → Lpc/fc(M) − → Sol(M)[1]

Shifted Poisson structure Υ : L(M) ⊗ L(M)

id⊗j

− → L(M) ⊗ Sol(M)[1]

ev

− → R[1]

Thm:

(i) ∃ (unique up to homotopy) contracting homotopy G± for Lpc/fc(M), e.g.

Ω0

pc/fc(M)

id
−G±

d

Ω1

pc/fc(M) −δ

id
G±
Ω1

pc/fc(M) δd

id
−δ G±
Ω0

pc/fc(M) −d

id
Ω0

pc/fc(M)

Ω1

pc/fc(M) −δ

Ω1

pc/fc(M) δd

Ω0

pc/fc(M) −d

(ii) j = ∂G± and Υ = ∂(something) are exact

Alexander Schenkel ∞AQFT York 19 10 / 13

SLIDE 79

Example: Linear Yang-Mills as a homotopy AQFT II

⋄ Every derived critical locus carries a [1]-shifted Poisson structure, explicitly:

Smooth dual L(M) =
(−1)

Ω0

c(M) (0)

Ω1

c(M) −δ

(1)

Ω1

c(M) δd

(2)

Ω0

c(M) −d

Canonical inclusion j : L(M)

⊆

− → Lpc/fc(M) − → Sol(M)[1]

Shifted Poisson structure Υ : L(M) ⊗ L(M)

id⊗j

− → L(M) ⊗ Sol(M)[1]

ev

− → R[1]

Thm:

(i) ∃ (unique up to homotopy) contracting homotopy G± for Lpc/fc(M), e.g.

Ω0

pc/fc(M)

id
−G±

d

Ω1

pc/fc(M) −δ

id
G±
Ω1

pc/fc(M) δd

id
−δ G±
Ω0

pc/fc(M) −d

id
Ω0

pc/fc(M)

Ω1

pc/fc(M) −δ

Ω1

pc/fc(M) δd

Ω0

pc/fc(M) −d

(ii) j = ∂G± and Υ = ∂(something) are exact

(iii) Difference G := G+ − G− defines unshifted Poisson structure τ : L(M) ⊗ L(M)

id⊗G

− → L(M) ⊗ Sol(M)

ev

− → R (unique up to homotopy τ + ∂ρ)

Alexander Schenkel ∞AQFT York 19 10 / 13

SLIDE 80

Example: Linear Yang-Mills as a homotopy AQFT II

⋄ Every derived critical locus carries a [1]-shifted Poisson structure, explicitly:

Smooth dual L(M) =
(−1)

Ω0

c(M) (0)

Ω1

c(M) −δ

(1)

Ω1

c(M) δd

(2)

Ω0

c(M) −d

Canonical inclusion j : L(M)

⊆

− → Lpc/fc(M) − → Sol(M)[1]

Shifted Poisson structure Υ : L(M) ⊗ L(M)

id⊗j

− → L(M) ⊗ Sol(M)[1]

ev

− → R[1]

Thm:

(i) ∃ (unique up to homotopy) contracting homotopy G± for Lpc/fc(M), e.g.

Ω0

pc/fc(M)

id
−G±

d

Ω1

pc/fc(M) −δ

id
G±
Ω1

pc/fc(M) δd

id
−δ G±
Ω0

pc/fc(M) −d

id
Ω0

pc/fc(M)

Ω1

pc/fc(M) −δ

Ω1

pc/fc(M) δd

Ω0

pc/fc(M) −d

(ii) j = ∂G± and Υ = ∂(something) are exact

(iii) Difference G := G+ − G− defines unshifted Poisson structure τ : L(M) ⊗ L(M)

id⊗G

− → L(M) ⊗ Sol(M)

ev

− → R (unique up to homotopy τ + ∂ρ) (iv) Quantization CCR : PoChR → AlgAs(ChC) preserves quasi-isomorphisms and homotopic Poisson structures, i.e. CCR(V, τ + ∂ρ) ≃ CCR(V, τ)

Alexander Schenkel ∞AQFT York 19 10 / 13

SLIDE 81

Example: Linear Yang-Mills as a homotopy AQFT II

⋄ Every derived critical locus carries a [1]-shifted Poisson structure, explicitly:

Smooth dual L(M) =
(−1)

Ω0

c(M) (0)

Ω1

c(M) −δ

(1)

Ω1

c(M) δd

(2)

Ω0

c(M) −d

Canonical inclusion j : L(M)

⊆

− → Lpc/fc(M) − → Sol(M)[1]

Shifted Poisson structure Υ : L(M) ⊗ L(M)

id⊗j

− → L(M) ⊗ Sol(M)[1]

ev

− → R[1]

Thm:

(i) ∃ (unique up to homotopy) contracting homotopy G± for Lpc/fc(M), e.g.

Ω0

pc/fc(M)

id
−G±

d

Ω1

pc/fc(M) −δ

id
G±
Ω1

pc/fc(M) δd

id
−δ G±
Ω0

pc/fc(M) −d

id
Ω0

pc/fc(M)

Ω1

pc/fc(M) −δ

Ω1

pc/fc(M) δd

Ω0

pc/fc(M) −d

(ii) j = ∂G± and Υ = ∂(something) are exact

(iii) Difference G := G+ − G− defines unshifted Poisson structure τ : L(M) ⊗ L(M)

id⊗G

− → L(M) ⊗ Sol(M)

ev

− → R (unique up to homotopy τ + ∂ρ) (iv) Quantization CCR : PoChR → AlgAs(ChC) preserves quasi-isomorphisms and homotopic Poisson structures, i.e. CCR(V, τ + ∂ρ) ≃ CCR(V, τ) (v) Loc ∋ M → AYM(M) := CCR(L(M), τ) defines a homotopy AQFT

Alexander Schenkel ∞AQFT York 19 10 / 13

SLIDE 82

Towards descent in homotopy AQFT

⋄ The inclusion Loc⋄ ֒ → Loc of “diamonds” defines Quillen adjunction ext : AQFT∞(Loc⋄) AQFT∞(Loc) : res

Alexander Schenkel

∞AQFT York 19 11 / 13

SLIDE 83

Towards descent in homotopy AQFT

⋄ The inclusion Loc⋄ ֒ → Loc of “diamonds” defines Quillen adjunction ext : AQFT∞(Loc⋄) AQFT∞(Loc) : res

⋄ Homotopical descent condition: A ∈ AQFT∞(Loc) is homotopy j-local iff

derived counit ǫA : Lext res A

∼

− → A is weak equivalence

Alexander Schenkel ∞AQFT York 19 11 / 13

SLIDE 84

Towards descent in homotopy AQFT

⋄ The inclusion Loc⋄ ֒ → Loc of “diamonds” defines Quillen adjunction ext : AQFT∞(Loc⋄) AQFT∞(Loc) : res

⋄ Homotopical descent condition: A ∈ AQFT∞(Loc) is homotopy j-local iff

derived counit ǫA : Lext 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

SLIDE 85

Towards descent in homotopy AQFT

⋄ The inclusion Loc⋄ ֒ → Loc of “diamonds” defines Quillen adjunction ext : AQFT∞(Loc⋄) AQFT∞(Loc) : res

⋄ Homotopical descent condition: A ∈ AQFT∞(Loc) is homotopy j-local iff

derived counit ǫA : Lext 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

SLIDE 86

Towards descent in homotopy AQFT

⋄ The inclusion Loc⋄ ֒ → Loc of “diamonds” defines Quillen adjunction ext : AQFT∞(Loc⋄) AQFT∞(Loc) : res

⋄ Homotopical descent condition: A ∈ AQFT∞(Loc) is homotopy j-local iff

derived counit ǫA : Lext 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

SLIDE 87

Towards descent in homotopy AQFT

⋄ The inclusion Loc⋄ ֒ → Loc of “diamonds” defines Quillen adjunction ext : AQFT∞(Loc⋄) AQFT∞(Loc) : res

⋄ Homotopical descent condition: A ∈ AQFT∞(Loc) is homotopy j-local iff

derived counit ǫA : Lext 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

SLIDE 88

Towards descent in homotopy AQFT

⋄ The inclusion Loc⋄ ֒ → Loc of “diamonds” defines Quillen adjunction ext : AQFT∞(Loc⋄) AQFT∞(Loc) : res

⋄ Homotopical descent condition: A ∈ AQFT∞(Loc) is homotopy j-local iff

derived counit ǫA : Lext 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 Manm be category of oriented m-manifolds and j : Diskm → Manm the full subcategory of m-disks.

Alexander Schenkel ∞AQFT York 19 11 / 13

SLIDE 89

Towards descent in homotopy AQFT

⋄ The inclusion Loc⋄ ֒ → Loc of “diamonds” defines Quillen adjunction ext : AQFT∞(Loc⋄) AQFT∞(Loc) : res

⋄ Homotopical descent condition: A ∈ AQFT∞(Loc) is homotopy j-local iff

derived counit ǫA : Lext 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 Manm be category of oriented m-manifolds and j : Diskm → Manm the full subcategory of m-disks. Let A : Diskm → AlgE∞(ChK) be functor that is weakly equivalent to a constant functor with value A ∈ AlgE∞(ChK).

Alexander Schenkel ∞AQFT York 19 11 / 13

SLIDE 90

Towards descent in homotopy AQFT

⋄ The inclusion Loc⋄ ֒ → Loc of “diamonds” defines Quillen adjunction ext : AQFT∞(Loc⋄) AQFT∞(Loc) : res

⋄ Homotopical descent condition: A ∈ AQFT∞(Loc) is homotopy j-local iff

derived counit ǫA : Lext 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 Manm be category of oriented m-manifolds and j : Diskm → Manm the full subcategory of m-disks. Let A : Diskm → AlgE∞(ChK) be functor that is weakly equivalent to a constant functor with value A ∈ AlgE∞(ChK). Then the derived extension Lj! A(M) = Sing(M)

L

⊗ A at M ∈ Manm is given by derived higher Hochschild chains on Sing(M) with values in A.

Alexander Schenkel ∞AQFT York 19 11 / 13

SLIDE 91

Toy-model: R-Chern-Simons theory on 2-surfaces

⋄ Linear observables for flat R-connections on oriented 2-manifold M are L(M) = Ω•

c(M)[1] =

(−1)

Ω2

c(M) (0)

Ω1

c(M) d

(1)

Ω0

c(M) d

Alexander Schenkel

∞AQFT York 19 12 / 13

SLIDE 92

Toy-model: R-Chern-Simons theory on 2-surfaces

⋄ Linear observables for flat R-connections on oriented 2-manifold M are L(M) = Ω•

c(M)[1] =

(−1)

Ω2

c(M) (0)

Ω1

c(M) d

(1)

Ω0

c(M) d

⋄ Define A ∈ AQFT∞(Man2

max) ∼

= Fun

Man2, AlgE∞(ChK)
by

A(M) = E∞

L(M)
[No quantization here!]

Alexander Schenkel ∞AQFT York 19 12 / 13

SLIDE 93

Toy-model: R-Chern-Simons theory on 2-surfaces

⋄ Linear observables for flat R-connections on oriented 2-manifold M are L(M) = Ω•

c(M)[1] =

(−1)

Ω2

c(M) (0)

Ω1

c(M) d

(1)

Ω0

c(M) d

⋄ Define A ∈ AQFT∞(Man2

max) ∼

= Fun

Man2, AlgE∞(ChK)
by

A(M) = E∞

L(M)
[No quantization here!]

⋄ Restriction j∗A to 2-disks is weakly equivalent to constant functor with value E∞(R[−1]) because

U : Ω•

c(U)[1] → R[−1] is quasi-iso, for all U ∈ Disk2

Alexander Schenkel ∞AQFT York 19 12 / 13

SLIDE 94

Toy-model: R-Chern-Simons theory on 2-surfaces

⋄ Linear observables for flat R-connections on oriented 2-manifold M are L(M) = Ω•

c(M)[1] =

(−1)

Ω2

c(M) (0)

Ω1

c(M) d

(1)

Ω0

c(M) d

⋄ Define A ∈ AQFT∞(Man2

max) ∼

= Fun

Man2, AlgE∞(ChK)
by

A(M) = E∞

L(M)
[No quantization here!]

⋄ Restriction j∗A to 2-disks is weakly equivalent to constant functor with value E∞(R[−1]) because

U : Ω•

c(U)[1] → R[−1] is quasi-iso, for all U ∈ Disk2

⋄ Compute derived extension at M ∈ Man2: (Lj! j∗ A)(M) ≃ Sing(M)

L

⊗ E∞

R[−1]
≃ Sing(M) ⊗ E∞
R[−1]
Alexander Schenkel

∞AQFT York 19 12 / 13

SLIDE 95

Toy-model: R-Chern-Simons theory on 2-surfaces

⋄ Linear observables for flat R-connections on oriented 2-manifold M are L(M) = Ω•

c(M)[1] =

(−1)

Ω2

c(M) (0)

Ω1

c(M) d

(1)

Ω0

c(M) d

⋄ Define A ∈ AQFT∞(Man2

max) ∼

= Fun

Man2, AlgE∞(ChK)
by

A(M) = E∞

L(M)
[No quantization here!]

⋄ Restriction j∗A to 2-disks is weakly equivalent to constant functor with value E∞(R[−1]) because

U : Ω•

c(U)[1] → R[−1] is quasi-iso, for all U ∈ Disk2

⋄ Compute derived extension at M ∈ Man2: (Lj! j∗ A)(M) ≃ Sing(M)

L

⊗ E∞

R[−1]
≃ Sing(M) ⊗ E∞
R[−1]
[Fresse]

≃ E∞

N∗(Sing(M), R) ⊗ R[−1]
Alexander Schenkel

∞AQFT York 19 12 / 13

SLIDE 96

Toy-model: R-Chern-Simons theory on 2-surfaces

⋄ Linear observables for flat R-connections on oriented 2-manifold M are L(M) = Ω•

c(M)[1] =

(−1)

Ω2

c(M) (0)

Ω1

c(M) d

(1)

Ω0

c(M) d

⋄ Define A ∈ AQFT∞(Man2

max) ∼

= Fun

Man2, AlgE∞(ChK)
by

A(M) = E∞

L(M)
[No quantization here!]

⋄ Restriction j∗A to 2-disks is weakly equivalent to constant functor with value E∞(R[−1]) because

U : Ω•

c(U)[1] → R[−1] is quasi-iso, for all U ∈ Disk2

⋄ Compute derived extension at M ∈ Man2: (Lj! j∗ A)(M) ≃ Sing(M)

L

⊗ E∞

R[−1]
≃ Sing(M) ⊗ E∞
R[−1]
[Fresse]

≃ E∞

N∗(Sing(M), R) ⊗ R[−1]
[de Rham]

≃ E∞

Ω•

c(M)[1]

= A(M)

Alexander Schenkel ∞AQFT York 19 12 / 13

SLIDE 97

Toy-model: R-Chern-Simons theory on 2-surfaces

⋄ Linear observables for flat R-connections on oriented 2-manifold M are L(M) = Ω•

c(M)[1] =

(−1)

Ω2

c(M) (0)

Ω1

c(M) d

(1)

Ω0

c(M) d

⋄ Define A ∈ AQFT∞(Man2

max) ∼

= Fun

Man2, AlgE∞(ChK)
by

A(M) = E∞

L(M)
[No quantization here!]

⋄ Restriction j∗A to 2-disks is weakly equivalent to constant functor with value E∞(R[−1]) because

U : Ω•

c(U)[1] → R[−1] is quasi-iso, for all U ∈ Disk2

⋄ Compute derived extension at M ∈ Man2: (Lj! j∗ A)(M) ≃ Sing(M)

L

⊗ E∞

R[−1]
≃ Sing(M) ⊗ E∞
R[−1]
[Fresse]

≃ E∞

N∗(Sing(M), R) ⊗ R[−1]
[de Rham]

≃ E∞

Ω•

c(M)[1]

= A(M)

⇒ This toy-model is homotopy j-local for j : Disk2

max → Man2 max

Alexander Schenkel ∞AQFT York 19 12 / 13

SLIDE 98

Conclusions and outlook

⋄ Describing gauge theories requires higher categorical structures, in particular stacks and derived stacks

Alexander Schenkel ∞AQFT York 19 13 / 13

SLIDE 99

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

SLIDE 100

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

SLIDE 101

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 ⋄ Our main achievements so far are:

Definition of a model category AQFT∞(C) of homotopy AQFTs

Alexander Schenkel ∞AQFT York 19 13 / 13

SLIDE 102

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 ⋄ Our main achievements so far are:

Definition of a model category AQFT∞(C) of homotopy AQFTs Formal study of derived adjunctions, including derived local-to-global

extensions, together with simple toy-models

Alexander Schenkel ∞AQFT York 19 13 / 13

SLIDE 103

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 ⋄ Our main achievements so far are:

Definition of a model category AQFT∞(C) of homotopy AQFTs Formal study of derived adjunctions, including derived local-to-global

extensions, together with simple toy-models

Construction of linear Yang-Mills theory as a homotopy AQFT via

quantization of (linear approximations of) derived stacks

Alexander Schenkel ∞AQFT York 19 13 / 13

SLIDE 104

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 ⋄ Our main achievements so far are:

Definition of a model category AQFT∞(C) of homotopy AQFTs Formal study of derived adjunctions, including derived local-to-global

extensions, together with simple toy-models

Construction of linear Yang-Mills theory as a homotopy AQFT via

quantization of (linear approximations of) derived stacks

⋄ What I would like to understand in the nearer future:

? Homotopical descent condition for linear Yang-Mills and similar examples

Alexander Schenkel ∞AQFT York 19 13 / 13

SLIDE 105

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 ⋄ Our main achievements so far are:

Definition of a model category AQFT∞(C) of homotopy AQFTs Formal study of derived adjunctions, including derived local-to-global

extensions, together with simple toy-models

Construction of linear Yang-Mills theory as a homotopy AQFT via

quantization of (linear approximations of) derived stacks

⋄ What I would like to understand in the nearer future:

? Homotopical descent condition for linear Yang-Mills and similar examples ? Toy-models of non-linear homotopy AQFTs from derived algebraic geometry

[Lurie; Calaque,Pantev,To¨ en,Vaqui´ e,Vezzosi; Pridham]

Alexander Schenkel ∞AQFT York 19 13 / 13