![Page 1: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/1.jpg)
QFT and Geometric Bordism CategoriesThe reasonable effectiveness of a mathematical definition
Dan Freed
University of Texas at AustinInstitute for Advanced Study
April 9, 2015
![Page 2: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/2.jpg)
What is Quantum Field Theory?
• An open question, even according to physicists (e.g. Seiberg 2015Breakthrough Prize lecture)
• Yet there is a mathematical definition (late ’80s), well-explored fortopological and conformal field theories
• Also crucial is the elaboration to extended field theory (early ’90s)
• Less widely appreciated is its power for scale-dependent fieldtheories (but see Stolz-Teichner, Costello, . . . )
Plan of Lecture: Two applications + questions
• Invertible topological theories and phases of matter
• Line operators in 4-dimensional gauge theory
• Formulate issues for scale-dependent theories
![Page 3: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/3.jpg)
What is Quantum Field Theory?
• An open question, even according to physicists (e.g. Seiberg 2015Breakthrough Prize lecture)
• Yet there is a mathematical definition (late ’80s), well-explored fortopological and conformal field theories
• Also crucial is the elaboration to extended field theory (early ’90s)
• Less widely appreciated is its power for scale-dependent fieldtheories (but see Stolz-Teichner, Costello, . . . )
Plan of Lecture: Two applications + questions
• Invertible topological theories and phases of matter
• Line operators in 4-dimensional gauge theory
• Formulate issues for scale-dependent theories
![Page 4: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/4.jpg)
What is Quantum Field Theory?
• An open question, even according to physicists (e.g. Seiberg 2015Breakthrough Prize lecture)
• Yet there is a mathematical definition (late ’80s), well-explored fortopological and conformal field theories
• Also crucial is the elaboration to extended field theory (early ’90s)
• Less widely appreciated is its power for scale-dependent fieldtheories (but see Stolz-Teichner, Costello, . . . )
Plan of Lecture: Two applications + questions
• Invertible topological theories and phases of matter
• Line operators in 4-dimensional gauge theory
• Formulate issues for scale-dependent theories
![Page 5: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/5.jpg)
What is Quantum Field Theory?
• An open question, even according to physicists (e.g. Seiberg 2015Breakthrough Prize lecture)
• Yet there is a mathematical definition (late ’80s), well-explored fortopological and conformal field theories
• A crucial elaboration to extended field theory (early ’90s)
• Less widely appreciated is the power of these ideas forscale-dependent field theories (but see Stolz-Teichner, Costello, . . . )
Plan of Lecture: Two applications + questions
• Invertible topological theories and phases of matter
• Line operators in 4-dimensional gauge theory
• Formulate issues for scale-dependent theories
![Page 6: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/6.jpg)
What is Quantum Field Theory?
• An open question, even according to physicists (e.g. Seiberg 2015Breakthrough Prize lecture)
• Yet there is a mathematical definition (late ’80s), well-explored fortopological and conformal field theories
• A crucial elaboration to extended field theory (early ’90s)
• Less widely appreciated is the power of these ideas forscale-dependent field theories
Plan of Lecture: Two applications + questions
• Invertible topological theories and phases of matter
• Line operators in 4-dimensional gauge theory
• Formulate issues for scale-dependent theories
![Page 7: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/7.jpg)
What is Quantum Field Theory?
• An open question, even according to physicists (e.g. Seiberg 2015Breakthrough Prize lecture)
• Yet there is a mathematical definition (late ’80s), well-explored fortopological and conformal field theories
• A crucial elaboration to extended field theory (early ’90s)
• Less widely appreciated is the power of these ideas forscale-dependent field theories
Plan of Lecture: Two applications + questions
• Invertible topological theories and phases of matter
• Line operators in 4-dimensional gauge theory
• Formulate issues for scale-dependent theories
![Page 8: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/8.jpg)
What is Quantum Field Theory?
• An open question, even according to physicists (e.g. Seiberg 2015Breakthrough Prize lecture)
• Yet there is a mathematical definition (late ’80s), well-explored fortopological and conformal field theories
• A crucial elaboration to extended field theory (early ’90s)
• Less widely appreciated is the power of these ideas forscale-dependent field theories
Plan of Lecture: Two applications + questions
• Invertible topological theories and phases of matter
• Line operators in 4-dimensional gauge theory
• Formulate issues for scale-dependent theories
![Page 9: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/9.jpg)
What is Quantum Field Theory?
• An open question, even according to physicists (e.g. Seiberg 2015Breakthrough Prize lecture)
• Yet there is a mathematical definition (late ’80s), well-explored fortopological and conformal field theories
• A crucial elaboration to extended field theory (early ’90s)
• Less widely appreciated is the power of these ideas forscale-dependent field theories
Plan of Lecture: Two applications + questions
• Invertible topological theories and phases of matter
• Line operators in 4-dimensional gauge theory
• Formulate issues for scale-dependent theories
![Page 10: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/10.jpg)
What is Quantum Field Theory?
• An open question, even according to physicists (e.g. Seiberg 2015Breakthrough Prize lecture)
• Yet there is a mathematical definition (late ’80s), well-explored fortopological and conformal field theories
• A crucial elaboration to extended field theory (early ’90s)
• Less widely appreciated is the power of these ideas forscale-dependent field theories
Plan of Lecture: Two applications + questions
• Invertible topological theories and phases of matter
• Line operators in 4-dimensional gauge theory
• Formulate issues for scale-dependent theories
![Page 11: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/11.jpg)
QFT as a Representation of Geometric Bordism
Definition: A field theory is a homomorphism (⊗-functor)
F : Bord〈n−1,n〉(F) −→ VecttopC
F sheaf of background fields (orientation, metric, . . . )VecttopC category of complex topological vector spaces
Bord〈n−1,n〉(F):
objects morphisms
![Page 12: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/12.jpg)
QFT as a Representation of Geometric Bordism
Definition: A field theory is a homomorphism (⊗-functor)
F : Bord〈n−1,n〉(F) −→ VecttopC
F sheaf of background fields (orientation, metric, . . . )VecttopC category of complex topological vector spaces
Bord〈n−1,n〉(F):
objects morphisms
![Page 13: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/13.jpg)
QFT as a Representation of Geometric Bordism
Definition: A field theory is a homomorphism (⊗-functor)
F : Bord〈n−1,n〉(F) −→ VecttopC
F sheaf of background fields (orientation, metric, . . . )VecttopC category of complex topological vector spaces
Bord〈n−1,n〉(F):
objects morphisms
![Page 14: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/14.jpg)
QFT as a Representation of Geometric Bordism
Definition: A fully extended field theory is a homomorphism
F : Bordn(F) −→ C
F sheaf of background fields (orientation, metric, . . . )C topological n-category
Bordn(F):
objects 1-morphisms 2-morphisms · · ·
![Page 15: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/15.jpg)
Longrange Effective TFT of a Gapped System
Low energy behavior: energy gap =⇒ topological field theory α
α is a topological invariant of “phase” in space of gapped systems
Applies to lattice systems, assuming existence of thermodynamic limit
Assume α is fully extended (strong locality)
Microscopic short-range entanglement =⇒ α invertible
Invertible topological field theories are maps of infinite loop spaces:
Bordn(F)α //
��
C
|Bordn(F)| α̃ // C×
OO
![Page 16: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/16.jpg)
Longrange Effective TFT of a Gapped System
Low energy behavior: energy gap =⇒ topological field theory α
α is a topological invariant of “phase” in space of gapped systems
Applies to lattice systems, assuming existence of thermodynamic limit
Assume α is fully extended (strong locality)
Microscopic short-range entanglement =⇒ α invertible
Invertible topological field theories are maps of infinite loop spaces:
Bordn(F)α //
��
C
|Bordn(F)| α̃ // C×
OO
![Page 17: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/17.jpg)
Longrange Effective TFT of a Gapped System
Low energy behavior: energy gap =⇒ topological field theory α
α is a topological invariant of “phase” in space of gapped systems
Applies to lattice systems, assuming existence of thermodynamic limit
Assume α is fully extended (strong locality)
Microscopic short-range entanglement =⇒ α invertible
Invertible topological field theories are maps of infinite loop spaces:
Bordn(F)α //
��
C
|Bordn(F)| α̃ // C×
OO
![Page 18: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/18.jpg)
Longrange Effective TFT of a Gapped System
Low energy behavior: energy gap =⇒ topological field theory α
α is a topological invariant of “phase” in space of gapped systems
Applies to lattice systems, assuming existence of thermodynamic limit
Assume α is fully extended (strong locality)
Microscopic short-range entanglement =⇒ α invertible
Invertible topological field theories are maps of infinite loop spaces:
Bordn(F)α //
��
C
|Bordn(F)| α̃ // C×
OO
![Page 19: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/19.jpg)
Longrange Effective TFT of a Gapped System
Low energy behavior: energy gap =⇒ topological field theory α
α is a topological invariant of “phase” in space of gapped systems
Applies to lattice systems, assuming existence of thermodynamic limit
Assume α is fully extended (strong locality)
Microscopic short-range entanglement =⇒ α invertible
Invertible topological field theories are maps of infinite loop spaces:
Bordn(F)α //
��
C
|Bordn(F)| α̃ // C×
OO
![Page 20: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/20.jpg)
Longrange Effective TFT of a Gapped System
Low energy behavior: energy gap =⇒ topological field theory α
α is a topological invariant of “phase” in space of gapped systems
Applies to lattice systems, assuming existence of thermodynamic limit
Assume α is fully extended (strong locality)
Microscopic short-range entanglement =⇒ α invertible
Invertible topological field theories are maps of infinite loop spaces:
Bordn(F)α //
��
C
|Bordn(F)| α̃ // C×
OO
![Page 21: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/21.jpg)
Classification of Invertible Topological Field Theories
Thm (Galatius-Madsen-Tillmann-Weiss): |Bordn | ' Σ∞+nMTOn
MTOn = Thom(−ξn −→ BOn)
(MTO1 −→MTO2 −→MTO3 −→ · · · ) −→MO
Codomain: universal C = C× = ΣnIC× (Pontrjagin dual to sphere)
Many explicit computations (arXiv:1406.7278)
Thm (F.-Hopkins): The abelian group of 4d unitary invertibletopological field theories/deformation with time-reversal (T ) is:
(i) (T 2 = (−1)F ) [Σ4MTPin+4 ,Σ
4IC×]unitary ∼= Z/16Z(ii) (T 2 = +1) [Σ4MTPin−4 ,Σ
4IC×]unitary ∼= Z/2Z
Takeaway: The geometric bordism definition of extended field theoryenables these computations.
![Page 22: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/22.jpg)
Classification of Invertible Topological Field Theories
Thm (Galatius-Madsen-Tillmann-Weiss): |Bordn | ' Σ∞+nMTOn
MTOn = Thom(−ξn −→ BOn)
(Σ1MTO1 −→ Σ2MTO2 −→ Σ3MTO3 −→ · · · ) −→MO
Codomain: universal C = C× = ΣnIC× (Pontrjagin dual to sphere)
Many explicit computations (arXiv:1406.7278)
Thm (F.-Hopkins): The abelian group of 4d unitary invertibletopological field theories/deformation with time-reversal (T ) is:
(i) (T 2 = (−1)F ) [Σ4MTPin+4 ,Σ
4IC×]unitary ∼= Z/16Z(ii) (T 2 = +1) [Σ4MTPin−4 ,Σ
4IC×]unitary ∼= Z/2Z
Takeaway: The geometric bordism definition of extended field theoryenables these computations.
![Page 23: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/23.jpg)
Classification of Invertible Topological Field Theories
Thm (Galatius-Madsen-Tillmann-Weiss): |Bordn | ' Σ∞+nMTOn
MTOn = Thom(−ξn −→ BOn)
(Σ1MTO1 −→ Σ2MTO2 −→ Σ3MTO3 −→ · · · ) −→MO
Codomain: universal C = C× = ΣnIC× (Pontrjagin dual to sphere)
Many explicit computations (arXiv:1406.7278)
Thm (F.-Hopkins): The abelian group of 4d unitary invertibletopological field theories/deformation with time-reversal (T ) is:
(i) (T 2 = (−1)F ) [Σ4MTPin+4 ,Σ
4IC×]unitary ∼= Z/16Z(ii) (T 2 = +1) [Σ4MTPin−4 ,Σ
4IC×]unitary ∼= Z/2Z
Takeaway: The geometric bordism definition of extended field theoryenables these computations.
![Page 24: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/24.jpg)
Classification of Invertible Topological Field Theories
Thm (Galatius-Madsen-Tillmann-Weiss): |Bordn | ' Σ∞+nMTOn
MTOn = Thom(−ξn −→ BOn)
(Σ1MTO1 −→ Σ2MTO2 −→ Σ3MTO3 −→ · · · ) −→MO
Codomain: universal C = C× = ΣnIC× (Pontrjagin dual to sphere)
Many explicit computations (arXiv:1406.7278)
Thm (F.-Hopkins): The abelian group of 4d unitary invertibletopological field theories/deformation with time-reversal (T ) is:
(i) (T 2 = (−1)F ) [Σ4MTPin+4 ,Σ
4IC×]unitary ∼= Z/16Z(ii) (T 2 = +1) [Σ4MTPin−4 ,Σ
4IC×]unitary ∼= Z/2Z
Takeaway: The geometric bordism definition of extended field theoryenables these computations.
![Page 25: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/25.jpg)
Classification of Invertible Topological Field Theories
Thm (Galatius-Madsen-Tillmann-Weiss): |Bordn | ' Σ∞+nMTOn
MTOn = Thom(−ξn −→ BOn)
(Σ1MTO1 −→ Σ2MTO2 −→ Σ3MTO3 −→ · · · ) −→MO
Codomain: universal C = C× = ΣnIC× (Pontrjagin dual to sphere)
Bordn(F)α //
��
C
|Bordn(F)|α̃ // C×
OO
Thm (F.-Hopkins): The abelian group of 4d unitary invertibletopological field theories/deformation with time-reversal (T ) is:
(i) (T 2 = (−1)F ) [Σ4MTPin+4 ,Σ
4IC×]unitary ∼= Z/16Z(ii) (T 2 = +1) [Σ4MTPin−4 ,Σ
4IC×]unitary ∼= Z/2Z
Takeaway: The geometric bordism definition of extended field theoryenables these computations.
![Page 26: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/26.jpg)
Classification of Invertible Topological Field Theories
Thm (Galatius-Madsen-Tillmann-Weiss): |Bordn | ' Σ∞+nMTOn
MTOn = Thom(−ξn −→ BOn)
(Σ1MTO1 −→ Σ2MTO2 −→ Σ3MTO3 −→ · · · ) −→MO
Codomain: universal C = C× = ΣnIC× (Pontrjagin dual to sphere)
Bordn(F)α //
��
C
|Bordn(F)|α̃ // C×
OO
Thm (F.-Hopkins): The abelian group of 4d unitary invertibletopological field theories/deformation with time-reversal (T ) is:
(i) (T 2 = (−1)F ) [Σ4MTPin+4 ,Σ
4IC×]unitary ∼= Z/16Z(ii) (T 2 = +1) [Σ4MTPin−4 ,Σ
4IC×]unitary ∼= Z/2Z
Takeaway: The geometric bordism definition of extended field theoryenables these computations.
![Page 27: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/27.jpg)
Towards 2nd Application: Relative Field Theories
Warmup: σ-model into manifold M with fundamental group π = π1M
π −→ M̃ −→M
Fiber bundle of fields: M̃ −→M −→ Bπ
Up one level: A→ G→ G covering of compact Lie groups, A finite
Fiber bundle of fields: B∇G −→ B∇G −→ B2A
Integrate over the fibers to define a relative theory f
![Page 28: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/28.jpg)
Towards 2nd Application: Relative Field Theories
Warmup: σ-model into manifold M with fundamental group π = π1M
π −→ M̃ −→M
Fiber bundle of fields: M̃ −→M −→ Bπ
Up one level: A→ G→ G covering of compact Lie groups, A finite
Fiber bundle of fields: B∇G −→ B∇G −→ B2A
Integrate over the fibers to define a relative theory f
![Page 29: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/29.jpg)
Towards 2nd Application: Relative Field Theories
Warmup: σ-model into manifold M with fundamental group π = π1M
π −→ M̃ −→M
Fiber bundle of fields: M̃ −→M −→ Bπ
Up one level: A→ G→ G covering of compact Lie groups, A finite
Fiber bundle of fields: B∇G −→ B∇G −→ B2A
Integrate over the fibers to define a relative theory f
![Page 30: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/30.jpg)
Towards 2nd Application: Relative Field Theories
Warmup: σ-model into manifold M with fundamental group π = π1M
π −→ M̃ −→M
Fiber bundle of fields: M̃ −→M −→ Bπ
Up one level: A→ G→ G covering of compact Lie groups, A finite
Fiber bundle of fields: B∇G −→ B∇G −→ B2A
Integrate over the fibers to define a relative theory f
![Page 31: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/31.jpg)
Towards 2nd Application: Relative Field Theories
Warmup: σ-model into manifold M with fundamental group π = π1M
π −→ M̃ −→M
Fiber bundle of fields: M̃ −→M −→ Bπ
Up one level: A→ G→ G covering of compact Lie groups, A finite
Fiber bundle of fields: B∇G −→ B∇G −→ B2A
Integrate over the fibers to define a relative theory f
![Page 32: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/32.jpg)
Towards 2nd Application: Relative Field Theories
Warmup: σ-model into manifold M with fundamental group π = π1M
π −→ M̃ −→M
Fiber bundle of fields: M̃ −→M −→ Bπ
Up one level: A→ G→ G covering of compact Lie groups, A finite
Fiber bundle of fields: B∇G −→ B∇G −→ B2A
Integrate over the fibers to define a relative theory f
![Page 33: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/33.jpg)
Towards 2nd Application: Relative Field Theories
Warmup: σ-model into manifold M with fundamental group π = π1M
π −→ M̃ −→M
Fiber bundle of fields: M̃ −→M −→ Bπ
Up one level: A→ G→ G covering of compact Lie groups, A finite
Fiber bundle of fields: B∇G −→ B∇G −→ B2A
Integrate over the fibers to define a relative theory f
![Page 34: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/34.jpg)
Towards 2nd Application: Relative Field Theories
Warmup: σ-model into manifold M with fundamental group π = π1M
π −→ M̃ −→M
Fiber bundle of fields: M̃ −→M −→ Bπ
Up one level: A→ G→ G covering of compact Lie groups, A finite
Fiber bundle of fields: B∇G −→ B∇G −→ B2A
Integrate over the fibers to define a relative theory f
![Page 35: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/35.jpg)
Towards 2nd Application: Relative Field Theories
Definition: Let α be an extended (n+ 1)-dimensional quantum fieldtheory. A field theory f relative to α is a homomorphism
f : 1 −→ τ≤nα
orf̃ : τ≤nα −→ 1
(see F.-Teleman arXiv:1212.1692)
Warning: Only the truncation of α need be defined
Boundary condition:
Given both f and f̃ we obtain an absolute n-dimensional theory f̃ ◦ f
Two boundary conditions:
![Page 36: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/36.jpg)
Towards 2nd Application: Relative Field Theories
Definition: Let α be an extended (n+ 1)-dimensional quantum fieldtheory. A field theory f relative to α is a homomorphism
f : 1 −→ τ≤nα
orf̃ : τ≤nα −→ 1
(see F.-Teleman arXiv:1212.1692)
Warning: Only the truncation of α need be defined
Boundary condition:
Given both f and f̃ we obtain an absolute n-dimensional theory f̃ ◦ f
Two boundary conditions:
![Page 37: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/37.jpg)
Towards 2nd Application: Relative Field Theories
Definition: Let α be an extended (n+ 1)-dimensional quantum fieldtheory. A field theory f relative to α is a homomorphism
f : 1 −→ τ≤nα
orf̃ : τ≤nα −→ 1
(see F.-Teleman arXiv:1212.1692)
Warning: Only the truncation of α need be defined
Boundary condition:
Given both f and f̃ we obtain an absolute n-dimensional theory f̃ ◦ f
Two boundary conditions:
![Page 38: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/38.jpg)
Towards 2nd Application: Relative Field Theories
Definition: Let α be an extended (n+ 1)-dimensional quantum fieldtheory. A field theory f relative to α is a homomorphism
f : 1 −→ τ≤nα
orf̃ : τ≤nα −→ 1
(see F.-Teleman arXiv:1212.1692)
Warning: Only the truncation of α need be defined
Boundary condition:
Given both f and f̃ we obtain an absolute n-dimensional theory f̃ ◦ f
Two boundary conditions:
![Page 39: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/39.jpg)
Towards 2nd Application: Relative Field Theories
Definition: Let α be an extended (n+ 1)-dimensional quantum fieldtheory. A field theory f relative to α is a homomorphism
f : 1 −→ τ≤nα
orf̃ : τ≤nα −→ 1
(see F.-Teleman arXiv:1212.1692)
Warning: Only the truncation of α need be defined
Boundary condition:
Given both f and f̃ we obtain an absolute n-dimensional theory f̃ ◦ f
Two boundary conditions:
![Page 40: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/40.jpg)
Line Operators in 4d Gauge Theories
Point (“local”) operators F (S3) Line operators F (S2)
![Page 41: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/41.jpg)
Line Operators in 4d Gauge Theories
For f : 1→ τ≤4α the 1-category of line operators is organized by the
2-category α(S2). For α the topological theory of A-gerbes we find
f(S2) = {Lm,e} m ∈ H2(S2;A) ∼= A e ∈ H0(S2;A)∨ ∼= A∨
Fix (A′, q) to define f̃ : τ≤4α→ 1 and absolute theory F = f̃ ◦ f
A′ ≤ A, q : A′ → Q/Z (quadratic) =⇒ b = e2πi ∂q : A′×A′ → C× (bilinear)
F is a gauge theory with gauge group G/A′. (Recall covering GA−−→ G)
Main assertion: A “higher Gauss law” implies
f(S2) =⊕m′∈A′
e∈A∨ such that e|A′
=b(m′)−1
Lm′,e
Reproduces Aharony-Seiberg-Tachikawa (arXiv:1305.0318); furtherwork in Gaiotto-Kapustin-Seiberg-Willet (arXiv:1412.5148)
![Page 42: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/42.jpg)
Line Operators in 4d Gauge Theories
For f : 1→ τ≤4α the 1-category of line operators is organized by the
2-category α(S2). For α the topological theory of A-gerbes we find
f(S2) = {Lm,e} m ∈ H2(S2;A) ∼= A e ∈ H0(S2;A)∨ ∼= A∨
Fix (A′, q) to define f̃ : τ≤4α→ 1 and absolute theory F = f̃ ◦ f
A′ ≤ A, q : A′ → Q/Z (quadratic) =⇒ b = e2πi ∂q : A′×A′ → C× (bilinear)
F is a gauge theory with gauge group G/A′. (Recall covering GA−−→ G)
Main assertion: A “higher Gauss law” implies
f(S2) =⊕m′∈A′
e∈A∨ such that e|A′
=b(m′)−1
Lm′,e
Reproduces Aharony-Seiberg-Tachikawa (arXiv:1305.0318); furtherwork in Gaiotto-Kapustin-Seiberg-Willet (arXiv:1412.5148)
![Page 43: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/43.jpg)
Line Operators in 4d Gauge Theories
For f : 1→ τ≤4α the 1-category of line operators is organized by the
2-category α(S2). For α the topological theory of A-gerbes we find
f(S2) = {Lm,e} m ∈ H2(S2;A) ∼= A e ∈ H0(S2;A)∨ ∼= A∨
Fix (A′, q) to define f̃ : τ≤4α→ 1 and absolute theory F = f̃ ◦ f
A′ ≤ A, q : A′ → Q/Z (quadratic) =⇒ b = e2πi ∂q : A′×A′ → C× (bilinear)
F is a gauge theory with gauge group G/A′. (Recall covering GA−−→ G)
Main assertion: A “higher Gauss law” implies
f(S2) =⊕m′∈A′
e∈A∨ such that e|A′
=b(m′)−1
Lm′,e
Reproduces Aharony-Seiberg-Tachikawa (arXiv:1305.0318); furtherwork in Gaiotto-Kapustin-Seiberg-Willet (arXiv:1412.5148)
![Page 44: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/44.jpg)
Line Operators in 4d Gauge Theories
For f : 1→ τ≤4α the 1-category of line operators is organized by the
2-category α(S2). For α the topological theory of A-gerbes we find
f(S2) = {Lm,e} m ∈ H2(S2;A) ∼= A e ∈ H0(S2;A)∨ ∼= A∨
Fix (A′, q) to define f̃ : τ≤4α→ 1 and absolute theory F = f̃ ◦ f
A′ ≤ A, q : A′ → Q/Z (quadratic) =⇒ b = e2πi ∂q : A′×A′ → C× (bilinear)
F is a gauge theory with gauge group G/A′. (Recall covering GA−−→ G)
Main assertion: A “higher Gauss law” implies
f(S2) =⊕m′∈A′
e∈A∨ such that e|A′
=b(m′)−1
Lm′,e
Aharony-Seiberg-Tachikawa (arXiv:1305.0318)Gaiotto-Kapustin-Seiberg-Willet (arXiv:1412.5148)
![Page 45: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/45.jpg)
Axiom System for Scale-Dependent Theories?
Topological aspects continue to be interesting and fruitful. But now alsotime to turn attention to the larger
Question: Does QFT=representation of geometric bordism categorydefinition (suitably enhanced) capture entire scale-dependent theories?
How to attack? “Construct examples!” Perhaps not the main focus now.
So focus on structural issues, especially to provoke enhancements.
Sample questions among many possible:
• Define mass gap. Construct infrared topological theory.
• Define free theories. Asymptotic freedom.
• Construct quantum moduli space. Infrared conformal theory.
• Reconstruction: field theory on Minkowski spacetime.
![Page 46: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/46.jpg)
Axiom System for Scale-Dependent Theories?
Topological aspects continue to be interesting and fruitful. But now alsotime to turn attention to the larger
Question: Does QFT=representation of geometric bordism categorydefinition (suitably enhanced) capture entire scale-dependent theories?
How to attack? “Construct examples!” Perhaps not the main focus now.
So focus on structural issues, especially to provoke enhancements.
Sample questions among many possible:
• Define mass gap. Construct infrared topological theory.
• Define free theories. Asymptotic freedom.
• Construct quantum moduli space. Infrared conformal theory.
• Reconstruction: field theory on Minkowski spacetime.
![Page 47: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/47.jpg)
Axiom System for Scale-Dependent Theories?
Topological aspects continue to be interesting and fruitful. But now alsotime to turn attention to the larger
Question: Does QFT=representation of geometric bordism categorydefinition (suitably enhanced) capture entire scale-dependent theories?
How to attack? “Construct examples!” Perhaps not the main focus now.
So focus on structural issues, especially to provoke enhancements.
Sample questions among many possible:
• Define mass gap. Construct infrared topological theory.
• Define free theories. Asymptotic freedom.
• Construct quantum moduli space. Infrared conformal theory.
• Reconstruction: field theory on Minkowski spacetime.
![Page 48: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/48.jpg)
Axiom System for Scale-Dependent Theories?
Topological aspects continue to be interesting and fruitful. But now alsotime to turn attention to the larger
Question: Does QFT=representation of geometric bordism categorydefinition (suitably enhanced) capture entire scale-dependent theories?
How to attack? “Construct examples!” Perhaps not the main focus now.
So focus on structural issues, especially to provoke enhancements.
Sample questions among many possible:
• Define mass gap. Construct infrared topological theory.
• Define free theories. Asymptotic freedom.
• Construct quantum moduli space. Infrared conformal theory.
• Reconstruction: field theory on Minkowski spacetime.
![Page 49: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/49.jpg)
Axiom System for Scale-Dependent Theories?
Topological aspects continue to be interesting and fruitful. But now alsotime to turn attention to the larger
Question: Does QFT=representation of geometric bordism categorydefinition (suitably enhanced) capture entire scale-dependent theories?
How to attack? “Construct examples!” Perhaps not the main focus now.
So focus on structural issues, especially to provoke enhancements.
Sample questions among many possible:
• Define mass gap. Construct infrared topological theory.
• Define free theories. Asymptotic freedom.
• Construct quantum moduli space. Infrared conformal theory.
• Reconstruction: field theory on Minkowski spacetime.
![Page 50: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/50.jpg)
Axiom System for Scale-Dependent Theories?
Topological aspects continue to be interesting and fruitful. But now alsotime to turn attention to the larger
Question: Does QFT=representation of geometric bordism categorydefinition (suitably enhanced) capture entire scale-dependent theories?
How to attack? “Construct examples!” Perhaps not the main focus now.
So focus on structural issues, especially to provoke enhancements.
Sample questions among many possible:
• Define mass gap. Construct infrared topological theory.
• Define free theories. Asymptotic freedom.
• Construct quantum moduli space. Infrared conformal theory.
• Reconstruction: field theory on Minkowski spacetime.
![Page 51: QFT and Geometric Bordism CategoriesBreakthrough Prize lecture) Yet there is a mathematical de nition (late ’80s), well-explored for topological and conformal eld theories A crucial](https://reader033.vdocuments.us/reader033/viewer/2022042006/5e702a1c9169362ac417168b/html5/thumbnails/51.jpg)
Axiom System for Scale-Dependent Theories?
Topological aspects continue to be interesting and fruitful. But now alsotime to turn attention to the larger
Question: Does QFT=representation of geometric bordism categorydefinition (suitably enhanced) capture entire scale-dependent theories?
How to attack? “Construct examples!” Perhaps not the main focus now.
So focus on structural issues, especially to provoke enhancements.
Sample questions among many possible:
• Define mass gap. Construct infrared topological theory.
• Define free theories. Asymptotic freedom.
• Construct quantum moduli space. Infrared conformal theory.
• Reconstruction: field theory on Minkowski spacetime.