Mathematics

Geometry of N=2 Supersymmetry 2 of 4

Speaker: 
Andy Neitzke
Date: 
Tue, Aug 24, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

Coulomb branches of N=2 supersymmetric field theories in four dimensions support a rich geometry. My aim in these lectures will be to explain some aspects of this geometry, and its relation to the physics of the N=2 theories themselves.

I will first describe various constructions of N=2 theories and the corresponding Coulomb branches. In this story the main geometry visible is that of a complex integrable system, fibered over the Coulomb branch; one nice class of examples is associated to the Hitchin integrable system (moduli space of Higgs bundles over a Riemann surface). Fundamental objects in the N=2 theory (local operators, line operators, surface operators) all have geometric counterparts in the integrable system, as I will explain.

Next I will discuss a deformation of the story, which arises in physics from the Nekrasov-Shatashvili Omega-background. In this deformation, the Coulomb branch is replaced by a closely related space; for instance, the base of the Hitchin integrable system is replaced by a space parameterizing opers over the Riemann surface. One can use this deformation to give a concrete picture of the space of opers; in so doing one meets Stokes phenomena which are governed by the BPS indices (Donaldson-Thomas invariants) of the N=2 theory. This turns out to be closely related to the "exact WKB method" in analysis of ODEs. It is also connected to Riemann-Hilbert problems of a sort recently investigated by Bridgeland, as I will describe if time permits.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

Geometry of N=2 Supersymmetry 1 of 4

Speaker: 
Andy Neitzke
Date: 
Mon, Aug 23, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

Coulomb branches of N=2 supersymmetric field theories in four dimensions support a rich geometry. My aim in these lectures will be to explain some aspects of this geometry, and its relation to the physics of the N=2 theories themselves.

I will first describe various constructions of N=2 theories and the corresponding Coulomb branches. In this story the main geometry visible is that of a complex integrable system, fibered over the Coulomb branch; one nice class of examples is associated to the Hitchin integrable system (moduli space of Higgs bundles over a Riemann surface). Fundamental objects in the N=2 theory (local operators, line operators, surface operators) all have geometric counterparts in the integrable system, as I will explain.

Next I will discuss a deformation of the story, which arises in physics from the Nekrasov-Shatashvili Omega-background. In this deformation, the Coulomb branch is replaced by a closely related space; for instance, the base of the Hitchin integrable system is replaced by a space parameterizing opers over the Riemann surface. One can use this deformation to give a concrete picture of the space of opers; in so doing one meets Stokes phenomena which are governed by the BPS indices (Donaldson-Thomas invariants) of the N=2 theory. This turns out to be closely related to the "exact WKB method" in analysis of ODEs. It is also connected to Riemann-Hilbert problems of a sort recently investigated by Bridgeland, as I will describe if time permits.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

Derived Geometry in Twists of Gauge Theories 4 of 4

Speaker: 
Tudor Dimofte
Date: 
Thu, Aug 26, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

These lectures will review and develop methods in algebraic geometry (in particular, derived algebraic geometry) to describe topological and holomorphic sectors of quantum field theories. A recurring theme will be the interaction of local and extended operators, and of QFT's in different dimensions. The main examples will come from twists of supersymmetric gauge theories, and will connect to a large body of recent and ongoing work on 3d Coulomb branches, 3d mirror symmetry (and geometric Langlands), logarithmic VOA's and non-semisimple TQFT's, and categorified cluster algebras.

The basic plan for the lectures is:

  • Lecture 1 (2d warmup): categories of boundary conditions, interfaces, and Koszul duality
  • Lectures 2 and 3 (3d): twists of 3d N=2 and N=4 gauge theories; vertex algebras, chiral categories, and braided tensor categories; d mirror symmetry; quantum groups at roots of unity and derived non-semisimple 3d TQFT's (compared and contrasted with Chern-Simons theory)
  • Lecture 4 (4d): line and surface operators in 4d N=2 gauge theory, the coherent Satake category, and relations to Schur indices and 4d N=2 vertex algebras
  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

Derived Geometry in Twists of Gauge Theories 3 of 4

Speaker: 
Tudor Dimofte
Date: 
Wed, Aug 25, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

These lectures will review and develop methods in algebraic geometry (in particular, derived algebraic geometry) to describe topological and holomorphic sectors of quantum field theories. A recurring theme will be the interaction of local and extended operators, and of QFT's in different dimensions. The main examples will come from twists of supersymmetric gauge theories, and will connect to a large body of recent and ongoing work on 3d Coulomb branches, 3d mirror symmetry (and geometric Langlands), logarithmic VOA's and non-semisimple TQFT's, and categorified cluster algebras.

The basic plan for the lectures is:

  • Lecture 1 (2d warmup): categories of boundary conditions, interfaces, and Koszul duality
  • Lectures 2 and 3 (3d): twists of 3d N=2 and N=4 gauge theories; vertex algebras, chiral categories, and braided tensor categories; d mirror symmetry; quantum groups at roots of unity and derived non-semisimple 3d TQFT's (compared and contrasted with Chern-Simons theory)
  • Lecture 4 (4d): line and surface operators in 4d N=2 gauge theory, the coherent Satake category, and relations to Schur indices and 4d N=2 vertex algebras
  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

Derived Geometry in Twists of Gauge Theories 2 of 4

Speaker: 
Tudor Dimofte
Date: 
Tue, Aug 24, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

These lectures will review and develop methods in algebraic geometry (in particular, derived algebraic geometry) to describe topological and holomorphic sectors of quantum field theories. A recurring theme will be the interaction of local and extended operators, and of QFT's in different dimensions. The main examples will come from twists of supersymmetric gauge theories, and will connect to a large body of recent and ongoing work on 3d Coulomb branches, 3d mirror symmetry (and geometric Langlands), logarithmic VOA's and non-semisimple TQFT's, and categorified cluster algebras.

The basic plan for the lectures is:

  • Lecture 1 (2d warmup): categories of boundary conditions, interfaces, and Koszul duality
  • Lectures 2 and 3 (3d): twists of 3d N=2 and N=4 gauge theories; vertex algebras, chiral categories, and braided tensor categories; d mirror symmetry; quantum groups at roots of unity and derived non-semisimple 3d TQFT's (compared and contrasted with Chern-Simons theory)
  • Lecture 4 (4d): line and surface operators in 4d N=2 gauge theory, the coherent Satake category, and relations to Schur indices and 4d N=2 vertex algebras
  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

Derived Geometry in Twists of Gauge Theories 1 of 4

Speaker: 
Tudor Dimofte
Date: 
Mon, Aug 23, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

These lectures will review and develop methods in algebraic geometry (in particular, derived algebraic geometry) to describe topological and holomorphic sectors of quantum field theories. A recurring theme will be the interaction of local and extended operators, and of QFT's in different dimensions. The main examples will come from twists of supersymmetric gauge theories, and will connect to a large body of recent and ongoing work on 3d Coulomb branches, 3d mirror symmetry (and geometric Langlands), logarithmic VOA's and non-semisimple TQFT's, and categorified cluster algebras.

The basic plan for the lectures is:

  • Lecture 1 (2d warmup): categories of boundary conditions, interfaces, and Koszul duality
  • Lectures 2 and 3 (3d): twists of 3d N=2 and N=4 gauge theories; vertex algebras, chiral categories, and braided tensor categories; d mirror symmetry; quantum groups at roots of unity and derived non-semisimple 3d TQFT's (compared and contrasted with Chern-Simons theory)
  • Lecture 4 (4d): line and surface operators in 4d N=2 gauge theory, the coherent Satake category, and relations to Schur indices and 4d N=2 vertex algebras
  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

Elliptic Fibrations and Singularities to Anomalies and Spectra 4 of 4

Speaker: 
Monica Jinwoo Kang
Date: 
Thu, Aug 26, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

Throughout my lectures I will explain the geometry of elliptic fibration which can gave rise to understanding the spectra and anomalies in lower-dimensional theories from the Calabi-Yau compactifications of F-theory. I will first explain what elliptic fibration is and explain Kodaira types, which gives rise an ADE classification. Utilizing Weierstrass model of elliptic fibrations, I will discuss Tate’s algorithm and Mordell-Weil group. By considering codimension one and two singularities and studying the geometry of crepant resolutions, we can define G-models that are geometrically-engineered models from F-theory. I will discuss the dictionary between the gauge theory and the elliptic fibrations and how to incorporate this to learn about topological invariants of the compactified Calabi-Yau that is one of the ingredient to understand spectra in the compactified theories. I will explain the more refined connection to understand the Coulomb branch of the 5d N=1 theories and 6d (1,0) theories and their anomalies from this perspective.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

Elliptic Fibrations and Singularities to Anomalies and Spectra 3 of 4

Speaker: 
Monica Jinwoo Kang
Date: 
Wed, Aug 25, 2021
Location: 
PIMS, University of Saskatchewan
Zoom
Online
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

Throughout my lectures I will explain the geometry of elliptic fibration which can gåve rise to understanding the spectra and anomalies in lower-dimensional theories from the Calabi-Yau compactifications of F-theory. I will first explain what elliptic fibration is and explain Kodaira types, which gives rise an ADE classification. Utilizing Weierstrass model of elliptic fibrations, I will discuss Tate’s algorithm and Mordell-Weil group. By considering codimension one and two singularities and studying the geometry of crepant resolutions, we can define G-models that are geometrically-engineered models from F-theory. I will discuss the dictionary between the gauge theory and the elliptic fibrations and how to incorporate this to learn about topological invariants of the compactified Calabi-Yau that is one of the ingredient to understand spectra in the compactified theories. I will explain the more refined connection to understand the Coulomb branch of the 5d N=1 theories and 6d (1,0) theories and their anomalies from this perspective.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

Elliptic Fibrations and Singularities to Anomalies and Spectra 2 of 4

Speaker: 
Monica Jinwoo Kang
Date: 
Tue, Aug 24, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

Throughout my lectures I will explain the geometry of elliptic fibration which can give rise to understanding the spectra and anomalies in lower-dimensional theories from the Calabi-Yau compactifications of F-theory. I will first explain what elliptic fibration is and explain Kodaira types, which gives rise an ADE classification. Utilizing Weierstrass model of elliptic fibrations, I will discuss Tate’s algorithm and Mordell-Weil group. By considering codimension one and two singularities and studying the geometry of crepant resolutions, we can define G-models that are geometrically-engineered models from F-theory. I will discuss the dictionary between the gauge theory and the elliptic fibrations and how to incorporate this to learn about topological invariants of the compactified Calabi-Yau that is one of the ingredient to understand spectra in the compactified theories. I will explain the more refined connection to understand the Coulomb branch of the 5d N=1 theories and 6d (1,0) theories and their anomalies from this perspective.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

Elliptic Fibrations and Singularities to Anomalies and Spectra 1 of 4

Speaker: 
Monica Jinwoo Kang
Date: 
Mon, Aug 23, 2021
Location: 
PIMS, University of Saskatchewan
Online
Zoom
Conference: 
2nd PIMS Summer School on Algebraic Geometry in High Energy Physics
Abstract: 

Throughout my lectures I will explain the geometry of elliptic fibration which can give rise to understanding the spectra and anomalies in lower-dimensional theories from the Calabi-Yau compactifications of F-theory. I will first explain what elliptic fibration is and explain Kodaira types, which gives rise an ADE classification. Utilizing Weierstrass model of elliptic fibrations, I will discuss Tate’s algorithm and Mordell-Weil group. By considering codimension one and two singularities and studying the geometry of crepant resolutions, we can define G-models that are geometrically-engineered models from F-theory. I will discuss the dictionary between the gauge theory and the elliptic fibrations and how to incorporate this to learn about topological invariants of the compactified Calabi-Yau that is one of the ingredient to understand spectra in the compactified theories. I will explain the more refined connection to understand the Coulomb branch of the 5d N=1 theories and 6d (1,0) theories and their anomalies from this perspective.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
Class: 

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