Time | Speaker | Title | Resources | |
---|---|---|---|---|
09:30 to 10:45 | Tony Pantev (University of Pennsylvania, USA) |
Modular spectral covers and Hecke eigensheaves on interesections of quadrics (Lecture 1) In these talks I will review the Geometric Langlands Conjecture in the unramified and tamely ramified cases and will connect it to the homological mirror correspondence for the moduli of Higgs bundles on a curve. I will outline a program which uses non-abelian Hodge theory and Fourier-Mukai duality on the Hitchin system to construct automorphic D-modules on the moduli of bundles and objects in the Fukaya category on the moduli of Higgs bundles. I will discuss specific examples of the construction building automorphic sheaves on moduli spaces of bundles that are realized as intersections of quadrics. I will explain the resulting algebraic geometric question and will show how it can be solved explicitly by a higher dimensional version of the spectral cover construction and some interesting calculations with parabolic Chern classes. The focus will be on the projective geometry of the moduli spaces involved, and on the singularities and geometric subtleties needed for the correct formulation of the correspondence. This is a joint work with Ron Donagi and Carlos Simpson. Background References: (a) R. Donagi, T. Pantev “Geometric Langlands and non-abelian Hodge the- ory”, Surveys in differential geometry. Vol. XIII. , 85–116, Int. Press, 2009. https://www.icmat.es/seminarios/langlands/school/handouts/pantev.pdf |
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10:45 to 11:15 | -- | Tea Break | ||
11:15 to 12:30 | Sergei Gukov (California Institute of Technology, USA) |
Lecture 2: VOA[M4] (Lecture 1) In the first lecture I will introduce new dualities of 3d and 2d theories related to topology. This will give us an opportunity to learn about certain aspects of strongly coupled quantum field theories, on the one hand, and about equivalence relations (Kirby moves) in construction of 3-manifolds and 4-manifolds, on the other hand. Then, we will proceed to the main subject of these lectures, which involves holomorphic twists of 3d N=2 theories and 2d N=(0,2) theories, separately and together. It provides a bridge between physics, algebra, and topology, which we explore from various perspectives and in many different examples. In particular, we will study the half-index of the combined 2d-3d system, first introduced in a joint work with A.Gadde and P.Putrov in 2013. Following that line of work, we also review the first non-trivial duality in a non-abelian 2d N=(0,2) gauge theory, the so-called triality of 2d N=(0,2) SQCD, and discuss how it leads to a triality symmetry of the corresponding VOAs. We then discuss various ways of "gluing" vertex operator algebras that correspond to different gluing operations in the world of smooth 4-manifolds and illustrate how topological invariants of 4-manifolds arise as chiral correlation functions in the resulting algebras VOA[M4]. |
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12:30 to 14:30 | -- | Lunch | ||
14:30 to 15:45 | Piotr Sulkowski (University of Warsaw, Poland) |
Topological strings, knots, and quivers (Lecture 2) In the past three decades intimate links between knot theory and theoretical physics have been discovered. They include interpretation of polynomial knot invariants as partition functions of statistical models or expectation values in Chern-Simons quan- tum field theory, generalization of these relations to brane systems in topological string theory, identification of knot homologies with spaces of BPS states, relations to matrix models and topological recursion, etc. In the first lecture I will summarize some of these relations, and report some recent results in those contexts. In the second lecture I will show how (some of) these relations, as well as various knot invariants, are unified by re- lating them to quiver representation theory, in a way that we refer to as the knots-quivers correspondence. This correspondence is motived by various string theory constructions involving BPS states, and its consequences include the proof of the famous Labastida- Marino-Ooguri-Vafa conjecture (for symmetric representations), explicit (and unknown before) formulas for colored HOMFLY polynomials for various knots, new viewpoint on knot homologies and categorification, new dualities between quivers, new links with topo- logical strings and statistical models, etc. While the knots-quivers correspondence has already led to surprising new results, at the same time it poses new deep and interesting questions, which I will also summarize. Background References: (a) Edward Witten, “Quantum field theory and the Jones polynomial”, Commun. Math.Phys. 121 (1989) 351. |
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15:45 to 16:15 | -- | Tea Break | ||
16:15 to 17:30 | Sergei Gukov (California Institute of Technology, USA) |
Lecture 3: VOA[M4] (Lecture 2) In the first lecture I will introduce new dualities of 3d and 2d theories related to topology. This will give us an opportunity to learn about certain aspects of strongly coupled quantum field theories, on the one hand, and about equivalence relations (Kirby moves) in construction of 3-manifolds and 4-manifolds, on the other hand. Then, we will proceed to the main subject of these lectures, which involves holomorphic twists of 3d N=2 theories and 2d N=(0,2) theories, separately and together. It provides a bridge between physics, algebra, and topology, which we explore from various perspectives and in many different examples. In particular, we will study the half-index of the combined 2d-3d system, first introduced in a joint work with A.Gadde and P.Putrov in 2013. Following that line of work, we also review the first non-trivial duality in a non-abelian 2d N=(0,2) gauge theory, the so-called triality of 2d N=(0,2) SQCD, and discuss how it leads to a triality symmetry of the corresponding VOAs. We then discuss various ways of "gluing" vertex operator algebras that correspond to different gluing operations in the world of smooth 4-manifolds and illustrate how topological invariants of 4-manifolds arise as chiral correlation functions in the resulting algebras VOA[M4]. |
Time | Speaker | Title | Resources | |
---|---|---|---|---|
09:30 to 10:45 | Tudor Dimofte (University of California, USA) |
Secondary products in SUSY QFT A mathematical treatment of TQFT, based on category theory, was initiated in the early 90’s. In more modern times the mathematics of TQFT has come to use advanced techniques from derived geometry and higher algebra (`a la Lurie). This series of talks is loosely aimed at explaining how some of these advanced techniques apply in familiar examples of supersymmetric field theory and its topological twists. For physics, this will lead to some surprising new structure, as well as some useful organizing principles. The first and second lectures are based on work with C. Beem, D. Ben-Zvi, M. Bullimore, and A. Neitzke. The first lecture will re-examine operator algebras in topological twists of supersymmetric field theories. In d dimension, the algebras naturally come equipped with a Lie bracket of degree 1-d, which can be realized very concretely via topological descent. We’ll look at examples of this bracket, in 2d, 3d, and 4d. We’ll also use topological descent to give a new interpretation of the statement that turning on on Omega background is a form of quantization. The second lecture focuses on another sort of homological/descent operation, this time in the context of SUSY quantum mechanics with G symmetry. We’ll find that Hilbert spaces in (de Rham) SQM come equipped with a natural homological action of G, i.e. an action of the exterior algebra H∗(G). This action controls the process of gauging the G symmetry. A nice physical way to understand the H∗(G) action and gauging/ungauging comes from considering SUSY boundary conditions for 2d G gauge theory; this will lead us to a definition of the “Fukaya category of point/G.” The mathematical structures involved come from work of Goresky, Kottwitz, and MacPherson. Given time, we’ll discuss some higher-dimensional examples, and the physics of Koszul duality. The third lecture applies some of the ideas from the first two in the specific context of 3d N=4 gauge theories with linear matter. It is joint work with N. Garner, M. Geracie, and J. Hilburn. Our goal will be to define the category of line operators in the A and B twists of these theories, and to understand – in a concrete and computational way – the algebras of local operators bound to a line. These algebras get quantized in an Omega background; and they act on modules defined by boundary conditions. In the special case of an A twist and the trivial (identity) line operator, we will recover the Braverman-FinkelbergNakajima construction of the Coulomb-branch chiral ring. Background References: (a) Getzler, “Batalin-Vilkovisky algebras and two-dimensional topological field theories” hep-th/9212043 |
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10:45 to 11:15 | -- | Tea Break | ||
11:15 to 12:30 | Tony Pantev (University of Pennsylvania, USA) |
Modular spectral covers and Hecke eigensheaves on interesections of quadrics (Lecture 2) In these talks I will review the Geometric Langlands Conjecture in the unramified and tamely ramified cases and will connect it to the homological mirror correspondence for the moduli of Higgs bundles on a curve. I will outline a program which uses non-abelian Hodge theory and Fourier-Mukai duality on the Hitchin system to construct automorphic D-modules on the moduli of bundles and objects in the Fukaya category on the moduli of Higgs bundles. I will discuss specific examples of the construction building automorphic sheaves on moduli spaces of bundles that are realized as intersections of quadrics. I will explain the resulting algebraic geometric question and will show how it can be solved explicitly by a higher dimensional version of the spectral cover construction and some interesting calculations with parabolic Chern classes. The focus will be on the projective geometry of the moduli spaces involved, and on the singularities and geometric subtleties needed for the correct formulation of the correspondence. This is a joint work with Ron Donagi and Carlos Simpson. Background References: (a) R. Donagi, T. Pantev “Geometric Langlands and non-abelian Hodge the- ory”, Surveys in differential geometry. Vol. XIII. , 85–116, Int. Press, 2009. https://www.icmat.es/seminarios/langlands/school/handouts/pantev.pdf |
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12:30 to 14:30 | -- | Lunch | ||
14:30 to 15:45 | Sergei Gukov (California Institute of Technology, USA) |
Lecture 4: VOA[M4] (Lecture 3) In the first lecture I will introduce new dualities of 3d and 2d theories related to topology. This will give us an opportunity to learn about certain aspects of strongly coupled quantum field theories, on the one hand, and about equivalence relations (Kirby moves) in construction of 3-manifolds and 4-manifolds, on the other hand. Then, we will proceed to the main subject of these lectures, which involves holomorphic twists of 3d N=2 theories and 2d N=(0,2) theories, separately and together. It provides a bridge between physics, algebra, and topology, which we explore from various perspectives and in many different examples. In particular, we will study the half-index of the combined 2d-3d system, first introduced in a joint work with A.Gadde and P.Putrov in 2013. Following that line of work, we also review the first non-trivial duality in a non-abelian 2d N=(0,2) gauge theory, the so-called triality of 2d N=(0,2) SQCD, and discuss how it leads to a triality symmetry of the corresponding VOAs. We then discuss various ways of "gluing" vertex operator algebras that correspond to different gluing operations in the world of smooth 4-manifolds and illustrate how topological invariants of 4-manifolds arise as chiral correlation functions in the resulting algebras VOA[M4]. |
||
15:45 to 16:15 | -- | Tea Break | ||
16:15 to 17:30 | Tony Pantev (University of Pennsylvania, USA) |
Modular spectral covers and Hecke eigensheaves on interesections of quadrics (Lecture 3) In these talks I will review the Geometric Langlands Conjecture in the unramified and tamely ramified cases and will connect it to the homological mirror correspondence for the moduli of Higgs bundles on a curve. I will outline a program which uses non-abelian Hodge theory and Fourier-Mukai duality on the Hitchin system to construct automorphic D-modules on the moduli of bundles and objects in the Fukaya category on the moduli of Higgs bundles. I will discuss specific examples of the construction building automorphic sheaves on moduli spaces of bundles that are realized as intersections of quadrics. I will explain the resulting algebraic geometric question and will show how it can be solved explicitly by a higher dimensional version of the spectral cover construction and some interesting calculations with parabolic Chern classes. The focus will be on the projective geometry of the moduli spaces involved, and on the singularities and geometric subtleties needed for the correct formulation of the correspondence. This is a joint work with Ron Donagi and Carlos Simpson. Background References: (a) R. Donagi, T. Pantev “Geometric Langlands and non-abelian Hodge the- ory”, Surveys in differential geometry. Vol. XIII. , 85–116, Int. Press, 2009. https://www.icmat.es/seminarios/langlands/school/handouts/pantev.pdf |
Time | Speaker | Title | Resources | |
---|---|---|---|---|
09:30 to 10:45 | Tony Pantev (University of Pennsylvania, USA) |
Modular spectral covers and Hecke eigensheaves on interesections of quadrics (Lecture 4) In these talks I will review the Geometric Langlands Conjecture in the unramified and tamely ramified cases and will connect it to the homological mirror correspondence for the moduli of Higgs bundles on a curve. I will outline a program which uses non-abelian Hodge theory and Fourier-Mukai duality on the Hitchin system to construct automorphic D-modules on the moduli of bundles and objects in the Fukaya category on the moduli of Higgs bundles. I will discuss specific examples of the construction building automorphic sheaves on moduli spaces of bundles that are realized as intersections of quadrics. I will explain the resulting algebraic geometric question and will show how it can be solved explicitly by a higher dimensional version of the spectral cover construction and some interesting calculations with parabolic Chern classes. The focus will be on the projective geometry of the moduli spaces involved, and on the singularities and geometric subtleties needed for the correct formulation of the correspondence. This is a joint work with Ron Donagi and Carlos Simpson. Background References: (a) R. Donagi, T. Pantev “Geometric Langlands and non-abelian Hodge the- ory”, Surveys in differential geometry. Vol. XIII. , 85–116, Int. Press, 2009. https://www.icmat.es/seminarios/langlands/school/handouts/pantev.pdf |
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10:45 to 11:15 | -- | Tea Break | ||
11:15 to 12:30 | Satoshi Nawata (Fudan University, Shanghai) |
Representations of DAHA from Hitchin moduli space I will talk about physics approach to understand representation theory of double affine Hecke algebra (DAHA). DAHA can be realized as an algebra of line operators in 4d N=2* theory and therefore it appears as quantization of coordinate ring of Hitchin moduli space over once-punctured torus. Using 2d A-model on the Hitchin moduli space, I will explain relationship between representation category of DAHA and Fukaya category of the Hitchin moduli space. Background References: (a) Gukov Witten, Branes and Quantization (0809.0305) |
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12:30 to 14:30 | -- | Lunch | ||
14:30 to 15:45 | Tudor Dimofte (University of California, USA) |
G actions in SUSY QM; or, the Fukaya category of point/G A mathematical treatment of TQFT, based on category theory, was initiated in the early 90’s. In more modern times the mathematics of TQFT has come to use advanced techniques from derived geometry and higher algebra (`a la Lurie). This series of talks is loosely aimed at explaining how some of these advanced techniques apply in familiar examples of supersymmetric field theory and its topological twists. For physics, this will lead to some surprising new structure, as well as some useful organizing principles. The first and second lectures are based on work with C. Beem, D. Ben-Zvi, M. Bullimore, and A. Neitzke. The first lecture will re-examine operator algebras in topological twists of supersymmetric field theories. In d dimension, the algebras naturally come equipped with a Lie bracket of degree 1-d, which can be realized very concretely via topological descent. We’ll look at examples of this bracket, in 2d, 3d, and 4d. We’ll also use topological descent to give a new interpretation of the statement that turning on on Omega background is a form of quantization. The second lecture focuses on another sort of homological/descent operation, this time in the context of SUSY quantum mechanics with G symmetry. We’ll find that Hilbert spaces in (de Rham) SQM come equipped with a natural homological action of G, i.e. an action of the exterior algebra H∗(G). This action controls the process of gauging the G symmetry. A nice physical way to understand the H∗(G) action and gauging/ungauging comes from considering SUSY boundary conditions for 2d G gauge theory; this will lead us to a definition of the “Fukaya category of point/G.” The mathematical structures involved come from work of Goresky, Kottwitz, and MacPherson. Given time, we’ll discuss some higher-dimensional examples, and the physics of Koszul duality. The third lecture applies some of the ideas from the first two in the specific context of 3d N=4 gauge theories with linear matter. It is joint work with N. Garner, M. Geracie, and J. Hilburn. Our goal will be to define the category of line operators in the A and B twists of these theories, and to understand – in a concrete and computational way – the algebras of local operators bound to a line. These algebras get quantized in an Omega background; and they act on modules defined by boundary conditions. In the special case of an A twist and the trivial (identity) line operator, we will recover the Braverman-FinkelbergNakajima construction of the Coulomb-branch chiral ring. Background References: For Lecture 1: |
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15:45 to 16:15 | -- | Tea Break | ||
16:15 to 17:30 | Lakshya Bhardwaj (PITP, Canada) |
State-sum construction of 3d (s)pin TQFTs and fermionic SPT phases We will use a well known state-sum construction of 3d TQFTs due to Turaev and Viro to obtain a state-sum construction for 3d spin-TQFTs. We will then generalize the Turaev-Viro construction to unoriented 3d TQFTs. Combining the above two ideas, we will be able to obtain a state-sum construction for 3d Pin^+ TQFTs. We will also discuss the application of these state-sum constructions to the problem of classification of fermionic SPT phases in 2+1 dimensions. |
Time | Speaker | Title | Resources | |
---|---|---|---|---|
09:30 to 10:45 | Du Pei (Aarhus University, Denmark) |
On mirror symmetry of (B, A, A)-branes Picking a real form G r of a complex Lie group G defines a “(B, A, A)-brane” inside the moduli space of G-Higgs bundles. Under mirror symmetry, this (B, A, A)-brane will be mapped to a hyperholomorphic sheaf – a (B, B, B)-brane – over the moduli space of G^{∨} -Higgs bundles, where G^{∨} is the Langlands dual group of G. In this talk, I will discuss how to construct these hyperholomorphic sheaves, and show how these proposals can be tested by computing equivariant indices. In particular, I will give computational evidence to Nigel Hitchin’s proposal for the case of G = GL_{2} and G_{r} = U (1, 1). This talk is based on joint work with Tamas Hausel and Anton Mellit. Background References: (a) D.Pei, T. Hausel and A. Mellit, “Mirror symmetry with branes by equivariant Ver-linde formulae”, arXiv:1712.04408 |
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10:45 to 11:15 | -- | Tea Break | ||
11:15 to 12:30 | Tudor Dimofte (University of California, USA) |
Line operators and geometry in 3d N=4 gauge theory A mathematical treatment of TQFT, based on category theory, was initiated in the early 90’s. In more modern times the mathematics of TQFT has come to use advanced techniques from derived geometry and higher algebra (`a la Lurie). This series of talks is loosely aimed at explaining how some of these advanced techniques apply in familiar examples of supersymmetric field theory and its topological twists. For physics, this will lead to some surprising new structure, as well as some useful organizing principles. The first and second lectures are based on work with C. Beem, D. Ben-Zvi, M. Bullimore, and A. Neitzke. The first lecture will re-examine operator algebras in topological twists of supersymmetric field theories. In d dimension, the algebras naturally come equipped with a Lie bracket of degree 1-d, which can be realized very concretely via topological descent. We’ll look at examples of this bracket, in 2d, 3d, and 4d. We’ll also use topological descent to give a new interpretation of the statement that turning on on Omega background is a form of quantization. The second lecture focuses on another sort of homological/descent operation, this time in the context of SUSY quantum mechanics with G symmetry. We’ll find that Hilbert spaces in (de Rham) SQM come equipped with a natural homological action of G, i.e. an action of the exterior algebra H∗(G). This action controls the process of gauging the G symmetry. A nice physical way to understand the H∗(G) action and gauging/ungauging comes from considering SUSY boundary conditions for 2d G gauge theory; this will lead us to a definition of the “Fukaya category of point/G.” The mathematical structures involved come from work of Goresky, Kottwitz, and MacPherson. Given time, we’ll discuss some higher-dimensional examples, and the physics of Koszul duality. The third lecture applies some of the ideas from the first two in the specific context of 3d N=4 gauge theories with linear matter. It is joint work with N. Garner, M. Geracie, and J. Hilburn. Our goal will be to define the category of line operators in the A and B twists of these theories, and to understand – in a concrete and computational way – the algebras of local operators bound to a line. These algebras get quantized in an Omega background; and they act on modules defined by boundary conditions. In the special case of an A twist and the trivial (identity) line operator, we will recover the Braverman-FinkelbergNakajima construction of the Coulomb-branch chiral ring. Background References: For Lecture 1: |
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12:30 to 14:30 | -- | Lunch | ||
15:45 to 16:15 | -- | Tea Break |