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Monday, 05 July 2021
Time Speaker Title Resources
13:50 to 14:00 Rajesh Gopakumar (ICTS-TIFR, India) Welcome Remarks
14:00 to 14:10 Aswin Balasubramanian (Rutgers University, USA) Introduction by organiser
14:10 to 15:30 Ana Peon-Nieto (University of Nice Sophia Antipolis, France and University of Birmingham, UK) Geometry of the Global Nilpotent Cone 1

Higgs bundles are central players in the current arena in mathematics and physics. Their applications to mirror symmetry and the geometric Langlands programme have captured a lot of attention in the past few years. In this mini-course, we will study some aspects of the geometry of the moduli space of these objects M_X , focusing on the nilpotent cone . Besides capturing the global topology of M_X , the nilpotent cone is at the heart of many applications that we will explore, always basing our discussion on manageable examples. More precisely, after a brief introduction, we will discuss Donagi-Pantev’s programme towards geometric Langlands through abelianisation, as well as discuss two important conjectures therein. One, due to the authors of the programme, concerns the equality of the wobbly and shaky loci. The other one, due to Drinfeld, concerns pure codimensionality of wobbly loci.

Outline

(1) Introduction to Higgs bundles and the non abelian Hodge correspondence. Refs: [Hi2, S1, S2]
1.1 The rank one case as a motivating example
1.2 Mirror symmetry for Hitchin systems. Refs: [HT, DP1]

(2) Applications in a nutshell
2.1 Geometric Langlands via abelianisation. Refs: [DP2, DP3]
2.2 The nilpotent cone in all this. Refs: [HT, BNR]

(3) The nilpotent cone. Refs:[BD, Fa, G, La, PaP]
3.1 Wobbly bundles and irreducible components. Drinfeld’s conjecture. Refs:[PaP]
3.2 Wobbly bundles and shaky bundles in geometric Langlands- Donagi-Pantev’s conjecture. Refs: [PPe, Z, HH, Pe]

References:

[BNR] A. Beauville, M.S. Narasimhan, and S. Ramanan, Spectral curves and the generalised  theta divisor. J. Reine Angew. Math. 398 (1989), 169–179.

[BD] A.A. Beilinson and V.G. Drinfeld, Quantization of Hitchin’s fibration and Langlands’ program. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 3–7, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996.

[DP1]  R.Y. Donagi and T.B. Pantev,Langlands duality for Hitchin systems. Invent. Math. 189  (2012), no. 3, 653–735.

[DP2]  R.Y. Donagi and T.B. Pantev,Geometric Langlands and non-abelian Hodge theory. Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, 85–116, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009.

[DP3] R.Y. Donagi and T.B. Pantev,Parabolic Hecke eigensheaves. Preprint, arXiv:1910.02357 [math.AG]

[Fa] G. Faltings, Theta functions on moduli spaces of G-bundles, J. Algebraic Geom. 18 (2009), 309-369. DOI: https://doi.org/10.1090/S1056-3911-08-00499-2.

[G]  V. Ginzburg,The global nilpotent variety is Lagrangian. Duke Math. J. 109 (2001), no. 3, 511–519 .

[HT]  T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153 (2003), no. 1, 197–229.

[HH] T. Hausel and N. Hitchin, Very stable Higgs bundles, equivariant multiplicity and mirror symmetry, Preprint, arXiv: 2101.08583.

[Hi1] N. Hitchin,Stable bundles and integrable systems. Duke Math. J. 54 (1987), no. 1, 91–114.

[Hi2] N. Hitchin,The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126.

[La] G. Laumon, Un analogue global du cône nilpotent, Duke Math. Jour. 57 (1988), 647–671. 

[NR1]  M.S. Narasimhan and S. Ramanan, Moduli of Vector Bundles on a Compact Riemann  Surface, Ann. Math., Second Series, Vol. 89, No. 1 (1969), 14–51.

[PaP] S. Pal and C. Pauly, Wobbly divisors in the moduli space of rank two bundle, to appear in Adv. in Geometry.

[PPe] C. Pauly and A. Peón-Nieto, Very stable bundles and properness of the Hitchin map,  A. Geom. Dedicata (2018). DOI:10.1007/s10711-018-0333-6. Preprint arXiv:1710.10152  [math.AG].

[Pe] A. Peón-Nieto, Wobbly and shaky bundles and resolutions of rational maps, arXiv:2007.13447.

[S1] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I and II. Inst. Hautes Études Sci. Publ. Math. No. 79 (1994), 47–129.

[S2] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. No. 80 (1994), 5–79 .

[Z] H. Zelaci On very stablity of principal G-bundles. Geom. Dedicata 204 (2020), 165–173.

15:30 to 17:00 Madalena Lemos (Durham University, UK) A mini-course on vertex operator algebras of N= 2 Superconformal Field Theories 1

Any four-dimensional N>=2 superconformal field theory possess a protected subsector isomorphic to a two-dimensional chiral algebra/vertex operator algebra [arxiv:1312.5344]. After a brief review of N=2 SCFTs we will describe the construction of this subsector, the consequences for four-dimensional physics, as well as a brief outlook of recent progress. Some basic familiarity with conformal field theories is desirable, for example with what is covered in [a] section 1,2,3 and the corresponding background material (the remaining sections of [a] provide details on 4d N=2 superconformal theories which is relevant for the mini-course, but detailed knowledge of that will not be assumed). Some familiarity with enhanced Virasoro algebra in 2d CFTs, along the lines of what is summarized in chapter 1 of [b] is also beneficial. The bulk of the lectures will proceed along the same lines as [c].

References 

[a] Eberhardt, Superconformal symmetry and representations, arxiv:2006.13280

[b] Ribault, Minimal lectures on two-dimensional conformal field theory, arxiv:1609.09523

[c] Lemos, Lectures on chiral algebras of N⩾2 superconformal field theories, arXiv:2006.13892

20:30 to 20:40 Aswin Balasubramanian (Rutgers University, USA) Introduction by organiser
20:40 to 22:00 Mathew Bullimore (Durham University, UK) A Mathematical Introduction to 3d N = 4 Gauge Theories 1

This mini course will introduce mathematical structures arising in 3d N = 4 gauge theories, with a focus on aspects that are important in geometric representation theory. The course will cover the supersymmetry algebra, central charges, mass and FI parameter deformations, walls and chambers, Higgs and Coulomb branch moduli spaces, and 3d mirror symmetry. This should provide a stepping stone to more advanced topics in the literature. I will try to explain how physicists think about these concepts in a way that is friendly to mathematicians.

References 

[a] A quick summary of some aspects of 3d N = 4 gauge theories can be found in section 2 of https://arxiv.org/pdf/1503.04817.pdf.

[b] For some motivation for mathematicians, see section 2 of https://arxiv.org/pdf/1706.05154.pdf.

22:00 to 23:30 Ana Peon-Nieto (University of Nice Sophia Antipolis, France and University of Birmingham, UK) Geometry of the Global Nilpotent Cone 1 (WP)

Higgs bundles are central players in the current arena in mathematics and physics. Their applications to mirror symmetry and the geometric Langlands programme have captured a lot of attention in the past few years. In this mini-course, we will study some aspects of the geometry of the moduli space of these objects M_X , focusing on the nilpotent cone . Besides capturing the global topology of M_X , the nilpotent cone is at the heart of many applications that we will explore, always basing our discussion on manageable examples. More precisely, after a brief introduction, we will discuss Donagi-Pantev’s programme towards geometric Langlands through abelianisation, as well as discuss two important conjectures therein. One, due to the authors of the programme, concerns the equality of the wobbly and shaky loci. The other one, due to Drinfeld, concerns pure codimensionality of wobbly loci.

Outline

(1) Introduction to Higgs bundles and the non abelian Hodge correspondence. Refs: [Hi2, S1, S2]
1.1 The rank one case as a motivating example
1.2 Mirror symmetry for Hitchin systems. Refs: [HT, DP1]

(2) Applications in a nutshell
2.1 Geometric Langlands via abelianisation. Refs: [DP2, DP3]
2.2 The nilpotent cone in all this. Refs: [HT, BNR]

(3) The nilpotent cone. Refs:[BD, Fa, G, La, PaP]
3.1 Wobbly bundles and irreducible components. Drinfeld’s conjecture. Refs:[PaP]
3.2 Wobbly bundles and shaky bundles in geometric Langlands- Donagi-Pantev’s conjecture. Refs: [PPe, Z, HH, Pe]

References:

[BNR] A. Beauville, M.S. Narasimhan, and S. Ramanan, Spectral curves and the generalised  theta divisor. J. Reine Angew. Math. 398 (1989), 169–179.

[BD] A.A. Beilinson and V.G. Drinfeld, Quantization of Hitchin’s fibration and Langlands’ program. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 3–7, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996.

[DP1]  R.Y. Donagi and T.B. Pantev,Langlands duality for Hitchin systems. Invent. Math. 189  (2012), no. 3, 653–735.

[DP2]  R.Y. Donagi and T.B. Pantev,Geometric Langlands and non-abelian Hodge theory. Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, 85–116, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009.

[DP3] R.Y. Donagi and T.B. Pantev,Parabolic Hecke eigensheaves. Preprint, arXiv:1910.02357 [math.AG]

[Fa] G. Faltings, Theta functions on moduli spaces of G-bundles, J. Algebraic Geom. 18 (2009), 309-369. DOI: https://doi.org/10.1090/S1056-3911-08-00499-2.

[G]  V. Ginzburg,The global nilpotent variety is Lagrangian. Duke Math. J. 109 (2001), no. 3, 511–519 .

[HT]  T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153 (2003), no. 1, 197–229.

[HH] T. Hausel and N. Hitchin, Very stable Higgs bundles, equivariant multiplicity and mirror symmetry, Preprint, arXiv: 2101.08583.

[Hi1] N. Hitchin,Stable bundles and integrable systems. Duke Math. J. 54 (1987), no. 1, 91–114.

[Hi2] N. Hitchin,The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126.

[La] G. Laumon, Un analogue global du cône nilpotent, Duke Math. Jour. 57 (1988), 647–671. 

[NR1]  M.S. Narasimhan and S. Ramanan, Moduli of Vector Bundles on a Compact Riemann  Surface, Ann. Math., Second Series, Vol. 89, No. 1 (1969), 14–51.

[PaP] S. Pal and C. Pauly, Wobbly divisors in the moduli space of rank two bundle, to appear in Adv. in Geometry.

[PPe] C. Pauly and A. Peón-Nieto, Very stable bundles and properness of the Hitchin map,  A. Geom. Dedicata (2018). DOI:10.1007/s10711-018-0333-6. Preprint arXiv:1710.10152  [math.AG].

[Pe] A. Peón-Nieto, Wobbly and shaky bundles and resolutions of rational maps, arXiv:2007.13447.

[S1] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I and II. Inst. Hautes Études Sci. Publ. Math. No. 79 (1994), 47–129.

[S2] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. No. 80 (1994), 5–79 .

[Z] H. Zelaci On very stablity of principal G-bundles. Geom. Dedicata 204 (2020), 165–173.

Tuesday, 06 July 2021
Time Speaker Title Resources
00:30 to 02:00 Madalena Lemos (Durham University, UK) A mini-course on vertex operator algebras of N= 2 Superconformal Field Theories 1 (WP)

Any four-dimensional N>=2 superconformal field theory possess a protected subsector isomorphic to a two-dimensional chiral algebra/vertex operator algebra [arxiv:1312.5344]. After a brief review of N=2 SCFTs we will describe the construction of this subsector, the consequences for four-dimensional physics, as well as a brief outlook of recent progress. Some basic familiarity with conformal field theories is desirable, for example with what is covered in [a] section 1,2,3 and the corresponding background material (the remaining sections of [a] provide details on 4d N=2 superconformal theories which is relevant for the mini-course, but detailed knowledge of that will not be assumed). Some familiarity with enhanced Virasoro algebra in 2d CFTs, along the lines of what is summarized in chapter 1 of [b] is also beneficial. The bulk of the lectures will proceed along the same lines as [c].

References:

[a] Eberhardt, Superconformal symmetry and representations, arxiv:2006.13280

[b] Ribault, Minimal lectures on two-dimensional conformal field theory, arxiv:1609.09523

[c] Lemos, Lectures on chiral algebras of N⩾2 superconformal field theories, arXiv:2006.13892

11:00 to 12:30 Mathew Bullimore (Durham University, UK) A Mathematical Introduction to 3d N = 4 Gauge Theories 1 (WP)

This mini course will introduce mathematical structures arising in 3d N = 4 gauge theories, with a focus on aspects that are important in geometric representation theory. The course will cover the supersymmetry algebra, central charges, mass and FI parameter deformations, walls and chambers, Higgs and Coulomb branch moduli spaces, and 3d mirror symmetry. This should provide a stepping stone to more advanced topics in the literature. I will try to explain how physicists think about these concepts in a way that is friendly to mathematicians.

References 

[a] A quick summary of some aspects of 3d N = 4 gauge theories can be found in section 2 of https://arxiv.org/pdf/1503.04817.pdf.

[b] For some motivation for mathematicians, see section 2 of https://arxiv.org/pdf/1706.05154.pdf.

15:30 to 17:00 Madalena Lemos (Durham University, UK) A mini-course on vertex operator algebras of N= 2 Superconformal Field Theories 2

Any four-dimensional N>=2 superconformal field theory possess a protected subsector isomorphic to a two-dimensional chiral algebra/vertex operator algebra [arxiv:1312.5344]. After a brief review of N=2 SCFTs we will describe the construction of this subsector, the consequences for four-dimensional physics, as well as a brief outlook of recent progress. Some basic familiarity with conformal field theories is desirable, for example with what is covered in [a] section 1,2,3 and the corresponding background material (the remaining sections of [a] provide details on 4d N=2 superconformal theories which is relevant for the mini-course, but detailed knowledge of that will not be assumed). Some familiarity with enhanced Virasoro algebra in 2d CFTs, along the lines of what is summarized in chapter 1 of [b] is also beneficial. The bulk of the lectures will proceed along the same lines as [c].

References:

[a] Eberhardt, Superconformal symmetry and representations, arxiv:2006.13280

[b] Ribault, Minimal lectures on two-dimensional conformal field theory, arxiv:1609.09523

[c] Lemos, Lectures on chiral algebras of N⩾2 superconformal field theories, arXiv:2006.13892

20:30 to 22:00 Mathew Bullimore (Durham University, UK) A Mathematical Introduction to 3d N = 4 Gauge Theories 2

This mini course will introduce mathematical structures arising in 3d N = 4 gauge theories, with a focus on aspects that are important in geometric representation theory. The course will cover the supersymmetry algebra, central charges, mass and FI parameter deformations, walls and chambers, Higgs and Coulomb branch moduli spaces, and 3d mirror symmetry. This should provide a stepping stone to more advanced topics in the literature. I will try to explain how physicists think about these concepts in a way that is friendly to mathematicians.

References 

[a] A quick summary of some aspects of 3d N = 4 gauge theories can be found in section 2 of https://arxiv.org/pdf/1503.04817.pdf.

[b] For some motivation for mathematicians, see section 2 of https://arxiv.org/pdf/1706.05154.pdf.

22:00 to 23:30 Miroslav Rapcak (University of California, Berkeley, USA) The Miura operator at the M2-M5 Intersection

M5-branes in the omega background give rise to a large class of vertex operator algebras. Analogously, M2-branes in the omega background lead to a large class of Higgs branch/Coulomb branch algebras. I am going to give a new interpretation to the Miura operator, an important object in the theory of integrable hierarchies and vertex operator algebra, as an operator living at the M5-M2 intersection and intertwining the action of the two algebras. 

References: 

[1] Gaiotto, Rapcak, Miura operators, degenerate fields and the M2-M5 intersection, 2012.04118

[2] Gaiotto, Oh, Aspects of \Omega-deformed M-theory, 1907.06495

[3] Prochazka, Rapcak, W-algebra modules, free fields, and Gukov-Witten defects, 1808.08837

Wednesday, 07 July 2021
Time Speaker Title Resources
00:30 to 02:00 Madalena Lemos (Durham University, UK) A mini-course on vertex operator algebras of N= 2 Superconformal Field Theories 2 (WP)

Any four-dimensional N>=2 superconformal field theory possess a protected subsector isomorphic to a two-dimensional chiral algebra/vertex operator algebra [arxiv:1312.5344]. After a brief review of N=2 SCFTs we will describe the construction of this subsector, the consequences for four-dimensional physics, as well as a brief outlook of recent progress. Some basic familiarity with conformal field theories is desirable, for example with what is covered in [a] section 1,2,3 and the corresponding background material (the remaining sections of [a] provide details on 4d N=2 superconformal theories which is relevant for the mini-course, but detailed knowledge of that will not be assumed). Some familiarity with enhanced Virasoro algebra in 2d CFTs, along the lines of what is summarized in chapter 1 of [b] is also beneficial. The bulk of the lectures will proceed along the same lines as [c].

References:

[a] Eberhardt, Superconformal symmetry and representations, arxiv:2006.13280

[b] Ribault, Minimal lectures on two-dimensional conformal field theory, arxiv:1609.09523

[c] Lemos, Lectures on chiral algebras of N⩾2 superconformal field theories, arXiv:2006.13892

11:00 to 12:30 Mathew Bullimore (Durham University, UK) A Mathematical Introduction to 3d N = 4 Gauge Theories 2 (WP)

This mini course will introduce mathematical structures arising in 3d N = 4 gauge theories, with a focus on aspects that are important in geometric representation theory. The course will cover the supersymmetry algebra, central charges, mass and FI parameter deformations, walls and chambers, Higgs and Coulomb branch moduli spaces, and 3d mirror symmetry. This should provide a stepping stone to more advanced topics in the literature. I will try to explain how physicists think about these concepts in a way that is friendly to mathematicians.

References 

[a] A quick summary of some aspects of 3d N = 4 gauge theories can be found in section 2 of https://arxiv.org/pdf/1503.04817.pdf.

[b] For some motivation for mathematicians, see section 2 of https://arxiv.org/pdf/1706.05154.pdf.

14:00 to 15:30 Ana Peon-Nieto (University of Nice Sophia Antipolis, France and University of Birmingham, UK) Geometry of the Global Nilpotent Cone 2

Higgs bundles are central players in the current arena in mathematics and physics. Their applications to mirror symmetry and the geometric Langlands programme have captured a lot of attention in the past few years. In this mini-course, we will study some aspects of the geometry of the moduli space of these objects M_X , focusing on the nilpotent cone . Besides capturing the global topology of M_X , the nilpotent cone is at the heart of many applications that we will explore, always basing our discussion on manageable examples. More precisely, after a brief introduction, we will discuss Donagi-Pantev’s programme towards geometric Langlands through abelianisation, as well as discuss two important conjectures therein. One, due to the authors of the programme, concerns the equality of the wobbly and shaky loci. The other one, due to Drinfeld, concerns pure codimensionality of wobbly loci.

Outline

(1) Introduction to Higgs bundles and the non abelian Hodge correspondence. Refs: [Hi2, S1, S2]
1.1 The rank one case as a motivating example
1.2 Mirror symmetry for Hitchin systems. Refs: [HT, DP1]

(2) Applications in a nutshell
2.1 Geometric Langlands via abelianisation. Refs: [DP2, DP3]
2.2 The nilpotent cone in all this. Refs: [HT, BNR]

(3) The nilpotent cone. Refs:[BD, Fa, G, La, PaP]
3.1 Wobbly bundles and irreducible components. Drinfeld’s conjecture. Refs:[PaP]
3.2 Wobbly bundles and shaky bundles in geometric Langlands- Donagi-Pantev’s conjecture. Refs: [PPe, Z, HH, Pe]

References:

[BNR] A. Beauville, M.S. Narasimhan, and S. Ramanan, Spectral curves and the generalised  theta divisor. J. Reine Angew. Math. 398 (1989), 169–179.

[BD] A.A. Beilinson and V.G. Drinfeld, Quantization of Hitchin’s fibration and Langlands’ program. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 3–7, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996.

[DP1]  R.Y. Donagi and T.B. Pantev,Langlands duality for Hitchin systems. Invent. Math. 189  (2012), no. 3, 653–735.

[DP2]  R.Y. Donagi and T.B. Pantev,Geometric Langlands and non-abelian Hodge theory. Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, 85–116, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009.

[DP3] R.Y. Donagi and T.B. Pantev,Parabolic Hecke eigensheaves. Preprint, arXiv:1910.02357 [math.AG]

[Fa] G. Faltings, Theta functions on moduli spaces of G-bundles, J. Algebraic Geom. 18 (2009), 309-369. DOI: https://doi.org/10.1090/S1056-3911-08-00499-2.

[G]  V. Ginzburg,The global nilpotent variety is Lagrangian. Duke Math. J. 109 (2001), no. 3, 511–519 .

[HT]  T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153 (2003), no. 1, 197–229.

[HH] T. Hausel and N. Hitchin, Very stable Higgs bundles, equivariant multiplicity and mirror symmetry, Preprint, arXiv: 2101.08583.

[Hi1] N. Hitchin,Stable bundles and integrable systems. Duke Math. J. 54 (1987), no. 1, 91–114.

[Hi2] N. Hitchin,The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126.

[La] G. Laumon, Un analogue global du cône nilpotent, Duke Math. Jour. 57 (1988), 647–671. 

[NR1]  M.S. Narasimhan and S. Ramanan, Moduli of Vector Bundles on a Compact Riemann  Surface, Ann. Math., Second Series, Vol. 89, No. 1 (1969), 14–51.

[PaP] S. Pal and C. Pauly, Wobbly divisors in the moduli space of rank two bundle, to appear in Adv. in Geometry.

[PPe] C. Pauly and A. Peón-Nieto, Very stable bundles and properness of the Hitchin map,  A. Geom. Dedicata (2018). DOI:10.1007/s10711-018-0333-6. Preprint arXiv:1710.10152  [math.AG].

[Pe] A. Peón-Nieto, Wobbly and shaky bundles and resolutions of rational maps, arXiv:2007.13447.

[S1] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I and II. Inst. Hautes Études Sci. Publ. Math. No. 79 (1994), 47–129.

[S2] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. No. 80 (1994), 5–79 .

[Z] H. Zelaci On very stablity of principal G-bundles. Geom. Dedicata 204 (2020), 165–173.

15:30 to 17:00 Miroslav Rapcak (University of California, Berkeley, USA) The Miura operator at the M2-M5 Intersection (WP)

M5-branes in the omega background give rise to a large class of vertex operator algebras. Analogously, M2-branes in the omega background lead to a large class of Higgs branch/Coulomb branch algebras. I am going to give a new interpretation to the Miura operator, an important object in the theory of integrable hierarchies and vertex operator algebra, as an operator living at the M5-M2 intersection and intertwining the action of the two algebras. 

References: 

[1] Gaiotto, Rapcak, Miura operators, degenerate fields and the M2-M5 intersection, 2012.04118

[2] Gaiotto, Oh, Aspects of \Omega-deformed M-theory, 1907.06495

[3] Prochazka, Rapcak, W-algebra modules, free fields, and Gukov-Witten defects, 1808.08837

20:30 to 22:00 Mathew Bullimore (Durham University, UK) A Mathematical Introduction to 3d N = 4 Gauge Theories 3

This mini course will introduce mathematical structures arising in 3d N = 4 gauge theories, with a focus on aspects that are important in geometric representation theory. The course will cover the supersymmetry algebra, central charges, mass and FI parameter deformations, walls and chambers, Higgs and Coulomb branch moduli spaces, and 3d mirror symmetry. This should provide a stepping stone to more advanced topics in the literature. I will try to explain how physicists think about these concepts in a way that is friendly to mathematicians.

References 

[a] A quick summary of some aspects of 3d N = 4 gauge theories can be found in section 2 of https://arxiv.org/pdf/1503.04817.pdf.

[b] For some motivation for mathematicians, see section 2 of https://arxiv.org/pdf/1706.05154.pdf.

22:00 to 23:30 Jacques Hurtubise (McGill University, Canada) Geometry of the Hitchin integrable systems, and some variations 1

The Narasimhan-Seshadri (-Hitchin-Kobayashi-Yau...) correspondence.

The basic idea (ASD Yang-Mills), examples of various non-compact cases and reductions: monopoles on various spaces, Yang-Mills-Higgs. The Yang-MIlls-Higgs moduli. Higgs bundles, symplectic structures and integrable systems. The spectral curve and abelianization. Parabolic generalisations, examples. Darboux coordinates and geometry of Poisson surfaces.

Variants: The Sklyanin brackets. Integrable systems and surfaces. Local geometry of integrable systems of Jacobians: rank two systems. Other groups: Cameral covers and Higgs. Sklyanin type systems;  the analogue of the rank two condition.

Thursday, 08 July 2021
Time Speaker Title Resources
00:30 to 02:00 Ana Peon-Nieto (University of Nice Sophia Antipolis, France and University of Birmingham, UK) Geometry of the Global Nilpotent Cone 2 (WP)

Higgs bundles are central players in the current arena in mathematics and physics. Their applications to mirror symmetry and the geometric Langlands programme have captured a lot of attention in the past few years. In this mini-course, we will study some aspects of the geometry of the moduli space of these objects M_X , focusing on the nilpotent cone . Besides capturing the global topology of M_X , the nilpotent cone is at the heart of many applications that we will explore, always basing our discussion on manageable examples. More precisely, after a brief introduction, we will discuss Donagi-Pantev’s programme towards geometric Langlands through abelianisation, as well as discuss two important conjectures therein. One, due to the authors of the programme, concerns the equality of the wobbly and shaky loci. The other one, due to Drinfeld, concerns pure codimensionality of wobbly loci.

Outline

(1) Introduction to Higgs bundles and the non abelian Hodge correspondence. Refs: [Hi2, S1, S2]
1.1 The rank one case as a motivating example
1.2 Mirror symmetry for Hitchin systems. Refs: [HT, DP1]

(2) Applications in a nutshell
2.1 Geometric Langlands via abelianisation. Refs: [DP2, DP3]
2.2 The nilpotent cone in all this. Refs: [HT, BNR]

(3) The nilpotent cone. Refs:[BD, Fa, G, La, PaP]
3.1 Wobbly bundles and irreducible components. Drinfeld’s conjecture. Refs:[PaP]
3.2 Wobbly bundles and shaky bundles in geometric Langlands- Donagi-Pantev’s conjecture. Refs: [PPe, Z, HH, Pe]

References:

[BNR] A. Beauville, M.S. Narasimhan, and S. Ramanan, Spectral curves and the generalised  theta divisor. J. Reine Angew. Math. 398 (1989), 169–179.

[BD] A.A. Beilinson and V.G. Drinfeld, Quantization of Hitchin’s fibration and Langlands’ program. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 3–7, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996.

[DP1]  R.Y. Donagi and T.B. Pantev,Langlands duality for Hitchin systems. Invent. Math. 189  (2012), no. 3, 653–735.

[DP2]  R.Y. Donagi and T.B. Pantev,Geometric Langlands and non-abelian Hodge theory. Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, 85–116, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009.

[DP3] R.Y. Donagi and T.B. Pantev,Parabolic Hecke eigensheaves. Preprint, arXiv:1910.02357 [math.AG]

[Fa] G. Faltings, Theta functions on moduli spaces of G-bundles, J. Algebraic Geom. 18 (2009), 309-369. DOI: https://doi.org/10.1090/S1056-3911-08-00499-2.

[G]  V. Ginzburg,The global nilpotent variety is Lagrangian. Duke Math. J. 109 (2001), no. 3, 511–519 .

[HT]  T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153 (2003), no. 1, 197–229.

[HH] T. Hausel and N. Hitchin, Very stable Higgs bundles, equivariant multiplicity and mirror symmetry, Preprint, arXiv: 2101.08583.

[Hi1] N. Hitchin,Stable bundles and integrable systems. Duke Math. J. 54 (1987), no. 1, 91–114.

[Hi2] N. Hitchin,The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126.

[La] G. Laumon, Un analogue global du cône nilpotent, Duke Math. Jour. 57 (1988), 647–671. 

[NR1]  M.S. Narasimhan and S. Ramanan, Moduli of Vector Bundles on a Compact Riemann  Surface, Ann. Math., Second Series, Vol. 89, No. 1 (1969), 14–51.

[PaP] S. Pal and C. Pauly, Wobbly divisors in the moduli space of rank two bundle, to appear in Adv. in Geometry.

[PPe] C. Pauly and A. Peón-Nieto, Very stable bundles and properness of the Hitchin map,  A. Geom. Dedicata (2018). DOI:10.1007/s10711-018-0333-6. Preprint arXiv:1710.10152  [math.AG].

[Pe] A. Peón-Nieto, Wobbly and shaky bundles and resolutions of rational maps, arXiv:2007.13447.

[S1] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I and II. Inst. Hautes Études Sci. Publ. Math. No. 79 (1994), 47–129.

[S2] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. No. 80 (1994), 5–79 .

[Z] H. Zelaci On very stablity of principal G-bundles. Geom. Dedicata 204 (2020), 165–173.

11:00 to 12:30 Mathew Bullimore (Durham University, UK) A Mathematical Introduction to 3d N = 4 Gauge Theories 3 (WP)

This mini course will introduce mathematical structures arising in 3d N = 4 gauge theories, with a focus on aspects that are important in geometric representation theory. The course will cover the supersymmetry algebra, central charges, mass and FI parameter deformations, walls and chambers, Higgs and Coulomb branch moduli spaces, and 3d mirror symmetry. This should provide a stepping stone to more advanced topics in the literature. I will try to explain how physicists think about these concepts in a way that is friendly to mathematicians.

References 

[a] A quick summary of some aspects of 3d N = 4 gauge theories can be found in section 2 of https://arxiv.org/pdf/1503.04817.pdf.

[b] For some motivation for mathematicians, see section 2 of https://arxiv.org/pdf/1706.05154.pdf.

14:00 to 15:30 Dan Xie (Tsinghua University, China) Affine Springer fiber and representation theory of W algebra

I will discuss the interesting map between fixed variety of affine Springer fiber and representation theory of W algebra. The corresponding affine Springer fiber and W algebra appear in the study of 4d Argyres-Douglas theory, and the above map might be thought of as a generalized 3d mirror symmetry.

References: 

1. Z.W Yun, Lectures on Springer theories and orbital integral, arXiv:1602.01451

2. V.Kac, Wakimoto, On rationality of W algebras, arXiv: 0711.2296

3. D. Xie, Generalized Argyres-Douglas theories, arXiv: 1204.2270

4. J. W. Song, D.Xie, W.B Yan, Vertex operator algebras of Argyres-Douglas theories from M5 branes,  arXiv: 1706.01607

15:30 to 17:00 Madalena Lemos (Durham University, UK) A mini-course on vertex operator algebras of N= 2 Superconformal Field Theories 3

Any four-dimensional N>=2 superconformal field theory possess a protected subsector isomorphic to a two-dimensional chiral algebra/vertex operator algebra [arxiv:1312.5344]. After a brief review of N=2 SCFTs we will describe the construction of this subsector, the consequences for four-dimensional physics, as well as a brief outlook of recent progress. Some basic familiarity with conformal field theories is desirable, for example with what is covered in [a] section 1,2,3 and the corresponding background material (the remaining sections of [a] provide details on 4d N=2 superconformal theories which is relevant for the mini-course, but detailed knowledge of that will not be assumed). Some familiarity with enhanced Virasoro algebra in 2d CFTs, along the lines of what is summarized in chapter 1 of [b] is also beneficial. The bulk of the lectures will proceed along the same lines as [c].

References:

[a] Eberhardt, Superconformal symmetry and representations, arxiv:2006.13280

[b] Ribault, Minimal lectures on two-dimensional conformal field theory, arxiv:1609.09523

[c] Lemos, Lectures on chiral algebras of N⩾2 superconformal field theories, arXiv:2006.13892

20:30 to 22:00 Edward Witten (Institute for Advanced Study, Princeton, USA) Quantization, Gauge Theory, and the Analytic Approach to Geometric Langlands 1

The first lecture will be a general introduction to the problem of quantization. Contrary to the impression that one may get from textbooks, there is not a satisfactory general answer to the problem of quantizing a symplectic manifold.  I will explain two partial answers: geometric quantization and ``quantization by branes.''    In the second lecture, I will give an introduction to the gauge theory approach to the geometric Langlands correspondence. The third lecture is part of a special program in memory of M. S. Narasimhan and C. S. Seshadri. Much of their work has provided important foundations for the sort of mathematical physics that is represented at this meeting -- including my lectures.   Finally, the fourth lecture will explain work with D. Gaiotto in which we use ``quantization by branes'' to elucidate a new twist on geometric Langlands that has been discovered by P. Etinghof, E. Frenkel, and D. Kazhdan.

References: 

On quantization by branes: https://arxiv.org/abs/0809.0305

For a brief introduction to the subject of lecture 2, see the first three sections of https://arxiv.org/abs/0809.0305

(If you have time, you can look at the more detailed paper https://arxiv.org/pdf/hep-th/0604151.pdf.)

It is hard to give references for my lecture at the event in honor of Narasimhan and Seshadri, but I do recommend this famous article by Atiyah and Bott where some of their work is important background:

http://www.math.toronto.edu/mgualt/Morse%20Theory/Atiyah-Bott.pdf

Finally, on my fourth lecture, the reference would be a paper that Gaiotto and I are writing that hopefully will appear before my lectures.

22:00 to 23:30 Dan Xie (Tsinghua University, China) Affine Springer fiber and representation theory of W algebra (WP)

I will discuss the interesting map between fixed variety of affine Springer fiber and representation theory of W algebra. The corresponding affine Springer fiber and W algebra appear in the study of 4d Argyres-Douglas theory, and the above map might be thought of as a generalized 3d mirror symmetry.

References: 

1. Z.W Yun, Lectures on Springer theories and orbital integral, arXiv:1602.01451

2. V.Kac, Wakimoto, On rationality of W algebras, arXiv: 0711.2296

3. D. Xie, Generalized Argyres-Douglas theories, arXiv: 1204.2270

4. J. W. Song, D.Xie, W.B Yan, Vertex operator algebras of Argyres-Douglas theories from M5 branes,  arXiv: 1706.01607

Friday, 09 July 2021
Time Speaker Title Resources
00:30 to 02:00 Madalena Lemos (Durham University, UK) A mini-course on vertex operator algebras of N= 2 Superconformal Field Theories 3 (WP)

Any four-dimensional N>=2 superconformal field theory possess a protected subsector isomorphic to a two-dimensional chiral algebra/vertex operator algebra [arxiv:1312.5344]. After a brief review of N=2 SCFTs we will describe the construction of this subsector, the consequences for four-dimensional physics, as well as a brief outlook of recent progress. Some basic familiarity with conformal field theories is desirable, for example with what is covered in [a] section 1,2,3 and the corresponding background material (the remaining sections of [a] provide details on 4d N=2 superconformal theories which is relevant for the mini-course, but detailed knowledge of that will not be assumed). Some familiarity with enhanced Virasoro algebra in 2d CFTs, along the lines of what is summarized in chapter 1 of [b] is also beneficial. The bulk of the lectures will proceed along the same lines as [c].

References:

[a] Eberhardt, Superconformal symmetry and representations, arxiv:2006.13280

[b] Ribault, Minimal lectures on two-dimensional conformal field theory, arxiv:1609.09523

[c] Lemos, Lectures on chiral algebras of N⩾2 superconformal field theories, arXiv:2006.13892

11:00 to 12:30 Jacques Hurtubise (McGill University, Canada) Geometry of the Hitchin integrable systems, and some variations 1 (WP)

The Narasimhan-Seshadri (-Hitchin-Kobayashi-Yau...) correspondence.

The basic idea (ASD Yang-Mills), examples of various non-compact cases and reductions: monopoles on various spaces, Yang-Mills-Higgs. The Yang-MIlls-Higgs moduli. Higgs bundles, symplectic structures and integrable systems. The spectral curve and abelianization. Parabolic generalisations, examples. Darboux coordinates and geometry of Poisson surfaces.

Variants: The Sklyanin brackets. Integrable systems and surfaces. Local geometry of integrable systems of Jacobians: rank two systems. Other groups: Cameral covers and Higgs. Sklyanin type systems;  the analogue of the rank two condition.

14:00 to 15:30 Ana Peon-Nieto (University of Nice Sophia Antipolis, France and University of Birmingham, UK) Geometry of the Global Nilpotent Cone 3

Higgs bundles are central players in the current arena in mathematics and physics. Their applications to mirror symmetry and the geometric Langlands programme have captured a lot of attention in the past few years. In this mini-course, we will study some aspects of the geometry of the moduli space of these objects M_X , focusing on the nilpotent cone . Besides capturing the global topology of M_X , the nilpotent cone is at the heart of many applications that we will explore, always basing our discussion on manageable examples. More precisely, after a brief introduction, we will discuss Donagi-Pantev’s programme towards geometric Langlands through abelianisation, as well as discuss two important conjectures therein. One, due to the authors of the programme, concerns the equality of the wobbly and shaky loci. The other one, due to Drinfeld, concerns pure codimensionality of wobbly loci.

Outline

(1) Introduction to Higgs bundles and the non abelian Hodge correspondence. Refs: [Hi2, S1, S2]
1.1 The rank one case as a motivating example
1.2 Mirror symmetry for Hitchin systems. Refs: [HT, DP1]

(2) Applications in a nutshell
2.1 Geometric Langlands via abelianisation. Refs: [DP2, DP3]
2.2 The nilpotent cone in all this. Refs: [HT, BNR]

(3) The nilpotent cone. Refs:[BD, Fa, G, La, PaP]
3.1 Wobbly bundles and irreducible components. Drinfeld’s conjecture. Refs:[PaP]
3.2 Wobbly bundles and shaky bundles in geometric Langlands- Donagi-Pantev’s conjecture. Refs: [PPe, Z, HH, Pe]

References:

[BNR] A. Beauville, M.S. Narasimhan, and S. Ramanan, Spectral curves and the generalised  theta divisor. J. Reine Angew. Math. 398 (1989), 169–179.

[BD] A.A. Beilinson and V.G. Drinfeld, Quantization of Hitchin’s fibration and Langlands’ program. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 3–7, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996.

[DP1]  R.Y. Donagi and T.B. Pantev,Langlands duality for Hitchin systems. Invent. Math. 189  (2012), no. 3, 653–735.

[DP2]  R.Y. Donagi and T.B. Pantev,Geometric Langlands and non-abelian Hodge theory. Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, 85–116, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009.

[DP3] R.Y. Donagi and T.B. Pantev,Parabolic Hecke eigensheaves. Preprint, arXiv:1910.02357 [math.AG]

[Fa] G. Faltings, Theta functions on moduli spaces of G-bundles, J. Algebraic Geom. 18 (2009), 309-369. DOI: https://doi.org/10.1090/S1056-3911-08-00499-2.

[G]  V. Ginzburg,The global nilpotent variety is Lagrangian. Duke Math. J. 109 (2001), no. 3, 511–519 .

[HT]  T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153 (2003), no. 1, 197–229.

[HH] T. Hausel and N. Hitchin, Very stable Higgs bundles, equivariant multiplicity and mirror symmetry, Preprint, arXiv: 2101.08583.

[Hi1] N. Hitchin,Stable bundles and integrable systems. Duke Math. J. 54 (1987), no. 1, 91–114.

[Hi2] N. Hitchin,The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126.

[La] G. Laumon, Un analogue global du cône nilpotent, Duke Math. Jour. 57 (1988), 647–671. 

[NR1]  M.S. Narasimhan and S. Ramanan, Moduli of Vector Bundles on a Compact Riemann  Surface, Ann. Math., Second Series, Vol. 89, No. 1 (1969), 14–51.

[PaP] S. Pal and C. Pauly, Wobbly divisors in the moduli space of rank two bundle, to appear in Adv. in Geometry.

[PPe] C. Pauly and A. Peón-Nieto, Very stable bundles and properness of the Hitchin map,  A. Geom. Dedicata (2018). DOI:10.1007/s10711-018-0333-6. Preprint arXiv:1710.10152  [math.AG].

[Pe] A. Peón-Nieto, Wobbly and shaky bundles and resolutions of rational maps, arXiv:2007.13447.

[S1] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I and II. Inst. Hautes Études Sci. Publ. Math. No. 79 (1994), 47–129.

[S2] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. No. 80 (1994), 5–79 .

[Z] H. Zelaci On very stablity of principal G-bundles. Geom. Dedicata 204 (2020), 165–173.

15:30 to 17:00 Edward Witten (Institute for Advanced Study, Princeton, USA) Quantization, Gauge Theory, and the Analytic Approach to Geometric Langlands 1 (WP)

The first lecture will be a general introduction to the problem of quantization. Contrary to the impression that one may get from textbooks, there is not a satisfactory general answer to the problem of quantizing a symplectic manifold.  I will explain two partial answers: geometric quantization and ``quantization by branes.''    In the second lecture, I will give an introduction to the gauge theory approach to the geometric Langlands correspondence. The third lecture is part of a special program in memory of M. S. Narasimhan and C. S. Seshadri. Much of their work has provided important foundations for the sort of mathematical physics that is represented at this meeting -- including my lectures.   Finally, the fourth lecture will explain work with D. Gaiotto in which we use ``quantization by branes'' to elucidate a new twist on geometric Langlands that has been discovered by P. Etinghof, E. Frenkel, and D. Kazhdan.

References: 

On quantization by branes: https://arxiv.org/abs/0809.0305

For a brief introduction to the subject of lecture 2, see the first three sections of https://arxiv.org/abs/0809.0305

(If you have time, you can look at the more detailed paper https://arxiv.org/pdf/hep-th/0604151.pdf.)

It is hard to give references for my lecture at the event in honor of Narasimhan and Seshadri, but I do recommend this famous article by Atiyah and Bott where some of their work is important background:

http://www.math.toronto.edu/mgualt/Morse%20Theory/Atiyah-Bott.pdf

Finally, on my fourth lecture, the reference would be a paper that Gaiotto and I are writing that hopefully will appear before my lectures.

20:30 to 22:00 Edward Witten (Institute for Advanced Study, Princeton, USA) Quantization by Branes and Geometric Langlands 2

The first lecture will be a general introduction to the problem of quantization. Contrary to the impression that one may get from textbooks, there is not a satisfactory general answer to the problem of quantizing a symplectic manifold.  I will explain two partial answers: geometric quantization and ``quantization by branes.''    In the second lecture, I will give an introduction to the gauge theory approach to the geometric Langlands correspondence. The third lecture is part of a special program in memory of M. S. Narasimhan and C. S. Seshadri. Much of their work has provided important foundations for the sort of mathematical physics that is represented at this meeting -- including my lectures.   Finally, the fourth lecture will explain work with D. Gaiotto in which we use ``quantization by branes'' to elucidate a new twist on geometric Langlands that has been discovered by P. Etinghof, E. Frenkel, and D. Kazhdan.

References: 

On quantization by branes: https://arxiv.org/abs/0809.0305

For a brief introduction to the subject of lecture 2, see the first three sections of https://arxiv.org/abs/0809.0305

(If you have time, you can look at the more detailed paper https://arxiv.org/pdf/hep-th/0604151.pdf.)

It is hard to give references for my lecture at the event in honor of Narasimhan and Seshadri, but I do recommend this famous article by Atiyah and Bott where some of their work is important background:

http://www.math.toronto.edu/mgualt/Morse%20Theory/Atiyah-Bott.pdf

Finally, on my fourth lecture, the reference would be a paper that Gaiotto and I are writing that hopefully will appear before my lectures.

22:00 to 23:30 Jacques Hurtubise (McGill University, Canada) Geometry of the Hitchin integrable systems, and some variations 2

The Narasimhan-Seshadri (-Hitchin-Kobayashi-Yau...) correspondence.

The basic idea (ASD Yang-Mills), examples of various non-compact cases and reductions: monopoles on various spaces, Yang-Mills-Higgs. The Yang-MIlls-Higgs moduli. Higgs bundles, symplectic structures and integrable systems. The spectral curve and abelianization. Parabolic generalisations, examples. Darboux coordinates and geometry of Poisson surfaces.

Variants: The Sklyanin brackets. Integrable systems and surfaces. Local geometry of integrable systems of Jacobians: rank two systems. Other groups: Cameral covers and Higgs. Sklyanin type systems;  the analogue of the rank two condition.

Saturday, 10 July 2021
Time Speaker Title Resources
00:30 to 02:00 Ana Peon-Nieto (University of Nice Sophia Antipolis, France and University of Birmingham, UK) Geometry of the Global Nilpotent Cone 3 (WP)

Higgs bundles are central players in the current arena in mathematics and physics. Their applications to mirror symmetry and the geometric Langlands programme have captured a lot of attention in the past few years. In this mini-course, we will study some aspects of the geometry of the moduli space of these objects M_X , focusing on the nilpotent cone . Besides capturing the global topology of M_X , the nilpotent cone is at the heart of many applications that we will explore, always basing our discussion on manageable examples. More precisely, after a brief introduction, we will discuss Donagi-Pantev’s programme towards geometric Langlands through abelianisation, as well as discuss two important conjectures therein. One, due to the authors of the programme, concerns the equality of the wobbly and shaky loci. The other one, due to Drinfeld, concerns pure codimensionality of wobbly loci.

Outline

(1) Introduction to Higgs bundles and the non abelian Hodge correspondence. Refs: [Hi2, S1, S2]
1.1 The rank one case as a motivating example
1.2 Mirror symmetry for Hitchin systems. Refs: [HT, DP1]

(2) Applications in a nutshell
2.1 Geometric Langlands via abelianisation. Refs: [DP2, DP3]
2.2 The nilpotent cone in all this. Refs: [HT, BNR]

(3) The nilpotent cone. Refs:[BD, Fa, G, La, PaP]
3.1 Wobbly bundles and irreducible components. Drinfeld’s conjecture. Refs:[PaP]
3.2 Wobbly bundles and shaky bundles in geometric Langlands- Donagi-Pantev’s conjecture. Refs: [PPe, Z, HH, Pe]

References:

[BNR] A. Beauville, M.S. Narasimhan, and S. Ramanan, Spectral curves and the generalised  theta divisor. J. Reine Angew. Math. 398 (1989), 169–179.

[BD] A.A. Beilinson and V.G. Drinfeld, Quantization of Hitchin’s fibration and Langlands’ program. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 3–7, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996.

[DP1]  R.Y. Donagi and T.B. Pantev,Langlands duality for Hitchin systems. Invent. Math. 189  (2012), no. 3, 653–735.

[DP2]  R.Y. Donagi and T.B. Pantev,Geometric Langlands and non-abelian Hodge theory. Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, 85–116, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009.

[DP3] R.Y. Donagi and T.B. Pantev,Parabolic Hecke eigensheaves. Preprint, arXiv:1910.02357 [math.AG]

[Fa] G. Faltings, Theta functions on moduli spaces of G-bundles, J. Algebraic Geom. 18 (2009), 309-369. DOI: https://doi.org/10.1090/S1056-3911-08-00499-2.

[G]  V. Ginzburg,The global nilpotent variety is Lagrangian. Duke Math. J. 109 (2001), no. 3, 511–519 .

[HT]  T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153 (2003), no. 1, 197–229.

[HH] T. Hausel and N. Hitchin, Very stable Higgs bundles, equivariant multiplicity and mirror symmetry, Preprint, arXiv: 2101.08583.

[Hi1] N. Hitchin,Stable bundles and integrable systems. Duke Math. J. 54 (1987), no. 1, 91–114.

[Hi2] N. Hitchin,The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126.

[La] G. Laumon, Un analogue global du cône nilpotent, Duke Math. Jour. 57 (1988), 647–671. 

[NR1]  M.S. Narasimhan and S. Ramanan, Moduli of Vector Bundles on a Compact Riemann  Surface, Ann. Math., Second Series, Vol. 89, No. 1 (1969), 14–51.

[PaP] S. Pal and C. Pauly, Wobbly divisors in the moduli space of rank two bundle, to appear in Adv. in Geometry.

[PPe] C. Pauly and A. Peón-Nieto, Very stable bundles and properness of the Hitchin map,  A. Geom. Dedicata (2018). DOI:10.1007/s10711-018-0333-6. Preprint arXiv:1710.10152  [math.AG].

[Pe] A. Peón-Nieto, Wobbly and shaky bundles and resolutions of rational maps, arXiv:2007.13447.

[S1] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I and II. Inst. Hautes Études Sci. Publ. Math. No. 79 (1994), 47–129.

[S2] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. No. 80 (1994), 5–79 .

[Z] H. Zelaci On very stablity of principal G-bundles. Geom. Dedicata 204 (2020), 165–173.

Monday, 12 July 2021
Time Speaker Title Resources
15:30 to 17:00 Vikraman Balaji (Chennai Mathematical Institute, India) Parahoric torsors and degeneration of moduli spaces

I will briefly introduce ideas from my joint work with C.S. Seshadri on "Parahoric torsors" and indicate how this leads to an understanding and solution to the problem of degenerations of the moduli spaces of principal $G$-bundles.

References:

V.BALAJI, C.S.SESHADRI, Moduli of parahoric $\mathcal G$--torsors on a compact Riemann surface, {\em Journal of Algebraic Geometry}, Volume 24, pages 1-49, (2015).

V. BALAJI, Torsors on semistable curves and degenerations, (math archiv) (to appear in the {\em Proceedings of the Indian Academy of Sciences}, 2021).

19:00 to 20:30 Jacques Hurtubise (McGill University, Canada) Geometry of the Hitchin integrable systems, and some variations 3

The Narasimhan-Seshadri (-Hitchin-Kobayashi-Yau...) correspondence.

The basic idea (ASD Yang-Mills), examples of various non-compact cases and reductions: monopoles on various spaces, Yang-Mills-Higgs. The Yang-MIlls-Higgs moduli. Higgs bundles, symplectic structures and integrable systems. The spectral curve and abelianization. Parabolic generalisations, examples. Darboux coordinates and geometry of Poisson surfaces.

Variants: The Sklyanin brackets. Integrable systems and surfaces. Local geometry of integrable systems of Jacobians: rank two systems. Other groups: Cameral covers and Higgs. Sklyanin type systems;  the analogue of the rank two condition.

20:30 to 22:00 Edward Witten (Institute for Advanced Study, Princeton, USA) On The Work Of Narasimhan and Seshadri 3

The first lecture will be a general introduction to the problem of quantization. Contrary to the impression that one may get from textbooks, there is not a satisfactory general answer to the problem of quantizing a symplectic manifold.  I will explain two partial answers: geometric quantization and ``quantization by branes.''    In the second lecture, I will give an introduction to the gauge theory approach to the geometric Langlands correspondence. The third lecture is part of a special program in memory of M. S. Narasimhan and C. S. Seshadri. Much of their work has provided important foundations for the sort of mathematical physics that is represented at this meeting -- including my lectures.   Finally, the fourth lecture will explain work with D. Gaiotto in which we use ``quantization by branes'' to elucidate a new twist on geometric Langlands that has been discovered by P. Etinghof, E. Frenkel, and D. Kazhdan.

References: 

On quantization by branes: https://arxiv.org/abs/0809.0305

For a brief introduction to the subject of lecture 2, see the first three sections of https://arxiv.org/abs/0809.0305

(If you have time, you can look at the more detailed paper https://arxiv.org/pdf/hep-th/0604151.pdf.)

It is hard to give references for my lecture at the event in honor of Narasimhan and Seshadri, but I do recommend this famous article by Atiyah and Bott where some of their work is important background:

http://www.math.toronto.edu/mgualt/Morse%20Theory/Atiyah-Bott.pdf

Finally, on my fourth lecture, the reference would be a paper that Gaiotto and I are writing that hopefully will appear before my lectures.

22:00 to 23:30 Shrawan Kumar (University of North Carolina at Chapel Hill, USA) Conformal blocks for Galois covers of algebraic curves

This is a joint work with Jiuzu Hong. We study the space of twisted conformal blocks attached to A-curves S with marked A-orbits and an action of A on a simple Lie algebra g, where A is a finite group. We prove that if A stabilizes a Borel subalgebra of g, then Propogation Theorem and Factorization Theorem hold. We endow a projectively flat connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed A-curves; in particular,it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. We further identify the space of twisted conformal blocks with the space of global sections of certain line bundles on the stack of A-equivariant principal G-bundles over the curve S, G being the simply-connected group with Lie algebra g. This generalizes the Verlinde theory of conformal blocks to the twisted setting.

Tuesday, 13 July 2021
Time Speaker Title Resources
15:30 to 17:00 Joerg Teschner (University of Hamburg and DESY, Germany) Analytic geometric Langlands-correspondence: Relations to conformal field theory and integrable models 1

In the first of my three lectures I plan to present a review of the approach of Beilinson and Drinfeld to the geometric Langlands correspondence (GL), based on ideas from conformal field theory and the representation theory of the affine Lie algebras at the critical level. The goal of my second lecture will be to review some aspects of a recent variant of the GL called the analytic Langlands correspondence. The third lecture will discuss a natural deformation of the analytic Langlands correspondence related to the H3-WZNW model, and its relation to a generalisation of the AGT-correspondence.

References:

Lecture 1:

E. Frenkel, Affine algebras, Langlands duality and Bethe ansatz, in Proc. of Int. Congress of Math. Phys. (Paris, 1994), ed. D. Iagolnitzer, pp. 606–642, International Press, 1995 (arXiv:qalg/9506003).

E. Frenkel, Lectures on the Langlands program and conformal field theory, in Frontiers in number theory, physics, and geometry. II (P. Cartier, ed.), pp. 387–533. Springer, Berlin, 2007. (arXiv:hep-th/0512172).

Lecture 2: 

J. Teschner, Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence, in Geometry and Physics, in honour of Nigel Hitchin, Vol. I, eds. Dancer, e.a., pp. 347–375, Oxford University Press, 2018 (arXiv:1707.07873).

P. Etingof, E. Frenkel, and D. Kazhdan, An analytic version of the Langlands correspondence for complex curves, in Integrability, Quantization, and Geometry, dedicated to Boris Dubrovin, Vol. II, eds. S. Novikov, e.a., pp. 137–202, Proc. Symp. Pure Math. 103.2, AMS, 2021 (arXiv:1908.09677).

Lecture 3:

Based on ideas going back to, J. Teschner, Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I, Adv. Theor. Math. Phys. 15 (2011), no. 2, 471–564. [arXiv:1005.2846] and some unpublished work. 

20:30 to 22:00 Edward Witten (Institute for Advanced Study, Princeton, USA) Quantization by Branes and Geometric Langlands 4

The first lecture will be a general introduction to the problem of quantization. Contrary to the impression that one may get from textbooks, there is not a satisfactory general answer to the problem of quantizing a symplectic manifold.  I will explain two partial answers: geometric quantization and ``quantization by branes.''    In the second lecture, I will give an introduction to the gauge theory approach to the geometric Langlands correspondence. The third lecture is part of a special program in memory of M. S. Narasimhan and C. S. Seshadri. Much of their work has provided important foundations for the sort of mathematical physics that is represented at this meeting -- including my lectures.   Finally, the fourth lecture will explain work with D. Gaiotto in which we use ``quantization by branes'' to elucidate a new twist on geometric Langlands that has been discovered by P. Etinghof, E. Frenkel, and D. Kazhdan.

References: 

On quantization by branes: https://arxiv.org/abs/0809.0305

For a brief introduction to the subject of lecture 2, see the first three sections of https://arxiv.org/abs/0809.0305

(If you have time, you can look at the more detailed paper https://arxiv.org/pdf/hep-th/0604151.pdf.)

It is hard to give references for my lecture at the event in honor of Narasimhan and Seshadri, but I do recommend this famous article by Atiyah and Bott where some of their work is important background:

http://www.math.toronto.edu/mgualt/Morse%20Theory/Atiyah-Bott.pdf

Finally, on my fourth lecture, the reference would be a paper that Gaiotto and I are writing that hopefully will appear before my lectures.

22:00 to 23:30 Sam Raskin (University of Texas, USA) Geometric Langlands and 3d mirror symmetry 1

In this series of talks, we will give an introduction to the geometric Langlands correspondence, emphasizing local aspects of the subject. We will describe expected compatibilities for the geometric Langlands conjectures in the language of boundary conditions. Many of these compatibility conjectures have emerged only recently, coming from work of Ben-Zvi, Braverman, Costello, Dimofte, Finkelberg, Gaiotto, Hilburn, Sakellaridis, Venkatesh, Witten, and Yoo (and probably others) on uniting 3d mirror symmetry with geometric Langlands. Finally, we will provide an overview of recent joint work with Justin Hilburn establishing one of these recent conjectures in the abelian case. 

References: 

Beilinson Drinfeld - Quantization of Hitchin's integrable system and Hecke eigensheaves link

Beilinson Drinfeld - Chiral algebras link

Gaitsgory - Outline of the proof of the geometric Langlands conjecture for GL_2 link

Gaiotto Witten - S-duality of boundary conditions in N = 4 super Yang–Mills theory link

Hilburn Raskin - Tate's thesis in the de Rham setting link

Wednesday, 14 July 2021
Time Speaker Title Resources
20:30 to 22:00 Heeyeon Kim (Rutgers University, USA) Moduli space of vortices and 3d N=4 gauge theories

I will discuss effective quantum mechanics obtained from a large class of 3d N=4 gauge theories compactified on a Riemann surface. In the first part of the talk, I will focus on the Witten index of the quantum mechanics, which can be identified as the virtual Euler characteristic of the moduli space of vortices on the Riemann surface. In particular, I will discuss 3d mirror symmetry that provides non-trivial relations among these invariants, and their wall-crossing formulae derived from a path integral point of view. Finally, I will comments on explicit construction of the space of supersymmetric ground states of these theories.

22:00 to 23:30 Sam Raskin (University of Texas, USA) Geometric Langlands and 3d mirror symmetry 2

In this series of talks, we will give an introduction to the geometric Langlands correspondence, emphasizing local aspects of the subject. We will describe expected compatibilities for the geometric Langlands conjectures in the language of boundary conditions. Many of these compatibility conjectures have emerged only recently, coming from work of Ben-Zvi, Braverman, Costello, Dimofte, Finkelberg, Gaiotto, Hilburn, Sakellaridis, Venkatesh, Witten, and Yoo (and probably others) on uniting 3d mirror symmetry with geometric Langlands. Finally, we will provide an overview of recent joint work with Justin Hilburn establishing one of these recent conjectures in the abelian case. 

References: 

Beilinson Drinfeld - Quantization of Hitchin's integrable system and Hecke eigensheaves link

Beilinson Drinfeld - Chiral algebras link

Gaitsgory - Outline of the proof of the geometric Langlands conjecture for GL_2 link

Gaiotto Witten - S-duality of boundary conditions in N = 4 super Yang–Mills theory link

Hilburn Raskin - Tate's thesis in the de Rham setting link

Thursday, 15 July 2021
Time Speaker Title Resources
14:00 to 15:30 Dario Beraldo (University College London, United Kingdom) On the Ramanujan conjecture in Geometric Langlands

After discussing the notion of temperedness arising in the geometric Langlands program, I’ll sketch a proof of a version of the Ramanujan conjecture in this setting. Essential ingredients for the formulation and the proof are the derived Satake equivalence and (geometric analogues of) the Deligne-Lusztig duality functors.

References: 

[1]  D. Arinkin, D. Gaitsgory, Singular support of coherent sheaves and the geometric Langlands conjecture. Selecta Math. (N.S.) 21 (2015), no. 1, 1-199.

[2]  D. Arinkin, D. Gaitsgory, The category of singularities as a crystal and global Springer fibers. J. Amer. Math. Soc. 31 (2018), no. 1, 135-214.

[3] D. Beraldo. On the geometric Ramanujan conjecture, https://arxiv.org/abs/2103.17211

15:30 to 17:00 Joerg Teschner (University of Hamburg and DESY, Germany) Analytic geometric Langlands-correspondence: Relations to conformal field theory and integrable models 2

In the first of my three lectures I plan to present a review of the approach of Beilinson and Drinfeld to the geometric Langlands correspondence (GL), based on ideas from conformal field theory and the representation theory of the affine Lie algebras at the critical level. The goal of my second lecture will be to review some aspects of a recent variant of the GL called the analytic Langlands correspondence. The third lecture will discuss a natural deformation of the analytic Langlands correspondence related to the H3-WZNW model, and its relation to a generalisation of the AGT-correspondence.

References:

Lecture 1:

E. Frenkel, Affine algebras, Langlands duality and Bethe ansatz, in Proc. of Int. Congress of Math. Phys. (Paris, 1994), ed. D. Iagolnitzer, pp. 606–642, International Press, 1995 (arXiv:qalg/9506003).

E. Frenkel, Lectures on the Langlands program and conformal field theory, in Frontiers in number theory, physics, and geometry. II (P. Cartier, ed.), pp. 387–533. Springer, Berlin, 2007. (arXiv:hep-th/0512172).

Lecture 2: 

J. Teschner, Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence, in Geometry and Physics, in honour of Nigel Hitchin, Vol. I, eds. Dancer, e.a., pp. 347–375, Oxford University Press, 2018 (arXiv:1707.07873).

P. Etingof, E. Frenkel, and D. Kazhdan, An analytic version of the Langlands correspondence for complex curves, in Integrability, Quantization, and Geometry, dedicated to Boris Dubrovin, Vol. II, eds. S. Novikov, e.a., pp. 137–202, Proc. Symp. Pure Math. 103.2, AMS, 2021 (arXiv:1908.09677).

Lecture 3:

Based on ideas going back to, J. Teschner, Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I, Adv. Theor. Math. Phys. 15 (2011), no. 2, 471–564. [arXiv:1005.2846] and some unpublished work. 

20:30 to 22:00 Amihay Hanany (Imperial College, UK) Magnetic Quivers and Phase Diagrams - New ways of thinking about moduli spaces of supersymmetric gauge theories 1

This lecture series will focus on two new developments which significantly change the way we think and analyze moduli spaces of supersymmetric gauge theories. The magnetic quiver is a tool to solve long standing problems on the dynamics of strongly coupled theories like 4d AD (Argyres Douglas) points, 5d fixed points and 6d tensionless strings.

The Hasse diagram neatly encodes phases of the theory with fixed sets of massless states and allows to understand the complexity of given moduli spaces. We will introduce the new concepts, look at simple examples and discuss the exciting ways they change our thinking of moduli spaces.

References:

Branes, Quivers and the Affine Grassmannian, arXiv:2102.06190

22:00 to 23:30 Emily Cliff (University of Sydney, Australia) Moduli spaces of principal 2-group bundles and a categorification of the Freed--Quinn line bundle

 A 2-group is a higher categorical analogue of a group, while a smooth 2-group is a higher categorical analogue of a Lie group. An important example is the string 2-group, defined by Schommer-Pries. We study the notion of principal bundles for smooth 2-groups, and investigate the moduli "space" of such objects. 

In particular in the case of flat principal bundles for a finite 2-group over a Riemann surface, we prove that the moduli space gives a categorification of the Freed--Quinn line bundle. This line bundle has as its global sections the state space of Chern--Simons theory for the underlying finite group. We can also use our results to better understand the notion of geometric string structures (as previously studied by Waldorf and Stolz--Teichner). 

This is joint work with Dan Berwick-Evans, Laura Murray, Apurva Nakade, and Emma Phillips. 

References on 2-groups

https://arxiv.org/abs/1407.6849 (Ganter, Usher: Representation and character theory of finite categorical groups)

https://arxiv.org/abs/math/0307200 (Baez, Lauda: Higher-dimensional algebra V: 2-groups)

Friday, 16 July 2021
Time Speaker Title Resources
15:30 to 17:00 Joerg Teschner (University of Hamburg and DESY, Germany) Analytic geometric Langlands-correspondence: Relations to conformal field theory and integrable models 3

In the first of my three lectures I plan to present a review of the approach of Beilinson and Drinfeld to the geometric Langlands correspondence (GL), based on ideas from conformal field theory and the representation theory of the affine Lie algebras at the critical level. The goal of my second lecture will be to review some aspects of a recent variant of the GL called the analytic Langlands correspondence. The third lecture will discuss a natural deformation of the analytic Langlands correspondence related to the H3-WZNW model, and its relation to a generalisation of the AGT-correspondence.

References:

Lecture 1:

E. Frenkel, Affine algebras, Langlands duality and Bethe ansatz, in Proc. of Int. Congress of Math. Phys. (Paris, 1994), ed. D. Iagolnitzer, pp. 606–642, International Press, 1995 (arXiv:qalg/9506003).

E. Frenkel, Lectures on the Langlands program and conformal field theory, in Frontiers in number theory, physics, and geometry. II (P. Cartier, ed.), pp. 387–533. Springer, Berlin, 2007. (arXiv:hep-th/0512172).

Lecture 2: 

J. Teschner, Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence, in Geometry and Physics, in honour of Nigel Hitchin, Vol. I, eds. Dancer, e.a., pp. 347–375, Oxford University Press, 2018 (arXiv:1707.07873).

P. Etingof, E. Frenkel, and D. Kazhdan, An analytic version of the Langlands correspondence for complex curves, in Integrability, Quantization, and Geometry, dedicated to Boris Dubrovin, Vol. II, eds. S. Novikov, e.a., pp. 137–202, Proc. Symp. Pure Math. 103.2, AMS, 2021 (arXiv:1908.09677).

Lecture 3:

Based on ideas going back to, J. Teschner, Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I, Adv. Theor. Math. Phys. 15 (2011), no. 2, 471–564. [arXiv:1005.2846] and some unpublished work. 

20:30 to 22:00 Amihay Hanany (Imperial College, UK) Magnetic Quivers and Phase Diagrams - New ways of thinking about moduli spaces of supersymmetric gauge theories 2

This lecture series will focus on two new developments which significantly change the way we think and analyze moduli spaces of supersymmetric gauge theories. The magnetic quiver is a tool to solve long standing problems on the dynamics of strongly coupled theories like 4d AD (Argyres Douglas) points, 5d fixed points and 6d tensionless strings.

The Hasse diagram neatly encodes phases of the theory with fixed sets of massless states and allows to understand the complexity of given moduli spaces. We will introduce the new concepts, look at simple examples and discuss the exciting ways they change our thinking of moduli spaces.

References:

Branes, Quivers and the Affine Grassmannian, arXiv:2102.06190

22:00 to 23:30 Sam Raskin (University of Texas, USA) Geometric Langlands and 3d mirror symmetry 3

In this series of talks, we will give an introduction to the geometric Langlands correspondence, emphasizing local aspects of the subject. We will describe expected compatibilities for the geometric Langlands conjectures in the language of boundary conditions. Many of these compatibility conjectures have emerged only recently, coming from work of Ben-Zvi, Braverman, Costello, Dimofte, Finkelberg, Gaiotto, Hilburn, Sakellaridis, Venkatesh, Witten, and Yoo (and probably others) on uniting 3d mirror symmetry with geometric Langlands. Finally, we will provide an overview of recent joint work with Justin Hilburn establishing one of these recent conjectures in the abelian case. 

References: 

Beilinson Drinfeld - Quantization of Hitchin's integrable system and Hecke eigensheaves link

Beilinson Drinfeld - Chiral algebras link

Gaitsgory - Outline of the proof of the geometric Langlands conjecture for GL_2 link

Gaiotto Witten - S-duality of boundary conditions in N = 4 super Yang–Mills theory link

Hilburn Raskin - Tate's thesis in the de Rham setting link

Monday, 19 July 2021
Time Speaker Title Resources
11:00 to 12:30 Hiraku Nakajima (Kavli IPMU / NAOJ, Tokyo, Japan) Geometric Satake correspondence for affine Lie groups 1

Let G be a complex reductive group, which is a complexification of a compact Lie group G_c. Let \mathrm{Gr}_G be the affine Grassmannian associated with G. It is a partial flag variety associated with the affine Lie group G_{\mathrm{aff}} for G, and related to moduli spaces of singular G_c-monopoles on \mathbb{R}^3. Geometric Satake correspondence gives a topological construction of representations of the Langlands dual group G^\vee via \mathrm{Gr}_G. If we replace G by G_{\mathrm{aff}}, we cannot naively consider \mathrm{Gr}_{G_{\mathrm{aff}}}. Nevertheless we can still consider moduli spaces of G_c-instantons on the Taub-NUT space divided by a finite cyclic group (multi-Taub-NUT space), and construct integrable representations of the Langlands dual group of G_{\mathrm{aff}}. (By technical reasons, a rigorous proof is given only in type A.)

References:

I will give a pedagogic presentation of materials, sacrificing mathematically rigorous introduction of basic tools, e.g., perverse sheaves, the definition of affine Grassmannian, and a definition of Coulomb branches via equivariant Borel-Moore homology. For mathematically oriented participants, I recommend two articles of Mark Andrea de Cataldo and Zinwen Zhu in Geometry of Moduli Spaces and Representation Theory, IAS Park City 24, AMS 2017 for the first two topics, and my expository article https://arxiv.org/abs/1706.05154 for Coulomb branches. An exposition of geometric Satake for affine Lie algebras is available at https://arxiv.org/abs/1812.11710.

20:30 to 22:00 Amihay Hanany (Imperial College, UK) Magnetic Quivers and Phase Diagrams - New ways of thinking about moduli spaces of supersymmetric gauge theories 3

This lecture series will focus on two new developments which significantly change the way we think and analyze moduli spaces of supersymmetric gauge theories. The magnetic quiver is a tool to solve long standing problems on the dynamics of strongly coupled theories like 4d AD (Argyres Douglas) points, 5d fixed points and 6d tensionless strings.

The Hasse diagram neatly encodes phases of the theory with fixed sets of massless states and allows to understand the complexity of given moduli spaces. We will introduce the new concepts, look at simple examples and discuss the exciting ways they change our thinking of moduli spaces.

References:

Branes, Quivers and the Affine Grassmannian, arXiv:2102.06190

22:00 to 23:30 Shlomo Razamat (Technion Israel Institute of Technology, Israel) SQCD and pairs of pants

We will discuss a geometric re-interpretation of N=1 SQCD with special unitary gauge groups. We will argue that the 4d SU(M) SQCD in the middle of the conformal window can be engineered by compactifying certain 6d SCFTs on three punctured spheres.  We will also discuss in this context the interplay between simple geometric and group theoretic considerations and field theoretic strong coupling phenomena. In particular we will show how many known and novel dualities and symmetry emergence phenomena of supersymmetric gauge theories, with  simple and semi-simple  special unitary gauge groups, are related to the Weyl group of D_{6+2K}. We will also comment on the relation of this construction to integrable models.

The talk will be based mainly on: 

https://arxiv.org/pdf/2006.03480.pdf 

https://arxiv.org/pdf/1906.05088.pdf

https://arxiv.org/pdf/2106.08335.pdf

https://arxiv.org/pdf/1910.03603.pdf

Tuesday, 20 July 2021
Time Speaker Title Resources
15:30 to 17:00 Claudia Scheimbauer (Technische Universität München, Deutschland) Extending topological field theories

Given the partition function of a topological field theory (TFT), following the approach by Atiyah and Segal, one may ask for a symmetric monoidal functor out of a suitable category of bordisms (spacetimes) which extends the partition function to manifolds with boundary and corners. For this, we need a suitable codomain generalizing (super) vector spaces and linear maps. Conversely, given a codomain, the celebrated Cobordism Hypothesis provides a classification of all fully local extended TFTs via dualizabilty conditions.
In this talk I will explain a commonly studied family of targets: the higher Morita categories of $E_n$-algebras and bimodules and discuss dualizability therein. Important examples are a 3-category of fusion categories and a 4-category of modular tensor categories. Then I will discuss why these do not suffice for Reshetikin-Turaev theories and I will give an outlook on work-in-progress with Freed and Teleman on how to remedy this.

20:30 to 22:00 Mina Aganagic (University of California, Berkeley, USA) Knot homologies from Mirror Symmetry 1

Khovanov showed, more than 20 years ago, that the Jones polynomial emerges as an Euler characteristic of a homology theory. The knot categorification problem is to find a uniform description of this theory, for all gauge groups, which originates from physics, or from geometry. In these lectures, I will describe two solutions to this problem, which originate in string theory, and which are related by a version of homological mirror symmetry. 

In the first lecture, I will give an overview of the problem, the first of its solutions. The first approach (based on a category of B-type branes on the moduli spaces of singular monopoles on R^3, or Coulomb branches of 3d N=4 quiver gauge theories) was recently proven by Ben Webster to be equivalent to his earlier, purely algebraic approach by KRLW algebras. In the second. lecture, I will describe the second approach (based on a category of A-branes in a Landau-Ginsburg model) and the (equivariant) homological mirror symmetry that relates it to the first. The second theory is solvable explicitly, in terms of a simpler cousin of KRLW algebras. This leads to a new, geometric and algebraic, understanding of link homologies. In the third lecture, I will describe the string theory origin of these approaches.

References: 

M. Aganagic, ``Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves,’' arXiv:2004.14518

M. Aganagic, ``Knot Categorification from Mirror Symmetry, Part II: Lagrangians,’' arXiv:2105.06039.

22:00 to 23:30 Justin Hilburn (Perimeter Institute for Theoretical Physics, Canada) 2-Categorical 3d Mirror Symmetry

 It is known that a 3d TQFT is completely determined by the 2-category of boundary conditions it assigns to a point. The 2-category assigned to B-twisted 3d N=4 gauge theories has been described in physics work of Kapustin, Rozansky, Saulina and mathematical work of Arinkin but the 2-category assigned to an A-twisted 3d N=4 theory has only been described in a few cases by Kapustin, Vyas, Setter and in the pure gauge theory case by Teleman. In this talk I will describe work in progress with Ben Gammage and Aaron Mazel-Gee which on a proof of (a version of) 2-categorical mirror symmetry for abelian gauge theories.

Wednesday, 21 July 2021
Time Speaker Title Resources
11:00 to 12:30 Hiraku Nakajima (Kavli IPMU / NAOJ, Tokyo, Japan) Geometric Satake correspondence for affine Lie groups 2

Let G be a complex reductive group, which is a complexification of a compact Lie group G_c. Let \mathrm{Gr}_G be the affine Grassmannian associated with G. It is a partial flag variety associated with the affine Lie group G_{\mathrm{aff}} for G, and related to moduli spaces of singular G_c-monopoles on \mathbb{R}^3. Geometric Satake correspondence gives a topological construction of representations of the Langlands dual group G^\vee via \mathrm{Gr}_G. If we replace G by G_{\mathrm{aff}}, we cannot naively consider \mathrm{Gr}_{G_{\mathrm{aff}}}. Nevertheless we can still consider moduli spaces of G_c-instantons on the Taub-NUT space divided by a finite cyclic group (multi-Taub-NUT space), and construct integrable representations of the Langlands dual group of G_{\mathrm{aff}}. (By technical reasons, a rigorous proof is given only in type A.)

References:

I will give a pedagogic presentation of materials, sacrificing mathematically rigorous introduction of basic tools, e.g., perverse sheaves, the definition of affine Grassmannian, and a definition of Coulomb branches via equivariant Borel-Moore homology. For mathematically oriented participants, I recommend two articles of Mark Andrea de Cataldo and Zinwen Zhu in Geometry of Moduli Spaces and Representation Theory, IAS Park City 24, AMS 2017 for the first two topics, and my expository article https://arxiv.org/abs/1706.05154 for Coulomb branches. An exposition of geometric Satake for affine Lie algebras is available at https://arxiv.org/abs/1812.11710.

20:30 to 22:00 Mina Aganagic (University of California, Berkeley, USA) Knot homologies from Mirror Symmetry 2

Khovanov showed, more than 20 years ago, that the Jones polynomial emerges as an Euler characteristic of a homology theory. The knot categorification problem is to find a uniform description of this theory, for all gauge groups, which originates from physics, or from geometry. In these lectures, I will describe two solutions to this problem, which originate in string theory, and which are related by a version of homological mirror symmetry. 

In the first lecture, I will give an overview of the problem, the first of its solutions. The first approach (based on a category of B-type branes on the moduli spaces of singular monopoles on R^3, or Coulomb branches of 3d N=4 quiver gauge theories) was recently proven by Ben Webster to be equivalent to his earlier, purely algebraic approach by KRLW algebras. In the second. lecture, I will describe the second approach (based on a category of A-branes in a Landau-Ginsburg model) and the (equivariant) homological mirror symmetry that relates it to the first. The second theory is solvable explicitly, in terms of a simpler cousin of KRLW algebras. This leads to a new, geometric and algebraic, understanding of link homologies. In the third lecture, I will describe the string theory origin of these approaches.

References: 

M. Aganagic, ``Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves,’' arXiv:2004.14518

M. Aganagic, ``Knot Categorification from Mirror Symmetry, Part II: Lagrangians,’' arXiv:2105.06039.

22:00 to 23:30 Alex Weekes (University of Saskatchewan, Canada) Coulomb branches for quiver gauge theories with symmetrizers

The Coulomb branches of 3d N=4 quiver gauge theories are very interesting from the perspective of representation theory because of their close relationship with affine Grassmannians and the Geometric Satake correspondence (see Prof. Nakajima's lectures.)  One limitation of this connection is that quiver gauge theories only make sense for symmetric Kac-Moody types (such as finite ADE), but not symmetrizable types (such as finite BCFG).  In this talk I will discuss joint work with Prof. Nakajima, which extends the (mathematical) definition of Coulomb branches to incorporate symmetrizable types.  I will also discuss some simple examples and properties.

References: 

  1. H. Nakajima, A. Weekes, Coulomb branches for quiver gauge theories with symmetrizers, arXiv:1907.06552
  2. A. Braverman, M. Finkelberg, H. Nakajima, Coulomb branches of 3d N=4 quiver gauge theories and slices in the affine Grassmannian, arXiv:1604.03625
Thursday, 22 July 2021
Time Speaker Title Resources
11:00 to 12:30 Philsang Yoo (Tsinghua University, China) Twisted S-duality

String theory is a framework of physics which among other things gives rise to various interesting predictions in mathematics. Some of these predictions are derived from S-duality that provides a non-trivial equivalence of two physical theories. The main aim of the talk is to explain how to mathematically understand aspects of string theory, focusing on topological strings, and discuss S-duality in this context. A large part of the talk will be devoted to providing a dictionary for string theory and mathematical objects in this restricted context of topological string theory. If time permits, I will also discuss its conjectural applications to geometric Langlands theory and the representation theory of the Yangian. This talk is based on a joint work in progress with Surya Raghavendran.

20:30 to 22:00 Mina Aganagic (University of California, Berkeley, USA) Knot homologies from Mirror Symmetry 3

Khovanov showed, more than 20 years ago, that the Jones polynomial emerges as an Euler characteristic of a homology theory. The knot categorification problem is to find a uniform description of this theory, for all gauge groups, which originates from physics, or from geometry. In these lectures, I will describe two solutions to this problem, which originate in string theory, and which are related by a version of homological mirror symmetry. 

In the first lecture, I will give an overview of the problem, the first of its solutions. The first approach (based on a category of B-type branes on the moduli spaces of singular monopoles on R^3, or Coulomb branches of 3d N=4 quiver gauge theories) was recently proven by Ben Webster to be equivalent to his earlier, purely algebraic approach by KRLW algebras. In the second. lecture, I will describe the second approach (based on a category of A-branes in a Landau-Ginsburg model) and the (equivariant) homological mirror symmetry that relates it to the first. The second theory is solvable explicitly, in terms of a simpler cousin of KRLW algebras. This leads to a new, geometric and algebraic, understanding of link homologies. In the third lecture, I will describe the string theory origin of these approaches.

References: 

M. Aganagic, ``Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves,’' arXiv:2004.14518

M. Aganagic, ``Knot Categorification from Mirror Symmetry, Part II: Lagrangians,’' arXiv:2105.06039.

Friday, 23 July 2021
Time Speaker Title Resources
11:00 to 12:30 Hiraku Nakajima (Kavli IPMU / NAOJ, Tokyo, Japan) Geometric Satake correspondence for affine Lie groups 3

Let G be a complex reductive group, which is a complexification of a compact Lie group G_c. Let \mathrm{Gr}_G be the affine Grassmannian associated with G. It is a partial flag variety associated with the affine Lie group G_{\mathrm{aff}} for G, and related to moduli spaces of singular G_c-monopoles on \mathbb{R}^3. Geometric Satake correspondence gives a topological construction of representations of the Langlands dual group G^\vee via \mathrm{Gr}_G. If we replace G by G_{\mathrm{aff}}, we cannot naively consider \mathrm{Gr}_{G_{\mathrm{aff}}}. Nevertheless we can still consider moduli spaces of G_c-instantons on the Taub-NUT space divided by a finite cyclic group (multi-Taub-NUT space), and construct integrable representations of the Langlands dual group of G_{\mathrm{aff}}. (By technical reasons, a rigorous proof is given only in type A.)

References:

I will give a pedagogic presentation of materials, sacrificing mathematically rigorous introduction of basic tools, e.g., perverse sheaves, the definition of affine Grassmannian, and a definition of Coulomb branches via equivariant Borel-Moore homology. For mathematically oriented participants, I recommend two articles of Mark Andrea de Cataldo and Zinwen Zhu in Geometry of Moduli Spaces and Representation Theory, IAS Park City 24, AMS 2017 for the first two topics, and my expository article https://arxiv.org/abs/1706.05154 for Coulomb branches. An exposition of geometric Satake for affine Lie algebras is available at https://arxiv.org/abs/1812.11710.