ICTS

Moduli spaces of stable parabolic connections on curves are very interesting objects which are related to different area of mathematics like algebraic geometry, integrable systems, mathematical physics and Geometric Langlands conjecture. In this lecture, we will explain about an explicit geometry of the moduli spaces of stable parabolic connections on curves introduced and constructed by Inaba, Iwasaki and Saito and Inaba. Then we will review on a work of Arinkin and Lysenko on a rank 2 connections on the projective line with 4 singular points, which is related to Geometric Langlands conjecture in this case. We then explain about the joint work on the moduli space of rank 2 parabolic bundles on the projective line with Simpson and Loray. If time permits, related works of Geometric Langlands conjecture in these cases may be discussed.

Background References:

(a) D. Arinkin, Orthogonality of natural sheaves on moduli stacks of SL(2)-bundles with connections on P 1 minus 4 points, Selecta Math. (N.S.) 7, no. 2, (2001), 213–239.
(b) D. Arinkin, S. Lysenko, On the moduli of SL 2 -bundles with connections on P 1 \{x 1 , . . . , x 4 }. Internat. Math. Res. Notices 1997, no. 19, 983–999.
(c) M. Inaba, K. Iwasaki, and M. -H. Saito, Moduli of stable parabolic connections,Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. I,Publ. Res. Inst. Math. Sci. 42 (2006), no. 4, 987–1089.
(d) M. Inaba, K. Iwasaki, and M. -H. Saito, Moduli of stable parabolic connections,Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. II,Moduli spaces and arithmetic geometry (Tokyo), Adv. Stud. Pure Math., vol. 45,Math. Soc. Japan, Tokyo, 2006, pp. 387–432.
(e) A. Komyo, M. -H. Saito, Explicit description of jumping phenomena on moduli spacesof parabolic connections and Hilbert schemes of points on surfaces to appar in KyotoJournal of Mathematics, arXiv:1611.00971
(f) F. Loray, M.-H. Saito, C. T. Simpson, Foliations on the moduli space of rank twoconnections on the projective line minus four points. to appear in ”Seminaires et Congre” of the Societe Mathematique de France (SMF), arXiv:1012.3612v2.
(g) F. Loray, M.-H. Saito, Lagrangian Fibrations in Duality on Moduli Spaces of Rank 2 Logarithmic Connections Over the Projective Line. Int. Math. Res. Not. (IMRN),No. 4 (2015), 995–1043.

I will explain how a certain quasi-topological N = (0,2) gauged sigma model physically realizes the mathematical theory of “Twisted Chiral Differential Operators”. In turn, I will give a non-gauge theoretic interpretation of the geometric Langlands correspondence for any simply-connected, simple, complex Lie group. I will also explain how worldsheet twisted-instantons can trivialize the chiral algebra of the sigma model completely, whence we would be able to connect the vanishing of the Witten genus on string manifolds with positive Ricci curvature to the conditions for the existence of Hecke eigensheaves in the geometric Langlands correspondence at genus zero. If time permits, I will also explain the connections to knot homologies and quantum groups as suggested by the physics of the sigma model.

References:

1. https://arxiv.org/abs/1111.0691
2. https://arxiv.org/abs/0810.4964
3. https://arxiv.org/abs/hep-th/0512172

In this talk I will give a relative moduli space of connections on smooth projective curves whose generic fiber is a moduli space of regular singular connections and whose special fiber is a moduli space of unramified irregular singular connections. In the construction, we avoid using the parabolic structure but we use another parametrization of the local exponents. If we fix a diagonal matrix N with the distinct eigenvalues, the data of local exponents is given by a polynomial in N. Using this idea, we define a space of local exponents and we can construct over this base space a family of moduli spaces of connections. On the moduli space of generic unframified irregular singular connections, we construct, joint with M-H. Saito, a generalized isomonodromic deformation via patching local forms of Jimbo-Miwa-Ueno equations. On an unfolded family of moduli spaces of connections, I expect to construct an unfolded generalized isomonodromic deformation based on the theory by Hurtubise, Lambert and Rousseau, but my proof is not completed yet. I will explain fundamental solutions of an unfolded linear differential equation with an asymptotic nature given in the theory by Hurtubise, Lambert and Rousseau.

Background References:

1. J. Hurtubise, C. Lambert and C. Rousseau, Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincar´e rank k, Mosc. Math. J. 14 (2) (2014) 309–338.
2. J. Hurtubise and C. Rousseau, Moduli space for generic unfolded differential linear systems, Adv. Math. 307 (2017), 1268–1323.
3. M. Inaba and M.-H. Saito, Moduli of unramified irregular singular parabolic connections on a smooth projective curve, Kyoto J. Math 53 (2013), no. 2, 433–482.

This will be a mini-course on the Kobayashi-Hitchin correspondence between wild harmonic bundles and good filtered Higgs bundles on compact Riemann surfaces.

Background References:

(a) T. Mochizuki, Harmonic bundles and Toda lattices with opposite sign, arXiv:1301.1718 
(b) C. T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), 867–918. 
(c) C. T. Simpson, Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3(1990), 713–770.

N = 2 supersymmetric field theories in D = 4 provide a fruitful ground for the intersection between physics and mathematics. A particularly interesting subset, theories of “Class-S,” arise via compactification from D = 6. Many properties of these theories are encoded in the geometry of the compactification. In these lectures, I will try to explain the preceding sentences and give an overview of the classification of such theories.

Background References:

(a) Chacaltana, Distler, Trimm, Zhu, “Tinkertoys for the E8 Theory”, arXiv:1802.09626
(b) Chalcaltana, Distler, Trimm, Zhu, “Tinkertoys for the E7 Theory”, arXiv:1704.07890
(c) Chacaltana, Distler, Tachikawa, “Nilpotent orbits and codimension-two defects of 6d N=(2,0) theories”, arXiv:1203.2930
(d) Chacaltana, Distler, “Tinkertoys for Gaiotto Duality”, arXiv:1008.5203

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. 
(b) Edward Witten, “Chern-Simons gauge theory as a string theory”, Prog. Math. 133(1995) 637, arXiv: hep-th/9207094. 
(c) Hirosi Ooguri, Cumrun Vafa, “Knot invariants and topological strings”, Nucl. Phys.B577 (2000) 419, arXiv: hep-th/9912123.
(d) J. Labastida, Marcos Marino, Hirosi Ooguri, Cumrun Vafa, “Knots, links and branesat large N”, JHEP 0011 (2000) 007, arXiv: hep-th/0010102. 
(e) Marcos Marino, “Chern-Simons theory and topological strings”, Rev. Mod.Phys. 77 (2005) 675-720, arXiv: hep-th/0406005. 
(f) Sergei Gukov, Albert Schwarz, Cumrun Vafa, “Khovanov-Rozansky homology andtopological strings”, Lett.Math.Phys. 74 (2005) 53-74, arXiv: hep-th/0412243. 
(g) Sergei Gukov, Ingmar Saberi, “Lectures on knot homology and quantum curves”,arXiv: 1211.6075 [hep-th]. 
(h) Sergei Gukov, Piotr Su lkowski, “A-polynomial, B-model, and Quantization”, JHEP1202 (2012) 070, arXiv: 1108.0002 [hep-th]. 
(i) Hiroyuki Fuji, Sergei Gukov, Piotr Su lkowski, “Super-A-polynomial for knots andBPS states”, Nucl. Phys. B867 (2013) 506, arXiv: 1205.1515 [hep-th]. 
(j) Stavros Garoufalidis, Piotr Kucharski, Piotr Su lkowski, “Knots, BPS states, and algebraic curves”, Commun. Math. Phys. 346 (2016) 75-113, arXiv: 1504.06327 [hep-th]. 
(k) Piotr Kucharski, Markus Reineke, Marko Stosic, Piotr Su lkowski, “BPS states, knotsand quivers”, Phys. Rev. D 95 (2017) 121902(R), arXiv: 1707.02991 [hep-th]. 
(l) Piotr Kucharski, Markus Reineke, Marko Stosic, Piotr Su lkowski, “Knots-quiverscorrespondence”, arXiv: 1707.04017 [hep-th]. 
(m) Mi losz Panfil, Marko Stosic, Piotr Su lkowski, “Donaldson-Thomas invariants, torusknots, and lattice paths”, arXiv: 1802.04573 [hep-th]

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 
(b) R. Donagi, T. Pantev “Langlands duality for Hitchin systems”, Invent. Math. 189(2012), no. 3, 653–735. https://arxiv.org/abs/math/0604617. 
(c) R. Donagi, T. Pantev, C. Simpson “Direct Images in Non Abelian Hodge Theory”,https://arxiv.org/abs/1612.06388. 
(d) C. Pauly, A. Peón-Nieto “Very stable bundles and properness of the Hitchin map”,https://arxiv.org/abs/1710.10152. 
(e) S. Pal, C. Pauly “The wobbly divisors of the moduli space of rank-2 vector bundles”,https://arxiv.org/abs/1803.11315.

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
(b) Freed, “The Cobordism Hypothesis” arXiv:1210.5100 (for some perspective on the categorical approach to TQFT)

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 
(b) R. Donagi, T. Pantev “Langlands duality for Hitchin systems”, Invent. Math. 189(2012), no. 3, 653–735. https://arxiv.org/abs/math/0604617. 
(c) R. Donagi, T. Pantev, C. Simpson “Direct Images in Non Abelian Hodge Theory”,https://arxiv.org/abs/1612.06388. 
(d) C. Pauly, A. Peón-Nieto “Very stable bundles and properness of the Hitchin map”,https://arxiv.org/abs/1710.10152. 
(e) S. Pal, C. Pauly “The wobbly divisors of the moduli space of rank-2 vector bundles”,https://arxiv.org/abs/1803.11315.

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 = GL₂ 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