**Venue**: Madhava Lecture Hall

Monday, 16 July 2018

**M. Inaba, M-H Saito**

Lecture 1 by M-H Saito on “Moduli spaces of parabolic connections and parabolic bundles and Geometric Langlands”

Abstract:

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.

Lecture 2 by M. Inaba on “Moduli space of regular singular parabolic connections and isomonodromic deformation.”

Abstract:

In the joint work with Iwasaki and Saito, we introduced the moduli space of regular singular parabolic connections on smooth projective curves and proved the properness of the Riemann-Hilbert morphism which is a holomorphic map from the moduli space of regular singular parabolic connections to the moduli space of the representations of a fundamental group. As a corollary we get the geometric Painlevé property of the isomonodromic deformation on the moduli space of regular singular parabolic connections. In this talk I will give a quick introduction to the moduli space of regular singular parabolic connections on a smooth projective curve and its properties. After that I will explain the isomonodromic deformation on the moduli space of regular singular parabolic connections and its geometric Painlevé property. In the case of rank two connections on the projective line with regular singularities at 4 points, the isomonodromic deformation coincides with the Painlevé equation of type VI. If time permits, I will give an algebraic construction of the isomonodromy equation on the moduli space of regular singular parabolic connections.

*Background References:*

(a) 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.

(b) M. Inaba, Moduli of parabolic connections on curves and the Riemann-Hilbert correspondence, J. Algebraic. Geom. 22 (2013), no. 3, 407–480.

Lecture 3 by M. Inaba on “ Unfolding of the moduli space of unramified irregular singular connections.”

Abstract:

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:*

- 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.
- J. Hurtubise and C. Rousseau, Moduli space for generic unfolded differential linear systems, Adv. Math. 307 (2017), 1268–1323.
- 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.

**Ron Donagi**

- Curves and their Jacobians
- Picard varieties
- Local systems
- Class field theory
- Hecke operators, correspondences
- Geometric Langlands
- Other Langlands
- G-bundles
- Hitchin's system
- Non abelian Hodge theory
- Ramified GLC
- GLC and HMS
- Abelianization via Hitchin's system
- Duality for Hitchin systems
- The gerbe of Higgs bundles
- GLC as deformation of abelianized GLC
- GLC as hyper Kahler rotation of abelianized GLC
- Open non abelian Hodge theory
- Wobbly bundles
- Outline of a program
- Prospects

*References:*

[Ari04], D.Arinkin, Moduli of connections with a small parameter on a curve, arXiv:math/0409373.

[Ari10] D.Arinkin, Autoduality of compactified Jacobians for curves with plane singularities, arXiv:1001.3868.

[AD12] D.Arinkin, D.Gaitsgory, Singular support of coherent sheaves and the geometric Langlands conjecture, arXiv:1201.6343.

[BD04] A. Beilinson and V. Drinfeld, Chiral Algebras, AMS Colloquium Publications, vol. 51, 2004, ISBN-10: 0-8218-3528-9.

[Don93] R. Donagi, Decomposition of spectral covers. Ast ́erisque, 218:145–175, 1993.

[Don95] R. Donagi, Spectral covers, In Current topics in complex algebraic geometry (Berkeley, CA, 1992/93), MSRI Publ. 28, pp. 65–86, Cambridge Univ. Press, Cambridge, 1995.

[DP06] R.Donagi and T.Pantev: Langlands duality for Hitchin systems, Inv Math, arXiv:math/0604617

[DDP07] D.E. Diaconescu, R. Donagi, and T. Pantev, Intermediate Jacobians and ADE Hitchin systems. Math. Res. Lett. 14 (2007), no. 5, 745–756.

[DG02] R. Donagi and D. Gaitsgory, The gerbe of Higgs bundles. Transform. Groups, 7(2):109–153, 2002.

[Dr83] V. Drinfeld, Two-dimensional l-adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2). Amer. J. Math. 105 (1983), no. 1, 85–114.

[FGV02] E. Frenkel, D. Gaitsgory and K. Vilonen, On the geometric Langlands conjecture, J. Amer. Math. Soc. 15 (2002), no. 2, 367-417

[G02] D. Gaitsgory, Geometric Langlands correspondence for GLn, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 571-582.

[GW07] S.Gukov and E.Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, arXiv:hep-th/0612073.

[Hit87] N. Hitchin, Stable bundles and integrable systems, Duke Math. J., 54(1):91–114, 1987.

[HT03] T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math., 153(1):197–229, 2003.

[KW07] A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1(2007), no. 1, 1–236.

[Lau87] G. Laumon, Correspondance de Langlands g ́eom ́etrique pour les corps de fonctions, Duke Math. J. 54, (1987), 309-359.

[Sim94] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety - I, Publications Math ́ematiques de l’I.H.E.S., 79: 47–129, 1994; II, Publications Math ́ematiques de l’I.H.E.S., 80: 5–79, 1995.

**Tuesday, 17 July 2018**

**Meng-Chwan Tan**

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:*

**Takuro Mochizuki**

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.

**Wednesday, 18 July 2018**

**Sergey Cherkis**

Yang-Mills instantons and Yang-Mills-Higgs monopoles play prominent role in quantum gauge theory. These lectures will describe a systematic way of constructing instantons and monopoles and explore their moduli spaces. We shall further explore a generalization producing Yang-Mills instantons on curved manifolds (called gravitational instantons).

*Background References:*

(a) M. Atiyah, “Geometry of Yang-Mills Fields” Pisa Lectures, 1979 (or in Michael Atiyah: Collected works, vol. 5* 75-173).

(b) W. Nahm, “Self-dual monopoles and calorons,” Lecture Notes in Physics 201 (1984).

(c) M.K. Prasad, “Instantons and Monopoles in Yang-Mills Gauge Theories,” Physica 1D (1980) 167–191.

(d) C.D.A. Blair and S.A. Cherkis “Singular Monopoles from Cheshire Bows,” Nucl.Phys. B845 (2011) 140-164; e-Print: arXiv:1010.0740 [hep-th]

(e) S.A. Cherkis, “Instantons on Gravitons,” Commun.Math.Phys. 306 (2011) 449-483; e-Print: arXiv:1007.0044 [hep-th].

**Thurday, 19 July 2018**

**Jacques Distler**

Abstract:

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

**Friday, 20 July 2018**

**Lotte Hollands**

First I try to explain the relation between four-dimensional N=2 quantum field theories and (quantum) Hitchin integrable systems, following references [a], [b] and [c]. Then I introduce spectral networks, and motivate why they are relevant in physics (for instance, in understanding the BPS spectrum of N=2 theories) as well as in mathemat- ics (for instance, in generating Darboux coordinate systems on Hitchin moduli spaces), following references [b] and [d] The second talk will be about “*Spectral problems for the E6 Minahan-Nemeschansky the-ory*”. According to Nekrasov and Shatashvili, the Coulomb vacua of four-dimensional N=2 theories of “class S”, subjected to the Omega background in two of the four dimen- sions, correspond to the eigenstates of a quantisation of a Hitchin integrable system. The vacua may be found as the intersection between two Lagrangian branes in the Hitchin moduli space, one of which is the space of opers (or quantum Hamiltonians) and one is defined in terms of a system of Darboux coordinates on the corresponding moduli space of flat connections. I will introduce such a system of Darboux coordinates on the mod- uli space of SL(3) flat connections on the three-punctured sphere through a procedure called abelianization and describe the spectral problem characterising the corresponding quantum Hitchin system. This talk is based on work to appear with Andrew Neitzke.

*Background References:*

(a) Neitzke, Hitchin systems in *N* = 2 theories, arXiv:1412.7145.

(b) Gaiotto, Moore and Neitzke, Wall-crossing, Hitchin Systems, and the WKB Approximation, arXiv:0907.3987v2.

(c) Nekrasov and Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, arXiv:0908.4052v1.

(d) Hollands and Neitzke, Spectral networks and Fenchel-Nielsen coordinates, 1312.2979v2.

**Monday, 23 July 2018**

**Piotr Sulkowski**

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].

**P. Ramadevi**

We will discuss the construction of knot polynomials from Chern-Simons field theory. We will indicate our computational status and limitations towards tackling classification of knots. We will briefly discuss the developments within topological strings and intersecting brane model.

*Background References:*

(a) R.K. Kaul, T.R. Govindarajan, P. Ramadevi, ‘Chern-Simons Theory as a theory ofknots and links,” Nucl. Phys. B402 (1993)548-566

(b) H. Ooguri, C. Vafa, “Knot Invariants and Topological Strings,”Nucl.Phys.B577:419-438,2000

(c) E.Witten, “Five branes and knots,” arXiv:1101.3216 (hep-th)

**Sergei Gukov**

_{4}]”

Lecture 1 : This will be an introduction to the physics perspective on Geometric Langlands. Mini-course : The main goal of these lectures is to see explicitly, ”in action” how holomor- phic twists of 3d N = 2 theories and 2d N = (0, 2) theories enjoy non-trivial equivalence relations (dualities) that a low-dimensional topologist can recognize as Kirby moves, i.e. equivalence relations in constructions of 3-manifolds and 4-manifolds, respectively. This provides a bridge between physics, algebra, and topology, which we explore from various perspectives and in many different examples. In particular, we will study the half- index of the combined 2d-3d system, first introduced in a joint work with A.Gadde and P.Putrov in 2013. Following that line of work, we also review the first non-trivial duality in a non-abelian 2d N = (0, 2) gauge theory, the so-called triality of 2d N = (0, 2) SQCD, and discuss how it leads to a triality symmetry of the corresponding VOAs. We then discuss various ways of ”gluing” vertex operator algebras that correspond to different gluing operations in the world of smooth 4-manifolds and illustrate how topolog- ical invariants of 4-manifolds arise as chiral correlation functions in the resulting algebras VOA[M4]

*Background Reference:*

(a) Introductory Chapters of Kapustin-Witten (“Electric-Magnetic Duality And The Ge- ometric Langlands Program”) and Gukov-Witten I (“Gauge Theory, Ramification, And The Geometric Langlands Program”) for Lecture 1.

(b) A. Gadde, S. Gukov, P. Putrov,“Fivebranes and 4-manifolds”

(c) S. Gukov, P. Putrov, C. Vafa, “Fivebranes and 3-manifold homology”

(d) M. Dedushenko, S. Gukov, P. Putrov, “ Vertex algebras and 4-manifold invariants”

**Sujay Ashok**

We study half-BPS surface operators in four dimensional ${\mathcal N}=2$ SU($N$) gauge theories. We calculate the ramified instanton partition function using equivariant localization and extract the low-energy effective action on the four dimensional Coulomb branch. We also study surface operators as coupled 2d/4d quiver gauge theories with an SU$(N)$ flavour symmetry. In this description, the same surface operator can be described by different quivers that are related to each other by two dimensional Seiberg duality. We argue that these dual quivers correspond, on the localization side, to distinct integration contours that can be determined by the relative magnitudes and signs of the Fayet-Iliopoulos parameters of the two dimensional gauge nodes. We verify the proposal by mapping the solutions of the twisted chiral ring equations of the 2d/4d quivers onto individual residues of the localization integrand.

*Background References:*

- S. Gukov, Surface Operators, [arXiv:1412.7127]
- D. Gaiotto, S. Gukov, and N. Seiberg, Surface Defects and Resolvents, [arXiv:1307.2578]

**Tuesday, 24 July 2018**

**Tony Pantev**

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.

**Wednesday, 25 July 2018**

**Tudor Dimofte**

Lecture 1: Secondary products in SUSY QFT

Lecture 2: G actions in SUSY QM; or, the Fukaya category of point/G

Lecture 3: Line operators and geometry in 3d N=4 gauge theory

A mathematical treatment of TQFT, based on category theory, was initiated in the early 90’s. In more modern times the mathematics of TQFT has come to use advanced techniques from derived geometry and higher algebra (`a la Lurie). This series of talks is loosely aimed at explaining how some of these advanced techniques apply in familiar examples of supersymmetric field theory and its topological twists. For physics, this will lead to some surprising new structure, as well as some useful organizing principles. The first and second lectures are based on work with C. Beem, D. Ben-Zvi, M. Bullimore, and A. Neitzke. The first lecture will re-examine operator algebras in topological twists of supersymmetric field theories. In d dimension, the algebras naturally come equipped with a Lie bracket of degree 1-d, which can be realized very concretely via topological descent. We’ll look at examples of this bracket, in 2d, 3d, and 4d. We’ll also use topological descent to give a new interpretation of the statement that turning on on Omega background is a form of quantization. The second lecture focuses on another sort of homological/descent operation, this time in the context of SUSY quantum mechanics with G symmetry. We’ll find that Hilbert spaces in (de Rham) SQM come equipped with a natural homological action of G, i.e. an action of the exterior algebra H∗(G). This action controls the process of gauging the G symmetry. A nice physical way to understand the H∗(G) action and gauging/ungauging comes from considering SUSY boundary conditions for 2d G gauge theory; this will lead us to a definition of the “Fukaya category of point/G.” The mathematical structures involved come from work of Goresky, Kottwitz, and MacPherson. Given time, we’ll discuss some higher-dimensional examples, and the physics of Koszul duality. The third lecture applies some of the ideas from the first two in the specific context of 3d N=4 gauge theories with linear matter. It is joint work with N. Garner, M. Geracie, and J. Hilburn. Our goal will be to define the category of line operators in the A and B twists of these theories, and to understand – in a concrete and computational way – the algebras of local operators bound to a line. These algebras get quantized in an Omega background; and they act on modules defined by boundary conditions. In the special case of an A twist and the trivial (identity) line operator, we will recover the Braverman-FinkelbergNakajima construction of the Coulomb-branch chiral ring.

*Background References:*

For Lecture 1:

(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)

For Lecture 2:

(a) Witten, “Supersymmetry and Morse theory,” J.Diff.Geom. 17 (1982) no.4, 661-692

(b) Goresky, Kottwitz, and MacPherson, “Equivariant cohomology, Koszul duality, and the localization theorem” Invent. Math. 131 (1997) 24-83

(c) Guillemin and Sternberg “Supersymmetry and equivariant de Rham theory”

(d) Gaiotto, Moore, and Witten, “An introduction to the web-based formalism” arXiv:1506.04086 (more advanced and beyond what we need here... but actually a really nice illustration of derived concepts!)

For Lecture 3:

(a)Braverman and Finkelberg, Lecture notes on Coulomb branches (available at http://www.salafrancesco.altervista.org/uploads/8/3/1/9/83195312/braverman_finkelberg__coulomb_branches_of_3-dimensional_gauge__theories_and_related_ structures.pdf

(b) Section 9-10 of Kapustin and Witten, “Electric-magnetic duality and the Geometric Langlands program” hep-th/0604151

(c) Assel and Gomis, “Mirror symmetry and loop operators” arXiv:1506.01718

(d) Bullimore, Dimofte, Gaiotto, Hilburn, and Kim, “Vortices and Vermas” arXiv:1609.04406 (a particular illustration of the framework I’ll discuss)

**Thurday, 26 July 2018**

**Satoshi Nawata**

I will talk about physics approach to understand representation theory of double affine Hecke algebra (DAHA). DAHA can be realized as an algebra of line operators in 4d N=2* theory and therefore it appears as quantization of coordinate ring of Hitchin moduli space over once-punctured torus. Using 2d A-model on the Hitchin moduli space, I will explain relationship between representation category of DAHA and Fukaya category of the Hitchin moduli space.

*Background References:*

(a) Gukov Witten, Branes and Quantization (0809.0305)

(b) Cherednik, Double Affine Hecke Algebra, London Mathematical Society

**Friday, 27 July 2018**

**Du Pei**

*B, A, A*)-branes

Picking a real form G r of a complex Lie group G defines a “(*B, A, A*)-brane” inside the moduli space of G-Higgs bundles. Under mirror symmetry, this (*B, A, A*)-brane will be mapped to a hyperholomorphic sheaf – a (*B, B, B*)-brane – over the moduli space of G^{∨} -Higgs bundles, where G^{∨} is the Langlands dual group of *G*. In this talk, I will discuss how to construct these hyperholomorphic sheaves, and show how these proposals can be tested by computing equivariant indices. In particular, I will give computational evidence to Nigel Hitchin’s proposal for the case of G = GL_{2} and G_{r} = *U* (1, 1). This talk is based on joint work with Tamas Hausel and Anton Mellit.

*Background References:*

(a) D.Pei, T. Hausel and A. Mellit, “Mirror symmetry with branes by equivariant Ver-linde formulae”, arXiv:1712.04408