Mini courses
Summary of the mini courses

Schedule

Aliakbar Daemi (Washington University)

Talk I
Title: moduli of flat connections on Riemann surfaces
Abstract: I will start with a brief review of basic notions in gauge theory: connections on principal bundles, their curvature, and the gauge group action on the space of connections.
I will then explain how the moduli space of all flat connections on a Riemann surface modulo the action of the gauge group determines a symplectic manifold. I will focus on some of the key features of thismoduli space that are relevant for the following talks.
References: [AB]
 
Talk II
Title: 3-manifold Lagrangians and symplectic instanton homology
Abstract: In the second lecture, we will shift our focus to 3-manifolds. In particular, I will explain how moduli of flat connections on a 3-manifold Y (with boundary) determine a Lagrangian in the symplectic manifold  associated to the boundary of  Y. Next, I will discuss how Lagrangian Floer homology of such Lagrangians can be used to define a 3-manifold invariant, called symplectic instanton homology.
References: [H], [WW] and [DFL]
 
Talk III
Title: An introduction to instanton Floer homology
Abstract: Instanton Floer homology is a topological invariant of 3-manifolds, which is obtained by applying methods of Morse homology to the Chern—Simons functional. This invariant, along with its variations for knots and links, has recently found many interesting applications in low dimensional topology. In this talk, I will review the definition of instanton homology and various algebraic structures on this invariant, which will be useful for the final talk.
References: [F], [D] and [DFL]
 
Talk IV
Title: Mixed equation and the Atiyah—Floer conjecture
Abstract: Atiyah—Floer conjecture predicts a connection between gauge theory and symplectic topology. More specifically, it proposes that instanton Floer homology and symplectic instanton homology are isomorphic invariants of 3-manifolds. In this talk, I will review the proof of the Atiyah—Floer conjecture for admissible bundles. In particular, this shows that framed instanton homology (introduced by Floer, Kronheimer and Mrowka) and its symplectic variant (defined by Wehrheim and Woodward) are isomorphic to each other. The key geometric ingredient in the proof is the mixed equation, relating ASD equation for connections and holomorphic curve equation.
References: [A], [DFL]
 
References:
[AB] M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615
[A] Michael Atiyah, New invariants of 3- and 4-dimensional manifolds, The mathematical heritage of Hermann Weyl (Durham, NC, 1987), 1988, pp. 285–299.
[D] S. K. Donaldson, Floer homology groups in Yang-Mills theory, Cambridge Tracts in Mathematics, vol. 147, Cambridge University Press, Cambridge, 2002.
[DFL] Aliakbar Daemi, Kenji Fukaya, and Maksim Lipyanskiy, Lagrangians, SO(3)-instantons and the Atiyah-Floer conjecture, arXiv, 2021
[F] Andreas Floer, An instanton-invariant for 3-manifolds, Comm. Math. Phys. 118 (1988), no. 2, 215–240.
[H] Christopher M. Herald, Legendrian cobordism and Chern-Simons theory on 3-manifolds with boundary, Comm. Anal. Geom. 2 (1994), no. 3, 337–413
[WW] Katrin Wehrheim and Chris Woodward, Floer field theory for coprime rank and degree, arXiv e-prints (2016Jan), arXiv:1601.04924

 

Siqi He (Chinese Academy of Sciences)

Title: Z/2 Harmonic 1-Forms and Related Problems in Geometry

Abstract: This mini-course will focus on Z/2 harmonic 1-forms and their connections to gauge theory, topology, and compactification problems. The lectures will be divided into four parts:

Gauge Theory with SL(2,C) structure group: Gauge-theoretic equations with SL(2,C) structure group, including flat connection equations and the Kapustin–Witten equations. We will discuss their basic properties and related geometric and topological problems.

Compactness of Flat SL(2,C) Connections: Taubes’ compactness theorem and the role of Z/2 harmonic 1-forms in describing the ideal boundary. Basic properties of Z/2 harmonic 1-forms will also be introduced.

Deformation of Z/2 Harmonic 1-Forms: Donaldson’s work on the deformation of Z/2 harmonic 1-forms, along with possible geometric applications of these deformations.

Relations to Low-Dimensional Topology: Connections between Z/2 harmonic 1-forms and classical objects in low-dimensional topology, including Thurston’s compactification of Teichmüller space, measured foliations, and the Morgan–Shalen compactification.

The lectures are intended for PhD students and early-career researchers with a background in differential geometry or gauge theory.

References:

S. K. Donaldson, Deformations of multivalued harmonic functions, arXiv:1912.08274.

C. H. Taubes, PSL(2,C) connections on 3-manifolds with L² bounds on curvature, arXiv:1205.0514.

C. H. Taubes, Compactness theorems for SL(2,C) generalizations of the 4-dimensional anti-self-dual equations, arXiv:1307.6447.

H. Siqi, R. Wentworth, B. Zhang, Z/2 harmonic 1-forms, R-trees, and the Morgan–Shalen compactification, arXiv:2409.04956.

 

Johanna Knapp (University of Melbourne)

Title: The Physical Mathematics of Gauged Linear Sigma Models

Abstract: Gauged linear sigma models (GLSMs), first introduced by Witten in 1993, are supersymmetric gauge theories in two dimensions. They provide a powerful tool to study properties of extra dimensions in string theory and the mathematical structures behind them. The aim of these lectures is to show how a physics analysis of GLSMs (vacuum configurations, low-energy effective theories, D-branes, path integrals etc.) leads to advanced mathematics (GIT quotients, categorical equivalences, enumerative invariants etc.). The main focus will be on GLSMs that are related to Calabi-Yau compactifications of string theory.

A rough outline of the lectures is as follows (we may not cover all of it):

1.    GLSMs: physics definition and phases

•    Field content and symmetries
•    Phases of GLSMs
•    Higgs vs Coulomb branches
•    Examples

2.    (B-type) D-branes in GLSMs

•    B-branes in GLSMs
•    D-brane transport and categorical equivalences (physics perspective)

3.    GLSM partition functions and what they compute

•    (Hemi-)sphere partition function
•    B-brane central charges and enumerative invariants

Literature: There aren't really any good recent reviews and premliminary reading is not necessary, but many of the GLSM papers of the past few years contain useful summaries. Some good older references are:

•    Witten "Phases of N=2 theories in two-dimensions" hep-th/9301042: Witten's original paper on GLSMs
•    Hori et el "Mirror symmetry" (available at http://www.claymath.org/library/monographs/cmim01.pdf): The big mirror symmetry book, Chapter 15 is on GLSMs; other chapters may be useful too
•    Herbst,Hori,Page "Phases Of N=2 Theories In 1+1 Dimensions With Boundary" arXiv:0803.2045[hep-th]: 265 page-long exhaustive discussion on B-branes in abelian GLSMs, Chapters 4,5,7,10 are the most relevant
•    Jockers et al "Two-Sphere Partition Functions and Gromov-Witten Invariants" arXiv:1208.6244[hep-th]: sphere partition function and enumerative invariants

Notes:

 

Yixuan Li (Australian National University)

Title: Mirror Symmetry of Type A Affine Grassmannian Slices

Abstract: This mini-course is about a mirror symmetry result central to Mina Aganagic’s ICM 2022 talk [1] on two categorifications of Jones polynomials. Recall that the Jones polynomial of a knot can be calculated via the fundamental representation V of the quantum group U_q(sl_2) roughly in the following way: First present the knot as the closure of a braid with n strands by Alexander’s theorem. Then associated to the n strands, we have the weight spaces of the tensor product of n copies of V. Associated to the braid, we have a product of R-matrices acting on each weight space. Jones polynomial is related to the trace of this product of R-matrices on the weight spaces of this tensor product. Thus to category this picture, we need to upgrade the weight spaces to categories and the R matrices to certain braid group actions on these categories. Taking trace would be interpreted as taking a homomorphism between certain objects in the category.

Via the geometric Satake equivalence[2][3], weight spaces of tensor products of fundamental representations of gl(m) are related to the geometry of certain slices in the affine grassmannian of Gl(m). These slices are conical symplectic singularities. There are two ways to smoothen this singularity: One can consider the semi-universal symplectic deformation or the symplectic resolution. Hence the weight spaces will be categorified into certain Fukaya categories of deformed affine grassmannian slices or the category of coherent sheaves on the symplectic resolution of these slices. The braid group action will in fact be provided by the monodromy action of the semi-universal symplectic deformation.

Mirror symmetry is a relation between Fukaya category of a symplectic manifold X and the derived category of coherent sheaves on a complex manifold X^. In fact these two categorifications will be related to each other by a conjectural homological mirror symmetry. In [4] we proved a partial result saying that the coherent side embeds into the symplectic side. In fact, as another evidence, we can show manually that both the quantum connection on X^ and a Gauss-Manin connection on X can be identified with certain Knizhnik-Zamolodchikov connections, following the work [5] of Danilenko.

These talks are prepared for PhD students and early career researchers with a background in representation theory or geometry/topology. Time permitting, I will mention the connection of these slices with certain monopole moduli spaces as predicted by [6]. What we need in order to prove these predictions is some control over Kapustin-Witten equations.

Talk 1: Examples of Homological Mirror Symmetry
 
In this talk we briefly discuss the framework of homological mirror symmetry: an equivalence of categories between (B-side) coherent sheaves on one variety X and (A-side) Fukaya-Seidel category of a symplectic manifold Y equipped with a superpotential W. We will also talk about how to add in a torus action on the B-side, which is mirror to considering a covering space on the A-side. We will work through the examples where the B-sides are coherent sheaves on C*, A^1, P^1 and T*P^1.
 
References:
[1] Auroux, A beginner's introduction to Fukaya categories
[2] Strominger-Yau-Zaslow, Mirror Symmetry is T-duality
[3] Kontsevich, Homological Algebra of Mirror Symmetry
[4] Fang-Liu-Treumann-Zaslow, T-Duality and Homological Mirror Symmetry of Toric Varieties
 
Talk 2: Examples of Homological Mirror Symmetry Equipped with Central Charge Functions
 
In this talk we will discuss numerical aspects of mirror symmetry, continuing on the examples in talk 1. To each object of a category in question, we will associate a central charge function which only depends on the K-theory class of the object. They further depend on complexified Kahler parameters on the B-side variety X and complex structure parameters on the A-side symplectic manifold Y, which will be exchanged under mirror symmetry. Mirror symmetry will also equate central charge functions of an object and its mirror. Moreover, these central charge functions will be sections of the same flat connection over the complexfied Kahler parameters (quantum connection: the fibers are K theory of X) and the complex structure parameters (Gauss-Manin connection: the fibers are middle-dimensional homology cycles of Y, on which W decays at infinity). We will mainly focus on defining and calculating the central charge functions, leaving the definition of the quantum connection to talk 3.
 
References:
[1] Givental, Homological Geometry and Mirror Symmetry
[2] Okounkov, 33 Lectures on Quasimaps and Elliptic Stable Envelopes
[3] Okounkov, Lectures on K-theoretic Computations in Enumerative Geometry
[3] Dedushenko-Nekrasov, Interfaces and Quantum Algebras, II: Cigar Partition Function
 
Talk 3: Quantum Connections on T*G/B and Springer Theory
 
In this talk we will focus on the example of cotangent bundle of the flag variety of a reductive group G. When G is SL(n), this is an example of a Coulomb branch of a 3D N=4 Type A quiver gauge theory, which is the focus of the last talk. We will discuss how the problem can be reduced to the case of SL(2), where it is essentially solved by applying the Aspinwall-Morrison multiple cover formula. T*G/B is also symplectic resolution (called the Springer resolution) of a conical symplectic singularity, which is the nilpotent cone (space of nilpotent elements of the Lie algebra of G). The nilpotent cone is an affine variety, so all maps from the genus zero curve P^1 will have image contained in a fiber of the Springer resolution. The quantum connection can be neatly expressed from formulas in Springer theory.
 
References:
[1] Braverman-Maulik-Okounkov,  Quantum Cohomology of the Springer Resolution
[2] Chriss-Ginzburg, Representation Theory and Complex Geometry
[3] Givental, A Tutorial on Quantum Cohomology
 
Talk 4: Mirror Symmetry of Type A Affine Grassmannian Slices, KZ Connections and Braid Group Actions
 
In this talk we will discuss the example of type A affine grassmannian slices, whose quantum connections can be identified with certain trigonometric Knizhnik-Zamolodchikov connections. The Gauss-Manin connections on the mirror side and their solutions by central charge functions recover integral formulas of solutions of KZ connections. In certain cases, it recovers Gauss’s integral formula of hypergeometric functions. The complex structure moduli and complexified Kahler moduli in these cases are certain hyperplane complements of a complex torus. The monodromy of these connections are related to certain affine braid groups, which act on the categories as well.
 
References:
[1] Etingof-Frenkel-Kirillov, Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations
[2] Cherednik, Affine Extensions of Knizhnik–Zamolodchikov Equations and Lusztig’s Isomorphisms
[3] Danilenko, Quantum Differential Equation for Slices of the Affine Grassmannian
[4] Aganagic, Knot Categorification from Mirror Symmetry, Parts I&II
[5] Aganagic-Danilenko-Li-Shende-Zhou, Quiver Hecke Algebras from Floer Homology in Coulomb Branches
[6] Bezrukavnikov-Riche, Affine Braid Group Actions on Derived Categories of Springer Resolutions

 

Emmanuel Scheidegger (BICMR, Peking University)

Title: TOPOLOGICAL STRINGS ON CALABI–YAU THREEFOLDS

Abstract: The goal of this minicourse is to give an introduction to topological string theory on Calabi–Yau threefolds. The lectures will be divided into four parts:

1. Motivation and expectations from physics: We introduce the ideas of Bershadsky, Cecotti, Ooguri, and Vafa on the topological string on Calabi–Yau threefolds [1]. The fundamental quantity to be studied is the partition function. We review its properties with a focus on the open problems in mathematics.

2. Basics of Calabi–Yau threefolds: We review invariants and constructions of Calabi–Yau threefolds [2], [3], in particular in toric varieties. This is closely related to the minicourse by Johanna Knapp. If time permits we also discuss singular and nongeometric Calabi–Yau spaces.

3. Moduli space of Calabi–Yau threefolds: We explain the complex structure moduli space and stringy Kähler moduli space of Calabi–Yau threefolds [2], [3] and their relation by mirror symmetry. We pass to the local description in terms of period domains and discuss period integrals and Picard–Fuchs equations [4]. The entire structure is governed what is known as special Kähler geometry [5].

4. Propagators and holomorphic anomaly equation: We review the holomorpic anomaly equation following [1], and their solution by introducing the socalled propagators. The latter form a finitely generated ring, called the BCOV ring, [6]. We discuss the properties of the partition function at the boundary of the moduli space, and how to determine it following e.g. [7].

References

[1] M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, “Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes,” Commun. Math. Phys. 165 (1994) 311–428, arXiv:hep-th/9309140.

[2] M. Gross, D. Huybrechts, and D. Joyce, Calabi-Yau manifolds and related geometries. Universitext. Springer-Verlag, Berlin, 2003.

[3] D. A. Cox and S. Katz, Mirror symmetry and algebraic geometry, vol. 68 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1999. https://doi.org/10.1090/surv/068.

[4] S. Hosono, A. Klemm, S. Theisen, and S.-T. Yau, “Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces,” Nuclear Phys. B 433 no. 3, (1995) 501–552, arXiv:hep-th/9406055 [hep-th].

[5] D. S. Freed, “Special Kähler manifolds,” Comm. Math. Phys. 203 no. 1, (1999) 31–52, arXiv:hep-th/9712042.

[6] S. Hosono, “BCOV ring and holomorphic anomaly equation,” in New developments in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008), vol. 59 of Adv. Stud. Pure Math., pp. 79–110. Math. Soc. Japan, Tokyo, 2010.

arXiv:0810.4795 [math.AG].

[7] M. Alim and E. Scheidegger, “Topological strings on elliptic fibrations,” Commun. Number Theory Phys. 8 no. 4, (2014) 729–800, arXiv:1205.1784 [hep-th].