Hiraku Nakajima：Quiver gauge theories with symmetrizers

Abstract: We generalize the mathematical definition of Coulomb branches of 3-dimensional N=4 SUSY quiver gauge theories in arXiv:1503.03676, arXiv:1601.03686, arXiv:1604.03625 to the cases with symmetrizers. We obtain generalized affine Grassmannian slices of type BCFG as examples of the construction, and their deformation quantizations via truncated shifted Yangians. Finally, we study modules over these quantizations and relate them to the lower triangular part of the quantized enveloping algebra of type ADE. (Based on the joint work with Alex Weekes, arXiv:1907.06522.)

Richard Thomas: Square root Euler classes and counting sheaves on Calabi-Yau 4-folds

Abstract: I will explain a nice characteristic class of SO(2n,C) bundles in both Chow cohomology and K-theory, and how to localise it to the zeros of an isotropic section. This builds on work of Edidin-Graham, Polishchuk-Vaintrob, Anderson and many others.This can be used to construct an algebraic virtual cycle (and virtual structure sheaf) on moduli spaces of stable sheaves on Calabi-Yau 4-folds.It recovers the real derived differential geometry virtual cycle of Borisov-Joyce but has nicer properties, like a torus localisation formula. Joint work with Jeongseok Oh (KIAS).Seminar slides:  

Kentaro Hori: D-brane central charges

Abstract: I will describe what are known as D-brane central charges in various contexts including string compactifications, tt^* geometry, hemisphere partition function, Gromov-Witten theory, and old matrix models. I will then discuss their relationships.

Shuai Guo：The Landau-Ginzburg/Calabi-Yau correspondence for the quintic threefold

Abstract: In this talk, we will first introduce the physical and mathematical versions of the Landau-Ginzburg/Calabi-Yau correspondence conjecture for the Calabi-Yau threefolds. Then we will explain our approach to prove this conjecture for the most simple Calabi-Yau threefold - the quintic threefold. This is a work in progress joint with Felix Janda and Yongbin Ruan. 

Huai-Liang Chang: BCOV Feynman structure in A side

Abstract:  Gromov Witten invariants Fg encodes the numbers of genus g curves in Calabi Yau threefolds. They play a role in enumerative geometry and are not easy to be determined. In 1993 Bershadsky, Cecotti, Ooguri, Vafa solved holomorphic anomaly equations for Fg's in B side and their solution exhibited a hidden ``Feynman structure” governing all Fg’s at once. The method was via path integral while its counterpart in mathematic has been missing for decades. In 2018, a large N topological string theory is developed in math and provides the wanted ``Feynman structure”. New features are (i) dynamically quantizing the Kaehler moduli parameterin Witten’s GLSM, and (ii) enhancing the Calabi Yau target to a large N bulk with Calabi Yau boundary. Both are achieved within a Landau Ginzburg type construction (P fields and cosections in math terms). This theory, called “N-Mixed Spin P field” in math, is thus responsible for BCOV theory in A side. In it the A model theory of the large N bulk encodes the B model propagators of the boundary’s mirror. This is a new phenomenon intertwining bulk-boundary correspondence, mirror symmetry, and Gauge-String(B-A) duality.In this talk we will see genuine ideas behind these features, along a new angle to realize NMSP from people familiar with Givental theory.  

Zhengyu Zong: All genus open-closed Crepant Transformation Conjecture for toric Calabi-Yau 3-orbifolds

Abstract: The Crepant Transformation Conjecture, proposed by Ruan, asserts certain equivalence between Gromov-Witten theory of two manifolds/orbifolds which are related by a crepant transformation. In general, the higher genus Crepant Transformation Conjecture is quite hard to study. Even the formulation of the Crepant Transformation Conjecture in the higher genus case is subtle. In this talk, I will explain the proof of the all genus Crepant Transformation Conjecture for general toric Calabi-Yau 3-orbifolds. We will consider the higher genus Gromov-Witten theory of toric Calabi-Yau 3-orbifolds in both open-string sector and closed-string sector. This talkis based on an ongoing project joint with Bohan Fang, Chiu-Chu Melissa Liu, and Song Yu.

Mauricio Romo：Hemisphere partition function, Landau-Ginzburg orbifolds and FJRW invariants

Abstract: We consider LG orbifolds and the central charges of their B-branes (equivariant matrix factorizations). We will focus on the hemisphere partition function on the gauged linear  sigma model (GLSM) extension of certain LG orbifolds (even though our proposal does not require a GLSM embedding), and how this provides information about their Gamma class, I/J-function and some predictions about FJRW invariants. 

Hans Jockers: Wilson Loop Algebras and Quantum K-Theory

Abstract: In this talk we review certain aspects of Wilson line operators in 3d N=2 supersymmetric gauge theories with a Higgs branch that is geometrically described by complex Grassmannians. We discuss the relationship between Wilson loop algebras of the gauge theory and the quantum K-theoretic ring of Schubert structure sheaves of the complex Grassmannians.

Du Pei：On Quantization of Coulomb Branches

Abstract: The study of Coulomb branches of supersymmetric quantum field theories have in recent years led to many interesting insights into geometry and representation theory. In this talk, I will discuss the A-models with Coulomb branch targets, and show how to use them to better understand the representation theory of the double affine Hecke algebra.

Hiroshi Iritani：Quantum cohomology and birational transformation

Abstract: A famous conjecture of Yongbin Ruan says that quantum cohomology of birational varieties becomes isomorphic after analytic continuation when the birational transformation preserves the canonical class (the so-called crepant transformation). When the transformation is not crepant, the quantum cohomology becomes non-isomorphic, but it is conjectured that one side is a direct summand of the other. In this talk, I will explain a conjecture that a semiorthogonal decomposition of topological K-groups (or derived categories) should induce a relationship between quantum cohomology. The relationship between quantum cohomology can be described in terms of solutions to a Riemann-Hilbert problem.Seminar note：https://www.math.kyoto-u.ac.jp/~iritani/talk_QC_birat.pdf

Qile Chen: The logarithmic gauged linear sigma model

Abstract:  We introduce the notion of log R-maps generalizing stable maps with p-fields, and develop a proper moduli stack of stable log R-maps in the case of a hybrid gauged linear sigma model. This moduli stack carries two virtual fundamental classes -- the canonical virtual cycle and the reduced virtual cycle. The main results are two comparison theorems:  (1) We identify the reduced virtual cycle with the Kiem-Li cosection localized virtual cycle which was shown to recover Gromov-Witten theory of certain critical locus.  (2) We relate the reduced virtual cycle to the canonical virtual cycle which can have larger symmetry in many interesting examples.  This is part of a project aiming at the foundation of a new technique for computing higher genus Gromov--Witten invariants of complete intersections. The talk consists of joint work with Felix Janda, Yongbin Ruan, and Adrien Sauvaget.

Bumsig Kim: Virtual Factorizations

Abstract: In this talk, based on joint work with David Favero, we explain how to define an A-side cohomological field theory for a given gauged linear sigma model (GLSM). For this, we follow the approach by Polishchuk -- Vaintrob, i.e., we construct a virtual factorization on a suitable smooth stack U containing the moduli space LGQ of Landau-Ginzburg stable (quasi)maps. The latter moduli space LGQ was introduced by Fan, Jarvis and Ruan. The state space taken is the hypercohomology of Hodge complex twisted by the superpotential of GLSM in the inertia stack of the associated DM stack of GLSM. We will use the notion of Atiyah class of global matrix factorizations from a joint work with A. Polishchuk and a simplified construction of the ambient stack U from a previous joint work with I. Ciocan-Fontanine, D. Favero, J. Guere, M. Shoemaker.

Hai Dong: Grassmanian via Dual Grassmanian

Abstract: Grassmannian Gr(r,n) is geometrically isomorphic to dual Grassmannian Gr(n-r,n). However, they have very different combinatorial structures, originated from their GIT presentations. It is a mysterious and yet highly nontrivial problem to match their combinatorial structures directly. A famous example is level-rank duality from physics. In this talk, I will examine the relation of I-functions of Grassmannian and its dual in both quantum cohomology and quantum K-theory cases. Furthermore, the twisted I-functions of vector bundles of Grassmannians in quantum cohomology and I-functions with the level structure in quantum K-theory, which is introduced by Yongbin Ruan and Ming Zhang, will also be examined.

Albrecht Klemm: Topological strings on genus one fibered Calabi-Yau 3-folds and string dualities

Abstract: We calculate the generating functions of BPS indices using their modular propertiesin Type II and M-theory compactifications on compact genus one fibered CY 3-foldswith singular fibers and additional rational sections or just N-sections, in order tostudy string dualities in four and five dimensions as well as rigid limits in which gravitydecouples. The generating functions are Jacobi-forms of Γ_1(N) with the complexifiedfiber volume as modular parameter. The string coupling λ, or the \epsilon_{±} parameters inthe rigid limit, as well as the masses of charged hypermultiplets and non-Abelian gaugebosons are elliptic parameters. To understand this structure, we show that specificauto-equivalences act on the category of topological B-branes on these geometries andgenerate an action of Γ1(N) on the stringy K¨ahler moduli space. We argue that theseactions can always be expressed in terms of the generic Seidel-Thomas twist with respectto the 6-brane together with shifts of the B-field and are thus monodromies. This impliesthe elliptic transformation law that is satisfied by the generating functions. We useHiggs transitions in F-theory to extend the ansatz for the modular bootstrap to genusone fibrations with N-sections and boundary conditions fix the all genus generatingfunctions for small base degrees completely. This allows us to study in depth a widerange of new, non-perturbative theories, which are Type II theory duals to the CHLZN orbifolds of the heterotic string on K3 × T2. In particular, we compare the BPSdegeneracies in the large base limit to the perturbative heterotic one-loop amplitudewith $R^2_+F_+^{2g−2}  insertions for many new Type II geometries. In the rigid limit we canrefine the ansatz and obtain the elliptic genus of superconformal theories in 5d.