Online Seminar

10212020
Hiraku Nakajima：Quiver gauge theories with symmetrizers
Abstract: We generalize the mathematical definition of Coulomb branches of 3dimensional 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.)

10092020
Richard Thomas: Square root Euler classes and counting sheaves on CalabiYau 4folds
Abstract: I will explain a nice characteristic class of SO(2n,C) bundles in both Chow cohomology and Ktheory, and how to localise it to the zeros of an isotropic section. This builds on work of EdidinGraham, PolishchukVaintrob, 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 CalabiYau 4folds.It recovers the real derived differential geometry virtual cycle of BorisovJoyce but has nicer properties, like a torus localisation formula. Joint work with Jeongseok Oh (KIAS).Seminar slides:

07282020
Kentaro Hori: Dbrane central charges
Abstract: I will describe what are known as Dbrane central charges in various contexts including string compactifications, tt^* geometry, hemisphere partition function, GromovWitten theory, and old matrix models. I will then discuss their relationships.

07212020
Shuai Guo：The LandauGinzburg/CalabiYau correspondence for the quintic threefold
Abstract: In this talk, we will first introduce the physical and mathematical versions of the LandauGinzburg/CalabiYau correspondence conjecture for the CalabiYau threefolds. Then we will explain our approach to prove this conjecture for the most simple CalabiYau threefold  the quintic threefold. This is a work in progress joint with Felix Janda and Yongbin Ruan.

07142020
HuaiLiang 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 “NMixed 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 bulkboundary correspondence, mirror symmetry, and GaugeString(BA) 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.

07072020
Abstract: The Crepant Transformation Conjecture, proposed by Ruan, asserts certain equivalence between GromovWitten 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 CalabiYau 3orbifolds. We will consider the higher genus GromovWitten theory of toric CalabiYau 3orbifolds in both openstring sector and closedstring sector. This talkis based on an ongoing project joint with Bohan Fang, ChiuChu Melissa Liu, and Song Yu.

06302020
Mauricio Romo：Hemisphere partition function, LandauGinzburg orbifolds and FJRW invariants
Abstract: We consider LG orbifolds and the central charges of their Bbranes (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/Jfunction and some predictions about FJRW invariants.

06232020
Hans Jockers: Wilson Loop Algebras and Quantum KTheory
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 Ktheoretic ring of Schubert structure sheaves of the complex Grassmannians.

06162020
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 Amodels with Coulomb branch targets, and show how to use them to better understand the representation theory of the double affine Hecke algebra.

06092020
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 socalled crepant transformation). When the transformation is not crepant, the quantum cohomology becomes nonisomorphic, 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 Kgroups (or derived categories) should induce a relationship between quantum cohomology. The relationship between quantum cohomology can be described in terms of solutions to a RiemannHilbert problem.Seminar note：https://www.math.kyotou.ac.jp/~iritani/talk_QC_birat.pdf

06022020
Qile Chen: The logarithmic gauged linear sigma model
Abstract: We introduce the notion of log Rmaps generalizing stable maps with pfields, and develop a proper moduli stack of stable log Rmaps 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 KiemLi cosection localized virtual cycle which was shown to recover GromovWitten 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 GromovWitten invariants of complete intersections. The talk consists of joint work with Felix Janda, Yongbin Ruan, and Adrien Sauvaget.

05262020
Bumsig Kim: Virtual Factorizations
Abstract: In this talk, based on joint work with David Favero, we explain how to define an Aside 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 LandauGinzburg 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. CiocanFontanine, D. Favero, J. Guere, M. Shoemaker.

05192020
Hai Dong: Grassmanian via Dual Grassmanian
Abstract: Grassmannian Gr(r,n) is geometrically isomorphic to dual Grassmannian Gr(nr,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 levelrank duality from physics. In this talk, I will examine the relation of Ifunctions of Grassmannian and its dual in both quantum cohomology and quantum Ktheory cases. Furthermore, the twisted Ifunctions of vector bundles of Grassmannians in quantum cohomology and Ifunctions with the level structure in quantum Ktheory, which is introduced by Yongbin Ruan and Ming Zhang, will also be examined.

05122020
Albrecht Klemm: Topological strings on genus one fibered CalabiYau 3folds and string dualities
Abstract: We calculate the generating functions of BPS indices using their modular propertiesin Type II and Mtheory compactifications on compact genus one fibered CY 3foldswith singular fibers and additional rational sections or just Nsections, in order tostudy string dualities in four and five dimensions as well as rigid limits in which gravitydecouples. The generating functions are Jacobiforms 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 nonAbelian gaugebosons are elliptic parameters. To understand this structure, we show that specificautoequivalences act on the category of topological Bbranes 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 SeidelThomas twist with respectto the 6brane together with shifts of the Bfield and are thus monodromies. This impliesthe elliptic transformation law that is satisfied by the generating functions. We useHiggs transitions in Ftheory to extend the ansatz for the modular bootstrap to genusone fibrations with Nsections and boundary conditions fix the all genus generatingfunctions for small base degrees completely. This allows us to study in depth a widerange of new, nonperturbative 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 oneloop 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.