Online Seminar
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03232023
Abstract. Assuming certain comparison between non-commutative Hodge structures with classical Hodge structures, we prove the CEI associated with a smooth projective family of Calabi-Yau's satisfy the holomorphic anomaly equation. This is based on a work in progress with Yefeng Shen.
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03232023
Junwu Tu: An introduction to categorical enumerative invariants (CEI)
Abstract. In this talk, we shall present the definition of CEI associated with smooth and proper Calabi-Yau categories. We also sketch an explicit combinatorial formula of CEI. In the end, we discuss about concrete computations as well as some interesting questions. The talk is based on joint works with Andrei Caldararu and Lino Amorim.
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03172023
Yalong Cao: From curve counting on Calabi-Yau 4-folds to quasimaps for quivers with potentials
Abstract: I will start by reviewing an old joint work with Davesh Maulik and Yukinobu Toda on relating Gromov-Witten, Gopakumar-Vafa and stable pair invariants on compact Calabi-Yau 4-folds. For non-compact CY4 like local curves, similar invariants can be studied via the perspective of quasimaps to quivers with potentials. In a joint work in progress with Gufang Zhao, we define a virtual count for such quasimaps and prove a gluing formula. Computations of examples will also be discussed.
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03102023
Denis Nesterov:Enumerative mirror symmetry for moduli spaces of Higgs bundles
Abstract: We discuss conjectures, called genus 1 Enumerative mirror symmetry for moduli spaces of Higgs bundles, which relate curve-counting invariants of moduli spaces of Higgs SL(r)-bundles to curve-counting invariants of moduli spaces of Higgs PGL(r)-bundles.
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02272023
Jie Gu:Resurgent structure in topological strings
Abstract. Topological string theory has (spacetime) instanton sectors, which the resurgence theory predicts to be completely controlled by the perturbative free energy via Stokes transformations. Recent results also suggest the Stokes constants are related to BPS invariants/DT invariants. To make this picture complete, one needs to first solve the instanton amplitudes and then calculate the Stokes constants. We demonstrate that the first problem can be solved exactly and completely through a transseries extension of the BCOV holomorphic anomaly equations. We also make progress in the second problem. We focus on examples in local Calabi-Yau threefold and comment that similar results can be obtained for quintic as well.
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02042023
We discuss techniques to calculate symplectic invariants of CY 3-folds $M$, namely Gromov-Witten (GW) invariants, Pandharipande-Thomas (PT) invariants, and Donaldson-Thomas (DT) invariants. Physicallythe latter are closely related to BPS brane bound states in type IIA string compactifications on $M$. We focus on the rank $r_{\bar 6}=1$ DT invariants that count $\bar D6-D2-D0$ brane bound states related to PT- and high genus GW invariants, which are calculable by mirror symmetry and topological string B-model methods modulo certain boundary conditions, and the rank zero DT invariants that count rank $r_4=1$ $D4-D2-D0$ brane bound states. It has been conjectured by Maldacena, Strominger, Witten and Yin that the latter are governed by an index that has modularity properties due to $S$-duality in string theory and extends to a mock modularity index of higher depth for $r_4>1$. Again the modularity allows to fix at least the $r_4=1$ index up to boundary conditions fixing their polar terms. Using Wall crossing formulas obtained by Feyzbakhsh certain PT invariants or Katz-Klemm-Vafa (KVV) invariants close to the Castelnuovo bound can be related to the $r_4=1$ $D4-D2-D0$ invariants. This provides further boundary conditions for the topological string B-model approach as well as for the $D4-D2-D0$ brane indices. The approach allows to prove the Castelnuovo bound and calculate the $r_{\bar 6}=1$ DT- invariants or the GW invariants to higher genus than hitherto possible, as was pointed out in https://arxiv.org/abs/2301.08066 by Alexandrov, Feyzbakhsh, Pioline, Schimannek and me. See also http://www.th.physik.uni-bonn.de/Groups/Klemm/data.php for concrete evaluations. Lecture I: ``Recursive solution of the perturbative topological String''In this lecture we explain how to solve the topological string recursively in terms of non holomorphic modular objects and discuss the integer invariants that it calculates.Lecture II: S-duality and the index of $D4-D2-D0$ bound states''We review the approach of Maldacena, Strominger, Witten and Yin to the calculate the abelian ($r_4=1$) $D4-D2-D0$ bound state degneracies from a modular index and the relation of these invariants to the PT and DT invariants discussed above ( if time permits we comment on the mock modularity of higher depth for $r_4>1$). Lecture III: Castelnuovo bound and Wall crossing'' In this lecture we define the above mentioned symplectic invariants more mathematically and discuss their stability and their Wall crossing behaviour and summarize the implication on concrete calculations of them.
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12312022
Gregory W. Moore: Supersymmetric QFT & Invariants Of Smooth Four-Manifolds
Abstract. After describing briefly some aspects of nonabelian Yang-Mills theory and Donaldson invariants the physicists' derivation of the "Witten conjecture" expressing Donaldson invariants in terms of Seiberg-Witten invariants will be sketched. Time permitting, more recent developments related to this physical approach will be sketched.
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12192022
Xueqing Wen:Topological mirror symmetry for parabolic Higgs moduli of type B/C—From local to global
Abstract. The moduli of Higgs bundles on curve was firstly studied by Hitchin in 1980’s and has been vastly studied during the last over 30 years. One of the most remarkable observation about Higgs moduli was proposed by Hausel and Thaddeus in 2003 that there is a mirror relation between the G-Higgs moduli and G^L-Higgs moduli, here G^L is the Langlands dual group of G. They conjectured that there is a topological mirror symmetry between SL_n/PGL_n Higgs moduli and this conjecture was proved by Groechenig, Wyss and Ziegler also by Maulik and Shen using different methods. The mirror phenomenon of Higgs moduli was also noticed by Gokov, Kapustin and Witten by the viewpoint of physics. Especially, Gokov and Witten proposed in physics that there should be a mirror relation of parabolic Higgs bundles for Langlands dual groups and nilpotent orbits inserted at the marked points. If one considers Higgs bundles with nilpotent orbits inserted, the most interesting case is type B/C. In this talk, we will show that how to relate nilpotent orbits in type B/C using Kazhdan-Lusztig map and loop Lie algebra, and then use these local computations to prove a topological mirror symmetry statement for parabolic Higgs moduli of type B/C using p-adic integration. This program was suggested by Prof. Ruan, and it is a joint work with Weiqiang He, Xiaoyu Su, Bin Wang and Yaoxiong Wen.
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12142022
Yang Zhou:A generalization of mixed-spin-P fields
Abstract. The theory of Mixed-Spin fields was introduced by Chang-Li-Li-Liu for the quintic threefold. Chang-Guo-Li has successfully applied it to prove famous conjectures on the higher-genus Gromov-Witten invariants. In this talk I will explain a generalization of the construction to more spaces. The key is the stability condition which guarantees the separatedness and properness of certain moduli spaces. It also generalizes the construction of the mathematical Gauged Linear Sigma Model by Fan-Jarvis-Ruan, removing their technique assumption about good lifitings.This is a joint work with Huai-Liang Chang, Shuai Guo, Jun Li and Wei-Ping Li.
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12072022
Abstract. In this talk we will discuss a construction of moduli space of stable parabolic vector bundles $\overline{\mathfrak{U}}_{_{g, n, r}}$ over $\overline{M}_{_{g, n}}$. The objects that appear over the boundary of $\overline{M}_{_{g, n}}$ i.e., over singular curves will remain vector bundles. We will also discuss about the singularity of the total space and the fiber over a marked stable curve which are "nicer" than their torsion free counterpart.
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12012022
Minxin Huang: Quantum Periods and TBA-like Equations for a Class of Calabi-Yau Geometries
Abstract. We continue the study of a novel relation between quantum periods and TBA(Thermodynamic Bethe Ansatz)-like difference equations, generalize previous works to a large class of Calabi-Yau geometries described by three-term quantum operators. We give two methods to derive the TBA-like equations. One method uses only elementary functions while the other method uses Faddeev’s quantum dilogarithm function. The two approaches provide different realizations of TBA-like equations which are nevertheless related to the same quantum period.
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11252022
Changjian Su: Motivic Chern classes of Schubert cells and applications
Abstract: Motivic Chern classes in K-theory are generalizations of the MacPherson classes in homology. I will talk about some recent developments about motivic Chern classes of the Schubert cells in the flag varieties, and their applications to representations of p-adic dual groups and the K-theoretic stable envelopes.
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11182022
Dan Xie: One classification of theories with eight supercharges
Abstract. We will discuss the classification of theories with eight supercharges by using the associated Coulomb branch geometry,which is described by a mixed Hodge module over Coulomb branch. The classification of rank one case is related to classification of rational elliptic surface,and the rank two is related to genus two Lefchetz fibration over P^1. Arithmetic aspects such as Mordeil-Weil lattice would also play an important role.
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11102022
Shizhuo Zhang:Categorical Torelli problem for Fano varieties
Abstract: It is well known that for a smooth Fano variety, bounded derived category of coherent sheaves determine their isomorphism classes. It is natural to ask whether it is possible to reconstruct them with less information, say a semi-orthogonal component, known as categorical Torelli problem. I will talk about recent progress on this problem, in particular for Fano threefolds and fourfolds.It is based on the work myself and the joint work with Zhiyu Liu, Augustinas Jacovskis and Soheyla Feyzbakhsh.