Program
 Geometry and Physics Seminar Student No.： 40 Time： Tue 10:30-13:00, 2017.9.15-2018.1.5 Instructor： Li Si, Zong Zhengyu, Chen Jingyue, Wen Hao Place： Conference room 1, Floor 1, Jin Chun Yuan West Building Starting Date： 2017-9-19 Ending Date： 2018-1-5

Date: 2017-10-24 Tue

Speaker: Aimhay Hanany[Imperial College London]

Time: 10:30-12:00

Place: Conference room 1, Floor 1, Jin Chun Yuan West Building

Title: Small instanton transition and Higgs branch at infinite coupling

Abstract: In a six dimensional field theory, a well known generalization of the Higgs mechanism is the small instanton transition.
In such a transition the Higgs branch increases significantly.
I will discuss the Higgs branch of an Sp gauge theory and show the two moduli spaces which arise at finite coupling and infinite coupling.

Date: 2017-10-17 Tue

Speaker: SUEN Yat-Hin [CUHK]

Time: 10:30-12:00

Place: Conference room 1, Floor 1, Jin Chun Yuan West Building

Title: On SYZ-transform of immersed Lagrangian multi-sections

Abstract: In this talk, I will talk about the SYZ-mirror symmetry of immersed Lagrangian multi-sections of a semi-flat torus fibration. I will introduce the notion of lifted-Hamiltonian isotopy, which is the mirror analog for isomorphism of holomorphic bundles. Furthermore, the Floer cohomology for a certain pair of immersed Lagrangian submanifolds turns out to be invariant under this new notion of equivalence. Finally, I will focus on the 2-torus and study the relation between Lagrangian surgery and extension of holomorphic bundles there.

Date: 2017-10-11 Wed

Speaker: Zhu Chenchang [YMSC, Tsinghua University]

Time: 16:00-17:00

Place: Conference room 1, Floor 1, Jin Chun Yuan West Building

Title: String principal bundles and Courant algebroids

Abstract: Just like Atiyah Lie algebroids encode the infinitesimal symmetries of principal bundles, exact Courant algebroids are believed to encode the infinitesimal symmetries of S1-gerbes. At the same time, transitive Courant algebroids may be viewed as the higher analogue of Atiyah Lie algebroids, and the non-commutative analogue of exact Courant algebroids. In this article, we explore what the "principal bundle" behind transitive Courant algebroids are, and they turn out to be principal 2-bundles of string groups. First, we construct the stack of principal 2-bundles of string groups with connection data. We prove a lifting theorem for the stack of string principal bundles with connections and show the multiplicity of the lifts once they exist. This is a differential geometrical refinement of what is known for string structures by Redden, Waldorf and Stolz-Teichner. We also extend the result of Bressler and Chen-Sti\'enon-Xu on extension obstruction involving transitive Courant algebroids to the case of transitive Courant algebroids with connections, as a lifting theorem with the description of multiplicity once liftings exist. At the end, we build a morphism between these two stacks. The morphism turns out to be neither injective nor surjective in general, which shows that the process of associating the "higher Atiyah algebroid" loses some information and at the same time, only some special transitive Courant algebroids come from string bundles.

Date: 2017-9-26

Speaker: Sun Zhe [YMSC, Tsinghua University]

Time: 10:30-12:00

Place: Conference room 1, Floor 1, Jin Chun Yuan West Building

Title: Deforming PSL(n,R) Hitchin component and Goldman symplectic form

Abstract: (This is joint work with Anna Wienhard and Zhang Tengren. ) Let S be a closed, connected, oriented surface of genus at least 2. It is well-known that on Teichmuller space, the twist flows along a pants decomposition of S is a maximal family of commuting Hamiltonian flows whose hamiltonian functions are Poisson commuting. We prove that any ideal triangulation on S determines a symplectic trivialization (with respect to the Goldman symplectic form) of the tangent bundle of the PSL(n,R) Hitchin component. One can then consider the parallel flows with respect to the flat structure given by this trivialization. We give a geometric description of all such flows in terms of explicit deformations of the associated Frenet curves, and prove that all such flows are Hamiltonian. Applying this to a particular ideal triangulation allows us compute the Goldman symplectic pairing explicitly. As a consequence, we find a maximal family of Poisson commuting Hamiltonian flows on the PSL(n,R) Hitchin component and a global Darboux coordinates for PSL(n,R) Hitchin component.

Date: 2017-9-19

Speaker: Jiang Xiaotian [University of Chinese Academy of Sciences]

Time: 10:30-12:00

Place: Conference room 1, Floor 1, Jinchun Yuan West Building

Title: D-brane superpotentials, SU(2) Ooguri-Vafa invariants and TypeII/F -theory duality

Abstract: In this paper, we focus on the phase transition of the D-brane system with multiple open-string deformation parameters in terms of toric geometry: parallel D- brane phase ←→ coincident D-brane phase. Several D-brane system with various closed-string moduli are studied. In parallel phase, the D-branes with different position parameters are separated corresponding to the Coulomb branch of the gauge theory with the U(1) × U(1) gauge symmetry group. While in the coincident phase, the coincidence of D-branes gives rise to the singularity in the corresponding 4-fold in terms of TypeII/F theory duality corresponding to the Higgs branch of the gauge theory with the non-Abelian gauge group SU(2). The enhancement of gauge group shows the phase transition to the Higgs branch. The off-shell superpotentials are obtained on the B-model side using the typeII/F theory duality for each phase, and the Ooguri- Vafa invariants are extracted from the expression of instanton expansion of them on the A-model side with the help of mirror symmetry. It comes out that in the parallel D-branes phase the disk invariants for individual D-brane in the D-branes system are the same as the results derived from the one which contains the only D-branes up to a transform of the index of invariants. We find the discrete Z2 symmetry superpotential in all the models which is a signal of decoupling of the parallel D-branes. Furthermore, as an evidence of the phase transition the disk invariants corresponding to the parallel phases and the coincident phase respectively are different in our calculation.

Date: 2017-7-4

Speaker: Dmitry Doryn [IBS Center for Geometry and Physics]

Time: Tue, 10:30-12:00

Place: Conference room 4, Floor 2, Jin Chun Yuan West Building

Title: Differential models for B-type open-closed topological Landau-Ginzburg theories.

Abstract: I will speak on two articles written with my co-authors on some differential (in contrast to algebraic) models for the open-closed topological LG theories. This work is motivated by and substantially improves the old result of Lazaroiu published 13 years ago. We can get a simple concrete description of the models is the case of Stein manifolds. I will also discuss the non-degeneracy of the constructed traces as part of the properties expected for these models.

Date: 2017-7-6

Speaker: Ezra Getzler [Northwestern University]

Time: Thu, 10:30-12:00

Place: Conference room 1, Floor 1, Jin Chun Yuan West Building

Title: Moment map for generalized geometry

Abstract: I explain the definition of moment maps in generalized geometry, and in particular for Courant algebroids and multisymplectic manifolds, using the notion of an L-infinity morphism.

Date: 2017-6-27

Speaker: Jingyu Zhao (Institute of Advanced Study)

Time: Tue 9:00-10:30

Place: Conference room 4, Floor 2, Jin Chun Yuan West Building

Title: Degeneration of the noncommutative Hodge-to-de Rham spectral sequences

Abstract: For a compact Kahler manifold, the Hodge-to-de Rham spectral sequence degenerates at E^1 page. This can be proved using Hodge theory which is highly transcendental. In 1982, Deligne and Illusie gave a purely algebraic proof of the degeneration of the Hodge-to-de Rham spectral sequences using reduction to characteristic p methods. For a DG category, there is a noncommutative Hodge-to-de Rham spectral sequence which relates the Hochschild homology and periodic cyclic homology of A. It is conjectured by Kontsevich and Soibelman and proved by Kaledin in 2016 that the noncommutative Hodge-to-de Rham spectral sequences degenerates for smooth and proper DG categories. In this lecture, we give an outline of the proof of degeneration following Kaledin’s work. The key step is to prove a non-commutative analogue of the Cartier isomorphism in the work of Deligne and Illusie.

Speaker: Yining Zhang (Indiana University)

Time: 10:30-12:00

Place: Conference room 4, Floor 2, Jin Chun Yuan West Building

Title: Dual Hodge decompositions and derived Poisson brackets

Abstract: After briefly recall the definition of cyclic homology and classical Hodge decomposition, I will state a theorem by B. Feigin and B. Tsygan. Using this theorem, I shall construct a Hodge type decomposition (dual Hodge decomposition) of (reduced) cyclic homology of universal enveloping algebras of Lie algebras. In fact this decomposition can be viewed as being Koszul dual to the Hodge decomposition of cyclic homology of cocommutative coalgebras. Next, I will introduce the notion of noncommutative Poisson bracket in the sense of Crawley-Boevey and its homological generalization derived Poisson bracket. As a consequence, if an algebra admits a derived Poisson structure, then its cyclic homology would be a graded Lie algebra. Lastly, I want to talk about the interactions between this Lie bracket on cyclic homology and dual Hodge decomposition. If time permits, I will mentions some applications to string topology. This is based on a joint work with Yuri Berest and Ajay C. Ramadoss.

Date: 2017-6-22

Speaker: Zhengfang Wang (BICMR)

Time: Thu 9:00-10:30

Place: Conference room 4, Floor 2, Jin Chun Yuan West Building

Title: A-infinity and L-infinity algebra structures arising from (dg) Frobenius algebras

Abstract: Given a (differential graded) Frobenius algebra, one can construct an unbounded complex, called Tate-Hochschild complex, which is defined as a mapping cone of a certain morphism from the Hochschild chain complex (with some shift) to the Hochschild cochain complex. We construct a cyclic (or called Calabi-Yau) A-infinity algebra structure, which extends the classical Hochschild cup and cap products, on the Tate-Hochschild complex. This A-infinity algebra has an explicit formula with m_n=0 for n>3. We will also construct an $L$-infinity algebra structure by extending the classical Gerstenhaber bracket. Moreover, we prove that the cohomology ring of the Tate-Hochschild complex is a Batalin-Vilkovisky (BV) algebra with the BV operator extending the Connes' boundary operator. At the end, we will talk about how to relate the Tate-Hochschild complex to string topology. This talk is based on the recent joint work with M. Rivera.

Speaker: Bohan Fang (BICMR)

Time: Thu 10:30-12:00

Place: Conference room 1, Floor 1, Jin Chun Yuan West Building

Title: Oscillatory integrals on the T-dual cycles

Abstract: The mirror of a complete toric variety is a Landau-Ginzburg model. The oscillatory integral of the superpotential function on a Lagrangian cycle mirror to a coherent sheaf is a B-model genus 0 invariant. I will describe a way to compute it, and the answer is a genus 0 Gromov-Witten descendant potential with certain Gamma class of the coherent sheaf inserted. This result is related to Iritani's result identifying the integral structures on both sides. When the toric variety is P^1, this result can be extended to compute all genus descendants from Eynard-Orantin's topological recursion theory on the mirror of P^1.

Date: 2017-6-19

Speaker: Daping Weng (Yale University)

Time: Mon, 9:00-10:30

Place: Conference room 4, Floor 2, Jin Chun Yuan West Building

Title: Cluster Donaldson-Thomas Transformation of Grassmannian.

Abstract: On the one hand, there is a 3d Calabi Yau category with stability conditions associated to a quiver without loops or 2-cycles with generic potential, and one can study its Donaldson-Thomas invariants. On the other hand, such a quiver also defines a cluster Poisson variety, which often has geometric realizations. In certain cases the Donaldson-Thomas invariants of the former can be captured by a birational automorphism of the latter. In this talk I will describe the cluster Poisson structure on the moduli space of configurations of points on a projective space, and state my result on the geometric realization of the corresponding cluster Donaldson-Thomas transformation.

Speaker: Ping Xu (Penn State University)

Time: Mon, 10:30-12:00

Place: Conference room 4, Floor 2, Jin Chun Yuan West Building

Title: Formal exponential maps and L-infinity structure

Abstract: Exponential maps arise naturally in the contexts of Lie theory and smooth manifolds. The infinite jets of these classical exponential maps are related to the Poincar\'e--Birkhoff--Witt isomorphism and the complete symbols of differential operators. We will investigate the question how to extend these maps to graded manifolds by introducing formal exponential maps in a purely algebraic way. As an application, we will show there is an L-infinity structure in connection with the Atiyah class of a dg manifold.

Date: 2017-6-13

Speaker: Yang Shen (Zhejiang University)

Time: Tue 9:00-10:30

Place: Conference room 3, Floor 2, Jin Chun Yuan West Building

Title: Moduli spaces as ball quotients

Abstract: In this talk, I introduce our recent work on Hodge theory, which gives a Hodge theoretic criterion to characterize the moduli spaces of certain projective manifolds to be ball quotients.

This is a joint work with Professor Kefeng Liu.

Speaker: Jingyu Zhao (Institute of Advanced Study)

Time: Tue 10:30-12:00

Place: Conference room 3, Floor 2, Jin Chun Yuan West Building

Title: Getzler’s Gauss-Manin connection on equivariant Floer cohomology

Abstract: For a family of DG (or A_infinity) algebra, Getzler defined a Gauss-Manin connection on the periodic cyclic homology of the family. Using Getzler’s definition, the recent work of Seidel constructed a q-connection on equivariant Floer cohomology. In this talk, we will describe a version of the periodic cyclic homology of the Fukaya category that recovers the q-connection on the quantum cohomology of a noncompact symplectic manifold.

Date: 2017-6-6

Speaker: Xuwen Zhu (Stanford University)

Time: Tue 10:30-12:00

Place: Conference room 4, Floor 2, Jin Chun Yuan West Building

Title: The deformation theory of constant curvature metrics with conical singularities

Abstract: We would like to understand the deformation theory of constant curvature metrics with prescribed conical singularities on a compact Riemann surface. In the positive curvature case, when some or all of the cone angles are bigger than $2\pi$, the analysis is much more complicated than the small angle case. We discover that one key ingredient of the obstructed deformation is related to splitting of cone points. We construct a resolution of the configuration space, and prove a new regularity result that the family of constant curvature conical metrics has a nice compactification as the cone points coalesce, and moreover, the fibrewise family of constant curvature metrics is polyhomogeneous on this compactification. And we hope to apply this new construction to describe the moduli space of spherical conic metrics with no angle constraints. This is joint work with Rafe Mazzeo.

Date: Tue 2017-4-25

Speaker: Jie Tu (CMS, Zhejiang University )

Time: 10:30-12:00

Place: Conference room 3, Floor 2, Jin Chun Yuan West Building

Title: The Deformation of Pairs (X,E) Lifting from Base Family

Abstract: In this talk, I will introduce my joint work with K.F. Liu on the analytic deformation theory of pairs (X,E), where X is a compact complex manifold and E is holomorphic vector bundle over X.

The splitting of holomorphic cotangent bundle via any integrable connection de- composes the Beltrami differential of pairs into the horizontal part and the vertical part. The horizontal part is the Beltrami differential of base family {Xt}. When the vertical part is vanishing under the decomposition by a Nakano semi-positive Chern connection ∇, i.e. {(Xt , Et )} is lifting from base family {Xt } via ∇, we get a infinitesimal extension of ∂ ̄-closed bundle valued (n, q)-form by the recursive method.

Date: Tue 2017-4-11

Speaker: Chunle Huang 黄春乐（Zhejiang University

Time: 10:30-12:00

Place: Conference room 3, Floor 2, Jin Chun Yuan West Building

Title: Logarithmic vanishing theorems on compact K\"{a}hler manifolds

Abstract: We will first establish an $L^2$-type Dolbeault isomorphism for logarithmic differential forms by H\"{o}rmander's $L^2$-estimates. By using this isomorphism and the construction of smooth Hermitian metrics, we obtain a number of new Akizuki-Kodaira-Nakano type vanishing theorems for sheaves of logarithmic differential forms on compact K\"ahler manifolds with simple normal crossing divisors, which generalize several classical vanishing theorems, including Norimatsu's vanishing theorem, Gibrau's vanishing theorem, Le Potier's vanishing theorem and a version of the Kawamata-Viehweg vanishing theorem.

Date: 2017-3-28

Speaker: Baosen Wu 吴宝森YMSC

Time: 10:30-12:00

PlaceConference room 3, Floor 2, Jin Chun Yuan West Building

Title: Construction of stable sheaves via Serre correspondence

Abstract: Serre correspondence is a classical tool to construct rank 2 vector bundles or coherent sheaves with prescribed Chern classes. However, testing the stability of such bundles is in general very difficult. We shall work on Calabi-Yau threefolds and use deformation to show the stability of such bundles or sheaves.

Date: 2017-3-14

Speaker: 曹亚龙

Time: 10:00-11:30

Place: Conference room3, Floor 2, Jin Chun Yuan West Building

Title: Gopakumar-Vafa type invariants for Calabi-Yau 4-folds

Abstract:

As an analogy of Gopakumar-Vafa conjecture for CY 3-folds, Klemm-Pandharipande proposed GV type invariants on CY 4-folds using GW theory and conjectured their integrality. In this talk, we propose a sheaf theoretical interpretation to these invariants using Donaldson-Thomas theory on CY 4-folds. This is a joint work with Davesh Maulik and Yukinobu Toda.

Date: 2017-3-7

Speaker: Sz-Sheng Wang 王赐圣（YMSC

Title: Towards decompositions of small transitions of Calabi-Yau threefolds

Abstract: Let \pi : \hat{X} \to X be a small projective resolution of a Calabi--Yau 3-fold X, which has terminal singularities. If X can be smoothed to a Calabi--Yau manifold \tilde{X}, then the process of going from \hat{X} to \tilde{X} is called a small transition.

In order to decompose small transitions, we introduce a subclass of small transitions which we call“primitive”small transitions and study such subclass. More precisely, we show that if the natural closed immersion of miniversal deformation spaces Def(\hat{X}) \hookrightarrow Def(X) is an isomorphism then X has only ordinary double points as singularities. A determinantal construction of conifold transition will also presented, if there is enough time.

History:

Spring,2016:  http://ymsc.tsinghua.edu.cn/sjcontent.asp?id=799

Autumn,2015:  http://ymsc.tsinghua.edu.cn/sjcontent.asp?id=731