Seminars

Generalized Complex Geometry Seminar Please note: double seminar on 5 December. Date: Thursday, 5 December 2024, at 14:00 -17:00 Venue: room 205, iGAP. First speaker: Pavol Severa (Geneva) Title: Poisson-Lie T-duality and Generalized Ricci flow Poisson-Lie T-duality is a curious equivalence of various 2-dimensional variational problems, which generalizes the more widely-known T-duality. I will talk about its relation to generalized geometry (i.e. Courant algebroids) and about the related generalization of the Ricci flow. Second speaker: Thomas Schick (Göttingen) Title: Equivariant T-duality T-duality is a concept inspired by certain aspects of stringtheory (the T stand for target space). The physics predicts that one can pass from a torus bundle over a base space, equipped with some background B-field to a dual torus bundle with dual background B-field without changing the physical content of the theory. Strominger-Yau-Zaslow even propose that mirror symmetry is a special case of T-duality. To understand the mathematical underpinnings of this duality, the concept of topological T-duality has been introduced. We present the fundamental ideas of this concept: what are the objects we put in duality? What is the precise meaning of “being dual”? Among the most striking mathematical consequences, we present the T-duality isomorphism between twisted K-theory of the two dual spaces. The last part of the talk will present new developments which concern the investigation of additional symmetries, encoded in the action of a compact group G. We discuss the meaning of equivariant T-duality and we show that one obtains a T-duality isomorphism in the full Atiyah-Segal equivariant K-theory.

Speaker: Konstantin Aleshkin (Mathematics Department, Columbia University) Date: Monday 11 March 16:30 Venue: room 005, SISSA. Title: Crossing the walls in GLSM Weighted extremal metrics, introduced by Lahdili, provide a vast and convenient generalisation of ordinary extremal Kähler metrics, and encompass many well-known examples of canonical metrics such as Gauged Linear Sigma Models (GLSM) have been an area of research and a useful tool in theoretical physics since the early 90s. In recent years, the mathematical theory of GLSM was formulated by various people. Mathematical theory is a Gromov-Witten-type curve-counting problem associated with a critical locus of a function on a Geometric Invariant Theory (GIT) quotient of vector space. In the talk, I plan to give an idea of the mathematical theory, define and compute specific generating series of invariants, and relate them to physical quantities called hemisphere and sphere partition functions. If time permits, I will discuss how variation of the GIT stability condition leads to a wall-crossing phenomenon for the invariants.

Due to personal reasons Michael Hallam has cancelled, and this seminar will not be taking place as previously scheduled .   Speaker: Michael Hallam (Aarhus) Date: Thursday 08 February 15:00 – 16:00 Venue: room 205: 1st floor in the IGAP/ IFPU building. Title: Weighted extremal metrics and stability Weighted extremal metrics, introduced by Lahdili, provide a vast and convenient generalisation of ordinary extremal Kähler metrics, and encompass many well-known examples of canonical metrics such as Kähler–Ricci solitons and extremal Sasaki metrics. Generalising work of Arezzo–Pacard–Singer and Székelyhidi, we show that the blowup of a weighted extremal manifold at a relatively stable point admits weighted extremal metrics in classes making the exceptional divisor small. In the vein of Stoppa–Székelyhidi, we then use this existence result to prove that manifolds with weighted cscK metrics are weighted K-polystable, sharpening the already known weighted K-semistability proven by Lahdili and Inoue.

IGAP Complex Differential Geometry Seminar organized by Jacopo Stoppa and Claudio Arrezzo   Thursday, 25 January 2024, 15:00 – 17:30, Room 205, 1st floor in the IGAP/ IFPU building.    15:00 – 16:00 Yalong Shi (Nanjing University) Compactness of cscK metrics near the canonical class We shall prove that the set of cscK metrics on minimal models constructed by Jian-Shi-Song is precompact with respect to the Gromov-Hausdorff topology. An important step is a uniform estimate for solutions of J-equations when the Kahler class approaches the boundary of the Kahler cone. This is joint work with B. Guo, W. Jian and J. Song. 16:30 – 17:30 Sun Jun (Wuhan University) Translating solitons to symplectic mean curvature flow In the talk, we will show that a symplectic translating soliton with mean curvature vector squared integral and some other natural assumptions must be a plane. This talk is based on joint work with Xiaoli Han and Jiayu Li.

Speaker: Prof. M. Bershtein (Edinburgh University) Date: Wednesday 13 December – 16:00 Venue: SISSA room 005 Title: Highest weight vectors in Coset Construction We revisit the classical Goddard Kent Olive coset construction. We find the formulas for the highest weight vectors in coset decomposition and calculate their norms. We also derive formulas for matrix elements of natural vertex operators between these vectors. This leads to relations on conformal blocks. Due to the AGT relation, these relations are equivalent to blowup relations on Nekrasov partition functions with the presence of the surface defect. These relations can be used to prove Kyiv formulas for the Painlevé tau-functions (following Nekrasov’s method). As another application we prove new Seiberg integral formulas.

Complex differential geometry seminar Speaker: Sohaib Khalid (SISSA) Date: 13 December 2023, 15:00-17:00 Venue: IFPU/IGAP room 205 Title: Semipositive line bundles and (1,1) classes (after Tosatti, Part I)

Complex differential geometry seminar Speaker: Pietro Ciusa (SISSA) Date: Wednesday 29 November, 15:00 – 17:00 Venue: IFPU/IGAP room 205 Title: Coupled Kähler-Ricci solitons on toric Fano manifolds (after Hultgren) I will explain and sketch the proof of a result of J. Hultgren which proves the existence of coupled Kähler Ricci solitons (in the sense of Hultgren and Witt-Nyström) on toric Fano manifolds, generalising a well-known theorem of Wang and Zhu in the uncoupled case.

Complex differential geometry seminar Speaker: Jacopo Stoppa (SISSA) Date: 15th November 2023, 3pm-4pm Venue: Room 205 Title: K-stability and large complex structure limits According to mirror symmetry, the geometry of a given Fano manifold endowed with some extra data, including an arbitrary Kähler class, should be reflected in a mirror Landau-Ginzburg model, i.e. a noncompact complex manifold endowed with a nonconstant holomorphic function. On the other hand, a fundamental notion for constructing moduli of Fano manifolds is K-polystability, i.e. positivity of the Donaldson-Futaki invariants for nonproduct test-configurations. In this talk I will introduce the problem of characterising K-polystable Kähler classes on a Fano in terms of their mirror Landau-Ginzburg models. I will then discuss some first concrete results in the case of slope stability for del Pezzo surfaces. The computations involve the particular “large complex structure limit” of the Landau-Ginzburg model corresponding to scaling the Kähler class on the Fano, which acts trivially on K-polystability.

Speaker: Sohaib Khalid (SISSA) Date: Wednesday, 8 November 2023, 15:00 – 17:00 Venue: Room 205, 1st floor in the IGAP/ IFPU building Title: Inverse Hessian and Monge-Ampere type equations We will discuss the recent work of Fang-Ma (arXiv:2309.15451) in the study of equations of the Monge-Ampere and inverse Hessian type, which links the solvability of such equations to the existence of subsolutions and to a certain numerical criterion, extending and generalising previous works on the deformed Hermitian Yang-Mills equations, the J-equation and many others. In order to state their results precisely, we will review notions of positivity of differential forms. If time permits, we will briefly sketch their strategy of proof and outline potential applications.

Speaker: Bruno Carneiro da Cunha (Pernambuco U.) Date: Tuesday 26/9 at 16.30 Venue: IGAP/IFPU room 205 – former “new sissa building” Miramare campus Title: Semiclassical conformal blocks and black hole scattering Conformal blocks are special functions appearing in the representation theory of the Virasoro algebra, the symmetry algebra of two-dimensional conformal field theories (2d CFTs). Since the beginning of the studies on 2d CFTs, the relation between the semiclassical limit of conformal blocks and the accessory parameter of Fuchsian differential equations has been outlined. In this talk, I will make use of some deep connections between conformal blocks and the isomonodromic problem to relate expansions of the semiclassical conformal blocks at different critical points. I will then apply these expansions to the study of particular Fuchsian equations arising in black hole perturbation theory.

Speaker: Kazunobu Maruyoshi (Seikei University – Tokyo) Date: Tuesday 19/9 at 16.00 Venue: IGAP/IFPU room 205 – former “new sissa building” Title: Dualities of Adjoint SQCD and Supersymmetry Enhancement We propose a new dual description of four-dimensional N=1 SU(N) gauge theory with one adjoint (X) and Nf fundamental matters with a superpotential W=Tr X^{p+1}. The dual theory consists of the D_p[SU(N)] Argyres-Douglas theory coupled to SU(N) gauge theory and Nf fundamentals with a superpotential deformation. We study renormalization group fixed points of the Argyres-Douglas dual theories with and without superpotential deformations, and identify the conditions for them to be dual to the fixed points of adjoint SQCD. We check our proposal via matching central charges, chiral operators and superconformal indices. We find that when Nf=2N and p=2, the dual theory flows to N=2 SU(N) superconformal QCD with 2N flavors upon suitable superpotential deformation, exhibiting supersymmetry enhancement.

Speaker: Thomas Nicosanti Date: Tuesday 19 September 14:30 Venue: Room 205 IGAP Via Beirut Title: SYMPLECTIC CUTS IN TOPOLOGICAL STRING THEORY Symplectic geometry is ubiquitous in physics, especially in string theory, and it plays a crucial role in explaining some key features of strings. Mirror symmetry might be one of the most famous examples of this interplay between theoretical physics and symplectic geometry. Concepts like symplectic reduction and Lagrangian submanifolds are well-known to theoretical physicists. However, a more recent construction called symplectic cutting, introduced by Lerman only in ’95, still has to find its application in physics. Being closely related to Lagrangian submanifolds, it is expected to become a relevant technique in string theory. After a brief introduction to equivariant volumes, we review a recent work by L. Cassia, P. Longhi and M. Zabzine on symplectic cuts of toric Calabi-Yau threefolds. The paper argues that the quantum equivariant volume associated with the symplectic cut plays the role of a measure for the original manifold and that it also captures the string superpotential of an A-brane supported on the Lagrangian submanifold associated with the cut. Using these facts, they predict a relation between the genus zero closed Gromov-Witten invariants of the original manifold and the genus zero open Gromov-Witten invariants of the Lagrangian submanifold. We conclude by considering a simple example, where the starting manifold is C^3, and we show all this machinery at work.

Speaker: Anton Shchechkin (SISSA and INFN) Date: Tuesday 16 May at 16:00 Venue: Room Dubrovin (136) of SISSA. Title: Painleve equations and spaces of initial data The spaces of initial data give a powerful tool for studying Painleve equations. Firstly introduced by Okamoto for differential Painleve equations, this notion was subsequently developed by Sakai, who constructed a geometric classification of Painleve equations, which includes both differential and difference ones. The latter classifies CP^2 blowed up in 9 points up to certain equivalency and each class corresponds to a certain Painleve equation. I will start my first talk from a toy example constructing the space of initial data for an order 1 LDEq by using the blowup procedure. Then we will construct this space for Painleve VI. Finally we will talk about geometrical properties of this space and how it is connected with properties of the corresponding Painleve equation. In the second talk we will speak about properties of spaces of initial data, but in a more wide context, which includes q-Painleve equations. Particularly we will pay attention on an example for q-Painleve VI.

Speaker: Reza Seyyedali (IPM, Teheran, Iran) Date: 12 May 2023 14:00 -15:00 Venue: IGAP lecture room 205 (former SISSA building, via Beirut 2 – Trieste) Title: Some a priori estimate for constant scalar curvature Kahler metrics The problem of existence of constant scalar curvature Kahler  metrics (cscK) on compact Kahler manifolds has been studied for many  years. In the case of complex one dimension, the answer is provided by  the uniformization theorem. It states that any compact Riemann surface  admits a metric of constant Gaussian curvature. One way to prove this  fact is to solve a semi linear elliptic equation. In higher dimensions, cscK metrics satisfy a fully nonlinear fourth  order elliptic equation. It is extremely difficult to deal with such  equations partially due to lack of maximum principle. In a recent breakthrough, Chen and Cheng proved some important a priori estimates for cscK metrics. More generally they showed that in a given Kahler class, the set of metrics with uniform bounded scalar curvature  and bounded relative entropy is compact in C^{\infty} topology.? In a joint work with Z. Lu, we generalized their results. Mainly, we replace the uniform boundedness assumption on scalar curvature with some L^p boundedness.

Speaker: Prof. Cumrum Vafa (Harvard University) Date: 4 May 2023 16:00 Venue: SISSA — room 128/129 Title: Black Holes, Species Scale, and Topological Strings

Speaker: Pavlo Gavrylenko (University of Geneva) Date: 24 and 31 Jan 2023 14:30 Venue: IGAP Room 205 Title: Introduction to isomonodromy/CFT correspondence Abstract: In the first talk I would like to tell about Fuchsian systems, their monodromy, isomonodromic deformations, and tau functions. In the second talk I’m going to tell some basic things about conformal field theory and its application to solve isomonodromic deformations equations, which was first found in the paper by Gamayun, Iorgov, Lisovyy https://arxiv.org/abs/1207.0787 .

Speaker: Li Chao (ICTP) Date: 14 Dec 2022 14:30 Venue: IGAP Room 205 Title: Submanifolds with special Kahler metrics of complex space forms Abstract: Submanifolds of complex space forms are common examples of Kahler manifolds. It is natural to study the ones whose induced metrics are KE, cscK, or more generally extremal. In this talk, I will first review some classic results, then introduce our recent work on KE and extremal Kahler submanifolds of complex projective spaces.

Speaker: Ruadhai Dervan (Glasgow) Date: 6 Dec 2022 15:00 Venue: IGAP Room 305 Title: Stability and Kähler-Einstein metrics in big classes Abstract: I will discuss an analogue of the Yau-Tian-Donaldson conjecture in the setting of big line bundles, and will explain progress towards proving it. In the setting of an ample line bundle (and hence a Kähler class), the Yau-Tian-Donaldson conjecture links the existence of Kähler-Einstein metrics with a variety of notions of algebro-geometric stability. The main point of my talk will be to motivate the relevance of considering big line bundles, to explain the definition of stability in this setting, and to speculate as to the relevance of these ideas to algebraic geometry. I will also explain parts of the analytic side of the story, which involves singular metrics. The main result will be that the existence of a Kähler-Einstein metric on a projective manifold with big anticanonical class implies a version of Ding stability. This is joint work with Rémi Reboulet, and is parallel to independent work of Darvas-Zhang and Trusiani.

Speaker: T. Hausel (IST, Austria) and A. Mellit (Univ. of Vienna) Date: 5 Dec 2022 10:00 – 6 Dec 2022 15:00 Venue: the IFPU Seminar Room (room number 205, new SISSA Building) Title:  A Proof of the P=W Conjecture Abstract: This short series of four lectures by T. Hausel (IST, Austria) and A. Mellit (Univ. of Vienna) will discuss the recent proof of the P=W conjecture in the theory of the moduli space of Higgs bundles on a curve. 1) Tamas Hausel Introduction to P=W and mirror symmetry Motivated by topological mirror symmetry for Langlands dual Higgs bundle moduli spaces, one can study the arithmetic of the corresponding character varieties to gain information on the weight filtration on their cohomology. In turn this lead to the P=W conjecture which identifies the weight filtration on the cohomology of the character variety with the perverse filtration of the Hitchin system on the cohomology of the Higgs moduli space. Mirror symmetry and big algebras First we discuss the mirror symmetry identification of the coordinate ring of certain very stable upward flows in the Hitchin system and the Kirillov algebra for the minuscule representation of the Langlands dual group via the equivariant cohomology of the cominuscule flag variety (e.g. complex Grassmannian). In turn we explain a conjectural extension of this picture to non-very stable upward flows in terms of a big commutative subalgebra of the Kirillov algebra, which also ringifies the equivariant intersection cohomology of the corresponding affine Schubert variety. 2) Anton Mellit P=W via H_2 By H_2 we denote the Lie algebra of polynomial hamiltonian vector fields on the plane. We consider the moduli space of stable twisted Higgs bundles on an algebraic curve of given coprime rank and degree. De Cataldo, Hausel and Migliorini proved in the case of rank 2 and conjectured in arbitrary rank that two natural filtrations on the cohomology of the moduli space coincide. One is the weight filtration W coming from the Betti realization, and the other one is the perverse filtration P induced by the Hitchin map. Motivated by computations of the Khovanov-Rozansky homology of links by Gorsky, Hogancamp and myself, we look for an action of H_2 on the cohomology of the moduli space. We find it in the algebra generated by two kinds of natural operations: on the one hand we have the operations of cup product by tautological classes, and on the other hand we have the Hecke operators acting via certain correspondences. We then show that both P and W coincide with the filtration canonically associated to the sl_2 subalgebra of H_2. For more details, visit https://indico.ictp.it/event/10145/

Speaker: Annamaria Ortu (SISSA) Date: 30 Nov 2022 14:00 Venue: IGAP Lecture Room (Old SISSA Building, Via Beirut 2) Title:  Special Kähler metrics on fibrations Abstract: Proper holomorphic submersions of Kähler manifolds can be thought of as both a generalisation of holomorphic vector bundles and as a way of studying the behaviour of Kähler manifolds in families. We will consider fibrations whose fibres are K-semistable deformations of Kähler manifolds with constant scalar curvature, in a way compatible with the fibration structure. On such fibrations, we will describe a canonical choice of a relatively Kähler metric, called an optimal symplectic connection, that provides a generalisation of the Hermite-Einstein condition on vector bundles and gives the existence of special Kähler metrics on the total space.

Speaker: Eloise Hamilton – IMJP-RG Date: 24 June at 16:30 Venue: Zoom Title:  Group actions and the cohomology of moduli spaces Abstract: Moduli spaces play a crucial role in solving classification problems: their geometry encapsulates all of the features of the objects which they classify, and thus questions about the objects can be answered by studying the geometry of the moduli space instead. The aim of this talk is to present some techniques for studying the cohomology of moduli spaces. These techniques require the presence of a group action: either a group action on the moduli space itself, or a group action on a parameter space whose quotient is the given moduli space. I will focus mostly on the latter case, which relates to Geometric Invariant Theory (GIT), and explain how to compute the topology of GIT quotients, both in the classical (reductive) setting and in the recently developed non-reductive setting.

Speaker: Veronica Fantini – IHES Date: 17 June at 16:30 Venue: Zoom Title:  On the recostructing problem in mirror symmetry Abstract: In the first part of this talk, I will discuss the reconstructing problem in mirror symmetry, namely how we can construct the mirror partner of a given variety. Although there are many examples in which the costruction is done explicitly, it is usually quite technical and I could not give it in full detail in this seminar. However, I’ll try to explain the main ideas behind Kontsevich–Soibelman’s construction and the Gross–Siebert’s program. In both approaches, a central role is played by scattering diagrams, and in the last part of the talk, I will discuss one of the main results of my thesis which is the definition of the extended tropical vertex group where scattering diagrams can be defined.

Speaker: Kazuya Yonekura (Tohoku U.) Date: 15 June at 11:00 Venue: Zoom Title:  Topological violation of global symmetries in quantum gravity Abstract: I will talk about a topological mechanism of global symmetry violation in quantum gravity, based on my own work https://arxiv.org/abs/2011.11868

Speaker: Sarah Harrison Date: 01 June at 16:00 Venue: Zoom Title:  Chaos and the spectrum on Moduli space Abstract: We analyze the spectrum of the Weil-Petersson Laplacian on the moduli space of a genus zero Riemann surface with four punctures via a perturbative expansion of the path integral of Liouville theory. Our numerical results furnish evidence that the eigenvalues obey the statistics of a random matrix in the Gaussian Orthogonal Ensemble. We comment on possible implications for the quantum geometry of Riemann surfaces and quantum gravity in anti–de Sitter space. Based on work with A. Maloney and T. Numasawa.

Speaker: Andrea Ricolfi – SISSA Date: Thursday, May 27 at 4.30pm (Rome) Venue: Zoom Title:  The d-critical structure on the Quot scheme of points on a 3-fold Abstract: D-critical schemes and Artin stacks were introduced by Joyce in 2015, and play a central role in Donaldson-Thomas theory. They typically occur as truncations of (-1)-shifted symplectic derived schemes, but the problem of constructing the d-critical structure on a “DT moduli space” without passing through derived geometry is wide open. We discuss this problem, and new results in this direction, when the moduli space is the Hilbert (or Quot) scheme of points on a Calabi-Yau 3-fold. Work in progress with Michail Savvas.

Speaker: Tudor Dimofte (UC Davis) Date: 18 May at 14:30 Venue: Zoom Title:  QFT’s for non-semisimple TQFT’s Abstract: Thirty years ago, work of Witten and Reshetikhin-Turaev activated the study of quantum invariants of links and three-manifolds. A cornerstone of subsequent developments was a three-pronged approach involving 1) quantum field theory (Chern-Simons); 2) rational VOA’s (WZW); and 3) semisimple representation theory of quantum groups. The second and third perspectives have since been extended, to logarithmic VOA’s and related non-semisimple quantum-group categories. I will propose a family of 3d quantum field theories that similarly extend the first perspective to a non-semisimple (and more so, derived) regime. The 3d QFT’s combine Chern-Simons theory with a topologically twisted supersymmetric theory. They support boundary VOA’s whose module categories are dual to modules for Feigin-Tipunin algebras and (correspondingly) to modules for small quantum groups at even roots of unity. The QFT is also compatible with deformations by flat connections, related to the center of quantum groups at roots of unity. This is joint work with T. Creutzig, N. Garner, and N. Geer. I will mention potential connections to recent work of Gukov-Hsin-Nakajima-Park-Pei-Sopenko

Speaker: Kazunobu Maruyoshi Date: 04 May at 11:00 Venue: Zoom Title:  Wilson-‘t Hooft lines as transfer matrices Abstract: In this talk, we will establish a correspondence between a class of Wilson-‘t Hooft lines in 4d N=2 supersymmetric gauge theories described by circular quivers and transfer matrices constructed from dynamical L-operators for trigonometric quantum integrable systems. We compute the vacuum expectation values of the Wilson-‘t Hooft lines in a twisted product space S^1×R_3 by supersymmetric localization and show that they are equal to the Wigner transforms of the transfer matrices. A variant of the AGT correspondence implies an identification of the transfer matrices with Verlinde operators in Toda theory, which we also verify. We explain how these field theory setups are related to 4d Chern-Simons theory via embedding into string theory and dualities. This is based on the collaboration with Toshihiro Ota and Junya Yagi; arXiv:2009.12391[hep-th].

Speaker: Umar Shahzad – SISSA Date: Thursday, April 29 at 4.30pm (Rome) Venue: Zoom Title:  Ricci-flat metrics on the canonical bundles. Abstract: In many cases, a crepant resolution of $\mathbb{C}^3/G$, where $G$ is a finite subgroup of  $SL(3,\mathbb{C})$, happens to be the total space of the canonical bundle over a Kahler manifold of complex dimension $2$. Since any such crepant resolution is a noncompact Calabi-Yau, there is a natural question of constructing a Ricci-flat metric on it. The existence of such metrics was proven by Joyce. In this talk, I will show how to construct a Ricci-flat metric when a crepant resolution is a space $X=tot(K_{M})$, where $K_{M}$ is the canonical bundle over a 2-dimensional Kahler manifold $M$.

Speaker: Michele Graffeo – SISSA Date: Thursday, April 22 at 4.30pm (Rome) Venue: Zoom Title:  Crepant resolutions of symplectic quotient singularities as  moduli spaces of constellations. Abstract: It is a well known fact that, given a finite subgroup G < SL(n,C) (n=2,3) there exists a crepant resolution of the singularity  C^n/G which parametrizes “scheme theoretic orbits of the G action”. In the seminar I will start by explaining the above sentence and then I will show how G-constellations help to fit the above statement ina  more general context. The key words of this seminar are: 1. Group theory 2. Resolution of singularities & birational transformations 3. Combinatorics 4. Commutative algebra 5. Stability conditions (following King) 6. Many many examples.

Speaker: Daniele Dorigoni (Durham U.) Date: 20 April at 14:30 Venue: Zoom Title: An exact integrated correlator in \mathcal{N} = 4 SU(N) SYM Abstract: Between all the magical properties of \mathcal{N} = 4 SU(N) super Yang-Mills perhaps one of the most important is Montonen-Olive electric-magnetic SL(2,Z) duality. In particular this leads to the constraint that observables must be invariant under inversion of the complex YM coupling \tau, i.e. under \tau -> -1 / \tau. In this talk we will focus on one such physical quantity, namely an integrated correlator of four super-conformal primaries of the stress-tensor multiplet.  I will firstly review how this correlator can be computed via supersymmetric localisation on S^4, and then discuss how this quantity can be rewritten in a manifestly SL(2,Z) invariant way for any number of colours N, and any value of the complex YM coupling \tau. Thanks to this novel expression we can explore various different regimes: perturbative SYM, large-N supergravity approximation, large-N ‘t Hooft expansion. All of these regimes are connected via a remarkable Laplace-difference equation relating the SU(N) to the SU(N+1) and SU(N−1) correlators.

Speaker: Matteo Gallet (RICAM) Date: 15 April at 16:30 (Rome) Venue: Zoom Title: Configuration curves of mobile hexapods Abstract: Hexapods are mechanical devices constituted of two rigid bodies, called base and platform, connected by six rods, called legs, which are anchored to the base and the platform via spherical joints. When the legs are in general position and of general length, these objects are rigid. Understanding mobile hexapods is one of the main questions in theoretical kinematics. I this talk, I will report some results from a joint work with Hans-Christian Graf von Bothmer and Josef Schicho concerning a partial classification of some configuration curves of mobile hexapods.

Speaker: Yasunori Lee (Tokyo U., IPMU) Date: 13 April at 11:00 Venue: Zoom Title: Some comments on 6d global gauge anomalies Abstract: Given a G gauge theory, there can be global (non-perturbative) gauge transformations under which the partition function is not invariant. In 6d, relevant cases include G = SU(2), SU(3), and G2, and the old computations utilizing homotopy groups affirmed that the anomalous phases can indeed arise in all three cases. On the other hand, from the modern point of view utilizing bordism groups, there should not be such global gauge anomalies in the first place. In this talk, I will describe how this apparent conflict is resolved by carefully examining the cancellation of perturbative gauge anomalies via 6d Green-Schwarz mechanism.

Speaker: Qiangru Kuang – SISSA Date: Thursday, April 8 at 4.30pm (Rome) Venue: Zoom Title: Excess bundle and blown formula Abstract: In this seminar we will define the excess bundle and use it to prove some formal properties of Gysin homomorphisms. In addition we will extend Gysin homomorphisms to lci morphisms and, as an application, derive the blowup formula.

Speaker: Yasuyuki Hatsuda (Rikkyo U.) Date: 30 March at 11:00 Venue: Zoom Title: Black Hole Quasinormal Modes and Seiberg-Witten Theory Abstract: Black hole perturbation theory leads to ordinary differential equations. Typically the resulting equations belong to Heun’s differential equation and its confluent cousins. The same equations turn out to be found in a “quantization” of Seiberg-Witten theory. I will explain what we can predict from this relation to black hole physics.

Speaker: Vitantonio Peragine – SISSA Date: 25 March at 16:30 Venue: Zoom Title: Cones and Segre classes-Part 2 Abstract: In this talk we will keep reviewing the contents of Chapter 4 of the book. In particular, we will see how Segre classes can be used to define the multiplicity of a variety along a sub-variety, and to compute some invariants of linear systems.

Speaker: Fei Yan (Rutgers) Date: 16 March at 14:30 Venue: Zoom Title: Exact WKB and quantum Seiberg-Witten curves Abstract: I will start with an overview on the exact WKB method for quantum Seiberg-Witten curves of 4d N=2 theories, in particular on different methods to compute the quantum periods. Then I will describe a geometric interpretation of exact WKB in the language of abelianization, and demonstrate the method in the example of 4d N=2 pure SU(3) Yang-Mills.

Speaker: Vitantonio Peragine – SISSA Date: 11 March at 16:30 Venue: Zoom Title: Cones and Segre classes Abstract: In this talk we will try to summarize the contents of Chapter 4 of the book. In particular, we will see how to generalize the notion of Segre class of a vector bundle to the case of an arbitrary cone (aka “singular vector bundle”). In the special case in which the cone is the normal cone to a closed subscheme, we will explain the behavior of the Segre class under the operations of pull-back and push-forward, and in particular the so called “birational invariance”. Segre classes will be used in one of Fulton’s constructions of intersection products. Moreover, they appear naturally in many areas of Algebraic Geometry, and we will try to present some examples of these occurrences.

Speaker: Vadym KURYLENKO (Leuven) Date: 04 March at 16:30 Venue: Zoom Title: Vector bundles and Chern classes Abstract: Next Thursday we will start by going through a part of the appendix. We will discuss things such as bundles, blow-ups and different types of cones. After that we will finish chapter 3 of the book and talk about Chow groups of bundles. In particular, I will sketch a proof of some isomorphisms between Chow groups of the bundle and the base space. We will finish by discussing the Gysin morphisms and considering a number of examples if time permits.

Speaker: Nikolay Bobev (Leuven) Date: 02 March at 14:30 Venue: Zoom Title: The Unreasonable Effectiveness of Higher-Derivative Supergravity in AdS_4 Holography Abstract: I will describe the four-derivative corrections to four-dimensional N=2 minimal gauged supergravity and show that they are controlled by two constants. Interestingly, the solutions of the equations of motion in the two-derivative theory are not modified by the higher-derivative corrections. I will use this to arrive at a general formula for the regularized on-shell action for any asymptotically locally AdS_4 solution of the theory and show how the higher-derivative corrections affect black hole thermodynamic quantities in a universal way. I will employ these results in the context of holography to derive new explicit results for the subleading corrections in the large N expansion of supersymmetric partition functions on various compact manifolds for a large class of three-dimensional SCFTs arising from M2- and M5-branes. I will also briefly discuss possible extensions and generalizations of these results.

Speaker: Asem Abdelraouf (SISSA) Date: 25 February at 4.30pm (Rome) Venue: Zoom Title: Vector bundles and Chern classes Abstract: This is the third seminar on intersection theory. It will cover the first two sections of the third chapter of Fulton’s book. I will explain the construction of the “Chern Class” operations associated to a vector bundle on a scheme, study their properties, and give some examples.

Speaker: Riccardo Ontani (SISSA) Date: 18 February at 4.30pm (Rome) Venue: Zoom Title: Rational equivalence Abstract: This is the second seminar on intersection theory. It will cover the second chapter of Fulton’s book. The aim is to describe how to intersect a divisor with a k-cycle on an integral variety. Some properties of this operation, such as its commutativity in particular cases, will be discussed during the talk.

Speaker: Michele GRAFFEO (SISSA) Date: 11 February at 4.30pm (Rome) Venue: Zoom Title: Rational equivalence Abstract: This is the first seminar on intersection theory. Firstly I will introduce the definition of k- cycles and rational equivalence, then I will explain how proper pushforward and flat pullback act on cycles. During the seminar there will be many examples.

Speaker: Michal Kapustka (IMPAN Krákow and University of Stavanger) Date: 04 February 2021, at 16:30 Venue: Zoom Title: Double mirrors of Calabi-Yau manifolds Abstract: It has been first observed by Rødland in 2000 that two different Calabi-Yau manifolds might have the same mirror. This is related to the fact that in the B-model the Picard Fuchs equation sometimes admits more than one point of Maximal Unipotent Monodromy. By homological mirror symmetry this is also related to the existence of derived equivalent pairs of Calabi-Yau manifolds.  We will discuss this phenomenon on some examples from different points of view. Beside derived categories, these will include, toric degenerations, gauged linear sigma models, stability conditions and motives.

Speaker: Andrea Ricolfi (SISSA) Date: 28 January 2021, at 16:30 Venue: Zoom Title: Mirror symmetry for the quintic 3-fold Abstract: We recall the “definition” of Mirror Symmetry and we show that the quintic 3-fold V has a mirror family V* in the sense of this definition. The main step is to identify complex moduli of V* and Kähler moduli of V locally around the most degenerate boundary points: this is the mirror map. We shall see that the mirror map identifies the A-model and B-model correlation functions, and how this led the physicists in the 90’s to make outstanding predictions on the number of rational curves on V.

Speaker: Lorenzo Sillari (SISSA) Date: 21 January 2021, at 16:30 Venue: Zoom Title: Degenerations of Hodge Structures Abstract: This talk brings many areas together: discrete geometry, statistics, intersection theory, classical algebraic geometry, geometric modeling, and physics. First, we recall the definition of the adjoint of a polytope given by Warren in 1996 in the context of geometric modeling. He defined this polynomial to generalize barycentric coordinates from simplices to arbitrary polytopes. Secondly, we show how this polynomial appears in statistics. It is the numerator of a generating function over all moments of the uniform probability distribution on a polytope. Thirdly, we prove the conjecture that the adjoint is the unique polynomial of minimal degree which vanishes on the non-faces of a simple polytope. In addition, we see that the adjoint appears as the central piece in Segre classes of monomial schemes, and in the study of scattering amplitudes in particle physics. Finally, we observe that adjoints of polytopes are special cases of the classical notion of adjoints of divisors. Since the adjoint of a simple polytope is unique, the corresponding divisors have unique canonical curves. In the case of three-dimensional polytopes, we show that these divisors are either K3 – or elliptic surfaces. This talk is based on joint works with Kristian Ranestad, Boris Shapiro and Bernd Sturmfels.

Speaker: Mario De Marco (SISSA) Date: 14 January 2021, at 16:30 Venue: Zoom Title: (1,2) Yukawa Couplings and Period Map for families of compact Calabi-Yau threefolds Abstract: In this seminar, I will give the definitions of  “(1,2) Yukawa couplings” and of “Period Map” for a family of compact CY threefolds. I will start recalling, briefly, what is a Hodge structure of weight  . Then, I will define the concept of “variation of Hodge structure”: I will first present the case in which the variation is associated with a family of compact Kähler manifolds, and then I will give an intrinsic definition. Finally, I will present two important objects associated with a family of compact CY threefolds: the “(1,2) Yukawa coupling”, and the “Period Map”. Both these objects will play an important role in the formulation of the mirror symmetry conjecture.

Speaker: Veronica Fantini (SISSA/IHES) Date: 17 December 2020, at 16:30 Venue: Zoom Title: Complexified Kähler Moduli Space & Pseudo-holomorphic Curves & Gromov-Witten Invariants and the (1,1) Yukawa Coupling Abstract: Mirror Symmetry has been a breakthrough both in physics and in mathematics: in particular Candelas, De La Ossa, Green, Parkes show that Mirror Symmetry predicts the number of rational curves on quintic 3-folds. In this talk we will introduce the tools needed to set up this counting problem in the symplectic side. First we will introduce pseudo-holomorphic curves and study some properties of their “moduli space”. This will lead us to the definition of Gromov–Witten invariants counting rational curves in a Calabi–Yau 3-fold. Then we will collect the invariants in a generating series (the so called (1,1) Yukawa coupling), whose formal parameter is a point in the complexified Kähler moduli.

Speaker: Solomiya Mizyuk (Virginia Tech) Date: 10 December 2020, at 16:30 Venue: Zoom Title: The Structure of the Kähler Cone & examples of Calabi-Yau manifolds Abstract: TBA

Speaker: Eric Sharpe (Virginia Tech) Date: 11 February 2020, at 14:00 Venue: SISSA Room 137 Title: A proposal for nonabelian mirrors in two-dimensional theories Abstract: In this talk we will describe a proposal for nonabelian mirrors to two-dimensional (2,2) supersymmetric gauge theories, generalizing the Hori-Vafa construction for abelian gauge theories. By applying this to spaces realized as symplectic quotients, one can derive B-twisted Landau-Ginzburg orbifolds whose classical physics encodes quantum cohomology rings of those spaces. The proposal has been checked in a variety of cases, but for sake of time the talk will focus on exploring the proposal in the special case of Grassmannians.

Speaker: Yegor Zenkevich (ITEP and Milano Bicocca University) Date: 21-23-28 January 2020, at 14:30 Venue: SISSA Room 126 Title: Macdonald polynomials technology Abstract: In this minicourse we will study the basics of Macdonald polynomials and quantum Ruijsenaars-Schneider integrable system associated with them. We will start with Schur symmetric functions and then try to understand how they are deformed into Macdonald polynomials. We will also consider some generalizations of the Ruijsenaars-Schneider system and its relations to supersymmetric gauge theories, representation theory and algebraic geometry.

Speaker: Veronica Fantini (SISSA) Date: Thursday 12 December 2019, at 14:30 Venue: IGAP (Institute for Geometry and Physics, the old SISSA building) Title: Kahler Meeting

Speaker: Mauricio Romo (Tsinghua University) Date: Tuesday 27 August 2019, at 14:30 Venue: IGAP (Institute for Geometry and Physics, the old SISSA building) Title: B-branes and Anomalous Gauged Linear Sigma Models Abstract: B-branes on Calabi-Yau (CY) X manifolds are in 1-1 correspondence with objects in the derived category of coherent sheaves. B-branes on Gauged Linear Sigma Models (GLSM) can be roughly interpreted as objects of the derived category of a CY stack. This interpretation can be used to generate equivalences between various categories associated to geometric and nongeometric phases of a CY. I will introduce these concepts and present some results for the case when the stack is not CY (physically, this means the GLSM is anomalous), that can be derived from physics. In particular I will explain how the equivalences between categories associated to phases are modified and comment on some consequence on K-theory using as a guide the example of Hirzebruch-Jung resolutions of cyclic surface singularities.

Speaker: Nadir Fasola (SISSA) Date: Friday 28 June 2019, at 10:00 Venue: IGAP (Institute for Geometry and Physics, the old SISSA building) Title: Nested instantons and punctual nested Hilbert schemes Abstract: Nested Hilbert schemes of points and curves on smooth projective surfaces carry interesting quantities for both geometry and physics. Their virtual fundamental classes have been shown to recover both the virtual classes of SW and reduced stable pair theories, while their obstruction theories can be used to obtain information about VW and reduced DT invariants. We show that the effective SUSY theory of a certain surface defect gives rise to a quiver GLSM which, in a particular case, models punctual nested Hilbert schemes on the complex plane. We will show how the partition function of such a theory naturally computes certain virtual invariants of these moduli spaces and how these results relate to a conjecture of Hausel, Letellier and Rodriguez-Villegas about the cohomology of character varieties.

Speaker: Du Pei (QGM-Caltech) Date: Tuesday, 28 May 2019, at 14:30. Venue: IGAP (Institute for Geometry and Physics, the old SISSA building) Title: Brane Quantization and Representations of DAHA Abstract: Double affine Hecke algebra (DAHA) is closely related to the algebra of line operators in 4d N=2* theory. In this talk, I will show how to use the A-model to the Coulomb branch of this theory to gain insight into the representation category of DAHA.

Speaker: Andrea Ricolfi (SISSA) Date: Thursday, 11 April 2019 at 11:30. Venue: IGAP (Institute for Geometry and Physics, the old SISSA building) Title: Enumerative Geometry of Quot schemes on 3-folds Abstract: We discuss the existence of a virtual fundamental class on Quot schemes of locally free sheaves on complex 3-folds. We compute Behrend’s virtual Euler characteristic of these Quot schemes, which in some cases gives rise to new examples of (higher rank) Donaldson-Thomas invariants. If time permits, we will sketch the parallel theory of motivic invariants.

Speaker: Fabrizio Del Monte (SISSA) Date: Tuesday, 26 March 2019 at 14.30. Venue: IGAP (Institute for Geometry and Physics, the old SISSA building) Title: Class S Theories, Free Fermions and Isomonodromic Deformations Beyond Genus Zero Abstract: Using arguments from Conformal Field Theory, Gamayun Iorgov and Lisovyy provided in 2012 an explicit expression for the tau function of the sixth Painlevé equation as a Fourier transform of Virasoro conformal blocks. This “Kiev formula” has been later generalized to more general isomonodromic problems on the sphere, with both regular and irregular punctures. Further, by using the AGT correspondence, one can see that the isomonodromic tau function is the dual partition function of an appropriate class S theory, for which the Painlevé equations (and generalizations thereof) can be thought of as renormalization group or, for the conformal case, conformal manifold deformation equations. In this talk we will first recap the role of Hitchin systems in the context of class S theories, and what the isomonodromy deformations mean from this perspective. Then, we will show how one can generalize the connection between gauge theory and isomonodromic deformations to the genus one case, by using an approach based on free fermions. This will provide a formula connecting the tau function to the dual partition function of a circular quiver gauge theory with both bifundamental and adjoint hypermultiplets, which has new features absent in the genus zero case.

Speaker: Lothar Göttsche (ICTP) Date: Tuesday, 12 March 2019 at 14:30. Venue: IGAP (Institute for Geometry and Physics, the old SISSA building) Title: Virtual topological invariants of moduli spaces of sheaves on surfaces II Abstract: Using arguments from theoretical physics, Vafa and Witten gave a generating function for the Euler numbers of moduli spaces of rank 2 coherent sheaves on algebraic surfaces. These moduli spaces are in general very singular, but they carry a perfect obstruction theory (they are virtually smooth). This gives virtual versions of many invariants of smooth projective varieties. Such virtual invariants occur everywhere in modern enumerative geometry, like Gromov-Witten invariants and Donaldson Thomas invariants, when attempting to make sense of the predictions from physics. We conjecture that the Vafa-Witten formula is true for the virtual Euler numbers. We confirm this conjecture in many examples. Then we give refinements of the conjecture. Our approach is based on Mochizuki’s formula which reduces virtual intersection numbers on moduli spaces of sheaves to intersection numbers on Hilbert schemes of points.

Speaker: Lothar Göttsche (ICTP) Date: Tuesday, 26 February 2019 at 14:30. Venue: IGAP (Institute for Geometry and Physics, the old SISSA building) Title: Virtual topological invariants of moduli spaces of sheaves on surfaces Abstract: Using arguments from theoretical physics, Vafa and Witten gave a generating function for the Euler numbers of moduli spaces of rank 2 coherent sheaves on algebraic surfaces. These moduli spaces are in general very singular, but they carry a perfect obstruction theory (they are virtually smooth). This gives virtual versions of many invariants of smooth projective varieties. Such virtual invariants occur everywhere in modern enumerative geometry, like Gromov-Witten invariants and Donaldson Thomas invariants, when attempting to make sense of the predictions from physics. We conjecture that the Vafa-Witten formula is true for the virtual Euler numbers. We confirm this conjecture in many examples. Then we give refinements of the conjecture. Our approach is based on Mochizuki’s formula which reduces virtual intersection numbers on moduli spaces of sheaves to intersection numbers on Hilbert schemes of points.

Speaker: Pavlo Gavrylenko (Skolkovo Institute of Science and Technology Moscow) Date: Tuesday, 5 February 2019 at 14:30. Venue: SISSA Room 136 Title: Introduction to Fredholm determinant representation of isomonodromic tau functions, II Abstract: In these two lectures I will try to give an elementary explanation of how one can get the representation of the general isomonodromic tau function on sphere with punctures as the Fredholm determinant of certain operator with matrix-valued integral kernel. I’m going to show the free-fermionic construction of the isomonodromic tau function, introduce generalized Wick theorem, and explain how to get the Fredholm determinant of our interest from the free-fermionic formulas. The second part will be devoted to the study of such obtained Fredholm determinant. It will be rewritten in terms of some projection operators, as matrix Toeplitz determinant, and after all differentiated explicitly in order to reproduce the definition of the Jimbo-Miwa-Ueno tau function.

Speaker: Pavlo Gavrylenko (Skolkovo Institute of Science and Technology Moscow) Date: Tuesday, 29 January 2019 at 14:30. Venue: SISSA Room 136 Title: Introduction to Fredholm determinant representation of isomonodromic tau functions, I Abstract: In these two lectures I will try to give an elementary explanation of how one can get the representation of the general isomonodromic tau function on sphere with punctures as the Fredholm determinant of certain operator with matrix-valued integral kernel. I’m going to show the free-fermionic construction of the isomonodromic tau function, introduce generalized Wick theorem, and explain how to get the Fredholm determinant of our interest from the free-fermionic formulas. The second part will be devoted to the study of such obtained Fredholm determinant. It will be rewritten in terms of some projection operators, as matrix Toeplitz determinant, and after all differentiated explicitly in order to reproduce the definition of the Jimbo-Miwa-Ueno tau function.

Speaker: Andrew Kels (SISSA) Date: Tuesday, 22 January 2019 at 16:00. Venue: SISSA Room 136 Title: On elliptic functions and tau-functions for the discrete Painleve equation of type E8, part II Abstract: In this talk I will introduce the concept of (Ohta-Ramani-Grammaticos) tau-functions on the E8 lattice. I will introduce an elliptic hypergeometric integral, and show how the latter can be used to construct hypergeometric solutions for these tau-functions. This talk is based on the speakers recent work (with Yamazaki), arXiv:1810.12103, which is in turn based on a work of Noumi, arXiv:1604.06869.

Speaker: Andrew Kels (SISSA) Date: Tuesday, 15 January 2019 at 14:30. Venue: SISSA Room 136 Title: On elliptic functions and tau-functions for the discrete Painleve equation of type E8, part I Abstract: In this talk I will introduce the concept of (Ohta-Ramani-Grammaticos) tau-functions on the E8 lattice. I will introduce an elliptic hypergeometric integral, and show how the latter can be used to construct hypergeometric solutions for these tau-functions. This talk is based on the speakers recent work (with Yamazaki), arXiv:1810.12103, which is in turn based on a work of Noumi, arXiv:1604.06869.