## Physikalische Mathematik

### Info

This seminar features recent results at the intersection of high-energy physics, string theory, and geometry and topology.

Prerequisites: None. Everyone is welcome. To receive announcements for this seminar, and for the advertisement of other talks at the intersection of physics and geometry in Heidelberg, you may subscribe to the Mailing List.

Time and Place: Regularly: Monday, 2 p.m.s.t., MATHEMATIKON SR 3
Alternatively: Tuesday or Thursday, 2 p.m.s.t., various locations (or as noted below).

Online seminar: During the coronavirus crisis the seminar is held online in cooperation with LMU Munich and University of Vienna. Please contact the organizers by e-mail to receive the link to the online seminar.

Standard time in the Summer 2021 is Monday, 3 p.m.c.t., unless indicated otherwise.

### Online Vorträge

Date Speaker Title, Abstract
October 25 Michele Schiavina (ETH - Zürich) BV-BFV approach to General Relativity
The BV-BFV formalism is a combination of the BV approach to quantisation of Lagrangian field theories with local symmetries and the BFV approach to quantisation of constrained Hamiltonian systems. It aims to assign compatible bulk-boundary cohomological data to a Lagrangian field theory on a manifold with boundary (and higher codimension strata), in view of a perturbative quantisation scheme that is compatible with cutting and gluing. General Relativity (GR), seen as a field theory, is a very important example to phrase within this setting, and one in which interesting new insight and complications emerge already at the classical level. In this talk I will present a summary of investigations on GR within the BV-BFV formalism, as well as other diffeomorphism-invariant theories, which have given access to rich and nontrivial information about the boundary structure of gravitational models. However, I will argue that the featured examples present unexpected complications for the program of quantisation with boundary (and higher strata). Indeed, I will show how the BV-BFV construction provides a filter to refine the notion of classical equivalence of field theories, which distinguishes theories in terms of their bulk-boundary behaviour, suggesting that some realisations — among the class of classically equivalent ones—may be more suitable for quantisation with boundary. This allows us to differentiate between, e.g., metric and coframe gravity as well as different string theory models and their 1d analogues. This is a summary of joint works with G. Canepa and A.S. Cattaneo.
November 8 Yongbin Ruan (IASM - Zhejiang University) TBA
TBA
November 15 Pavel Putrov (ICTP) TBA
TBA
November 29 Kevin Costello (Perimeter Institute) TBA
TBA
December 6 Mykola Dedushenko (Caltech) TBA
TBA
December 17 Daniel Roggenkamp (Mannheim Universität) TBA
TBA

### Veranstalter

Prof. J. Walcher, walcher@uni-heidelberg.de
Dr. Simone Noja, noja@mathi.uni-heidelberg.de

## Vergangene Semester

### Sommer 2021

Date Speaker Title, Abstract
April 12 Eiichiro Komatsu (Max-Plank-Institute for Astrophysics - Münich) Three tales of de Sitter
Based on works (to appear) with Lorenzo Di Pietro (Trieste) and Victor Gorbenko (Stanford). I will discuss several novel perspectives on quantum field theory in de Sitter spacetime and dS/CFT. The topics to be discussed include 1. Description of the dual CFT that directly produces the late-time correlation functions (rather than the wave functions). 2. New connection to theories in AdS. 3. Unitarity and analyticity of the late-time correlation functions.
April 19 Francesco Benini (SISSA - Trieste) Superconformal Index and Gravitational Path Integral
AdS/CFT provides a consistent non-perturbative definition of quantum gravity in asymptotically AdS spacetimes. Black holes should correspond to ensembles of states in the boundary field theory. By performing a careful analysis of the superconformal index of 4d N=4 SU(N) Super-Yang-Mills theory, with the help of a Bethe Ansatz type formula, we are able to exactly reproduce the Bekenstein-Hawking entropy of BPS black holes in AdS5 x S5. The large N limit exhibits many competing contributions, that we are able to identify with complex saddles of the (putative) gravitational path-integral. Along the way we propose a necessary condition for complex saddles to contribute, based on the size of their non-perturbative corrections. Such a prescription exactly matches the field theory analysis.
April 26 Mathew Bullimore (Durham University) Towards a Mathematical Definition of the 3d Superconformal Index
The aim of this talk is to give a mathematical definition of the superconformal index counting local operators in gauge theories with 3d N = 2 supersymmetry. This can be computed exactly using supersymmetric localisation, which leads to an explicit contour integral formula involving infinite q-Pochhammer symbols. I will explain how this result can be interpreted as the Witten index of a supersymmetric quantum mechanics, or index of a twisted Dirac operator on a certain infinite-dimensional space. To illustrate the essential points, I will focus on a concrete example of supersymmetric Chern-Simons theory.
May 3 Miguel Montero (Harvard University) Cobordisms, Anomalies, and the Swampland
The Swampland program aims at constraining the EFTs that can be consistently coupled to quantum gravity from general principles. In particular cases, absence of global symmetries can lead to strong constraints at low energies. In this talk I will explain how this works and illustrate this in the particular context of supersymmetric theories in d>6. In particular, vanishing of certain cobordism classes requires existence of singular defects, which we call "I-folds". I-fold compactifications can have anomalies, rendering the theory inconsistent. In this way, we find additional constraints in the rank of the gauge group. The resulting constraints establish Coulomb branch string universality: The only consistent supergravities in 8d and 9d are precisely those that arise from string compactifications. I will also briefly comment on applications of these techniques to other interesting setups.
May 10 Lukas Woike (University of Copenhagen) Higher structures from modular categories
Modular categories form a class of categories relevant in the representation theory of Hopf algebras and conformal field theory, where they can be obtained from vertex operator algebras. Thanks to a result of Bartlett, Douglas, Schommer-Pries and Vicary, a semisimple modular category is equivalent to a once-extended (anomalous) three-dimensional topological field theory. Handling non-semisimple modular categories is more involved, especially if one wants to take their non-trivial homological algebra into account. In my talk, I will explain how to give a meaning to the homological algebra of a modular category from the point of view of low-dimensional topology. In particular, I will discuss the construction of differential graded conformal blocks with homotopy coherent mapping class group representations, but also higher multiplicative structures and the connection to modified traces. As a motivating example, I will discuss these higher structures for Dijkgraaf-Witten theory, a discrete version of Chern-Simons theory. This is joint work with Christoph Schweigert.
May 17 Cyril Closset (University of Birmingham) Rank-1 5d SCFTs: Mapping out the U-plane
The simplest non-trivial 5d superconformal field theories are arguably the famous rank-one theories with En global symmetry, first discovered by Seiberg, which are ultraviolet completions of 5d N=1 supersymmetric SU(2) gauge theories with n-1 flavors. In a work to appear with Horia Magureanu, we revisit various aspects of their Coulomb branch physics upon compactification on a circle, using the known En Seiberg-Witten curves. The total space of the En curve fibered over the U-plane (the Coulomb branch) can be described as a rational elliptic surface. These surfaces were classified long ago by Persson and Miranda, and the 5d perspective gives us an interesting physical way to look at that classic mathematical result. I will describe in some detail the various massless points' in the parameters space of the field theory, where the flavor symmetry enhances from the generic U(1)n to a semi-simple Lie algebra, and/or where Argyres-Douglas theories live. I will also discuss the interesting cases when the U-plane is a modular curve. Finally, I will briefly talk about the 5d BPS quivers that can be associated to special points on the U-plane. Our results shed some interesting five-dimensional light on the study of topological strings on local Calabi-Yau threefolds.
June 14 John Huerta (University of Lisbon) Bundle gerbes on Lie supergroups
Bundle gerbes are analogues of line bundles important for conformal field theory, anomalies, and obstruction theory. Among bundle gerbes, a central role is played by the basic bundle gerbe, an essentially unique gerbe on any compact, simple and simply-connected Lie group. In this talk, we describe our work constructing the basic bundle gerbe for a large family of simple Lie supergroups, and show how the basic gerbe on a Lie supergroup decomposes into a tensor product of gerbes on the underlying Lie group and an auxiliary 2-form.
June 21 Johanna Knapp (University of Melbourne) Genus 1 fibered Calabi-Yau 3-folds with 5-sections - A GLSM perspective
Elliptic and genus one fibered Calabi-Yau spaces play a prominent role in string theory and mathematics. In this talk we will discuss examples and properties of a class of genus one fibered Calabi-Yau threefolds with 5-sections. These Calabi-Yaus cannot be constructed by means of toric geometry. One way to obtain them is as vacuum manifolds of gauged linear sigma models (GLSMs) with non-abelian gauge groups. This approach makes it possible to find connections between different genus one fibrations with 5-sections that fit into the framework of homological projective duality. Furthermore we briefly discuss applications in topological string theory and M-/F-theory. This is joint work with Emanuel Scheidegger and Thorsten Schimannek.
July 5 Jan Manschot (Trinity College - Dublin) Topological correlators of N=2* Yang-Mills theory
N=2* Yang-Mills theory is a mass deformation of N=4 Yang-Mills, which preserves N=2 supersymmetry. I will consider the topological twist of this theory with gauge group SU(2) on a smooth, compact four-manifold X. A consistent formulation requires coupling of the theory to a Spin-c structure, which is necessarily non-trivial if X is non-spin. I will discuss the contribution from the Coulomb branch to correlation functions in terms of the low energy effective field theory coupled to a Spin-c structure, and present how these are evaluated using mock modular forms. Upon varying the mass, the correlators can be shown to reproduce correlators of Donaldson-Witten theory as well as Vafa-Witten theory. Based on joint work with Greg Moore, arXiv:2104.06492.

### Winter 2020/21

Date Speaker Title, Abstract
October 12 Ezra Getzler (Northwestern) Gluing local gauge conditions in BV quantum field theory
In supersymmetric sigma models, there is frequently no global choice of Lagrangian submanifold for BV quantization. I will take the superparticle, a toy version of the Green-Schwarz superstring, as my example, and show how to extend the light-cone gauge to the physically relevant part of phase space. This involves extending a formula of Mikhalkov and A. Schwarz that generalizes the prescription of Batalin and Vilkovisky for the construction of the functional integral. This is joint work with S. Pohorence
October 19 Christopher Beem (Oxford) Hall-Littlewood Chiral Rings and Derived Higgs Branches
I will discuss a relatively novel algebraic structure arising in four-dimensional N=2 superconformal field theories: the Hall-Littlewood Chiral Ring. The HLCR is in a refinement of the more familiar Higgs branch chiral ring which encodes the Higgs branch of the moduli space of vacua as an algebraic variety. The HLCR in gauge theories is constructed as the cohomology of a kind of BRST complex, which allows it to be identified with the ring of functions on the derived Higgs branch (in the sense of derived algebraic geometry). I will describe the solution of the HLCR cohomology problem for a large class of Lagrangian theories (the class S theories of type A1), which illustrate a number of key features.
October 26 Theo Johnson-Freyd (Dalhousie) 3+1d topological orders with (only) an emergent fermion
There are exactly two bosonic 3+1d topological orders whose only nontrivial quasiparticle is an emergent fermion (and exactly one whose only nontrivial quasiparticle is an emergent boson). I will explain the meaning of this sentence: I will explain what a "3+1d topological order" is, and how I know that these are the complete list. Time permitting, I will you some details about these specific topological orders, and say what this classification has to do with "minimal modular extensions".
November 2 Matthias Traube (LMU - München) Cardy Algebras, Sewing-Constraints and String-Nets
In this talk I will bring together three different concepts surrounding categorial description for RCFTs. Firstly, Cardy algebras were introduced by Kong in order to describe the genus zero and one part of full open-closed RCFTs. Secondly, string-nets were shown by Kirillov to compute the state of the Reshetikhin-Turaev three dimensional topological field theory. I will bring the two ingredients together, in order to show the third one. That is, I will show how Cardy algebra colored string-nets solve the sewing constraints, thereby giving rise to consistent correlators in full open-close RCFTs and vice versa. The talk is based on the preprint: arXiv:2009.11895.
November 9 Ilka Brunner (LMU - München) Flow Defects and Phases of gauged linear sigma models
I will discuss a special class of defects in two dimensional supersymmetric theories. These "flow defects" connect UV and IR theories. They can in particular be used in the context of gauged linear sigma models, where they connect different phases. Here, they can be regarded as functors between brane categories and provide a new point of view on the "grade restriction rule" initially proposed by Herbst, Hori and Page.
November 16 Owen Gwilliam (Amherst) Centers of higher enveloping algebras and bulk-boundary systems
The universal enveloping algebra of a Lie algebra plays a key role in representation theory (for obvious reasons) and in physics, particularly in encoding symmetries of quantum systems. But it is just one in a family of higher enveloping algebras: each dg Lie algebra g has an enveloping E_n algebra U_n(g). (Here E_n refers to "n-dimensional algebras" in the sense of the little n-disks operad.) This construction admits a nice presentation via factorization algebras, by work of Knudsen, and we will discuss how it relates to symmetries of quantum field theories. We will discuss a model for the *center* of U_n(g) and how this framework encodes the observables of a bulk-boundary system where the bulk is topological BF theory for Lie algebra g and the boundary encodes "topological currents." (This is joint work with Greg Ginot, Brian Williams, and Mahmoud Zeinalian.)
November 30 Jörg Teschner (Universität Hamburg / Desy) Proposal for a Geometric Characterisation of Topological String Partition Functions
We propose a geometric characterisation of the topological string partition functions associated to the local Calabi-Yau (CY) manifolds used in the geometric engineering of d = 4, N = 2 supersymmetric field theories of class S. A quantisation of these CY manifolds defines differential operators called quantum curves. The partition functions are extracted from the isomonodromic tau-functions associated to the quantum curves by expansions of generalised theta series type. It turns out that the partition functions are in one-to-one correspondence with preferred coordinates on the moduli spaces of quantum curves defined using the Exact WKB method. The coordinates defined in this way jump across certain loci in the moduli space. The changes of normalisation of the tau-functions associated to these jumps define a natural line bundle.
December 7 Susanne Reffert (Universität Bern) The Large Charge Expansion
It has become clear in recent years that working in sectors of large global charge of strongly coupled and otherwise inaccessible CFTs leads to important simplifications. It is indeed possible to formulate an effective action in which the large charge appears as a control parameter. In this talk, I will explain the basic notions of the large-charge expansion using the simple example of the O(2) model and then generalize to the non-Abelian case which has a richer structure and exhibits new effects.
December 14 Dmitri Bykov (LMU - Munchen) Sigma models as Gross-Neveu models
I will show that there is a wide class of integrable sigma models, which includes CP^{n-1}, Grassmannian, flag manifold models, that are equivalent to bosonic (and mixed bosonic/fermionic) chiral Gross-Neveu models. The established equivalence allows to effortlessly construct trigonometric/elliptic deformations, provides a new look on the supersymmetric theory and on the cancellation of anomalies in the integrability charges. Using this formalism, we develop criteria for constructing quantum integrable models related to quiver varieties. Based on arXiv:2006.14124 and arXiv:2009.04608.
January 11 Kasia Rejzner (University of York) BV-BFV formalism in perturbative AQFT
BV-BFV formalism is a general framework for quantising gauge theories on manifolds with boundary. In this talk I will present some ideas on how to incorporate this framework into perturbative algebraic quantum field theory (pAQFT), which is a mathematically rigorous approach to QFT. After discussing general ideas, I will focus on their application to the study of asymptotic structure of quantum electrodynamics.
January 25 Simeon Hellerman (IPMU) The Large Quantum Number Expansion: Some Recent Developments
This is a continuation, with a view toward applications, of the talk by Susanne Reffert given on 7/12/2020
February 2 Andrea Brini (University of Sheffield) Quantum Geometry and Physics of Looijenga Pairs
A Looijenga pair is a pair (X,D) with X a smooth complex projective surface and D a singular anticanonical divisor in X. I will describe a series of correspondences relating five different classes of string-theory motivated invariants specified by the geometry of (X,D): - the log Gromov--Witten theory of (X,D), - the Gromov--Witten theory of X twisted by the sum of the dual line bundles to the irreducible components of D, - the open Gromov--Witten theory of special Lagrangians in a toric Calabi--Yau 3-fold determined by (X,D) - the Donaldson--Thomas theory of a symmetric quiver specified by (X,D), and - a class of BPS invariants considered in different contexts by Klemm--Pandharipande, Ionel--Parker, and Labastida--Marino--Ooguri--Vafa. I will also show how the problem of computing all these invariants is closed-form solvable. This is joint work with P. Bousseau (Saclay) and M. van Garrel (Warwick).

### Sommer 2020

Date Speaker Title, Abstract
February 17 Navid Nabijou (Glasgow) Tangent curves, degenerations, and blowups
It is well-known that every smooth plane cubic E supports precisely 9 flex lines. By analogy, we may ask: "How many degree d curves intersect E in a single point?" The problem of calculating such numbers of tangent curves has fascinated enumerative geometers for decades. Despite being an extremely classical and concrete problem, it was not until the advent of Gromov-Witten invariants in the 1990s that a general method was discovered. The resulting theory is incredibly rich, and the curve counts satisfy a suite of remarkable properties, some proven and some still conjectural.
In this talk, I will discuss two distinct projects which take inspiration from this geometry. In the first, joint with Lawrence Barrott, we study the behaviour of tangent curves as the cubic E degenerates to a cycle of lines. Using the machinery of logarithmic Gromov-Witten theory, we obtain detailed information concerning how the tangent curves degenerate along with E. The resulting theorems are purely classical, with no reference to Gromov-Witten theory, but they do not appear to admit a classical proof. In a separate project, joint with Dhruv Ranganathan, we perform iterated blowups of moduli spaces to prove the so-called local-logarithmic conjecture for hyperplane sections; this gives access to a large number of previously unknown enumerative theories.
No prior knowledge of Gromov-Witten theory will be assumed.
Mai 5 Ivo Sachs (Munich) From BV to string theory and back
abstract forthcoming...
Mai 18 Nils Carqueville (Vienna) An introduction to functorial TQFT with defects
abstract forthcoming...
Mai 25 Christoph Chiaffrino (Munich) Planar Quantum A-Infinity Algebras
abstract forthcoming...
June 8 Ingmar Saberi (Heidelberg) Holomorphic field theories and higher symmetries
abstract forthcoming...
June 15 N.N. TBA
abstract forthcoming...
June 22 Simone Noja (Milano) On Some Global and Local Problems in Supergeometry
abstract forthcoming...
June 29 Alberto Cattaneo (Zurich) Hamilton-Jacobi and Quantum Chern-Simons on Cylinders
A quantum field theory on a cylinder (“time” interval x “space”) is an evolution operator. Semiclassically it is given by (the exponential of) the Hamilton-Jacobi action, i.e., the action functional evaluated on the solution with the given initial and final “positions.” In the functional integral formalism, this is easily obtained by observing that one can expand around such a solution. For gauge theories in the BV formalism, however, this expansion is incompatible with a correct splitting of the bulk BV and the boundary BFV fields. In this case, as we will show in several examples, the Hamilton-Jacobi action is recovered from the lowest-order Feynman graphs. In this talk I will focus on Chern-Simons theories. In this case, as for every topological field theory, the evolution operator on the quantization of the reduced phase space is, projectively, the identity operator, so the only important information is a phase, which we will prove to be the Hamilton-Jacobi action by dint of the above method. Actually, we work perturbatively before reduction, so there are even more interesting terms. Moreover, one can play this game with a variety of different polarizations. Among the results, gauged Wess-Zumino-Witten naturally arises from 3d nonabelian Chern-Simons theory with complex polarizations and the Kodaira-Spencer action from 7d abelian Chern-Simons theory with a Calabi-Yau boundary (and appropriate linear-nonlinear polarizations on the boundary components). The latter result was obtained semiclassically by Shatashvili and Gerasimov. In our approach there is a choice of quantization for which there are no further corrections. This talk is based on work in progress with Pavel Mnev and Konstantin Wernli.
July 6 (cancelled) Francesca Ferrari (Trieste) Quantum modularity of 3-manifold invariants
Recently, a new homological invariant of 3-manifolds - which categorifies the Witten-Reshetikhin-Turaev invariant - has been discovered. This is known as the homological block. In this talk, I will explain the importance of quantum modular forms in the study of 3-manifold invariants and describe, in particular, how to predict topological data of 3-manifolds via the quantum modular properties of the associated homological block. The talk will be based on the article 1809.10148 and work in progress with Cheng, Chun, Feigin, Gukov, and Harrison.
July 13 Thomas Creutzig (Alberta) Algebraic blow-up
I will introduce a novel translation functor for W-algebras. W-algebras are related via the AGT-correspondence to four dimensional gauge theories and instanton partition functions of the gauge theories enjoy blow-up equations. I will explain that the translation functor provides a derivation of these blow-up equations from a W-algebra perspective. Moreover there are many more algebraic blow-up equations that should also have a geometric interpretation.
July 20 Thorsten Schimmanek (Wien) The quantum geometry of genus one fibered Calabi-Yau threefolds
The talk will consist of three parts. First we are going to review basic notions from topological string theory on general Calabi-Yau threefolds, most notably Gopakumar-Vafa invariants, the stringy Kähler moduli space and topological B-branes.
In the second part we are then equipped to discuss the beautiful relation between the enumerative geometry of genus one fibered Calabi-Yau threefolds, certain auto-equivalences of the associated brane categories as well as the theory of weak Jacobi forms.
Finally, we will describe how a particular subset of the enumerative invariants encodes the structure of the fibration and classical techniques from mirror symmetry can be applied to determine the types of reducible fibers.
If time permits, we will discuss an example where the corresponding physics via F-theory can then be used to identify transitions to singular Calabi-Yau threefolds with enumerative geometry related to $\Gamma(2)$.

### Winter 2019/20

Date Speaker Title, Abstract
September 16 Vivek Singh (Warschau) Chern-Simons invariants, Multi-boundary Entanglement and Knot-Quiver Correspondence
First, I will give a brief introduction to knot theory and its connection to Chern-Simons quantum field theory. Then I will briefly review my research works and particular focus on the recent developments on entanglement entropy associated with multi-boundary link complements. My plan to highlight the interesting relationship between the volume of hyperbolic link complements to non-zero entanglement negativity. In the end, I will briefly discuss Knot-Quiver correspondence.
Wednesday, October 2
Hörsaal, 2 p.m.
Mauricio Romo (Beijing) Enumerative invariants from exponential networks
I will review some aspects of exponential networks associated to CY 3-folds described by conic bundles. In particular I will focus on the relation between exponential networks and the so-called nonabelianization map which serves as a tool to connect counting of 3d-5d BPS states with certain classes of generalized DT invariants.
October 14 N.N. TBA
abstract forthcoming
October 21 Andrew Bruce (Luxemburg) (Z2)n-manifolds: recent developments and ongoing work
A locally ringed space approach to coloured supermanifolds was recently initiated by Grabowski and Poncin together with a collection of collaborators. We must comment that the theory, while a natural extension of the theory of supermanifolds, is not always straightforward nor trivial. While the basic ideas are now in place, we will report on some recent developments in the general theory. I will then discuss applications of (Z2)n-manifolds to mixed symmetry tensors as found in string theory.
October 28
(Philosophenweg)
Alex Altland (Köln) TBA
abstract forthcoming
November 4 N.N. TBA
abstract forthcoming
November 11 N.N. TBA
abstract forthcoming
December 16 Paul Norbury (Melbourne/München) Enumerative geometry via the moduli space of super Riemann surfaces
It was conjectured by Witten and proven by Kontsevich that a generating function for intersection numbers on the moduli space of curves is a tau function of the KdV hierarchy, now known as the Kontsevich-Witten tau function. Mirzakhani reproved this theorem via the study of Weil-Petersson volumes of moduli spaces of hyperbolic surfaces. In this lecture I will describe another collection of intersection numbers on the moduli space of curves whose generating function is a tau function of the KdV hierarchy. The proof uses an analogue of Mirzakhani's argument applied to the moduli space of super Riemann surfaces due to recent work of Stanford and Witten. This appearance of the moduli space of super Riemann surfaces to solve a problem over the classical moduli space is deep and surprising.
Tuesday, January 7
2:30pm, SR 2
Sara Tukachinsky (IAS) Defining open Gromov-Witten invariants via Fukaya $A_\infty$ algebras
Given a symplectic manifold $X$, let $L$ be a Lagrangian submanifold. A Fukaya $A_\infty$ algebra associated with $L$ is a deformation of the dg algebra of smooth differential forms on $L$ by pseudoholomorphic disks. This language turns out to be useful in open Gromov-Witten theory, as demonstrated in joint works with Jake Solomon. In particular, we develop a way of defining disk-counting invariants and show that they satisfy some general properties. For the projective space $(X,L)=({\mathbb C\mathbb P}^n, {\mathbb R\mathbb P}^n)$, these properties are sufficient to calculate all invariants explicitly. No prior knowledge of the notions above will be assumed.
Wednesday, January 8
11:15am, SR 9
Sara Tukachinsky (IAS) Quantum product on relative cohomology
The quantum product on the cohomology of a symplectic manifold X is a deformation of the cup product, or wedge product in the de Rham model. The deformation is given by adding contributions from pseudoholomorphic spheres. Adding a Lagrangian submanifold L, one might consider the relative cohomology $H^*(X,L)$. In a joint work with Jake Solomon, we define a quantum product on $H^*(X,L)$ that combines deformations of the wedge products of differential forms on $X$ and $L$, with corrections coming from pseudoholomorphic spheres as well as disks with boundary conditions in $L$. The associativity of this product is equivalent to the open WDVV equations, a PDE in the generating functions of the closed and open Gromov-Witten invariants.
January 13 Pierrick Bousseau (ETH Zürich) Quasimodular forms from Betti number
This talk will be about refined curve counting on local P2, the noncompact Calabi-Yau 3-fold total space of the canonical line bundle of the projective plane. I will explain how to construct quasimodular forms starting from Betti numbers of moduli spaces of one-dimensional coherent sheaves on P2. This gives a proof of some stringy predictions about the refined topological string theory of local P2 in the Nekrasov-Shatashvili limit. Partly based on work in progress with Honglu Fan, Shuai Guo, and Longting Wu.
Thursday, February 6
2:15pm, SR 5
Francesca Carocci (Edinburgh) Donaldson-Thomas invariants from numbers to MMHM (and back)
Donaldson-Thomas (DT) invariants were first introduced by Donaldson and Thomas as an enumerative theory for the Hilbert scheme of curves in Calabi-Yau 3-folds. In the past 20 years, they have been generalised and refined in various ways. In this talk, we concentrate on the DT invariants, first considered by Katz, arising from the mathematical incarnation of the moduli space of D-branes supported on curves. I will explain how these invariants are expected to be related to the genus 0 Gromov-Witten invariants, and then talk about a possible way to prove such a relation passing from numerical to categorical DT invariants. The latter will be a good excuse to talk about BPS Lie algebras for quiver with potentials and their use in DT theory.

### Sommer 2019

Date Speaker Title, Abstract
March 18 Luca Battistella (Bonn) Reduced Gromov-Witten theory in genus one and singular curves
The moduli space of genus 0 stable maps to projective space is a smooth orbifold. The quantum hyperplane principle allows us to compute the invariants of a hypersurface as twisted invariants of projective space, hence e.g. by torus localisation. In higher genus the moduli space can be arbitrarily singular. The genus 1 case has been particularly studied: J. Li, R. Vakil, and A. Zinger have desingularised the main component, defined reduced invariants, and compared them with standard ones, providing the first mathematical proof of the BCOV mirror symmetry prediction. Ten years later, we understand their construction in terms of log geometry and singular (worse than nodal) curves, thanks to work of D. Ranganathan, K. Santos-Parker, and J. Wise. I will describe some results in this direction, jointly obtained with F. Carocci and C. Manolache, and N. Nabijou and D. Ranganathan.
Tuesday, April 23
Hörsaal, 2 p.m.
Martin Cederwall (Göteborg) Pure spinors and supersymmetry
I will describe how pure spinors, suitably defined, arise from traditional superspace. In cases of maximal supersymmetry, such as D=10 super-Yang-Mills theory and D=11 supergravity, pure spinor superspace solves the old problem with off-shell formulations, and gives Batalin-Vilkovisky actions. I will also mention some related applications of pure spinors and minimal orbits.
April 29 N.N. T.B.A.
abstract forthcoming
May 6 Xujia Chen (Stony Brook) Bounding chains for Welschinger's invariants
abstract forthcoming
May 13 no seminar T.B.A.
abstract forthcoming
May 20 Francesca Ferrari (Trieste) False Theta Functions, Log VOA's and 3-Manifold Invariants
Since the 1980s, the study of invariants of 3-dimensional manifolds has benefited from the connections between topology, physics and number theory. Recently, a new topological invariant that categorifies the Witten-Reshetikhin-Turaev invariant has been discovered. This is known as the homological block. When the 3-manifold is a Seifert manifold given by a negative-definite plumbing the homological block turned out to be related to false theta functions and characters of logarithmic VOA's. In this talk, I describe the relations between this topological invariant, certain number theoretical objects and the representation theory of logarithmic VOA's.
Tuesday, May 21
Philosophenweg 19, 2pm
Du Pei (Aarhus/Caltech) Taming the Non-Unitary Zoo with Wild Higgs Bundles
We propose a new link between the geometry of moduli spaces of Higgs bundles and quantum topology. The construction goes through a class of four-dimensional quantum field theories that are said to satisfy "property F". Each such theory gives rise to a family of modular tensor categories, whose algebraic structures are intimately related to the geometry of the Coulomb branch. This is based on joint work with Mykola Dedushenko, Sergei Gukov, Hiraku Nakajima and Ke Ye.
Tuesday, June 4
Hörsaal, 2pm
Marc-Antoine Fiset (Oxford) Interpolating stringy geometry: from Spin(7) and G2 to Virasoro N=2
Spectral flow, topological twists, chiral rings related to a refinement of the de Rham cohomology and to marginal deformations, spacetime supersymmetry, mirror symmetry. These are some examples of features arising from the N=2 Virasoro chiral algebra of superstrings compactified on Calabi-Yau manifolds. To various degrees of certainty, similar features were also established for compactifications on 7- and 8-dimensional manifolds with exceptional holonomy group $G_2$ and Spin(7) respectively. In this talk, I will explain that these are more than analogies: I will flesh out the underlying symmetry connecting exceptional holonomy to Calabi-Yau surfaces (K3) via a limiting process.
June 10 No Seminar (Whit Monday)
June 17 Ezra Getzler (Northwestern) The Batalin-Vilkovisky formalism and supersymmetric particles I
Recently, I have been studying one-dimensional toy models of the superstring within the BV formalism: these are known respectively as the spinning particle (analogous to the GSO superstring) and the Superparticle (analogous to the Green-Schwarz superstring). The spinning particle turns out to be an AKSZ model, and exhibits some very interesting pathologies: it appears to be the first model to have been investigated that exhibits BV cohomology in all negative degrees. By contrast, the superparticle has a very well-behaved BV cohomology. The action for the superparticle contains a term of topological type (the dimensional reduction of a super-WZWN term), and to handle this, we borrow some ideas from Sullivan's approach to rational homotopy theory. In my third talk, I will turn to quantization: I will show how to generalize Lagrangians in the BV formalism to "flexible Lagrangians", defined chart-by-chart with families of homotopies on the intersections of charts. The work on the superparticle is joint with my graduate student Sean Pohorence.
Tuesday, June 18
SR 00. 200, 2pm
Ezra Getzler (Northwestern) The Batalin-Vilkovisky formalism and supersymmetric particles II
Recently, I have been studying one-dimensional toy models of the superstring within the BV formalism: these are known respectively as the spinning particle (analogous to the GSO superstring) and the superparticle (analogous to the Green-Schwarz superstring). The spinning particle turns out to be an AKSZ model, and exhibits some very interesting pathologies: it appears to be the first model to have been investigated that exhibits BV cohomology in all negative degrees. By contrast, the superparticle has a very well-behaved BV cohomology. The action for the superparticle contains a term of topological type (the dimensional reduction of a super-WZWN term), and to handle this, we borrow some ideas from Sullivan's approach to rational homotopy theory. In my third talk, I will turn to quantization: I will show how to generalize Lagrangians in the BV formalism to "flexible Lagrangians", defined chart-by-chart with families of homotopies on the intersections of charts. The work on the superparticle is joint with my graduate student Sean Pohorence.
Wednesday, June 19
SR 9, 9am ct
Ezra Getzler (Northwestern) The Batalin-Vilkovisky formalism and supersymmetric particles III
Recently, I have been studying one-dimensional toy models of the superstring within the BV formalism: these are known respectively as the spinning particle (analogous to the GSO superstring) and the superparticle (analogous to the Green-Schwarz superstring). The spinning particle turns out to be an AKSZ model, and exhibits some very interesting pathologies: it appears to be the first model to have been investigated that exhibits BV cohomology in all negative degrees. By contrast, the superparticle has a very well-behaved BV cohomology. The action for the superparticle contains a term of topological type (the dimensional reduction of a super-WZWN term), and to handle this, we borrow some ideas from Sullivan's approach to rational homotopy theory. In my third talk, I will turn to quantization: I will show how to generalize Lagrangians in the BV formalism to "flexible Lagrangians", defined chart-by-chart with families of homotopies on the intersections of charts. The work on the superparticle is joint with my graduate student Sean Pohorence.
Wednesday, August 21
SR 3, 2 p.m.
Alexey Basalaev (Skoltech) Open WDVV equation and ADE singularities F-manifolds
abstract forthcoming

### Winter 2018/19

Date Speaker Title, Abstract
Friday, January 11
SR 1, 2:00 p.m.s.t.
Philsang Yoo (Yale) Symplectic duality and geometric Langlands duality
It is believed that certain physical dualities of 3d N=4 theory and 4d N=4 gauge theory underlie symplectic duality and geometric Langlands duality, respectively. By setting up a mathematical model for those physical theories and studying their physical relationship as studied by Gaiotto and Witten in our framework, we suggest a program that finds new relationships between symplectic duality and geometric Langlands theory. In this talk, we aim to provide an introduction to our program. This is based on a joint work in progress with Justin Hilburn.
January 28 Lóránt Szegedy (Bonn) Combinatorial models of r-spin surfaces
Following a general introduction to topological field theories on $r$-spin surfaces, wee give a combinatorial model for $r$-spin surfaces with parametrised boundary, following Nowak (2015). We then use the combinatorial model to give a state sum construction of two-dimensional topological field theories on r-spin surfaces. An example of such a topological field theory computes the Arf-invariant of an $r$-spin surface as introduced by Randall-Willians, and Geiges and Gonazalo.

### Sommer 2018

Date Speaker Title, Abstract
Tuesday, March 13
2:00 p.m.s.t.
Piotr Kucharski (Uppsala) Extremal $A$-polynomials of knots
In my talk, I will explain in an elementary way what extremal $A$-polynomials are, and show how to obtain them from the usual ($a$-deformed) $A$-polynomials of knots. Then I will try to reward your attention by showing applications in knot theory, string theory, and contact geometry.
Friday, March 16
SR 3, 2:00 p.m.s.t.
Piotr Kucharski (Uppsala) Knots–Quivers Correspondence
This is a continuation of Piotr's previous talk, which will focus on recent work with Sułkowski, Reinecke, and Stošić.
Monday, July 9
SR C, 2:00 p.m.s.t.
Eric Sharpe (Virginia) A proposal for nonabelian mirrors in two-dimensional theories
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. Specifically, we will describe a construction of B-twisted Landau-Ginzburg orbifolds whose classical physics encodes Coulomb branch relations (quantum cohomology), excluded loci, and correlation functions of A-twisted gauge theories. The proposal has been checked in a wide variety of cases, but the talk will focus on exploring the proposal in two examples: Grassmannians (constructed as ${\it U}(k)$ gauge theories with fundamental matter), and SO(2k) gauge theories. If time permits, we will also discuss how this mirror proposal can be applied to test and refine recent predictions for IR behavior of pure supersymmetric ${\it SU}(n)$ gauge theories in two dimensions.
Monday, July 16, 2pm
Philosophenweg 19!
Michael Gutperle (UCLA) Holographic description of 5-dimensional conformal field theories.
This is the abstract.
Wednesday, August 8
SR 3, 2:00 p.m.s.t.
Makiko Mase (Tokyo Metropolitan University) On duality of families of K3 surfaces
Since an introduction to mathematical world from physicians, many concepts of mirror symmetry has been studied. In my talk, we will discuss a mirror of polytopes due to Batyrev, and that of Picard lattices of families of K3 surfaces due to Dolgachev. We conclude that these mirror symmetries correspond when we consider families that are obtained by a strange duality of bimodal singularities due to Ebeling-Takahashi, and Ebeling-Ploog.

### Winter 2017/18

Date Speaker Title, Abstract
September 25 Du Pei (QGM Aarhus and Caltech) Can one hear the shape of a drum?
Much like harmonics of musical instruments, spectra of quantum systems contain wealth of interesting information. In this talk, I will introduce new invariants of three- and four-manifolds using BPS spectra of quantum field theories. While most of them are completely novel, some of the new invariants categorify well-known old invariants such as the WRT invariant of 3-manifolds and the Donaldson invariant of 4-manifolds. This talk is based on arXiv:1701.06567 and ongoing work with Sergei Gukov, Pavel Putrov and Cumrun Vafa.
October 16 Laura Schaposnik (UIC) On Cayley and Langlands type correspondences for Higgs bundles.
The Hitchin fibration is a natural tool through which one can understand the moduli space of Higgs bundles and its interesting subspaces (branes). After reviewing the type of questions and methods considered in the area, we shall dedicate this talk to the study of certain branes which lie completely inside the singular fibres of the Hitchin fibrations. Through Cayley and Langlands type correspondences, we shall provide a geometric description of these objects, and consider the implications of our methods in the context of representation theory, Langlands duality, and within a more generic study of symmetries on moduli spaces.
November 6 Natalie Paquette (Caltech) Dual boundary conditions in 3d N=2 QFTs
We will study half-BPS boundary conditions in 3d N=2 field theories that preserve 2d (0,2) supersymmetry on the boundary. We will construct simple boundary conditions and study their local operator content using a quantity called the half-index. Using the half-index as a guide, we study the actions of a variety of 3d dualities on the boundary conditions, including level-rank duality, mirror symmetry, and Seiberg-like duality. Identifying the dual pairs of boundary conditions, in turn, helps lead to the construction of duality interfaces. This talk is based on work in progress with T. Dimofte and D. Gaiotto.
November 7
Tuesday, 2pm, 02/104
Arnav Tripathy (Harvard) Special cycles and BPS jumping loci
I'll sketch an attempt to bring the theory of special cycles, a deep part of number theory, into the domain of supersymmetric string compactifications. I'll describe a construction based on jumping loci for BPS state counts -- a separate phenomenon from the better-known wall-crossing! -- and explain in what cases these jumping loci generalize some parts of the theory of special cycles. Finally, I'll conclude with a host of physical and mathematical conjectures raised by this line of investigation.
November 20 Minhyong Kim (Oxford) Gauge theory in arithmetic geometry I
Number-theorists have been implicitly using gauge theory for perhaps 350 years, and explicitly for about 50 years. However, they did not use the terminology at all. I will review some of this story and explain why it's a good idea do to so now. In particular, we will describe some of the ideas of Diophantine gauge theory and arithmetic Chern-Simons theory.
November 21
Tuesday, 2pm, 02/104
Minhyong Kim (Oxford) Gauge theory in arithmetic geometry II
This is a continuation of the previous lecture
December 4 Lukas Hahn (Heidelberg) Super Riemann surfaces and their moduli
Abstract forthcoming
January 15 Helge Ruddat (Mainz) Tropical construction of Lagrangian submanifolds
Homological mirror symmetry suggests that complex submanifolds of a Calabi-Yau manifold match Lagrangian submanifolds of the mirror dual Calabi-Yau. In practice, a maximal degeneration needs to be chosen and then the submanifolds are identified by a duality of their degeneration data which is tropical geometry. Cheuk Yu Mak and I carry out this construction for lines on the quintic threefold which become spherical Lagrangians in the quintic mirror. Our construction applies more generally for Calabi-Yau threefolds in the Batyrev construction and probably even more generally at some point in the future. Quite surprisingly, many exotic Lagrangian threefolds can be constructed this way, many for the first time in a compact symplectic 6-manifold.
February 19 Guglielmo Lockhart (Amsterdam) Universal features of 6d self-dual string CFTs
BPS strings are the fundamental objects on the tensor branch of 6d $(1,0)$ SCFTs. They can be thought of as the instantons of the 6d gauge group, and are the building blocks for computing the "instanton piece" of the $\mathbb R^4\times T^2$ partition function of the parent 6d SCFT. The goal of this talk is to rephrase their properties from the point of view of a worldsheet $\mathcal N=(0,4)$ NLSM. This reveals that, despite their superficial differences, self-dual strings of arbitrary 6d SCFTs share many universal features. Along the way, this leads to a better understanding of the flavor symmetry of the parent 6d SCFTs. Moreover, the constraints from modularity and these universal features are strong enough that one can fix the elliptic genus of one self-dual string for a wide variety of SCFTs.
February 26 Jan Swoboda (München) The Higgs bundle moduli space and its asymptotic geometry
The Theorem of Narasimhan and Seshadri states a correspondence between the moduli space of stable holomorphic vector bundles over a Riemann surface $X$ and that of irreducible unitary connections of constant central curvature. This is one instance of a much more general correspondence due to Kobayashi and Hitchin. Higgs bundles come into play when the compact Lie group $\operatorname{SU}(r)$ is replaced by $\operatorname{SL}(r,\mathbb C)$. A suitable generalization of the constant central curvature connections in the former case is found in the solutions to Hitchin's self-duality equations. Due to the noncompactness of the Higgs bundle moduli space, a set of new questions revolving around its `geometry at infinity'' arises. In this talk I will focus on the asymptotics of the natural $L^2$-metric $G_{L^2}$ on the moduli space $\mathcal M$ of rank-$2$ Higgs bundles. I will show that on the regular part of the Hitchin fibration $(A,\Phi)\mapsto\det\Phi$ this metric is well-approximated by the semiflat metric $G_{\operatorname{sf}}$ coming from the completely integrable system on $\mathcal M$. This also reveals the asymptotically conic structure of $G_{L^2}$, with (generic) fibres of the above fibration being asymptotically flat tori. This result confirms some aspects of a more general conjectural picture made by Gaiotto, Moore and Neitzke. Its proof is based on a detailed understanding of the ends structure of $\mathcal M$. The analytic methods used here in addition yield a complete asymptotic expansion of the difference $G_{L^2}-G_{\operatorname{sf}}$ between the two metrics, with leading order term having polynomial decay and a rather explicit description. The results presented here are from recent joint work with Rafe Mazzeo, Hartmut Weiss and Frederik Witt.

### Sommer 2017

Date Speaker Title, Abstract
April 10 Ingmar Saberi (Heidelberg) Holographic lattice field theories
Recent developments in tensor network models (which are, roughly speaking, quantum circuits designed to produce analogues of the ground state in a conformal field theory) have led to speculation that such networks provide a natural discretization of the AdS/CFT correspondence. This raises many questions: just to begin, is there any sort of lattice field theory model underlying this connection? And how much of the usual AdS/CFT dictionary really makes sense in a discrete setting? I'll describe some recent work that proposes a setting in which such questions can perhaps be addressed: a discrete spacetime whose bulk isometries nevertheless match its boundary conformal symmetries. Many of the first steps in the AdS/CFT dictionary carry over without much alteration to lattice field theories in this background, and one can even consider natural analogues of BTZ black hole geometries.
Tuesday, April 11
2 p.m.s.t.
Michael Gekhtman (Notre Dame) Higher pentagram maps via cluster mutations and networks on surfaces
The pentagram map that associates to a projective polygon a new one formed by intersections of short diagonals was introduced by R. Schwartz and was shown to be integrable by V. Ovsienko, R. Schwartz and S. Tabachnikov. M. Glick demonstrated that the pentagram map can be put into the framework of the theory of cluster algebras, a new and rapidly developing area with many exciting connections to diverse fields of mathematics. In this talk I will explain that one possible family of higher-dimensional generalizations of the pentagram map is a family of discrete integrable systems intrinsic to a certain class of cluster algebras that are related to weighted directed networks on a torus and a cylinder. After presenting necessary background information on Poisson geometry of cluster algebras, I will show how all ingredients necessary for integrability - Poisson brackets, integrals of motion - can be recovered from combinatorics of a network. The talk is based on a joint project with M. Shapiro, S. Tabachnikov and A. Vainshtein.
May 29 Jon Keating (Bristol) The Riemann hypothesis and physics — a perspective
I will give an overview of some connections, mostly speculative, between the Riemann Hypothesis, random matrix theory, and quantum chaos.
July 3 Alex Turzillo (Caltech) Spin TQFT and fermionic gapped phases
I will discuss state sum constructions of two-dimensional Spin TQFTs and their G-equivariant generalizations. These models are related to tensor network descriptions of ground states of fermionic topological matter systems. We will revisit the classification of fermionic short range entangled phases enriched by a finite symmetry G and derive a group law for their stacking.
July 10 Markus Banagl (Heidelberg) Intersection homology and the conifold transition
Abstract forthcoming

### Winter 2016/17

Date Speaker Title, Abstract
November 28 John Alexander Cruz Morales (Bonn) On Stokes matrices for Frobenius manifolds
In this talk we will discuss how to compute the Stokes matrices for some semisimple Frobenius manifolds by using the so-called monodromy identity. In addition, we want to discuss the case when we get integral matrices and their relations with mirror symmetry. This is part of an ongoing project with M. Smirnov and previous joint work with Marius van der Put.
Frobenius manifolds, introduced by Dubrovin, are objects which know about many different things in mathematics, for example, the enumeration of rational curves, or the list of platonic solids. We will introduce a generalization of Frobenius manifolds which know about real (or open) enumerative geometry. What else do these extended Frobenius Manifolds know?
December 12 Pietro Longhi (Uppsala) Probing the geometry of BPS states with spectral networks
In presence of defects the Hilbert space of a quantum field theory can change in interesting ways. Surface defects in 4d N=2 theories introduce a class of 2d-4d BPS states, which the original 4d theory does not possess. For theories of class S, spectral networks count 2d-4d BPS states, and through the 2d-4d wall-crossing phenomenon the 4d BPS spectrum can be obtained. Adopting this physical viewpoint on spectral networks, I will illustrate some recent and ongoing developments based on this framework, with applications to the study of 2d (2,2) BPS spectra, and of 4d N=2 BPS monodromies.
Tuesday, December 13 Michael Bleher (Heidelberg) Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory
In the last decade it was realized that supersymmetric boundary conditions in super Yang-Mills theories can provide invaluable insight into several areas of current mathematical research. Motivated especially by their appearance in a categorification of knot invariants, I will give a short overview on half-BPS boundary conditions in 4d N=4 SYM theory. Of particular interest is the Nahm-pole boundary condition and the main goal will be to review the corresponding moduli space of supersymmetric vacua.
January 30 Chris Elliott (IHÉS) Algebraic Structures for Kapustin-Witten Twisted Gauge Theories
Topological twisting is a technique for producing topological field theories from supersymmetric field theories -- one exciting application is Kapustin and Witten's 2006 discovery that the categories appearing in the geometric Langlands conjecture can be obtained as topological twists of N=4 supersymmetric gauge theories, and that these two categories are interchanged by S-duality. There are, however, several incompatibilities between Kapustin and Witten's construction and the geometric representation theory literature. First, their techniques do not produce the right algebraic structures on the moduli spaces appearing in geometric Langlands, and secondly, their construction doesn't explain the singular support conditions Arinkin and Gaitsgory introduced in order to make the geometric Langlands correspondence possible. In this talk I'll explain joint work with Philsang Yoo addressing both of these issues: how to understand topological twisting in (derived) algebraic geometry, and how to interpret singular support conditions as arising from the choice of a vacuum state.
Wednesday, February 1
SR B, 2 p.m.s.t.
Chris Elliott (IHÉS) An Introduction to the Batalin-Vilkovisky Quantization Formalism
Abstract forthcoming
February 13 Steven Sivek (Bonn) The augmentation category of a Legendrian knot
Given a Legendrian knot in R3, Shende, Treumann, and Zaslow defined a category of constructible sheaves on the plane with singular support controlled by the front projection of the knot. This category turns out to be equivalent to a unital A category, called the augmentation category, which is defined in terms of a holomorphic curve invariant (Legendrian contact homology) of the knot. In this talk I will describe the construction of these categories and outline a proof that they are equivalent. This is joint work with Lenny Ng, Dan Rutherford, Vivek Shende, and Eric Zaslow.
February 20 Ben Knudsen (Harvard) A local-to-global approach to configuration spaces
I will describe how ideas borrowed from functorial field theory, the theory of chiral algebras, and BV theory may be profitably adapted to the purely topological problem of calculating Betti numbers of configuration spaces. These methods lead to improvements of classical results, a wealth of computations, and a new and combinatorial proof of homological stability.
February 27 Owen Gwilliam (Bonn) Chiral differential operators and the curved beta-gamma system
Chiral differential operators (CDOs) are a vertex algebra analog of the associative algebra of differential operators. They were originally introduced by mathematicians using just sheaf theory and vertex algebraic machinery. Later, Witten explained how CDOs on a complex manifold X arise as the perturbative part of the curved beta-gamma system with target X. I will describe recent work with Gorbounov and Williams in which we construct the BV quantization of this theory and use a combination of factorization algebras and formal geometry to recover CDOs. At the end, I hope to discuss how the techniques we developed apply to a broad class of nonlinear sigma models, including source manifolds of higher dimension.
March 20 Michele Cirafici (I.S.T. Lisboa) Framed BPS quivers and line defects
I will discuss a certain class of line defects in four dimensional supersymmetric theories with N=2. I will show that many properties of these operators can be rephrased in terms of quiver representation theory. In particular one can study BPS invariants of a new kind, the so-called framed BPS states, which correspond to bound states of ordinary BPS states with the defect. I will discuss how these invariants arise from framed quivers. Time permitting I will also discuss a formalism to study these quantities based on cluster algebras.

### Sommer 2016

Date Speaker Title, Abstract
April 25 Zhentao Lu (Oxford) Quantum sheaf cohomology on Grassmannians
I will give a brief introduction to quantum sheaf cohomology and correlation functions. I will talk about the computation of the classical cohomology ring $\sum H^q(X, \wedge^p E^*)$ for a vector bundle $E$ over the Grassmannian $X$, also known as the polymology. Then I will talk about the conjectural quantum sheaf cohomology derived from the Coulomb branch argument of the physics theory of gauged linear sigma models.
May 2 Seung-Joo Lee (Virginia Tech) Witten Index for Noncompact Dynamics
Many of the gauged dynamics motivated by string theory come with gapless asymptotic directions. In this talk, we focus on d=1 GLSM's of such kind and their Witten indices, having in mind of the associated D-brane bound state problems. Upon illustrating by examples that twisted partition functions can be misleading, we proceed to explore how physical Witten indices can sometimes be embedded therein. There arise further subtleties when gapless continuum sectors come from a gauge multiplet, as in non-primitive quiver or pure Yang-Mills theories. For such theories, the twisted partition functions tend to involve fractional expressions. We point out that these are tied to the notion of rational invariant in the wall-crossing formulae, offering a general mechanism of extracting the Witten index directly from the twisted partition function.
May 12
Thursday, 2pm, SR 11
Satoshi Nawata (Caltech/Aarhus) Various formulations of knot homology
This lecture is preceded by a talk entitled "Knot Homology from String Theory" in the Oberseminar Conformal field theory on Wednesday May 11, 2pm, Philosophenweg 12
May 23 N.N. T.B.A.
June 13 Nicolò Piazzalunga (Trieste/Madrid) Real Topological String Theory
June 23 Thursday, 2pm
RZ Statistik 02.104
Matt Young (Hong Kong) Algebra and geometry of orientifold Donaldson-Thomas theory
I will give an overview of the orientifold Donaldson-Thomas theory of quivers. Roughly speaking, orientifold Donaldson-Thomas theory is a virtual counting theory for principal G-bundles in three dimensional Calabi-Yau categories, where G is an orthogonal or symplectic group. This theory is best formulated in terms of geometrically defined representations of cohomological Hall algebras. I will explain how this set-up leads naturally to a categorification of the orientifold wall-crossing formula appearing in the string theory literature, a proof of the orientifold variant of the integrality conjecture of Kontsevich-Soibelman and a geometric interpretation of orientifold Donaldson-Thomas invariants. Partially based on joint work with Hans Franzen (Bonn).
July 11 N.N. T.B.A.
July 25 Mauricio Romo (IAS) Complex Chern-Simons and cluster algebras
I'll describe the cluster partition function (CPF), a computational tool that uses elements from algebra and representation theory to describe the partition function of Chern-Simons theory with gauge group SL(N,C) given some prescribed boundary conditions. The CPF allows for a perturbative expansion that can be used to compute invariants of a large class of hyperbolic knots. I'll comment on its relation with M-theory and further applications such as insertion of line operators, if time allows.

### Winter 2015/16

Date Speaker Title
October 20 Thomas Reichelt (Heidelberg) Semi-infinite Hodge structures. An introduction
November 2 N.N. T.B.A.
November 16 Dominik Wrazidlo (Heidelberg) Positive Topological Quantum Field Theories and Fold Maps
December 7 Dmytro Shklyarov (Chemnitz) On an interplay between Hodge theoretic and categorical invariants of singularities
December 14 Cornelius Schmidt-Colinet (München) Conformal perturbation defects between 2d CFTs
We consider some examples and properties of interfaces implementing renormalisation group flows between two-dimensional conformal field theories, including general perturbative results for entropy and transmissivity, and the explicit constructions for flows between coset models.
January 11
INF 227 (KIP) SB1.107
Renato Vianna (Cambridge) Infinitely many monotone Lagrangian tori in del Pezzo surfaces
In 2014, we showed how the Chekanov torus arises as a fiber of an almost toric fibration and how this perspective enables us to describe an infinite range of monotone Lagrangian tori. More precisely, for any Markov triple of integers $(a,b,c)$ -- satisfying $a^2+b^2+c^2=3abc$ -- we get a monotone Lagrangian torus $T(a^2,b^2,c^2)$ in ${\mathbb C} P^2$. Using neck-stretching techniques we are able to get enough information on the count of Maslov index $2$ pseudo-holomorphic disks that allow us to show that for $(d,e,f)$ a Markov triple distinct from $(a,b,c)$, $T(d^2,e^2,f^2)$ is not Hamiltonian isotopic to $T(a^2,b^2,c^2)$.
In this talk we will describe how to get almost toric fibrations for all del Pezzo surfaces, in particular for ${{\mathbb C} P}^2{\#{k{\overline{{{\mathbb C}P}^2}}}}$ for $4\le k\le 8$, where there is no toric fibrations (with monotone symplectic form). From there, we will be able to construct infinitely many monotone Lagrangian tori. Some Markov like equations appear. They are the same as the ones appearing in the work of Haking-Porokhorov regarding degeneration of surfaces to weighted projective spaces.
January 18 N.N. T.B.A.
January 25 Penka Georgieva (Jussieu) Real Gromov-Witten theory in all genera
We construct positive-genus analogues of Welschingers invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the quintic threefold. Our approach to the orientability problem is based entirely on the topology of real bundle pairs over symmetric surfaces. This allows us to endow the uncompactified moduli spaces of real maps from symmetric surfaces of all topological types with natural orientations and to verify that they extend across the codimension-one boundaries of these spaces. In reasonably regular cases, these invariants can be used to obtain lower bounds for counts of real curves of arbitrary genus. Joint work with A. Zinger.
February 1 N.N. T.B.A.