## Physical Mathematics

### 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).

### Schedule

Date Speaker Title, Abstract
Monday, November 5 TBA
Monday, November 12 TBA
Monday, November 26 TBA
Monday, December 3 TBA
Friday, January 11
SR 1, 2:00 p.m.s.t.
Philsang Yoo (Yale) TBA
This is the abstract.
Monday, January 21 TBA
Monday, January 28 TBA

### Organizers

Prof. J. Walcher, walcher@uni-heidelberg.de
Dr. Ingmar Saberi, saberi@mathi.uni-heidelberg.de

## Past Semesters

### Summer 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, April 30 TBA
Monday
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.

### Summer 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 16/17

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.

### Summer 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 15/16

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.