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 |
---|---|---|
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. | ||
Tuesday, March 13, 2018 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, 2018 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. |
Organizers
Prof. J. Walcher, walcher@uni-heidelberg.de
Dr. Richard Eager, eager@mathi.uni-heidelberg.de
Dr. Ingmar Saberi, saberi@mathi.uni-heidelberg.de