### 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 2022 is Monday, 4 p.m.c.t., unless indicated otherwise.

### Online Vorträge

Date | Speaker | Title, Abstract |
---|---|---|

April 25 | Minhyong Kim (University of Warwick) | Quantum Field Theory as Mathematical Formalism: The Case of Arithmetic Geometry |

Quantum field theory clearly has its origins in the largely successful attempt to classify the fundamental building blocks of matter and the interactions between them. On the other hand, a number of practitioners have suggested that it should gradually develop into a general purpose mathematical toolkit, following an evolution roughly similar to calculus. I will describe in this talk applications of this general philosophy to arithmetic geometry. | ||

May 2 | Murad Alim (University of Hamburg) | Non-perturbative quantum geometry, resurgence and BPS structures |

BPS invariants of certain physical theories correspond to Donaldson-Thomas (DT) invariants of an associated Calabi-Yau geometry. BPS structures refer to the data of the DT invariants together with their wall-crossing structure. On the same Calabi-Yau geometry another set of invariants are the Gromov-Witten (GW) invariants. These are organized in the GW potential, which is an asymptotic series in a formal parameter and can be obtained from topological string theory. A further asymptotic series in two parameters is obtained from refined topological string theory which contains the Nekrasov-Shatashvili (NS) limit when one of the two parameters is sent to zero. I will discuss in the case of the resolved conifold how all these asymptotic series lead to difference equations which admit analytic solutions in the expansion parameters. A detailed study of Borel resummation allows one to identify these solutions as Borel sums in a distinguished region in parameter space. The Stokes jumps between different Borel sums encode the BPS invariants of the underlying geometry and are captured in turn by another set of difference equations. I will further show how the Borel analysis of the NS limit connects to the exact WKB study of quantum curves. This is based on various joint works with Lotte Hollands, Arpan Saha, Iván Tulli and Jörg Teschner. | ||

May 9 | Urs Schreiber (Czech Academy of Science) | Anyonic Defect Branes and Conformal Blocks in Twisted Equivariant Differential K-Theory |

We demonstrate that twisted equivariant differential K-theory of transverse complex curves accommodates exotic charges of the form expected of codimension=2 defect branes, such as of D7-branes in IIB/F-theory on A-type orbifold singularities, but also of their dual 3-brane defects of class-S theories on M5-branes. These branes have been argued, within F-theory and the AGT correspondence, to carry special SL(2)-monodromy charges not seen for other branes, at least partially reflected in conformal blocks of the sl_2-WZW model over their transverse punctured complex curve. Indeed, it has been argued that all "exotic" branes of string theory are defect branes carrying such U-duality monodromy charges – but none of these had previously been identified in the expected brane charge quantization law given by K-theory. Here we observe that it is the subtle (and previously somewhat neglected) twisting of equivariant K-theory by flat complex line bundles appearing inside orbi-singularities (“inner local systems”) that makes the secondary Chern character on a punctured plane inside an A-type singularity evaluate to the twisted holomorphic de Rham cohomology which Feigin, Schechtman & Varchenko showed realizes sl_2-conformal block , here in degree 1 – in fact it gives the direct sum of these over all admissible fractional levels l = -2 + k /r. The remaining higher-degree sl_2-conformal blocks appear similarly if we assume our previously discussed “Hypothesis H” about brane charge quantization in M-theory. Since conformal blocks – and hence these twisted equivariant secondary Chern characters – solve the Knizhnik-Zamolodchikov equation and thus constitute representations of the braid group of motions of defect branes inside their transverse space, this provides a concrete first-principles realization of anyon statistics of – and hence of topological quantum computation on – defect branes in string/M-theory. | ||

May 16 | Fabian Hahner (Heidelberg University) | Derived Pure Spinor Superfields |

The pure spinor superfield formalism is a systematic way to construct supersymmetric multiplets from modules over the ring of functions on the nilpotence variety. After a short review of the technique, I present its derived generalization and explain how the derived formalism yields an equivalence of dg categories between multipets and modules over the Chevalley--Eilenberg algebra of supertranslations. This equivalence of categories is closely related to Koszul duality. If time permits, I will comment on applications to six-dimensional supersymmetry. | ||

May 23 | Leonardo Rastelli (Stony Brook University) | On the 4D SCFTs/VOAs correspondence |

I will describe some recent progress on the correspondence between four-dimensional \({\cal N=2}\) superconformal field theories (SCFTs) and two-dimensional vertex operator algebras (VOAs). In particular I will introduce the notion of the “Higgs scheme”, an extension by nilpotent elements of the standard Higgs variety of an \({\cal N=2}\) SCFT, which plays a natural role in the associated VOA. Unlike the Higgs variety, theHiggs scheme appears to be a perfect invariant, i.e. it conjecturally fully characterizes the SCFT. | ||

May 30 | Maxim Zabzine (Uppsala University) | The index of M-theory and equivariant volumes |

Motivated by M-theory, I will review rank-\(n\) K-theoretic Donaldson-Thomas theory on a toric threefold and its factorisation properties in the context of 5d/7d correspondence. In the context of this discussion I will revise the use of the Duistermaat-Heckman formula for non-compact toric Kahler manifolds, pointing out some mathematical and physical puzzles. | ||

June 20 | Heeyeon Kim (Rutgers University) | Path integral derivations of K-theoretic Donaldson invariants |

We discuss path integral derivations of topologically twisted partition functions of 5d \({\it SU}(2)\) supersymmetric Yang-Mills theory on \(M^4 \times S^1\), where M4 is a smooth closed four-manifold. Mathematically, they can be identified with the K-theoretic version of the Donaldson invariants. In particular, we provide two different path integral derivations of their wall-crossing formula for \(b_2^+(M4)=1\), first in the so-called U-plane integral approach, and in the perspective of instanton counting. We briefly discuss the generalization to \(b_2^+(M4)>1\). | ||

June 27 | Sara Pasquetti (University Milano-Bicocca) | Rethinking mirror symmetry as a local duality on fields |

We introduce an algorithm to piecewise dualise linear quivers into their mirror dual. The algorithm uses two basic duality moves and the properties of the S-wall which can all be derived by iterative applications of Seiberg-like dualities. | ||

July 4 | N.N. | T.B.A. |

abstract forthcoming | ||

July 11 | N.N. | T.B.A. |

abstract forthcoming | ||

July 18 | Hossein Movasati (IMPA) | Modular and automorphic forms & beyond |

I will talk on a project which aims to develop a unified theory of modular and automorphic forms. It encompasses most of the available theory of modular forms in the literature, such as those for congruence groups, Siegel and Hilbert modular forms, many types of automorphic forms on Hermitian symmetric domains, Calabi-Yau modular forms, with its examples such as Yukawa couplings and topological string partition functions, and even go beyond all these cases. Its main ingredient is the so-called ‘Gauss-Manin connection in disguise’. The talk is bases on the author's book with the same title, available in my webpage. |

### Veranstalter

Prof. J. Walcher, walcher@uni-heidelberg.de

Dr. Simone Noja,
noja@mathi.uni-heidelberg.de