Workshop on Supermoduli

[Ugo Bruzzo]

2021 WORKSHOP ON ALGEBRAIC GEOMETRY AND PHYSICS: SUPERMODULI AND SUPERSTRINGS

Organized by IGAP (Institute for Geometry and Physics) and SISSA (Trieste) and Universidad de Salamanca

The Workshop on Supermoduli and Superstrings will take place in an online format in the afternoons (CEST) of July 8th and 9th, 2021.

Schedule:

Thursday, July 8th

2:30 pm	Rita Fioresi (Univ. Bologna), Unitary Representations of real Lie Supergroups
		and Hermitian Symmetric Super Spaces
3:30 pm	break
3:45 pm	Pietro A. Grassi (Univ. Piemonte Orientale), New cohomologies for Lie superalgebras
4:45 pm	break
5:00 pm	A. Voronov (Univ. Minnesota), The moduli space of genus-zero super Riemann surfaces with Ramond punctures

Friday, July 9th

2:30 pm	Simone Noja (Univ. Heidelberg), On differential and  integral forms and total cotangent bundle supermanifolds
3:30 pm	break
3:45 pm	Giovanni Felder (ETH Zurich), Regularity of the superstring measure up to genus 11
4:45 pm	break
5:00 pm	A. Polishchuk (Univ. Oregon), The Mumford isomorphism and the superperiods near the boundary of the moduli of stable supercurves

Zoom coordinates:

Topic: Workshop on Supermoduli
Time: From Jul 8, 2021 02:00 PM Rome

Join Zoom Meeting
https://sissa-it.zoom.us/j/89309160261?pwd=QVZFdXVhYXhvR3M0bU1EYVJHbnZ4dz09 <https://sissa-it.zoom.us/j/89309160261?pwd=QVZFdXVhYXhvR3M0bU1EYVJHbnZ4dz09>

Meeting ID: 893 0916 0261
Passcode: WAGP2021

The Workshop is open to anybody who is interested in participating.

U. Bruzzo (Trieste) and D. Hernández Ruipérez (Salamanca)

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Abstracts:

G. Felder. In this talk I will review some aspects and subtleties of perturbative superstring theory in the Ramond-Neveu-Schwarz formalism.
The integral of the superstring measure over moduli of supercurves of genus g gives the genus g contribution to the vacuum
amplitude. One ingredient constituting the superstring measure is the super analogue of the Riemann period map, which is singular on the
locus of supercurves with even spin structures admitting non-trivial sections. We show that, in spite of this singularity, the superstring
measure is regular at least up to genus 11. The talk is based on joint work with David Kazhdan and Alexander Polishchuk (arXiv:1905.12805 and arXiv: 2006.13271).

R. Fioresi. In 1955 Harish-Chandra developed the theory of representations of real semisimple Lie group, which are infinitesimally highest weight modules. These modules were constructed globally in the spaces of sections of holomorphic vector bundles on the associated hermitian symmetric space. He also verified square integrability of the matrix element and gave conditions for unitarizability of these modules, leading to the structure of discrete series for Lie supergroups. In this talk we prove the super analogue of Harish-Chandra theory. We first show that given a supergroup G, any Harish-Chandra unitary representation of G_0 can be extended to a unitary representation of G and that such representations can be realized in the space of section of holomorphic line bundles on an hermitian superspace. In the end, we also shall give the geometric quantizationpoint of view on this supersymmetric construction (work in progress). This is a joint work with: C. Carmeli (Univ, Genoa), V.S.Varadarajan (UCLA), M.-K. Chuah (NTHU, Taiwan).

P.A. Grassi. We present new results in the computations of cohomologies for Lie superalgebras.  In particular, we will give the first explicit example of an invariant pseudoform representing a cohomology  class for the Chevalley-Eilenberg cohomology for osp(2|2) superalgebra emerging in N=2 string theory. 

S. Noja. A characteristic feature of supergeometry is that the de Rham complex of a supermanifold is not bounded from above, in particular there is no notion of a top differential form. This has led to the introduction of the so-called complex of integral forms, where sections of the Berezinian bundle play the role of top integral forms. In this talk I will introduce a double complex of non-commutative sheaves relating differential and integral forms on a generic supermanifold. If time permits, I will discuss how this framework specializes to total cotangent bundle supermanifolds in the context of odd symplectic (super)geometry.

A. Polishchuk. This talk is based on a joint work (some of it in progress) with Giovanni Felder and David Kazhdan. For purposes of integrating the superstring measure it is important to understand the polar behavior of the Mumford isomorphism and of the superperiods near the boundary of the compactification of the moduli space of supercurves. As in the classical case, the latter compactification is provided by the moduli space of stable supercurves. I will review the relevant facts about the geometry of stable supercurves and will present our results on this behaviour.

A. Voronov. This is a report on my work with Nadia Ott on an explicit construction of the moduli space of genus-zero super Riemann surfaces with Ramond punctures.  We give a new twist to the construction of this supermoduli space, giving first an explicit quotient description of the Artin supermoduli stack of genus-zero supercurves with no SUSY structure, and then describing the (n-3 | n/2-2)-dimensional supermoduli stack of genus-zero super Riemann surfaces with n ≥ 4 Ramond punctures as an explicit Deligne-Mumford quotient superstack. This addresses a question that Edward Witten posed in 2010.