**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.**

Local Organizers: U. Bruzzo (Trieste) and D.
Hernández Ruipérez (Salamanca)

The Workshop on
Supermoduli and Superstrings will take place in an online format
in the afternoons (CEST) of July 8th and 9th, 2021, and it is open to anybody
who is interested in participating.

The lectures will be recorded and added to this webpage at
the end of each session.

**Schedule:**

#### Thursday, July 8th

2:30 pm

Rita
Fioresi (Univ. Bologna),

Unitary
Representations of real Lie Supergroups and
Hermitian Symmetric Super Spaces (video)

3:30 pm break

3:45 pm

Pietro
A. Grassi (Univ. Piemonte Orientale),

New cohomologies for Lie
superalgebras (video)

4:45 pm break

5:00 pm

A.
Voronov (Univ. Minnesota),

The moduli space of
genus-zero super Riemann surfaces with Ramond punctures (video)

#### Friday, July 9th

2:30 pm

Simone
Noja (Univ. Heidelberg),

On differential and
integral forms and total cotangent bundle supermanifolds (video)

3:30 pm break

3:45 pm

Giovanni
Felder (ETH Zurich),

Regularity of the
superstring measure up to genus 11 (video)
4:45 pm break

5:00 pm

A.
Polishchuk (Univ. Oregon), T

he Mumford isomorphism and
the superperiods near the boundary of the moduli of stable
supercurves (video)

**Zoom
coordinates:**

Topic: Workshop on
Supermoduli

Meeting ID: 840 7541 0826

%%%%%%

**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.