NSR measures on hyperelliptic locus and non-renormalization of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math>-point functions

Abstract:We apply a definition of generalised super Calabi-Yau variety (SCY) to supermanifolds of complex dimension one. One of our results is that there are two SCY's having reduced manifold equal to P 1 , namely the projective super space P 1|2 and the weighted projective super space WP 1|1(2) . Then we compute the corresponding sheaf cohomology of superforms, showing that the cohomology with picture number one is infinite dimensional, while the de Rham cohomology, which is what matters from a physical point of view, remains finite dimensional. Moreover, we provide the complete real and holomorphic de Rham cohomology for generic projective super spaces P n|m . We also determine the automorphism groups: these always match the dimension of the projective super group with the only exception of P 1|2 , whose automorphism group turns out to be larger than the projective super group. By considering the cohomology of the super tangent sheaf, we compute the deformations of P 1|m , discovering that the presence of a fermionic structure allows for deformations even if the reduced manifold is rigid. Finally, we show that P 1|2 is self-mirror, whereas WP 1|1 (2) has a zero dimensional mirror. Also, the mirror map for P 1|2 naturally endows it with a structure of N = 2 super Riemann surface.

One-dimensional super Calabi-Yau manifolds and their mirrors

Noja

Cacciatori

Piazza

et al. 2017

Extending the Belavin-Knizhnik “wonderful formula” by the characterization of the Jacobian

Matone

2012

A long-standing question in string theory is to find the explicit expression of the bosonic measure, a crucial issue also in determining the superstring measure. Such a measure was known up to genus three. Belavin and Knizhnik conjectured an expression for genus four which has been proved in the framework of the recently introduced vector-valued Teichmüller modular forms. It turns out that for g ≥ 4 the bosonic measure is expressed in terms of such forms. In particular, the genus four Belavin-Knizhnik "wonderful formula" has a remarkable extension to arbitrary genus whose structure is deeply related to the characterization of the Jacobian locus. Furthermore, it turns out that the bosonic string measure has an elegant geometrical interpretation as generating the quadrics in P g−1 characterizing the Riemann surface. All this leads to identify forms on the Siegel upper half-space that, if certain conditions related to the characterization of the Jacobian are satisfied, express the bosonic measure as a multiresidue in the Siegel upper half-space. We also suggest that it may exist a super analog on the super Siegel half-space.

Finite strings from non-chiral Mumford forms

Matone

2012

We show that there is an infinite class of partition functions with world-sheet metric, space-time coordinates and first order systems, that correspond to volume forms on the moduli space of Riemann surfaces and are free of singularities at the Deligne-Mumford boundary. An example is the partition function with 4 = 2(c2 + c3 + c4 − c5) space-time coordinates, a b-c system of weight 3, one of weight 4 and a β-γ system of weight 5. Such partition functions are derived from the mapping of the Mumford forms to non-factorized scalar forms on Mg introduced in arXiv:1209.6049