# The complex world of superstrings: On semichiral sigma models

## TL;DR

This paper explores the geometrical structures of semichiral sigma models with extended supersymmetry, revealing conditions for off-shell supersymmetry and the role of T-duality in relating different models.

## Contribution

It constructs N=(4,4) supersymmetry for semichiral sigma models and analyzes their geometric constraints, including the impact of T-duality on supersymmetry transformations.

## Key findings

- Off-shell supersymmetry requires target space dimension > 4.
- Four-dimensional models can have on-shell hyperkähler geometry.
- T-duality relates chiral/twisted chiral models to semichiral models, explaining transformation differences.

## Abstract

Non-linear sigma models with extended supersymmetry have constrained target space geometries, and can serve as effective tools for investigating and constructing new geometries. Analyzing the geometrical and topological properties of sigma models is necessary to understand the underlying structures of string theory. The most general two-dimensional sigma model with manifest N=(2,2) supersymmetry can be parametrized by chiral, twisted chiral and semichiral superfields. In the research presented in this thesis, N=(4,4) (twisted) supersymmetry is constructed for a semichiral sigma model. It is found that the model can only have additional supersymmetry off-shell if the target space has a dimension larger than four. For four-dimensional target manifolds, supersymmetry can be introduced on-shell, leading to a hyperk\"ahler manifold, or pseudo-supersymmetry can be imposed off-shell, implying a target space which is neutral hyperk\"ahler. Different sigma models and corresponding geometries can be related to each other by T-duality, obtained by gauging isometries of the Lagrangian. The semichiral vector multiplet and the large vector multiplet are needed for gauging isometries mixing semichiral superfields, and chiral and twisted chiral superfields, respectively. We find transformations that close off-shell to a N=(4,4) supersymmetry on the field strengths and gauge potentials of the semichiral vector multiplet, and show that this is not possible for the large vector multiplet. A sigma model parametrized by chiral and twisted chiral superfields can be related to a semichiral sigma model by T-duality. The N=(4,4) supersymmetry transformations of the former model are linear and close off-shell, whereas those of the latter are non-linear and close only on-shell. We show that this discrepancy can be understood from T-duality, and find the origin of the non-linear terms in the transformations.

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Source: https://tomesphere.com/paper/1705.04124