# Advantage of the second-order formalism in double space T-dualization of type II superstring

## TL;DR

This paper demonstrates the advantages of using a second-order formalism in double space T-dualization of type II superstring theory, providing a more direct derivation of T-dual background fields without additional assumptions.

## Contribution

It introduces a second-order formalism with integrated fermionic momenta for bosonic T-dualization in double space, simplifying the derivation of T-dual background fields.

## Key findings

- Full T-dualization as permutation of coordinates
- T-dual background fields expressed in terms of initial fields
- Simplified derivation without additional assumptions

## Abstract

In this article we present bosonic T-dualization in double space of the type II superstring theory in pure spinor formulation. We use the action with constant background fields obtained from the general case under some physically and mathematically justified assumptions. Unlike Ref.\cite{bnbstip2}, where we used the first order theory, in this article fermionic momenta are integrated out. Full T-dualization in double space is represented as permutation of initial $x^\mu$ and T-dual coordinates $y_\mu$. Demanding that T-dual transformation law of the T-dual double coordinate ${}^\star Z^M=(y_\mu,x^\mu)$ is of the same form as for initial one $Z^M=(x^\mu,y_\mu)$, we obtain the form of the T-dual background fields in terms of the initial ones. Advantage of using the action with integrated fermionic momenta is that it gives all T-dual background fields in terms of the initial ones. In the case of the first order theory \cite{bnbstip2} T-dual R-R field strength was obtained out of double space formalism under additional assumptions.