Non-linear Structures in Non-critical NSR String

TL;DR

This paper explores the Ward identities of $ ext{W}_ ext{infinity}$ symmetry in super-Liouville theory coupled with matter, revealing that the constraints form $ ext{W}_q$ or $ ext{W}_p$ algebra equations depending on chirality, with non-linear terms from boundary anomalies.

Contribution

It demonstrates that the Ward identities in super-Liouville theory can be expressed as $ ext{W}_q$ algebra constraints, extending the understanding of non-linear structures in non-critical superstring theories.

Findings

Ward identities are expressed as $ ext{W}_q$ algebra constraints.

Constraints differ for positive and negative chirality, involving $ ext{W}_q$ and $ ext{W}_p$ algebras.

Non-linear terms originate from boundary anomalies in moduli space.

Abstract

We investigate the Ward identities of the $\W_{\infty}$ symmetry in the super-Liouville theory coupled to the super-conformal matter of central charge ${\hat c}_M = 1-2(p-q)^2 /pq$. The theory is classified into two chiralities. For the positive chirality, all gravitationally dressed scaling operators are generated from the $q-1$ gravitational primaries by acting one of the ring generators in the R-sector on them repeatedly. After fixing the normalizations of the dressed scaling operators, we find that the Ward identities are expressed in the form of the {\it usual} $\W_q$ algebra constraints as in the bosonic case: $\W^{(k+1)}_n \tau =0$, $(k=1,\cdots,q-1 ;~ n \in {\bf Z}_{\geq 1-k})$, where the equations for even and odd $n$ come from the currents in the NS- and the R-sector respectively. The non-linear terms come from the anomalous contributions at the boundaries of moduli space. The negative chirality is defined by interchanging the roles of $p$ and $q$. Then we get the $\W_p $ algebra constraints.