Supersymmetric IIB Solutions with Schrödinger Symmetry

TL;DR

This work constructs and analyzes supersymmetric IIB supergravity backgrounds with Schrödinger symmetry by applying the generalized null Melvin twist to AdS$_5\times X_5$, where $X_5$ is Sasaki–Einstein with at least a $U(1)^3$ isometry. The twist generically preserves two real supercharges, with four possible in certain cases such as $X_5=S^5$, and yields dynamical exponent $n=2$ for Schrödinger invariance; more general, scale-invariant solutions with $n\neq 2$ are generated by vector harmonics on $X_5$, with $n(n+2)=\lambda$ determined by vector-Laplacian eigenvalues. The paper provides explicit analyses for $S^5$, $T^{1,1}$, and the $Y^{p,q}$ and $L^{p,q,r}$ families, and develops a new class of solutions from vector harmonics that couple a one-form on $X_5$ to the nontrivial B-field, yielding a rich spectrum of exponents and potential holographic duals with Galilean invariance. The results establish a concrete bridge between geometric eigenvalue problems on internal manifolds and the dynamical data of non-relativistic holographic backgrounds, with broad implications for non-relativistic SUSY field theories and possible extensions to other dimensions and finite-temperature contexts.

Abstract

We find a class of non-relativistic supersymmetric solutions of IIB supergravity with non-trivial B-field that have dynamical exponent n=2 and are invariant under the Schrodinger group. For a general Sasaki-Einstein internal manifold with U(1)^3 isometry, the solutions have two real supercharges. When the internal manifold is S^5, the number of supercharges can be four. We also find a large class of non-relativistic scale invariant type IIB solutions with dynamical exponents different from two. The explicit solutions and the values of the dynamical exponents are determined by vector eigenfunctions and eigenvalues of the Laplacian on an Einstein manifold.