Ramond States of the D1-D5 CFT Away from the Free Orbifold Point

Abstract

The free orbifold point of the D1-D5 CFT must be deformed with a scalar marginal operator driving it to the region in moduli space where the holographic supergravity description of fuzzball microstates becomes available. We discuss the effects of the deformation operator on the twisted Ramond ground states of the CFT by computing four-point functions. One can thus extract the OPEs of the deformation operator with these Ramond fields to find the conformal dimensions of intermediate non-BPS states and the relevant structure constants. We also compute the anomalous dimensions at second order in perturbation theory, and find that individual single-cycle Ramond fields are renormalized, while the full multi-cycle ground states of the $S_N$ orbifold remain protected at leading order in the large-N expansion.

Appendix

The $({\mathbb T}^4)^N/S_N$ orbifold has N copies of a ‘seed’ $\mathcal N = (4,4)$ SCFT with central charge $c_\textrm{seed} = 6$. The total central charge is $c = 6 N$. Each copy, labeled by an index $I = \{1,\dots ,N\}$, has 4 real bosons and $(4 + \tilde{4})$ real fermions, all free, which can be gathered into SU(2) doublets. The holomorphic Ramond fields in the text are constructed from the bosonized fermions $\psi ^{\alpha \dot{1}}_I (z) = [ e^{- i \phi _{2,I}(z)} , e^{- i \phi _{1,I}(z)} ]^T$ and $\psi ^{\alpha \dot{2}}_I (z) = [ e^{ i \phi _{2,I}(z)} , - e^{i \phi _{1,I}(z)} ]^T$. The indices $\alpha = \pm$ correspond to the holomorphic R-symmetry group $\textrm{SU}(2)_L$ and $\dot{A} = \dot{1}, \dot{2}$ to the factor $\textrm{SU}(2)_2$ of the global symmetry.