Deforming the D1D5 CFT away from the orbifold point

TL;DR

This work analyzes how the D1D5 CFT departs from its orbifold point under a marginal twist deformation that fuses two copies into a doubly wound circle. By mapping to a covering space and using a squeezed-state ansatz, the authors derive closed-form expressions for the bosonic and fermionic pair coefficients γ^B_{mn} and γ^F_{mn}, after a sequence of spectral flows and a final supercharge action. The resulting final state is a highly structured squeezed state, indicating a broad, power-law distribution of excitations rather than a simple finite set, and providing a concrete framework to extend D1D5 microstate analyses away from the orbifold point. These results facilitate integrating the deformation over twist locations and initial excitations, advancing the program of connecting CFT deformations to black hole microstate geometries.

Abstract

The D1D5 brane bound state is believed to have an `orbifold point' in its moduli space which is the analogue of the free Yang Mills theory for the D3 brane bound state. The supergravity geometry generated by D1 and D5 branes is described by a different point in moduli space, and in moving towards this point we have to deform the CFT by a marginal operator: the `twist' which links together two copies of the CFT. In this paper we find the effect of this deformation operator on the simplest physical state of the CFT -- the Ramond vacuum. The twist deformation leads to a final state that is populated by pairs of excitations like those in a squeezed state. We find the coefficients characterizing the distribution of these particle pairs (for both bosons and fermions) and thus write this final state in closed form.

Figures (4)

• Figure 1: The effect of the twist contained in the deformation operator: two circles at earlier times get joined into one circle after the insertion of the twist.
• Figure 2: Before the twist insertion we have boson and fermion modes on two copies of the $c=6$ CFT. These modes are labeled with superscripts $(1), (2)$ respectively. The twist inserted at $w_0$ joins these to one copy for $\tau>\tau_0$; the modes here do not carry a superscript. The branch cut above $w_0$ indicates that we have two sets of fields at any given $\sigma$; these two sets go smoothly into each other as we go around the cylinder, giving a continuous field on a doubly wound circle.
• Figure 3: (a) The supercharge in the deformation operator is given by integrating $G^-_{\dot A}$ around the insertion at $w_0$. (b) We can stretch this contour as shown, so that we get a part above the insertion and a part below, joined by vertical segments where the contributions cancel. (c) The part above the insertion gives the zero mode of the supercharge on the doubly wound circle, while the parts below give the sum of this zero mode for each of the two initial copies of the CFT.
• Figure 4: The $z$ plane is mapped to the covering space -- the $t$ plane -- by the map $z=z_0+t^2$. The point $z=0$ corresponds to $\tau\rightarrow-\infty$ on the cylinder, and the two copies of the CFT there correspond to the points $t=\pm ia$. The location of the twist operator maps to $t=0$. The top the cylinder $\tau\rightarrow\infty$ maps to $t\rightarrow \infty$. After all maps and spectral flows, we have the NS vacuum at $t=0, \pm ia$, and so we can smoothly close all these punctures. The state $|\chi\rangle$ is thus just the $t$ plane vacuum; we must write this in terms of the original cylinder modes and apply the supercharge to get the final state $|\psi\rangle$.