Excitations in the deformed D1D5 CFT

TL;DR

This work analyzes the first-order deformation of the D1D5 CFT away from the orbifold point, showing that the deformation operator generates a universal squeezed-exponential structure in the final state. By decomposing the deformation into a twist $\sigma_2^+$ and a supercharge, and mapping through covering spaces, the authors derive how single and multiple initial excitations transform into linear combinations of final modes on a doubly wound circle, including Wick contractions. The results provide explicit coefficients (involving Gamma functions and the parameter $z_0=e^{w_0}$) that govern the mixing of bosonic and fermionic modes and establish a method to compute the deformation's effect on arbitrary initial states. This framework clarifies how energy from colliding left-right excitations spreads into many quanta, driving evolution toward the infrared and shedding light on the microscopic dynamics relevant to near-extremal D1D5 black holes.

Abstract

We perform some simple computations for the first order deformation of the D1D5 CFT off its orbifold point. It had been shown earlier that under this deformation the vacuum state changes to a squeezed state (with the further action of a supercharge). We now start with states containing one or two initial quanta and write down the corresponding states obtained under the action of deformation operator. The result is relevant to the evolution of an initial excitation in the CFT dual to the near extremal D1D5 black hole: when a left and a right moving excitation collide in the CFT, the deformation operator spreads their energy over a larger number of quanta, thus evolving the state towards the infrared.

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 $|\psi\rangle$.