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Spectral Flow in AdS(3)/CFT(2)

Gaston Giribet, Ari Pakman, Leonardo Rastelli

TL;DR

This work extends AdS$_3$/CFT$_2$ by constructing explicit 1/2 BPS vertex operators in spectral-flowed sectors of the H$_3^+$ WZW model, thereby completing the bulk-to-boundary dictionary for flowed states. It systematizes the spectral flow of both the bosonic current algebras and the free fermions, assembling flowed operators ${\cal O}_{h,w}^{(\epsilon)}(x,y)$ and their Ramond/NS sectors in the x,y basis, with detailed expressions for their conformal data. A partial computation of flowed three-point functions is performed, yielding explicit factorized structure constants $C_H$ and $C_S$ and revealing nontrivial relations among fermionic couplings; in particular, RR-NS correlators lead to a concrete prediction $C_H(w_i,h_i) = \frac{\sqrt{q_{h_1} q_{h_2} q_{h_3}}}{C_S(h_i-1)} \left(\frac{(w_1+w_2)!}{w_1! w_2!}\right)^{2}$ for extremal kinematics. These results support the expectation that the bulk-boundary correspondence persists in flowed sectors, while also highlighting the need for additional $H_3^+$ three-point couplings to complete the check and offering CFT identities that tie flowed and descendant sectors together.

Abstract

We study the spectral flowed sectors of the H3 WZW model in the context of the holographic duality between type IIB string theory in AdS(3)x S^3 x T^4 with NSNS flux and the symmetric product orbifold of T^4. We construct explicitly the physical vertex operators in the flowed sectors that belong to short representations of the superalgebra, thus completing the bulk-to-boundary dictionary for 1/2 BPS states. We perform a partial calculation of the string three-point functions of these operators. A complete calculation would require the three-point couplings of non-extremal flowed operators in the H3 WZW model, which are at present unavailable. In the unflowed sector, perfect agreement has recently been found between the bulk and boundary three-point functions of 1/2 BPS operators. Assuming that this agreement persists in the flowed sectors, we determine certain unknown three-point couplings in the H3 WZW model in terms of three-point couplings of affine descendants in the SU(2) WZW model.

Spectral Flow in AdS(3)/CFT(2)

TL;DR

This work extends AdS/CFT by constructing explicit 1/2 BPS vertex operators in spectral-flowed sectors of the H WZW model, thereby completing the bulk-to-boundary dictionary for flowed states. It systematizes the spectral flow of both the bosonic current algebras and the free fermions, assembling flowed operators and their Ramond/NS sectors in the x,y basis, with detailed expressions for their conformal data. A partial computation of flowed three-point functions is performed, yielding explicit factorized structure constants and and revealing nontrivial relations among fermionic couplings; in particular, RR-NS correlators lead to a concrete prediction for extremal kinematics. These results support the expectation that the bulk-boundary correspondence persists in flowed sectors, while also highlighting the need for additional three-point couplings to complete the check and offering CFT identities that tie flowed and descendant sectors together.

Abstract

We study the spectral flowed sectors of the H3 WZW model in the context of the holographic duality between type IIB string theory in AdS(3)x S^3 x T^4 with NSNS flux and the symmetric product orbifold of T^4. We construct explicitly the physical vertex operators in the flowed sectors that belong to short representations of the superalgebra, thus completing the bulk-to-boundary dictionary for 1/2 BPS states. We perform a partial calculation of the string three-point functions of these operators. A complete calculation would require the three-point couplings of non-extremal flowed operators in the H3 WZW model, which are at present unavailable. In the unflowed sector, perfect agreement has recently been found between the bulk and boundary three-point functions of 1/2 BPS operators. Assuming that this agreement persists in the flowed sectors, we determine certain unknown three-point couplings in the H3 WZW model in terms of three-point couplings of affine descendants in the SU(2) WZW model.

Paper Structure

This paper contains 21 sections, 225 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review of 1/2 BPS operators and their three-point functions
  3. The symmetric product orbifold
  4. The $AdS_3 \times S^3 \times T^4$ worldsheet
  5. Spectral Flow in $SL(2,R)$ and $SU(2)$
  6. Spectral Flow in Bosonic
  7. Spectral Flow in Bosonic $SU(2)$
  8. Spectral Flow for the Free Fermions
  9. Fermionic Multiplets
  10. $SU(2)$ Fermionic Multiplets
  11. The Ramond Sector
  12. Interactions of Fermionic Multiplets
  13. 1/2 BPS Flowed Spectrum
  14. The ADE Series
  15. Three-point Functions of 1/2 BPS Flowed Operators
  16. ...and 6 more sections

Figures (2)

  • Figure 1: Weight diagram of $SL(2,R)_{k'}$. The points in the $j_0^3$ axis are affine primaries in a $d_h^+$ representation. The spectral flow of the state with $\tilde{m}=h$ by $w=2$ units gives a state at level $-2h-k'$, which is the lowest weight state of a $d_H^+$ representation of the global algebra with $H=h + k'$. All the states of this $d_H^+$ representation are Virasoro primaries.
  • Figure 2: Weight diagram of a spin $j$ representation of affine $SU(2)_{k"}$ and the multiplet obtained from spectral flowing the $k_0^3 =-j$ affine primary by $w=2$ units. The points in the $k_0^3$ axis are $2j\!+\!1$ affine primaries. The affine descendants of the horizontal line at level $k"+2j$ are Virasoro primaries which form a representation of the global $SU(2)$ symmetry with spin $J=j+k"$.