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Superstrings in AdS(2)xS(2)xT(6)

D. Sorokin, A. Tseytlin, L. Wulff, K. Zarembo

TL;DR

The paper studies Green–Schwarz superstrings on AdS2×S2×T6 supported by RR flux, showing classical integrability to quadratic order in fermions and a consistent classical truncation to the PSU(1,1|2)/SO(1,1)×U(1) supercoset. It constructs a Lax connection from Noether currents to prove integrability, analyzes explicit IIA/IIB backgrounds, and derives a quadratic fermion and coset Lagrangian consistent with a Z4-graded coset structure. It further develops a finite-gap, classical framework for the coset sector and conjectures asymptotic Bethe-ansatz equations describing a subset of the quantum spectrum, while acknowledging the massless T6 modes not captured by the coset subsector. The work highlights non-decoupling of torus directions due to RR flux, discusses BMN limits and pp-wave spectra, and outlines steps toward an exact solution via TBA/Y-system approaches, signaling a rich integrable structure with caveats tied to massless modes.

Abstract

We consider the type IIB Green-Schwarz superstring theory on AdS(2)xS(2)xT(6) supported by homogeneous Ramond-Ramond 5-form flux and its type IIA T-duals. One motivation is to understand the solution of this theory based on integrability. This background is a limit of a 1/4 supersymmetric supergravity solution describing four intersecting D3-branes and represents a consistent embedding of AdS(2)xS(2) into critical superstring theory. Its AdS(2)xS(2) part with corresponding fermions can be described by a classically integrable PSU(1,1|2)/SO(1,1)xU(1) supercoset sigma-model. We point out that since the RR 5-form field has non-zero components along the 6-torus directions one cannot, in general, factorize the 10d superstring theory into the supercoset part plus 6 bosons and 6 additional massless fermions. Still, we demonstrate that the full superstring model (i) is classically integrable, at least to quadratic order in fermions, and (ii) admits a consistent classical truncation to the supercoset part. Following the analogy with other integrable backgrounds and starting with the finite-gap equations of the PSU(1,1|2)/SO(1,1)xU(1) supercoset we propose a set of asymptotic Bethe ansatz equations for a subset of the quantum string states.

Superstrings in AdS(2)xS(2)xT(6)

TL;DR

The paper studies Green–Schwarz superstrings on AdS2×S2×T6 supported by RR flux, showing classical integrability to quadratic order in fermions and a consistent classical truncation to the PSU(1,1|2)/SO(1,1)×U(1) supercoset. It constructs a Lax connection from Noether currents to prove integrability, analyzes explicit IIA/IIB backgrounds, and derives a quadratic fermion and coset Lagrangian consistent with a Z4-graded coset structure. It further develops a finite-gap, classical framework for the coset sector and conjectures asymptotic Bethe-ansatz equations describing a subset of the quantum spectrum, while acknowledging the massless T6 modes not captured by the coset subsector. The work highlights non-decoupling of torus directions due to RR flux, discusses BMN limits and pp-wave spectra, and outlines steps toward an exact solution via TBA/Y-system approaches, signaling a rich integrable structure with caveats tied to massless modes.

Abstract

We consider the type IIB Green-Schwarz superstring theory on AdS(2)xS(2)xT(6) supported by homogeneous Ramond-Ramond 5-form flux and its type IIA T-duals. One motivation is to understand the solution of this theory based on integrability. This background is a limit of a 1/4 supersymmetric supergravity solution describing four intersecting D3-branes and represents a consistent embedding of AdS(2)xS(2) into critical superstring theory. Its AdS(2)xS(2) part with corresponding fermions can be described by a classically integrable PSU(1,1|2)/SO(1,1)xU(1) supercoset sigma-model. We point out that since the RR 5-form field has non-zero components along the 6-torus directions one cannot, in general, factorize the 10d superstring theory into the supercoset part plus 6 bosons and 6 additional massless fermions. Still, we demonstrate that the full superstring model (i) is classically integrable, at least to quadratic order in fermions, and (ii) admits a consistent classical truncation to the supercoset part. Following the analogy with other integrable backgrounds and starting with the finite-gap equations of the PSU(1,1|2)/SO(1,1)xU(1) supercoset we propose a set of asymptotic Bethe ansatz equations for a subset of the quantum string states.

Paper Structure

This paper contains 25 sections, 190 equations, 1 figure.

Table of Contents

  1. Introduction
  2. GS superstrings in RR backgrounds and their classical integrability
  3. GS superstrings in RR backgrounds, equations of motion and conserved currents
  4. Lax connection built from conserved currents
  5. Superstrings in $AdS_2\times S^2\times T^6$ backgrounds
  6. Type IIA $AdS_2\times S^2\times T^6$ with $F_2$ and $F_4$ flux
  7. Type IIB $AdS_2\times S^2\times T^6$ with $F_5$ flux
  8. Consistent truncation of the $D=10$ superstring to the $PSU(1,1|2)/SO(1,1)\times U(1)$ supercoset sigma--model
  9. Quadratic fermionic part of the superstring Lagrangian
  10. Lagrangian of the supercoset model
  11. Supercoset truncation beyond the quadratic approximation
  12. BMN limits of the $AdS_2\times S^2\times T^6$ superstring
  13. BMN limit along geodesic in $S^2$
  14. Supersymmetries of the pp-wave background
  15. BMN limit of the superstring action
  16. ...and 10 more sections

Figures (1)

  • Figure 1: The Dynkin diagram for the Bethe equations. The Dynkin links correspond to the first term in (\ref{['classbethe']}). The inversion-symmetry links originate from the second term ($A\cdot S$), and in our case are proportional to the Dynkin links.