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Monodromy Bootstrap for SU(2|2) Quantum Spectral Curves: From Hubbard model to AdS3/CFT2

Simon Ekhammar, Dmytro Volin

TL;DR

This work develops a monodromy bootstrap framework to construct quantum spectral curves (QSCs) from algebraic Q-systems, producing four new low-rank QSCs: two from SU(2|2) and two from SU(2|2)×SU(2|2). A key result is that the SU(2|2) B-type case yields a Hubbard-model QSC when branch points are square roots, while the SU(2|2)×SU(2|2) cases A–D lead to a unique AdS3/CFT2-compatible QSC at zero central charge, with a PSU-like real form; this is supported by matching the massive-sector asymptotic Bethe equations. The analysis reveals that, unlike AdS5/CFT4, the AdS3 candidate requires non-square-root branch points and exhibits intricate monodromy data encoded in Pμ- and Qω-systems, dressed by a function F that acts as a source term. Overall, the monodromy bootstrap provides a principled route to derive QSCs for AdS/CFT-type integrable systems and delineates a landscape that includes Hubbard-like and AdS3/CFT2-like curves, with clear avenues for further checks including massless modes and higher-rank generalizations.

Abstract

We propose a procedure to derive quantum spectral curves of AdS/CFT type by requiring that a specially designed analytic continuation around the branch point results in an automorphism of the underlying algebraic structure. In this way we derive four new curves. Two are based on SU(2|2) symmetry, and we show that one of them, under the assumption of square root branch points, describes Hubbard model. Two more are based on SU(2|2) x SU(2|2). In the special subcase of zero central charge, they both reduce to the unique nontrivial curve which furthermore has analytic properties compatible with PSU(1,1|2) x PSU(1,1|2) real form. A natural conjecture follows that this is the quantum spectral curve of AdS/CFT integrable system with AdS3 x S3 x T4 background supported by RR-flux. We support the conjecture by verifying its consistency with the massive sector of asymptotic Bethe equations in the large volume regime. For this spectral curve, it is compulsory that branch points are not of the square root type which qualitatively distinguishes it from the previously known cases.

Monodromy Bootstrap for SU(2|2) Quantum Spectral Curves: From Hubbard model to AdS3/CFT2

TL;DR

This work develops a monodromy bootstrap framework to construct quantum spectral curves (QSCs) from algebraic Q-systems, producing four new low-rank QSCs: two from SU(2|2) and two from SU(2|2)×SU(2|2). A key result is that the SU(2|2) B-type case yields a Hubbard-model QSC when branch points are square roots, while the SU(2|2)×SU(2|2) cases A–D lead to a unique AdS3/CFT2-compatible QSC at zero central charge, with a PSU-like real form; this is supported by matching the massive-sector asymptotic Bethe equations. The analysis reveals that, unlike AdS5/CFT4, the AdS3 candidate requires non-square-root branch points and exhibits intricate monodromy data encoded in Pμ- and Qω-systems, dressed by a function F that acts as a source term. Overall, the monodromy bootstrap provides a principled route to derive QSCs for AdS/CFT-type integrable systems and delineates a landscape that includes Hubbard-like and AdS3/CFT2-like curves, with clear avenues for further checks including massless modes and higher-rank generalizations.

Abstract

We propose a procedure to derive quantum spectral curves of AdS/CFT type by requiring that a specially designed analytic continuation around the branch point results in an automorphism of the underlying algebraic structure. In this way we derive four new curves. Two are based on SU(2|2) symmetry, and we show that one of them, under the assumption of square root branch points, describes Hubbard model. Two more are based on SU(2|2) x SU(2|2). In the special subcase of zero central charge, they both reduce to the unique nontrivial curve which furthermore has analytic properties compatible with PSU(1,1|2) x PSU(1,1|2) real form. A natural conjecture follows that this is the quantum spectral curve of AdS/CFT integrable system with AdS3 x S3 x T4 background supported by RR-flux. We support the conjecture by verifying its consistency with the massive sector of asymptotic Bethe equations in the large volume regime. For this spectral curve, it is compulsory that branch points are not of the square root type which qualitatively distinguishes it from the previously known cases.

Paper Structure

This paper contains 30 sections, 160 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction and main results
  2. Algebraic structure of Q-system
  3. Geometric construction
  4. Symmetries
  5. Monodromy bootstrap
  6. Conventions on analytic continuation
  7. Half-plane analyticity
  8. Hodge-dual system.
  9. Connecting upper and lower half-planes
  10. Monodromy bootstrap
  11. Exploring analytic properties
  12. Function F and PSU vs SU cases
  13. Discontinuity relations for P and Q
  14. Discontinuity relations for mu_ab and omega_ij
  15. The most general Pmu-system
  16. ...and 15 more sections

Figures (2)

  • Figure 1: Analytic continuation perspective on the monodromy bootstrap. Crosses represent the Q-system, there are two them to emphasise the non-locality of the equations. Q-system can be parameterised using the physical or the mirror kinematics, and an UHPA or a LHPA basis. Along the contour of continuation $abcd$, only two out of four descriptions are suitable for each of the segments $a,b,c,d$. For instance, while on $a$, we should use an UHPA basis but the kinematics is irrelevant; while on $b$, we should use the physical kinematics and either an UHPA or a LHPA basis are acceptable. Before moving from a segment to the next one, we should use a continuous symmetry transformation to switch to the description suitable for both segments.
  • Figure :