Quantum Spectral Curve and Analytic Bethe Ansatz

A typical feature of quantum integrable models is the possibility to construct and/or describe Bethe Algebra—a large family of commuting operators whose simultaneous diagonalisation fully controls the spectrum of the theory. We introduce rich combinatorial and geometrical features of the Bethe Algebra which are encoded through a variety of relations between Baxter Q-functions and which are universal for a variety of different models. The relations themselves turn out to be meaningful also in the situations when their derivation from the commutative algebra is not known or even possible. Being supplemented with the ansatz on the analytical structure of Q-functions, explicit equations can then be derived to encode the spectra of concrete physical models. Two examples are discussed in detail: One is rational spin chains where the obtained techniques turned out to be an extremely powerful computation tool which also, in certain cases, can be used to rigorously prove completeness of Bethe equations. The second one is AdS/CFT integrability - we will show how our general approach allows one to derive Quantum Spectral Curve that encodes the exact spectrum of planar N=4 SYM.

Lecturer

Date

29th May ~ 10th July, 2024

Schedule

Weekday | Time | Venue | Online |
---|---|---|---|

Monday,Wednesday | 13:30 - 15:15 | A3-4-101 | - |

Prerequisite

Hamiltonian mechanics, quantum mechanics. Complex analysis. Lie groups, Lie algebras and root systems, supersymmetry, basics of commutative rings (brief recap will be offered). Symmetric polynomials and related combinatorics. Familiarity with algebraic Bethe Ansatz and construction of transfer matrices.

Syllabus

Part 1 - Bethe Algebra for rational GL(M|N) spin chains

[1] Introduction: computing spectrum of Heisenberg spin chain, with the help of integrability and without

[2] Fused flag structure of GL(M|N) QQ-system

[3] Proof of completeness of Wronskian Bethe equations. Fast analytic Bethe solver. Lazy numerical Bethe solver.

[4] Classification of unitary representatinos of SU(p,q|N) and non-compact Young diagrams. One-loop spectrum of AdS/CFT

[5] Separation of Variables for rational GL(M) spin chains

Part 2 - Bethe Algebra and fused flag geometry

[6] Recap of Lie algebras and root systems. Mutation games with Bethe equations and Weyl group.

[7] ODE/IM correspondence on the example of anharmonic oscillator

[8] ODE/IM correspondence, extended QQ-system, opers, and fused flags

[9] Bethe algebra for D-series using pure spinors. Beyond ADE.

Part 3 - AdS/CFT integrability

[10] History and main results

[11] Quantum Spectral Curve

[12] Monodromy bootstrap

[1] Introduction: computing spectrum of Heisenberg spin chain, with the help of integrability and without

[2] Fused flag structure of GL(M|N) QQ-system

[3] Proof of completeness of Wronskian Bethe equations. Fast analytic Bethe solver. Lazy numerical Bethe solver.

[4] Classification of unitary representatinos of SU(p,q|N) and non-compact Young diagrams. One-loop spectrum of AdS/CFT

[5] Separation of Variables for rational GL(M) spin chains

Part 2 - Bethe Algebra and fused flag geometry

[6] Recap of Lie algebras and root systems. Mutation games with Bethe equations and Weyl group.

[7] ODE/IM correspondence on the example of anharmonic oscillator

[8] ODE/IM correspondence, extended QQ-system, opers, and fused flags

[9] Bethe algebra for D-series using pure spinors. Beyond ADE.

Part 3 - AdS/CFT integrability

[10] History and main results

[11] Quantum Spectral Curve

[12] Monodromy bootstrap

Video Public

Yes

Notes Public

Yes

Language

English