Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 047, 17 pages      arXiv:1701.03057      https://doi.org/10.3842/SIGMA.2017.047
Contribution to the Special Issue on Combinatorics of Moduli Spaces: Integrability, Cohomology, Quantisation, and Beyond

Check-Operators and Quantum Spectral Curves

Andrei Mironov abcd and Alexei Morozov bcd
a) Lebedev Physics Institute, Moscow, 119991, Russia
b) ITEP, Moscow, 117218, Russia
c) Institute for Information Transmission Problems, Moscow, 127994, Russia
d) National Research Nuclear University MEPhI, Moscow, 115409, Russia

Received January 29, 2017, in final form June 19, 2017; Published online June 26, 2017

Abstract
We review the basic properties of effective actions of families of theories (i.e., the actions depending on additional non-perturbative moduli along with perturbative couplings), and their description in terms of operators (called check-operators), which act on the moduli space. It is this approach that led to constructing the (quantum) spectral curves and what is now nicknamed the EO/AMM topological recursion. We explain how the non-commutative algebra of check-operators is related to the modular kernels and how symplectic (special) geometry emerges from it in the classical (Seiberg-Witten) limit, where the quantum integrable structures turn into the well studied classical integrability. As time goes, these results turn applicable to more and more theories of physical importance, supporting the old idea that many universality classes of low-energy effective theories contain matrix model representatives.

Key words: matrix models; check-operators; Seiberg-Witten theory; modular kernel in CFT.

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