For WLZZ two-matrix models, integration contours describe only a finite-N subspace of Ward identity solutions, and the full solution space is recovered only in the N-to-infinity limit.
Check-Operators and Quantum Spectral Curves
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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
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Phases in WLZZ Matrix Models
For WLZZ two-matrix models, integration contours describe only a finite-N subspace of Ward identity solutions, and the full solution space is recovered only in the N-to-infinity limit.