Check-Operators and Quantum Spectral Curves

Understand

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