# Check-Operators and Quantum Spectral Curves

## TL;DR

This paper reviews the role of check-operators in constructing quantum spectral curves and their connection to topological recursion, modular kernels, and classical integrability in effective theories.

## Contribution

It introduces the framework of check-operators for effective actions, linking quantum spectral curves to algebraic and geometric structures in physical theories.

## Key findings

- Check-operators form a non-commutative algebra related to modular kernels.
- Quantum spectral curves are constructed via topological recursion.
- Classical limits recover Seiberg-Witten integrability.

## 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