Check-Operators and Quantum Spectral Curves

TL;DR

<3-5 sentence high-level summary>Check-operators provide a unifying framework to study families of theories by acting on their moduli; this leads to construction of (quantum) spectral curves and their topological recursion, and connects to SW geometry and modular kernels in CFT. The approach ties Virasoro constraints, loop equations, and DV/Whitham integrable hierarchies into a common language, enabling SW-like descriptions and quantization via main check-operators. It further shows that quantum spectral curves arise naturally from DV, AGT-deformed conformal blocks, and β-ensembles, and that modular kernels can be derived from period integrals of check-operator objects, aligning conformal blocks with integrable systems. The results illuminate deep links between matrix models, CFT, and quantum geometry, with potential broad applicability to universality classes of low-energy theories and topological string frameworks.

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