Comments on Class S(YK)

We present a DSSYK-like interpretation of the Schur half-indices of $\mathcal{N}=2$ $SU(2)$ gauge theories with matter, in the presence of fundamental Wilson lines. The Schur half-indices of these theories can be understood as transition amplitudes in a non-vacuum sector of ordinary DSSYK. In the language of chord diagrams, the half-indices are obtained by summing over diagrams with special segments, which correspond to coherent states of the $q$-oscillator algebra. In addition, we show that the Schur half-index of $SU(2)$ gauge theory with $n_F=4$ fundamental half-hypermultiplets corresponds to the partition function of a particle on the quantum disk.

💡 Research Summary

In this paper the authors extend the recently observed correspondence between the Schur half‑indices of four‑dimensional 𝒩=2 SU(2) gauge theories with fundamental Wilson lines and the double‑scaled Sachdev‑Ye‑Kitaev (DSSYK) model. The key claim is that the half‑index of any asymptotically free or conformal SU(2) theory with up to four fundamental hypermultiplets (and also the 𝒩=2* theory with an adjoint hypermultiplet) can be interpreted as a transition amplitude in a non‑vacuum sector of ordinary DSSYK.

The paper begins with a concise review of DSSYK. In the double‑scaling limit (p→∞, N→∞ with λ=2p²/N fixed) the model is exactly solvable by a combinatorial sum over chord diagrams. Each chord diagram contributes a factor q² for every crossing, and the full partition function can be written as a vacuum‑to‑vacuum matrix element ⟨0|Hⁿ|0⟩ of a transfer matrix H = a + a†, where a, a† obey the q‑deformed harmonic‑oscillator algebra