Surface Operators
Sergei Gukov
TL;DR
This work surveys surface operators as two‑dimensional defects in four‑dimensional $\mathcal N=2$ gauge theories, detailing their definitions, classifications, and roles in the AGT correspondence where they map to degenerate Liouville insertions via $Z^{\text{inst}}$ and $Z^{\text{Liouv}}$ with $\mathcal W$ related to the Seiberg–Witten differential by $\partial_a\mathcal W=\eta+\tau\alpha$ and $\mathcal W=\int_{p_*}^p \lambda_{SW}$. It synthesizes higher‑dimensional constructions (6d $(0,2)$, branes, geometric engineering) and geometric interpretations (open/closed BPS invariants) to connect 4d defects with 3d/5d theories and integrable systems via 3d‑3d correspondence and the NS limit, where Baxter equations and Bethe equations arise. The discussion of line operators within surface operators reveals non‑commutative OPE algebras governed by affine Hecke algebras and DAHA, tied to autoequivalences of derived categories and monodromies in the Kähler moduli space. Finally, the paper links these structures to the superconformal index and holographic duals, showing how surface operators function as order parameters that diagnose confining vs deconfined phases through geometric realizations on branes and gravity duals.
Abstract
This is the seventh article in the collection of reviews "Exact results on N=2 supersymmetric gauge theories", ed. J.Teschner. It discusses an interesting class of observables localised on surfaces that attracts steadily growing attention. In the correspondence to conformal field theory some of these observables get related to a class of fields in two dimensions called degenerate fields. These fields satisfy differential equations that can be used to extract a lot of information on the correlation functions. Understanding the origin of these differential equations within gauge theory may help explaining the AGT-correspondence itself.