The Omega Deformation From String and M-Theory

TL;DR

This work provides a string-theory realization of the four-dimensional $\Omega$-deformation with generic $\epsilon_1$ and $\epsilon_2$ via fluxtrap backgrounds, linking Nekrasov’s instanton partition function to a brane construction and its M-theory lift. A $9$-$11$ flip exposes a noncommutative description of the Ω-deformed geometry, yielding a geometric origin for the quantum spectral curve of the associated integrable system. The analysis spans multiple limits, including the topological-string and NS regimes, and connects the deformation to the gauge/Bethe correspondence, offering explicit deformed couplings from D-brane actions and a concrete M5-brane perspective. Overall, the paper unifies Omega-deformations, noncommutativity, and quantum integrable structures within a single string-theoretic framework, with potential implications for AGT and refined topological strings.

Abstract

We present a string theory construction of Omega-deformed four-dimensional gauge theories with generic values of ε_1 and ε_2. Our solution gives an explicit description of the geometry in the core of Nekrasov and Witten's realization of the instanton partition function, far from the asymptotic region of their background. This construction lifts naturally to M-theory and corresponds to an M5-brane wrapped on a Riemann surface with a selfdual flux. Via a 9-11 flip, we finally reinterpret the Omega deformation in terms of non-commutative geometry. Our solution generates all modified couplings of the Ω-deformed gauge theory, and also yields a geometric origin for the quantum spectral curve of the associated quantum integrable system.

Figures (1)

• Figure 1: The tn interpolates between $\mathbb{R}^4$ and $\mathbb{R}^3 \times S^1$, while the corresponding ttrap interpolates between the fluxtrap in flat space (tip of the cigar) and $\mathbb{R}^3 \times S^1$ with a constant $B$ field.