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String theory of the Omega deformation

Simeon Hellerman, Domenico Orlando, Susanne Reffert

TL;DR

The paper constructs a string-theoretic realization of the Omega deformation by embedding D2-branes in the fluxtrap background, the T-dual of a supersymmetric NS fluxbrane, which implements a SUSY-preserving real mass for adjoint chiral multiplets with $\varepsilon_1 = -\varepsilon_2$. This fluxtrap serves as a unifying framework that simultaneously reproduces the 3D real-mass deformation and the 4D Omega-deformation context via D3$'$-brane and D4$'$-brane perspectives, connecting to the Nekrasov–Shatashvili gauge/Bethe correspondence. The authors derive BPS saturated rotating brane solutions, compute the low-energy effective action with twisted masses, and discuss how the closed-string background encodes the deformed dynamics in both brane configurations. They further explore the emergence of a nonabelian SU(2) symmetry across vacua in brane setups and comment on potential extensions to SU(k), highlighting the broader implications for the interplay between gauge theories, spin chains, and string theory. Overall, the work provides a concrete, solvable string-theory model for Omega-deformed gauge theories and supplies tools for analyzing their nonperturbative structure.

Abstract

In this article, we construct a supersymmetric real mass deformation for the adjoint chiral multiplets in the gauge theory describing the dynamics of a stack of D2-branes in type II string theory. We do so by placing the D2-branes into the T-dual of a supersymmetric NS fluxbrane background. We furthermore note that this background is the string theoretic realization of an Omega-deformation of flat space in the directions transverse to the branes where the deformation parameters satisfy ε_1 = - ε_2. This Ω-deformation therefore serves to give supersymmetric real masses to the chiral multiplets of the 3D gauge theory. To illustrate the physical effect of the real mass term, we derive BPS-saturated classical solutions for the branes rotating in this background. Finally, we reproduce all the same structure in the presence of NS fivebranes and comment on the relationship to the gauge theory/spin-chain correspondence of Nekrasov and Shatashvili.

String theory of the Omega deformation

TL;DR

The paper constructs a string-theoretic realization of the Omega deformation by embedding D2-branes in the fluxtrap background, the T-dual of a supersymmetric NS fluxbrane, which implements a SUSY-preserving real mass for adjoint chiral multiplets with . This fluxtrap serves as a unifying framework that simultaneously reproduces the 3D real-mass deformation and the 4D Omega-deformation context via D3-brane and D4-brane perspectives, connecting to the Nekrasov–Shatashvili gauge/Bethe correspondence. The authors derive BPS saturated rotating brane solutions, compute the low-energy effective action with twisted masses, and discuss how the closed-string background encodes the deformed dynamics in both brane configurations. They further explore the emergence of a nonabelian SU(2) symmetry across vacua in brane setups and comment on potential extensions to SU(k), highlighting the broader implications for the interplay between gauge theories, spin chains, and string theory. Overall, the work provides a concrete, solvable string-theory model for Omega-deformed gauge theories and supplies tools for analyzing their nonperturbative structure.

Abstract

In this article, we construct a supersymmetric real mass deformation for the adjoint chiral multiplets in the gauge theory describing the dynamics of a stack of D2-branes in type II string theory. We do so by placing the D2-branes into the T-dual of a supersymmetric NS fluxbrane background. We furthermore note that this background is the string theoretic realization of an Omega-deformation of flat space in the directions transverse to the branes where the deformation parameters satisfy ε_1 = - ε_2. This Ω-deformation therefore serves to give supersymmetric real masses to the chiral multiplets of the 3D gauge theory. To illustrate the physical effect of the real mass term, we derive BPS-saturated classical solutions for the branes rotating in this background. Finally, we reproduce all the same structure in the presence of NS fivebranes and comment on the relationship to the gauge theory/spin-chain correspondence of Nekrasov and Shatashvili.

Paper Structure

This paper contains 30 sections, 155 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Supersymmetric real masses in field theory and string theory
  3. Real mass deformation of the D2--brane theory
  4. Lift to ${\cal N} = 1$ theories in 3+1 dimensions on a circle
  5. String embedding of the twisted mass for ${\cal N}=8$ SYM in $D=3$.
  6. Closed string fluxbrane and fluxtrap backgrounds
  7. Bulk fields and T--duality transformations
  8. Fluxbrane.
  9. Fluxtrap.
  10. Supersymmetry of the closed string background
  11. Fluxtrap solution as the string theory of the $\Omega$--background.
  12. Open strings
  13. Action and fermionic symmetries
  14. Supersymmetries preserved by the static embedding.
  15. BPS bound for the DBI action
  16. ...and 15 more sections

Figures (2)

  • Figure 1: Gauge and string theory interpretation of the real mass in three and four dimensions. The reduction in gauge theory is realized as a T--duality in string theory.
  • Figure 2: Realization of the $SU(2)$ symmetry via dualities in string theory.