The gravity dual of supersymmetric gauge theories on a squashed $S^1 \times S^3$

TL;DR

The paper constructs a one-parameter family of supersymmetric, asymptotically locally AdS solutions in five-dimensional minimal gauged supergravity whose conformal boundary is a squashed $S^3$ with a nontrivial R-symmetry background. The solution, analyzed via a combination of analytic (UV/IR expansions, linearised) and numerical methods, smoothly connects an AdS-like UV region to an IR cap, controlled by an IR parameter $oldsymbol{\xi}$ that sets the boundary squashing $v$. Holographic renormalisation yields an exact, gauge-dependent on-shell action that reduces to a boundary term, while holographic charges $E$, $J$, and $Q$ are computed in terms of the boundary data and satisfy BPS-type relations with the on-shell action. In the dual field theory, the background realizes rigid ${ m N}=1$ supersymmetry on a Hopf surface, and the gravity results are connected to the supersymmetric index and a Casimir-energy-like quantity, with a detailed discussion of scheme ambiguities and finite counterterms. These results illuminate the role of background geometry and R-symmetry fields in holographic descriptions of supersymmetric theories on curved spaces and provide concrete correspondences between gravity computations and field-theoretic indices on nontrivial manifolds.

Abstract

We present a new one-parameter family of supersymmetric solutions deforming AdS_5. This is constructed as an asymptotically locally anti de Sitter (AlAdS) solution of five-dimensional minimal gauged supergravity, with topology R x R^4 and a non-trivial graviphoton field, and can be uplifted to ten or eleven dimensional supergravities. An analytic continuation of this solution yields the gravity dual to a class of four-dimensional N=1 supersymmetric gauge theories on a curved manifold with topology S^1 x S^3, comprising an SU(2) x U(1)-symmetric squashed three-sphere, with a non-trivial background gauge field coupling to the R-symmetry current. We compute the holographically renormalised on-shell action and interpret it in terms of the Casimir energy of the dual field theory. We also determine the holographic conserved charges of the solution and discuss relations between them.

Figures (5)

• Figure 1: The solution $a$ to the sixth-order equation, rescaled by $e^{-\rho}$. Its square equals $g_{\theta\theta}/g_{\rho\rho}\,$. Asymptotically, it gives the parameter $a_0$, controlling the overall size of the boundary. The different values of the IR parameter $\xi$ are indicated on the curves.
• Figure 2: Ratio between the $g_{\psi\psi}$ and $g_{\theta\theta}$ components of the metric. Asymptotically, this gives the value of the parameter $v^2$, controlling the squashing of the boundary $S^3$.
• Figure 3: Other metric components and the gauge field $A$.
• Figure 4: Relation between the IR parameter $\xi$ and the UV parameter $v^2$ controlling the squashing of the boundary $S^3$. The squashing ranges between 0 and $\infty$ for $4.2 \gtrsim \xi \gtrsim -0.7$. The dots are effectively calculated values while the blue, continuous line is an interpolation. The red, dashed line represents the relation \ref{['UVparFromIRpar']} obtained from the linearised analysis around the AdS solution at $\xi =0$ (which is denoted by the slightly larger, red dot).
• Figure 5: The other UV parameters in terms of the squashing $v^2$.