Equivariant localization and holography

Abstract

We discuss the theory of equivariant localization focussing on applications relevant for holography. We consider geometries comprising compact and non-compact toric orbifolds, as well as more general non-compact toric Calabi-Yau singularities. A key object in our constructions is the equivariant volume, for which we describe two methods of evaluation: the Berline-Vergne fixed-point formula and the Molien-Weyl formula, supplemented by the Jeffrey-Kirwan prescription. We present two applications in supersymmetric field theories. Firstly, we describe a method for integrating the anomaly polynomial of SCFTs on compact toric orbifolds. Secondly, we discuss equivariant orbifold indices that are expected to play a key role in the computation of supersymmetric partition functions. In the context of supergravity, we propose that the equivariant volume can be used to characterise universally the geometry of a large class of supersymmetric solutions. As an illustration, we employ equivariant localization to prove the factorization in gravitational blocks of various supergravity free energies, recovering previous results as well as obtaining generalizations.

Figures (10)

• Figure 1: The spindle $\mathbbl{\Sigma}$ as a circle fibration over a segment.
• Figure 2: Fixed points are associated with vertices $p_a$ of the polytope. The vectors in the fan are orthogonal to the facets. For each vertex there is a corresponding cone $(v^a,v^{a+1})$ in the fan. The two inwards normals $u_a^i$, $1=1,2$ to the cone, which lie along the edges of the polytope, enter in the fixed point formula through the quantities $\epsilon_i^a=\frac{\epsilon \cdot u_a^i}{d_{a,a+1}}$.
• Figure 3: The fan and the polytope for a generic quarilateral.
• Figure 4: The JK prescription for the $\mathbbl{\Sigma}_{n_+,n_-} \to \mathbbl{\Sigma}_{m_+,m_-}$ bundle.
• Figure 5: The fan and the polytope for ${\cal O}(-p)\to \mathbbl{\Sigma}$. The resolution introduces a compact facet in the polytope that is orthogonal to $v^2$. The corresponding segment represents the toric polytope of a spindle $\mathbbl{\Sigma}$.