IIB an equivariantly localized puncture

TL;DR

The paper develops and applies an odd-dimensional localization framework to AdS$_3$ solutions in type IIB supergravity with $ ext{N}=(2,2)$ supersymmetry, by reducing along a shrinking circle to a boundary and then performing BVAB localization on the resulting even-dimensional space. It leverages a $U(1)^2$ R-symmetry to localize integrals of equivariant polyforms, enabling computation of flux quantization, central charges, and operator dimensions without requiring explicit supergravity solutions. The authors apply the method to various geometries—S^3- and S^5-bundled internal spaces over bases and Riemann surfaces—and derive central charges and BPS spectra, including a detailed puncture analysis that produces orbifold and toric data and clarifies the role of defects in holography. These results provide a robust, solution-independent route to extract CFT data from holographic setups with punctures/defects, highlighting how GK-type structures organize the global geometry and fluxes in AdS$_3$/2d CFT duals, and offering new checks against field-theory extremization concepts.

Abstract

We use equivariant localization to compute various observables for $\mathcal{N}=(2,2)$ preserving AdS$_3$ solutions in type IIB supergravity. Our method for localizing the odd-dimensional internal space is to perform a dimensional reduction along a shrinking circle which introduces a boundary in the remaining even-dimensional spacetime. Canonical even-dimensional localization can then be performed on this space after taking into account contributions from the boundary. We illustrate with a number of supersymmetric solutions. One of the novel aspects of this work is applying these techniques to study punctures and defects using localization. We obtain new results for the central charge of 2d SCFTs arising from compactifying $\mathcal{N}=4$ SYM on a punctured Riemann surface.

Figures (4)

• Figure 1: A toric diagram for the (partial) resolution we will glue in. There is a $\mathbb{C}^2/\mathbb{Z}_{k_a}$ singularity at each $y_a$, where $\xi$ acts on the total space with weights $(\epsilon_1^a,\epsilon_2^a,\epsilon_3^a)=(- b_1 l_{a+1}/k_a, b_1 l_a/k_a, b_2)$. At ${y}_0=0$ the unfibred R-symmetry U$(1)$ shrinks. The toric vectors are $v_a=(1,l_a)$ where $l_a=\sum_{b=a}^{d}k_b$, such that $k_a=l_a-l_{a+1}$.
• Figure 2: The toric diagram for the puncture geometry giving rise to a $\mathbb{C}^3/\mathbb{Z}_K$ puncture. The boundary of the diagram comes from two disconnected regions; the front turquoise line where $y=0$ and the factored out circle shrinks, and the plum back-side which is the $S^5\times S^1_{D}$ boundary of the excised region that we glue in upon reintroducing the factored out U$(1)$.
• Figure 3: Example of a (partial) resolution of a $\mathbb{C}^3/\mathbb{Z}_K$ singularity.
• Figure 4: A sub-diagram of part of a larger toric diagram giving rise to a compact four-cycle in orange. To each vector $v_a$ we associate a possibly non-compact cycle $D_a$. Of interest to us here are also the two-cycles $S_a=D_0\cap D_a$, which are represented by the red lines in the figure.