Localizing punctures in M-theory

TL;DR

This paper develops a holographic, localization-based framework to study 4d $\mathcal{N}=1$ SCFTs arising from M5-branes wrapped on punctured Riemann surfaces. By exploiting equivariant localization with a Killing vector $\xi$, the authors compute the central charge and BPS operator dimensions from fixed-point data, bypassing the need to solve full 11d supergravity. They recover known results for locally $\mathcal{N}=2$ punctures and derive new central-charge contributions for genuinely locally $\mathcal{N}=1$ punctures, including detailed analysis for $\mathbb{C}^2/\mathbb{Z}_K$ and $\mathbb{C}^3/\mathbb{Z}_K$ orbifold geometries (e.g., $Z_3$ and $Z_5$). The work demonstrates that puncture data decomposes into bulk plus localized pieces and establishes a concrete procedure to extremize over R-symmetry parameters and remaining geometric moduli, linking holographic localization to field-theory $a$-maximization and paving the way for broader defect constructions.

Abstract

We use equivariant localization and holography to study four-dimensional $\mathcal{N}=1$ superconformal field theories arising from M5-branes wrapped on a punctured Riemann surface. We explain how, given a Riemann surface with marked points, one can glue in a ``puncture geometry'' locally around each point. Using equivariant localization we show that the central charge consists of a bulk contribution plus localized puncture contributions. We recover and generalize the known results for locally $\mathcal{N}=2$ preserving punctures, and derive new results for genuinely locally $\mathcal{N}=1$ preserving punctures.

Figures (12)

• Figure 1: Toric diagram for $\mathbb{C}^2/\mathbb{Z}_K$.
• Figure 2: Partial resolution of the $\mathbb{C}^2/\mathbb{Z}_K$ singularity. To each compact face (blue line between two fixed points) one can associate a compact divisor. The Poincaré dual to these divisors (discussed further in appendix \ref{['app:explicitG']}) has support in a small neighbourhood of the blue line. For the divisor $\mathcal{D}_2$ we have plotted the support in red for illustration.
• Figure 3: Embedding of the $A_{K-1}$ singularity into $M$. Over each interior point of the shaded region we have a copy of $\mathrm{U}(1)\times \mathrm{U}(1)_R\times S^2_R$. The first U$(1)$ rotates the Riemann surface coordinate $w$, with $\Sigma_\epsilon=\{|w|\leq \epsilon\}$, and is not an isometry of the full solution. Note: (i) the U$(1)$'s collapse along the blue edges, as per the toric diagram, (ii) $S^2_R$ collapses along the red edge, which is $|z_1|=1$, (iii) the dotted black line is the locus $|w|=\epsilon$, which is a copy of $\partial\Sigma_\epsilon\times S^4$, with the first factor being a small $S^1$ around the orbifold point $x\in\Sigma_{g,n}$. The region shown is hence the $S^4$ orbibundle over $\Sigma_\epsilon$, and this geometry is glued into $\Sigma_{\mathrm{bulk}}$ along the dotted black line.
• Figure 4: Local $\mathcal{N}=2$ puncture geometry. Again, over each interior point of the shaded region is a copy of $\mathrm{U}(1)\times \mathrm{U}(1)_R\times S^2_R$, with the U$(1)$'s collapsing along the blue edges as per the toric diagram. The finite blue line segments now represent four-cycles $D_a \equiv\mathcal{D}_a\times S^2_R\cong \mathbb{WCP}^1_{[k_a,k_{a-1}]}\times S^2_R$, associated to new fluxes $n_a\in\mathbb{Z}$ for the M-theory four-form $G$ as described in the main text. The shaded red region is the support of the bulk part of the flux, as discussed in appendix \ref{['app:explicitG']}.
• Figure 5: The 3d version of the toric diagram for the $\mathcal{N}=2$ puncture. The dark blue lines and the red line are the same as in figure \ref{['fig:toric']}, with the former obtained by cutting the diagram vertically. The red face denotes the locus where $\partial_{\varphi_2}\rightarrow 0$. The four-cycle $D_{\epsilon}^{[1]}$ consists of the blue face between the black dashed line and the first green line from the right. The four-cycle $D_{\epsilon}^{[2]}$ consists of all of the red face. We can also easily identify compact two-cycles in the diagram, they are simply lines in the diagram between two nodes. The green lines are the $S^2_{R,a}$ used in equation \ref{['eq:intYN2']} and \ref{['DeltaS2']}.