Tao Probing the End of the World

TL;DR

The paper introduces a novel Type IIB 5-brane (p,q) Tao web for the E-string theory, realized as the world-volume of an M5-brane probing the M9 boundary, and shows that its spiral, cyclic structure encodes the 6d uplift via KK modes. By applying the topological vertex to this Tao web, the authors derive a closed-form generating function for the E-string elliptic genera, identifying it with the topological-string partition function on local $ frac{1}{2}$K3 and demonstrating exact agreement with independent elliptic-genus calculations. A detailed combinatorial construction of the partition function is given, including perturbative and instanton pieces, with explicit results up to four instantons and matching Nekrasov results. The work suggests a broader framework (class $ ext{T}$) for encoding 6d SCFTs via spiral Tao webs and paves the way for refinements, higher-rank generalizations, and connections to broader dualities and AGT-type correspondences. Overall, the Tao-web approach provides a powerful, geometrically intuitive route to exact partition functions and the 6d UV structure of E-string and related theories.

Abstract

We introduce a new type IIB 5-brane description for the E-string theory which is the world-volume theory on the M5-brane probing the end of the world M9-brane. The E-string in the new realization is depicted as spiral 5-branes web equipped with the cyclic structure which is key to uplifting to six dimensions. Utilizing the topological vertex to the 5-brane web configuration enables us to write down a combinatorial formula for the generating function of the E-string elliptic genera, namely the full partition function of topological strings on the local 1/2 K3 surface.

Figures (15)

• Figure 1: (a) A 5-brane web for $SU(2)$ pure Yang--Mills theory that corresponds to the local first del Pezzo $\mathbb{CP}^1\times \mathbb{CP}^1$. Introducing 7-branes (b) regularizes the configuration. The Hanany--Witten effect leads to (c) where the 7-brane background ${\bf B}{\bf C}{\bf B}{\bf C}$ is probed by a 5-brane loop.
• Figure 2: On the left-hand side, the starting configuration with 12 7-branes in the 5-brane loop. Moving them outside leads to the middle diagram (up to flop transition). On the right-hand side, we can move two 7-branes (blue dots) to infinity, which makes a diagram with two arms that rotate in a spiral infinitely many times.
• Figure 3: Similarity between a Tao web diagram and a Taoism symbol
• Figure 4: Straight 5-branes crossing a 5-brane are introduced to denote the 5-branes that are actually jumping over a 5-brane, for simplicity.
• Figure 5: A 5-brane web configuration that gives the local ninth del Pezzo surface.