E-Strings and Four-Manifolds

TL;DR

The paper develops a framework linking 6d SCFTs to smooth 4-manifold topology via the partition function of the E-string theory on $M_4\times T^2$, demonstrating integrality, modularity, and a TMF lift. By dissecting the Higgs and Coulomb branch contributions and leveraging Seiberg–Witten geometry, the authors derive and test modular/Jacobi structures governed by 4-manifold invariants and anomaly data, including the simple-type conjecture. The results show that the E-string partition function encapsulates subtle smooth-structure information and, when extended to the $E_8\times E_8$ theory, can yield fully topological invariants even for $b_2^+\le1$, opening avenues for detecting exotic smooth structures. The work also outlines multiple future directions, such as defects, torsion-valued invariants, categorification, and refinements via $E_8$ topological Jacobi forms, indicating a rich interplay between TMF, modularity, and 4-manifold topology. Overall, the study establishes a concrete, computable bridge between high-dimensional QFT and 4-manifold topology with substantial implications for both mathematics and physics.

Abstract

We investigate the physics of the E-string theory and its compactifications as well as their applications to four-dimensional topology. In particular, we compute the partition function of the topologically twisted theory on $M_4\times T^2$, where $M_4$ is a four-manifold. In a range of examples, we verify that this partition function, as a $q$-series, 1) has integral coefficients, 2) is modular, and 3) can be lifted to a topological modular form. Remarkably, the E-string theory "knows" about various subtle aspects of the world of smooth 4-manifolds, as the (topological) modularity of the partition function is contingent on a collection of properties of 4-manifolds and their Seiberg-Witten invariants, including, notably, the simple-type conjecture. Furthermore, both theoretical and empirical evidences indicate that this partition function defines a genuine smooth invariant, even when $b_2^+\le 1$. Therefore, the E-string theory may offer powerful new tools for exploring regions in the geography of 4-manifolds that have been inaccessible to existing invariants obtained from gauge theory and quantum field theory.

Figures (11)

• Figure 1: Phases of the E-string theory. The E-string theory as two different phases, which can be understood from the M5-M9 system. (a) For the configuration on the left, the M5 is constraint to live (and "dissolve") on the M9, where it can be view as an $E_8$ instanton. This gives the Higgs branch of the E-string theory, and the SCFT is realized as the zero-size limit for the instanton at the origin of the Higgs branch. (b) The configuration on the right gives vacua on the tensor branch. In this phase, the E-strings, realized as M2-branes suspended between the M5 and M9, acquire tension proportional to the displacement $l$. From the viewpoint of this phase, the SCFT is at the tensionless limit $l\rightarrow0$.
• Figure 2: Moduli space of the $T^2$-compactified E-string theory. The Coulomb branch has a cigar shape with three singularities near the tip. The Higgs branch, which is the minimal nilpotent orbit of $E_8$, opens up at the $E_8$ singularity.
• Figure 3: Seiberg--Witten geometry of the E-string theory. On the left is the massless limit where black points indicate type I$_1$ singularities while the purple point indicates the type II$^*$ singularities. On the right is the schematic geometry with 12 I$_1$ singularities for generic masses.
• Figure 4: Duality from the spacetime torus.$T^2_\sigma$ can also be parameterized by a parallelogram with lengths $R_5,R_6$ and angle $\varphi$. Under a $T$-transformation, the $S^1_{R_{6}}$ winds over $S^1_{R_5}$ once, while under an $S$-transformation, the two radii exchange values leaving the shape of the torus unchanged.
• Figure 5: Coulomb branch of the KK-compactified E-string theory with special holonomies. We illustrate a one-parameter family of holonomies preserving a $D_7\subset E_8$ subgroup. It start with the trivial holonomy (Hol$=1$) and ends with "Hol$=-1$" which refers to the unique element of $E_8$ with $D_8\simeq Spin(16) /\mathbb{Z}_2$ stabilizer. The "$-1$" holonomy breaks $\mathrm{SL}(2,\mathbb{Z})$ into $\Gamma_0(2)$, which is the subgroup that leaves the configuration invariant. The other two configurations in the SL$(2,\mathbb{Z})$ orbit can be reached by turning on the holonomy in different ways, which are also illustrated with dashed lines. When the holonomy is "generic" (still required to be $D_7$-preserving), only a $\mathbb{Z}$ subgroup generated by $T$ is expected to be preserved.

Theorems & Definitions (1)

• Definition 1