T-Branes and Geometry

TL;DR

<3-5 sentences>The paper addresses how T-branes, which are nilpotent, non-abelian bound states on seven-branes, leave a geometric imprint in six-dimensional F-theory. It develops a framework based on limiting mixed Hodge structures to track the remnants of three-form moduli in singular Calabi–Yau limits, revealing an emergent Hitchin-like system with defects that captures the RR moduli localized on discriminants. By relating the intermediate Jacobian of the smooth threefold to the spectral-cover data in heterotic duals, the authors provide a unified description that ties closed-string moduli to open-string Hitchin data through Deligne cohomology in singular limits. The results are checked in compact F-theory models with heterotic duals and are argued to extend to four-dimensional vacua, offering a road map to incorporate T-brane data into global compactifications.

Abstract

T-branes are a non-abelian generalization of intersecting branes in which the matrix of normal deformations is nilpotent along some subspace. In this paper we study the geometric remnant of this open string data for six-dimensional F-theory vacua. We show that in the dual M-theory / IIA compactification on a smooth Calabi-Yau threefold X, the geometric remnant of T-brane data translates to periods of the three-form potential valued in the intermediate Jacobian of X. Starting from a smoothing of a singular Calabi-Yau, we show how to track this data in singular limits using the theory of limiting mixed Hodge structures, which in turn directly points to an emergent Hitchin-like system coupled to defects. We argue that the physical data of an F-theory compactification on a singular threefold involves specifying both a geometry as well as the remnant of three-form potential moduli and flux which is localized on the discriminant. We give examples of T-branes in compact F-theory models with heterotic duals, and comment on the extension of our results to four-dimensional vacua.

Figures (2)

• Figure 1: Depiction of the moduli space of F-theory in six dimensions. The intermediate Jacobian $J(X_{\mathrm{smth}})$ fibers over the complex structure moduli $M_{\mathrm{cplx}}$. At singular points of the complex structure, the classical intermediate Jacobian description breaks down and is replaced by an emergent Hitchin system which captures the T-brane data. At the singular point in moduli space where both the complex structure and T-brane data are switched off, one can instead perform a Kähler resolution of the geometry, moving onto the Coulomb branch of the low energy theory. The two branches only meet at singular loci in the moduli space.
• Figure 2: Depiction of the spectral curve $\widetilde{C}$ and some possible degenerations. In the figure, we illustrate in the case of a three-sheeted cover of a curve $C$. Possible degenerations include a non-reduced scheme of length three, as indicated by $\widetilde{C}\rightarrow3\widetilde{C}_{1}^{\text{nr}}$, by a factorization into two smooth reducible components, as indicated by $\widetilde{C}\rightarrow\widetilde{C}_{1} \cup \widetilde{C}_{2}$, and a factorization into two reducible components, one of which is a non-reduced scheme, as indicated by $\widetilde{C}\rightarrow2\widetilde{C}_{1}^{\text{nr}} \cup \widetilde{C}_{2}$.