tt* Geometry in 3 and 4 Dimensions

TL;DR

The work develops a unified tt* framework that generalizes the 2d tt* geometry to 3d and 4d theories with four supercharges on flat torii. By treating compactified dimensions as infinite towers of 2d fields, the authors derive generalized monopole and instanton equations—hyperholomorphic connections in higher dimensions—whose topological data are encoded in line- and surface-operator rings. They connect these structures to elongated-space partition functions (S^2×S^1, S^3, S^2×T^2, S^3_b) and reveal a Nahm/NFM-type transform acting on the vacuum bundles, producing Verlinde-like algebras and spectral Lagrangian manifolds. The paper provides explicit 3d and 4d examples (free chiral, CP^1, SQCD, class R theories) and develops tools to compute brane amplitudes, S-matrices, and CFIV indices in these higher-dimensional tt* geometries, establishing deep links between tt* data, topological field theories, and holographic-brane constructions. Overall, the results extend the tt* program to a broader, highly structured landscape where monopole/instanton equations, Nahm transforms, and line/surface operators control the protected vacua geometry across dimensions, with concrete predictions for partition functions and Verlinde-type algebras.

Abstract

We consider the vacuum geometry of supersymmetric theories with 4 supercharges, on a flat toroidal geometry. The 2 dimensional vacuum geometry is known to be captured by the $tt^*$ geometry. In the case of 3 dimensions, the parameter space is $(T^{2}\times {\mathbb R})^N$ and the vacuum geometry turns out to be a solution to a generalization of monopole equations in $3N$ dimensions where the relevant topological ring is that of line operators. We compute the generalization of the 2d cigar amplitudes, which lead to $S^2\times S^1$ or $S^3$ partition functions which are distinct from the supersymmetric partition functions on these spaces, but reduce to them in a certain limit. We show the sense in which these amplitudes generalize the structure of 3d Chern-Simons theories and 2d RCFT's. In the case of 4 dimensions the parameter space is of the form $(T^3\times {\mathbb R})^M\times T^{3N}$, and the vacuum geometry is a solution to a mixture of generalized monopole equations and generalized instanton equations (known as hyper-holomorphic connections). In this case the topological rings are associated to surface operators. We discuss the physical meaning of the generalized Nahm transforms which act on all of these geometries.

Figures (23)

• Figure 1: 1+1 dimensional geometry with circle of length $\beta$ as the space and vacuum $|\alpha\rangle$.
• Figure 2: A holomorphic basis for states can be produced by topologically twisted path-integral on an infinitely long cigar, with chiral fields inserted at the tip of the cigar.
• Figure 3: The topologically twisted two point function $\eta_{ij}$ can be computed by topologically twisted path-integeral on $S^2$, where we insert the chiral operators on the two ends of the sphere. The path-integral respects supersymmetry for arbitrary choice of metric on $S^2$.
• Figure 4: The hermitian metric, which is induced from the hermitian inner product on the Hilbert space in the ground states of the theory can be obtained by path-integral on an infinitely elongated $S^2$ where on one half we have a topologically twisted theory with chiral fields inserted and on the other the anti-topological twisted theory with anti-chiral fields inserted.
• Figure 5: The overlap of the vacuum states with the D-branes give rise to $\Pi^a_i$ which are flat sections of the improved $tt^*$ connection.