Geometric Engineering, Mirror Symmetry and 6d (1,0) -> 4d, N=2

TL;DR

This work develops a comprehensive framework for obtaining 4d ${\cal N}=2$ SCFTs by toroidal compactification of 6d $(1,0)$ theories using F-theory geometric engineering and mirror symmetry. By mapping torus compactifications to Landau-Ginzburg mirrors, the authors derive vacuum geometries that include SW curves, LG-type vacua, and emergent class ${\cal S}$ structures, revealing two distinct routes to 4d fixed points: one with exact marginal couplings and $SL(2,\mathbb{Z})$ duality, and another where the 4d theory arises from an emergent punctured Riemann surface whose dualities are governed by the surface's mapping class group. The paper provides explicit realizations for orbifold 6d SCFTs, conformal matter, and non-Higgsable sectors, connecting 4d theories to $(E_n^{(1,1)},G)$ and $D_p(G)$ constructions, and clarifying the 6d origin of moduli spaces of affine ADE quivers. Overall, the work demonstrates that a single 6d theory can yield a rich zoo of 4d ${\cal N}=2$ theories, including class ${\cal S}$ theories, LG-vacua, and self-dual elliptic models, with profound implications for dualities and the geometric engineering dictionary.

Abstract

We study compactification of 6 dimensional (1,0) theories on T^2. We use geometric engineering of these theories via F-theory and employ mirror symmetry technology to solve for the effective 4d N=2 geometry for a large number of the (1,0) theories including those associated with conformal matter. Using this we show that for a given 6d theory we can obtain many inequivalent 4d N=2 SCFTs. Some of these respect the global symmetries of the 6d theory while others exhibit SL(2,Z) duality symmetry inherited from global diffeomorphisms of the T^2. This construction also explains the 6d origin of moduli space of 4d affine ADE quiver theories as flat ADE connections on T^2. Among the resulting 4d N=2 CFTs we find theories whose vacuum geometry is captured by an LG theory (as opposed to a curve or a local CY geometry). We obtain arbitrary genus curves of class S with punctures from toroidal compactification of (1,0) SCFTs where the curve of the class S theory emerges through mirror symmetry. We also show that toroidal compactification of the little string version of these theories can lead to class S theories with no punctures on arbitrary genus Riemann surface.

Figures (8)

• Figure 1: Type IIB mirror toric webs for compactification of 6d conformal theories of $A$-type. Generic situation which upon compactification gives rise to a Seiberg-Witten curve on a genus $g= M + (M-1)(N-1)$ Riemann surface with $2N$ punctures.
• Figure 2: up: brane web limit (no torus) Hanany:1997ghBrunner:1997gfdown: corresponding degeneration limit of a sphere with $M$ simple punctures and 2 full punctures.
• Figure 3: Horizontal/vertical (fiber/base) duality and corresponding degeneration limit: class $\mathcal{S}$$A_{M-1}$ theory on a torus with $N$ simple punctures.
• Figure 4: Brane web configuration giving the 5d version of a theory of class $\mathcal{S}[A_{k-1}]$ on a genus $g=g^\prime + (g^\prime-1)(p-1)$ Riemann surface with $2p$ full punctures, where $M=g^\prime k$ and $N=p k$
• Figure 5: Examples of 5d versions of class $\mathcal{S}[A_{k-1}]$ theories. left: 5d $T_k$ theory; right: 5d lift of class $\mathcal{S}[A_{k-1}]$ on a sphere with 4 full punctures and corresponding realization of it as the glueing of two $T_k$ theories.