4d N=1 from 6d N=(1,0) on a torus with fluxes

TL;DR

This work develops a bridge between six-dimensional ${ m N}=(1,0)$ theories on ${ m C}^2/{ m Z}_k$ and four-dimensional ${ m N}=1$ theories obtained by torus compactification with flux. By computing the 6d anomaly polynomial ${I_8}$ and integrating over the torus, the authors derive precise 4d anomaly data and map 6d charges to 4d line bundles, validating the Lagrangian quiver constructions for a wide class of flux choices. The 4d theories are organized as toric quivers with singlets, built from free trinions, and their anomaly and symmetry structures are checked against 6d predictions; several case studies (notably $N=k=2$, and $N=3$, $k=2$) reveal symmetry enhancements and nontrivial conformal-manifold dimensions that sometimes exceed naive 6d expectations due to accidental IR effects. Furthermore, the paper analyzes discrete twists via Stiefel-Whitney classes, showing how global-form data in 6d translates into 4d quiver twists and fluxes, with consistent anomaly matching across the two pictures. Overall, the work provides a comprehensive framework for predicting and verifying 4d ${ m N}=1$ theories from 6d compactifications with flux, including discrete fluxes, and offers explicit checks via indices and anomalies that support the proposed dictionary between dimensions and flux data.

Abstract

Compactifying N=(1,0) theories on a torus, with additional fluxes for global symmetries, we obtain N=1 supersymmetric theories in four dimensions. It is shown that for many choices of flux these models are toric quiver gauge theories with singlet fields. In particular we compare the anomalies deduced from the description of the six-dimensional theory and the anomalies of the quiver gauge theories. We also give predictions for anomalies of four-dimensional theories corresponding to general compactifications of M5-branes probing C_2/Z_k singularities.

Figures (8)

• Figure 1: On the left we have the free trinion. This is a collection of $N^2(2k)$ free fields. We organize these as bifundamentals of two copies of $\mathfrak{su}(N)^k$. In the picture the circles are $\mathfrak{su}(N)$ groups and one has $k$ groups winding around a cylinder. The trinion is associated to a compactification on a sphere with two maximal punctures (of different color) and a minimal puncture. On the right we glue trinions together to triangulate a torus. We have $lk$ trinions combined with every $\Phi$ gluing introducing bi-fundamental fields which appear as vertical lines in the diagram.
• Figure 2: We get rid of all the minimal punctures by giving vacuum expectation values to baryons. In every free trinion we give a vacuum expectation value to one of the baryons. The choice of the baryons is related to the flux and in general different choices lead to different theories in the IR. The baryons which do not receive a vev but are charged with same charge under the minimal puncture symmetry as the baryon which does receive vev, are flipped. In the diagrams the fields with a cross are the baryons which are flipped. In the picture the baryons which receive vacuum expectation value are weighed as $t^{\frac{N}{2}}\beta_1^N/\rho^N$, $t^{\frac{N}{2}}\gamma_2^N\alpha^N$, $t^{\frac{N}{2}}\gamma_2^N\epsilon^N$.
• Figure 3: On the left is the quiver diagram for the theory one gets when gluing together two free trions along all maximal punctures for $k=2$. On the right is a table summarizing the charges of the fields under all the non $R$-symmetries: the internal $\mathfrak{u}(1)_{\beta}, \mathfrak{u}(1)_{\gamma}, \mathfrak{u}(1)_{t}$, and the minimal puncture ones $\mathfrak{u}(1)_{\alpha}, \mathfrak{u}(1)_{\delta}$. Additionally there is a cubic superpotential for every triangle. Alternatively it is given by the most general cubic superpotential that is gauge invariant and consistent with the symmetry allocation in the table. All fields have the free $R$-charge $\frac{2}{3}$.
• Figure 4: The quiver diagram for the 4d class ${\cal S}_2$ theory corresponding to a torus with flux $(1,0,0)$. Next to the fields are their charges summarized through fugacities. We use mostly standard notation except for two points: lines from a group to itself represent $N^2$ hypermultiplets forming the adjoint plus singlet representations of the group; we write an X over a field to represent the fact that the baryon of that field is flipped. The theory has a cubic superpotential for every triangle which can also be derived by considering the most general cubic superpotential that is gauge invariant and consistent with the symmetry allocation. There is also the superpotential term which is not generally cubic coming from the flipping. All fields, save the flipping fields, have the free $R$-charge $\frac{2}{3}$.
• Figure 5: The quiver diagram for the 4d class ${\cal S}_2$ theory corresponding to a torus with flux $(\frac{1}{2},\frac{1}{2},0)$. Next to the fields are their charges summarized through fugacities. The theory has a quartic superpotential involving the four bifundamentals as well as the superpotential coming from the flipping. There is also an $R$-symmetry where it is convenient to give $R$-charge $\frac{1}{2}$ to the four bifundamentals $Q_i$ and $R$-charge $2-\frac{N}{2}$ to the flipping fields $\psi, \phi$.