Contributor info: Mr Osama Khlaif

Cyril Closset, Elias Furrer, Osama Khlaif

SciPost Phys. 18, 066 (2025) · published 24 February 2025 | Toggle abstract · pdf

We study three-dimensional $\mathcal{N}=2$ supersymmetric Chern–Simons-matter gauge theories with a one-form symmetry in the $A$-model formalism on $\Sigma_g× S^1$. We explicitly compute expectation values of topological line operators that implement the one-form symmetry. This allows us to compute the topologically twisted index on the closed Riemann surface $\Sigma_g$ for any real compact gauge group $G$ as long as the ground states are all bosonic. All computations are carried out in the effective $A$-model on $\Sigma_g$, whose $S^1$ ground states are the so-called Bethe vacua. We discuss how the 3d one-form symmetry acts on the Bethe vacua, and also how its 't Hooft anomaly constrains the vacuum structure. In the special case of the $SU(N)_K$ $\mathcal{N}=2$ Chern–Simons theory, we obtain results for the $(SU(N)/\mathbb{Z}_r)^{\theta}_K$ $\mathcal{N}=2$ Chern–Simons theories, for all non-anomalous $\mathbb{Z}_r \subseteq \mathbb{Z}_N$ subgroups of the centre of the gauge group, and with a $\mathbb{Z}_r$ $\theta$-angle turned on. In the special cases with $N$ even, ${N\over r}$ odd and ${K\over r}$ even, we find a mixed 't Hooft anomaly between gravity and the $\mathbb{Z}_r^{(1)}$ one-form symmetry of the $SU(N)_K$ theory, and the infrared 3d TQFT after gauging is spin. In all cases, we count the Bethe states and the higher-genus states in terms of refinements of Jordan's totient function. This counting gives us the twisted indices if and only if the infrared 3d TQFT is bosonic. Our results lead to precise conjectures about integrality of indices, which appear to have a strong number-theoretic flavour. Note: this paper directly builds upon unpublished notes by Brian Willett from 2020.

Cyril Closset, Osama Khlaif

SciPost Phys. 15, 085 (2023) · published 8 September 2023 | Toggle abstract · pdf

We consider unitary SQCD, a three-dimensional $\mathcal{N}=2$ supersymmetric Chern-Simons-matter theory consisting of one $U(N_c)_{k, k+l N_c}$ vector multiplet coupled to $n_f$ fundamental and $n_a$ antifundamental chiral multiplets, where $k$ and $l$ parameterise generic CS levels for $U(N_c)=(SU(N_c)\times U(1))/\mathbb{Z}_{N_c}$. We study the moduli space of vacua of this theory with $n_a=0$, for generic values of the parameters $N_c, k, l, n_f$ and with a non-zero Fayet-Ilopoulos parameter turned on. We uncover a rich pattern of vacua including Higgs, topological and hybrid phases. This allows us to derive a closed-form formula for the flavoured Witten index of unitary SQCD for any $n_f\neq n_a$, generalising previously known results for either $l=0$ or $n_f=n_a$. Finally, we analyse the vacuum structure of recently proposed infrared-dual gauge theories and we match vacua across the dualities, thus providing intricate new checks of those dualities. Incidentally, we also discuss a seemingly new level/rank duality for pure CS theories with $U(N)\times U(N')$ gauge group.