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Register a new userMacdonald Indices for Four-dimensional $mathcal N=3$ Theories
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We brute-force evaluate the vacuum character for $mathcal N=2$ vertex operator algebras labelled by crystallographic complex reflection groups $G(k,1,1)=mathbb Z_k$, $k=3,4,6$, and $G(3,1,2)$. For $mathbb Z_{3,4}$ and $G(3,1,2)$ these vacuum characters have been conjectured to respectively reproduce the Macdonald limit of the superconformal index for rank one and rank two S-fold $mathcal N=3$ theories in four dimensions. For the $mathbb Z_3$ case, and in the limit where the Macdonald index reduces to the Schur index, we find agreement with predictions from the literature.
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We compute the supersymmetric partition function of $mathcal{N}{=}1$ supersymmetric gauge theories with an $R$-symmetry on $mathcal{M}_4 cong mathcal{M}_{g,p}times S^1$, a principal elliptic fiber bundle of degree $p$ over a genus-$g$ Riemann surface, $Sigma_g$. Equivalently, we compute the generalized supersymmetric index $I_{mathcal{M}_{g,p}}$, with the supersymmetric three-manifold ${mathcal{M}_{g,p}}$ as the spatial slice. The ordinary $mathcal{N}{=}1$ supersymmetric index on the round three-sphere is recovered as a special case. We approach this computation from the point of view of a topological $A$-model for the abelianized gauge fields on the base $Sigma_g$. This $A$-model---or $A$-twisted two-dimensional $mathcal{N}{=}(2,2)$ gauge theory---encodes all the information about the generalized indices, which are viewed as expectations values of some canonically-defined surface defects wrapped on $T^2$ inside $Sigma_g times T^2$. Being defined by compactification on the torus, the $A$-model also enjoys natural modular properties, governed by the four-dimensional t Hooft anomalies. As an application of our results, we provide new tests of Seiberg duality. We also present a new evaluation formula for the three-sphere index as a sum over two-dimensional vacua.
Superconformal indices (SCIs) of 4d ${mathcal N}=4$ SYM theories with simple gauge groups are described in terms of elliptic hypergeometric integrals. For $F_4, E_6, E_7, E_8$ gauge groups this yields first examples of integrals of such type. S-duality transformation for G_2 and F_4 SCIs is equivalent to a change of integration variables. Equality of SCIs for SP(2N) and SO(2N+1) group theories is proved in several important special cases. Reduction of SCIs to partition functions of 3d $mathcal{N}=2$ SYM theories with one matter field in the adjoint representation is investigated, corresponding 3d dual partners are found, and some new related hyperbolic beta integrals are conjectured.
A class of 4d $mathcal{N}=3$ SCFTs can be obtained from gauging a discrete subgroup of the global symmetry group of $mathcal{N}=4$ Super Yang-Mills theory. This discrete subgroup contains elements of both the $SU(4)$ R-symmetry group and the $SL(2,mathbb{Z})$ S-duality group of $mathcal{N}=4$ SYM. We give a prescription for how to perform the discrete gauging at the level of the superconformal index and Higgs branch Hilbert series. We interpret and match the information encoded in these indices to known results for rank one $mathcal{N}=3$ theories. Our prescription is easily generalised for the Coloumb branch and the Higgs branch indices of higher rank theories, allowing us to make new predictions for these theories. Most strikingly we find that the Coulomb branches of higher rank theories are generically not-freely generated.
Using the off-shell formulation for ${mathcal N}=2$ conformal supergravity in four dimensions, we propose superconformal higher-spin multiplets of conserved currents and their associated unconstrained gauge prepotentials. The latter are used to construct locally superconformal chiral actions, which are demonstrated to be gauge invariant in arbitrary conformally flat backgrounds.
We consider supergravity theories with 16 supercharges in Minkowski space with dimensions $d>3$. We argue that there is an upper bound on the number of massless modes in such theories depending on $d$. In particular we show that the rank of the gauge symmetry group $G$ in $d$ dimensions is bounded by $r_Gleq 26-d$. This in particular demonstrates that 4 dimensional ${cal N}=4$ SYM theories with rank bigger than 22, despite being consistent and indeed finite before coupling to gravity, cannot be consistently coupled to ${cal N}=4$ supergravity in Minkowski space and belong to the swampland. Our argument is based on the swampland conditions of completeness of spectrum of defects as well as a strong form of the distance conjecture and relies on unitarity as well as supersymmetry of the worldsheet theory of BPS strings. The results are compatible with known string constructions and provide further evidence for the string lamppost principle (SLP): that string theory lamppost seems to capture ${it all}$ consistent quantum gravitational theories.
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