No Arabic abstract
Strings in $mathcal{N}=2$ supersymmetric ${rm U}(1)^N$ gauge theories with $N$ hypermultiplets are studied in the generic setting of an arbitrary Fayet-Iliopoulos triplet of parameters for each gauge group and an invertible charge matrix. Although the string tension is generically of a square-root form, it turns out that all existing BPS (Bogomolnyi-Prasad-Sommerfield) solutions have a tension which is linear in the magnetic fluxes, which in turn are linearly related to the winding numbers. The main result is a series of theorems establishing three different kinds of solutions of the so-called constraint equations, which can be pictured as orthogonal directions to the magnetic flux in ${rm SU}(2)_R$ space. We further prove for all cases, that a seemingly vanishing Bogomolnyi bound cannot have solutions. Finally, we write down the most general vortex equations in both master form and Taubes-like form. Remarkably, the final vortex equations essentially look Abelian in the sense that there is no trace of the ${rm SU}(2)_R$ symmetry in the equations, after the constraint equations have been solved.
A formula was recently proposed for the perturbative partition function of certain $mathcal N=1$ gauge theories on the round four-sphere, using an analytic-continuation argument in the number of dimensions. These partition functions are not currently accessible via the usual supersymmetric-localisation technique. We provide a natural refinement of this result to the case of the ellipsoid. We then use it to write down the perturbative partition function of an $mathcal N=1$ toroidal-quiver theory (a double orbifold of $mathcal N=4$ super Yang-Mills) and show that, in the deconstruction limit, it reproduces the zero-winding contributions to the BPS partition function of (1,1) Little String Theory wrapping an emergent torus. We therefore successfully test both the expressions for the $mathcal N=1$ partition functions, as well as the relationship between the toroidal-quiver theory and Little String Theory through dimensional deconstruction.
We study weak coupling perturbative series in 4d N=2 and 5d N=1 supersymmetric gauge theories with Lagrangians. We prove that the perturbative series of these theories in zero instanton sector are Borel summable for various observables. Our result for 4d $mathcal{N}=2$ case supports an expectation from a recent proposal on a semiclassical realization of infrared renormalons in QCD-like theories, where the semiclassical solution does not exist in N=2 theories and the perturbative series are unambiguous, namely Borel summable. We also prove that the perturbative series in arbitrary number of instanton sector are Borel summable for a wide class of theories. It turns out that exact results can be obtained by summing over the Borel resummations in each number of instanton sector.
A formulation of (non-anticommutative) N=1/2 supersymmetric U(N) gauge theory in noncommutative space is studied. We show that at one loop UV/IR mixing occurs. A generalization of Seiberg-Witten map to noncommutative and non-anticommutative superspace is employed to obtain an action in terms of commuting fields at first order in the noncommutativity parameter tetha. This leads to abelian and non-abelian gauge theories whose supersymmetry transformations are local and non-local, respectively.
A solution to the infinite coupling problem for N=2 conformal supersymmetric gauge theories in four dimensions is presented. The infinitely-coupled theories are argued to be interacting superconformal field theories (SCFTs) with weakly gauged flavor groups. Consistency checks of this proposal are found by examining some low-rank examples. As part of these checks, we show how to compute new exact quantities in these SCFTs: the central charges of their flavor current algebras. Also, the isolated rank 1 E_6 and E_7 SCFTs are found as limits of Lagrangian field theories.
We consider minimally supersymmetric QCD in 2+1 dimensions, with Chern-Simons and superpotential interactions. We propose an infrared $SU(N) leftrightarrow U(k)$ duality involving gauge-singlet fields on one of the two sides. It shares qualitative features both with 3d bosonization and with 4d Seiberg duality. We provide a few consistency checks of the proposal, mapping the structure of vacua and performing perturbative computations in the $varepsilon$-expansion.