N = 1 $$ \mathcal{N}=1 $$ theories of class S k $$ {\mathcal{S}}_k $$

On the Rational Recursive Sequence X_{N+1}=ƔX_{N-K}+(AX_N+BX_{N-K})⁄(CX_N-DX_{N-K})

Higgs branch localization of N = 1 theories on S 3 × S 1

We apply the method of Higgs branch localization to the N = 1 supersymmetric partition function on S3 × S1. As a result, we show that it can be written as the product of an elliptic vortex and anti-vortex partition function summed over a finite number of Higgs vacua.

Dynamics of higher order rational difference equation $x_{n+1}=(alpha+beta x_{n})/(A + Bx_{n}+ Cx_{n-k})$

The main goal of this paper is to investigate the periodic character, invariant intervals, oscillation and global stability and other new results of all positive solutions of the equation$$x_{n+1}=frac{alpha+beta x_{n}}{A + Bx_{n}+ Cx_{n-k}},~~ n=0,1,2,ldots,$$where the parameters $alpha$, $beta$, $A$, $B$ and $C$ are positive, and the initial conditions $x_{-k},x_{-k+1},ldots,x_{-1},x_{0}$ are...

on the rational recursive sequence x_{n+1}=ɣx_{n-k}+(ax_n+bx_{n-k})⁄(cx_n-dx_{n-k})

The moduli space of vacua of N = 2 class S theories

We develop a systematic method to describe the moduli space of vacua of four dimensional N = 2 class S theories including Coulomb branch, Higgs branch and mixed branches. In particular, we determine the Higgs and mixed branch roots, and the dimensions of the Coulomb and Higgs components of mixed branches. They are derived by using generalized Hitchin’s equations obtained from twisted compactifi...