Deconfinement on ℝ 2 × S L 1 × S β 1 $$ {mathrm{mathbb{R}}}^2times {S}_L^1times {S}_{beta}^1 $$ for all gauge groups and duality to double Coulomb gas

I study finite-temperature $$ mathcal{n}=1 $$ super yang-mills for any gauge group g = a n , b n , c n , d n , e 6,7,8, f 4, g 2, compactified from four dimensions on a torus, $$ {mathrm{mathbb{r}}}^2times {s}_l^1times {s}_{beta}^1 $$ . i examine in particular the low temperature regime l ≪ β = 1/t, where l is the length of the spatial circle with periodic boundary conditions and with anti-periodic boundary conditions for the adjoint gauginos along the thermal cycle s 1 . for small such l we are in a regime were semiclassical calculations can be performed and a transition occurs at t c much smaller than 1/nl. the transition is mediated by the competition between non-perturbative objects including ‘exotic’ topological molecules: neutral and magnetic bions composed of bps and kk monopole constituents, with r = rank(g) different charges in the co-root lattice of the gauge group g, and the perturbative electrically charged w-bosons (along with their wino superpartners). the difference from non-susy theories here is that the higgsing along the thermal cycle gives rise to a light modulus scalar field which couples to both bion-instantons and the w-bosons, and mediates a transition near t c where the bions and w-bosons compete with equal strengths. the transition is seen to be similar to previous studies on $$ {mathrm{mathbb{r}}}^3times {s}_l^1 $$ [1, 2, 12, 13] with general gauge group where a first order transition was found for all groups, but a second order one for the case of su(2) on the torus $$ {mathrm{mathbb{r}}}^2times {s}_l^1times {s}_{beta}^1 $$ , which was subjected to lattice studies in [1]. i determine a duality to a double coulomb gas of neutral and magnetic bions of different charges of their constituent monopole-instantons, and w-bosons of both scalar and electric charges. aharanov-bohm interactions exist between magnetic bions and w-bosons, and scalar charges of w-bosons and neutral bions attract like charges, as opposed to the magnetic and electric charges where like charges repel. it is hoped in the future that lattice studies of this coulomb gas can be done as in [1] for all gauge groups. it is hoped that a dual lattice ‘affine’ xy model with symmetry breaking perturbations can also be found in future studies of general gauge group as done in [1] for su(2).