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First order deconfinement transition in large<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math>Yang-Mills theory on a small<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>S</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:math>
We give an analytic demonstration that the $3+1$-dimensional large $N$ $SU(N)$ pure Yang-Mills theory, compactified on a small ${S}^{3}$ so that the coupling constant at the compactification scale is very small, has a first order deconfinement transition as a function of temperature. We do this by explicitly computing the relevant …