Towards a Dual Representation of Lattice QCD

Our knowledge about the QCD phase diagram at finite baryon chemical potential $\mu_{B}$ is limited by the well known sign problem. The path integral measure, in the standard determinantal approach, becomes complex at finite $\mu_{B}$ so that standard Monte Carlo techniques cannot be directly applied. As the sign problem is representation dependent, by a suitable choice of the fundamental degrees of freedom that parameterize the partition function, it can get mild enough so that reweighting techniques can be used. A successful formulation, capable to tame the sign problem, is known since decades in the limiting case $\beta\to 0$, where performing the gauge integration first, gives rise to a dual formulation in terms of color singlets (MDP formulation). Going beyond the strong coupling limit represents a serious challenge as the gauge integrals involved in the computation are only partially known analytically and become strongly coupled for $\beta>0$. We will present explict formulae for all the integral relevant for ${\rm SU}(N)$ gauge theories discretised \`a la Wilson, and will discuss how they can be used to obtain a positive dual formulation, valid for all $\beta$, for pure Yang Mills theory.