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.