  We find explicit SLE(8) partition functions for the scaling limits of Peano
curves in the uniform spanning tree (UST) in topological polygons with general
boundary conditions. They are given in terms of Coulomb gas integral formulas,
which can also be expressed in terms of determinants involving a-periods of a
hyperelliptic Riemann surface. We also identify the crossing probabilities for
the UST Peano curves as ratios of these partition functions. The partition
functions are interpreted as correlation functions in a logarithmic conformal
field theory (log-CFT) of central charge $c= -2$. Indeed, it is clear from our
results that this theory is not a minimal model and exhibits logarithmic
phenomena -- the limit functions have logarithmic asymptotic behavior, that we
calculate explicitly. General fusion rules for them could also be inferred from
the explicit formulas. The discovered algebraic structure matches the known
Virasoro staggered module classification, so in this sense, we give a direct
probabilistic construction for correlation functions in a log-CFT of central
charge $-2$ describing the UST model.
