  We investigate the large-scale behaviour of the Self-Repelling Brownian
Polymer (SRBP) in the critical dimension $d=2$. The SRBP is a model of
self-repelling motion, which is formally given by the solution a stochastic
differential equation driven by a standard Brownian motion and with a drift
given by the negative gradient of its own local time. As with its discrete
counterpart, the "true" self-avoiding walk (TSAW) of [D.J. Amit, G. Parisi, &
L. Peliti, Asymptotic behaviour of the "true" self-avoiding walk, Phys. Rev. B,
1983], it is conjectured to be logarithmically superdiffusive, i.e. to be such
that its mean-square displacement grows as $t(\log t)^\beta$ for $t$ large and
some currently unknown $\beta\in(0,1)$. The main result of the paper is an
invariance principle for the SRBP under the weak coupling scaling, which
corresponds to scaling the SRBP diffusively and simultaneously tuning down the
strength of the self-interaction in a scale-dependent way. The diffusivity for
the limiting Brownian motion is explicit and its expression provides compelling
evidence that the $\beta$ above should be $1/2$. Further, we derive the scaling
limit of the so-called environment seen by the particle process, which formally
solves a non-linear singular stochastic PDE of transport-type, and prove this
is given by the solution of a stochastic linear transport equation with
enhanced diffusivity.
