As one might know that topology plays an important role in the classical O(2) model in d=2 (i.e. x-y model in two dimension) becuase one can have non-trivial excitations, namely, vortices. These excitations, while having a huge energy, can contribute to the partition function at high temperatures becuase the entropy associated with them is also proportionately large. One can easily verify that perturbative epsilon-expansion (i.e. an expansion in no. of dimensions) fails here. The reason which is often cited is that such an expansion destroys all information related to topology since atleast till now there is no homotopy theory for fractional dimensions.
Considering an even simpler case, the ising transition in d=2 could be looked upon as condensation of domain walls which are again topological objects. My question is: why does an epsilon-expansion in this case works out quite nicely? (note: here one does an expansion near the lower critical dimension which is one). Is it that notion of domain walls can be extended to even fractional dimension in some way?
Btw, one might like to visit the website http://www.math.princeton.edu/~aizenman/OpenProblems.iamp/index.html which contains a collection of open problems in mathematical physics. Coming to a problem related to this post (though somewhat remotely), one might be surprised to find through this link that the perturbative result of O(N > 2) model in d=2 being always in a single disordered phase at all temperatures has actually not been rigourosly proved. Infact, it is even disputed by some numerical results (though I don’t claim any understanding of these). See for example, http://prola.aps.org/abstract/PRB/v54/i10/p7177_1.