I hope to compute a functional integral Z=∫Dϕe−S[ϕ] with an action
S[ϕ]=∫d2x√g((∇ϕ)2+1gM2(x)ϕ2)
The scalar field ϕ is defined on a two-dimensional curved sphere.. The mass-like term 1gM2(x)ϕ2 depends explicitly on x and I'm interested in limit g→0. Formally the result is the functional determinant logZ=logdet(−Δ+1gM2(x)) and I'm interested the small g expansion as a functional of M2(x).
I'm not very familiar with functional determinants but I've tried to apply the heat kernel method here without much success. The small g expansion here does not seem to reduce to the conventional large mass expansion. Moreover the heat kernel coefficients have the form like a2=∫d2x√g(16R−1gM2(x)) while I naively expect that the leading order in small g limit should be
logdet(−Δ+1gM2(x))∼logdet(1gM2(x))=Trlog(1gM2(x))∼∫d2z√glog(1gM2(x))
Where the last line is my guess for what the functional trace of a function (diff operator of order 0) should be. Heat kernel method does not seem to produce logarithms like that.
Any comments and pointers to the literature are welcome.