Path integral for free fermion on torus If one will consider free fermion on torus,one will face with different spin structures.
There are four spin structures, usually labeled ±±. The ++ spin
structure has a single positive chirality zero-mode (the “constant”
mode of χ) and the other spin structures have none. 
Could somebody provide references where partition functions for fermions on torus were calculated using path integral for different versions of spin structures?
(Without using Hamiltonian)
 A: The standard reference for torus parttion functions is E. and H. Verlinde  Chiral  Bosonization, Determinants  and the String Partition Function, Nuc Phys B 288 (1987) 357-396. They show that
 partition function for the $++$ spin structure is zero. 
The correlators are not necessarily zero as the zero eigenvalue  in the denominator of the fermion Green function can cancel the zero factor in the determinant.  If I  remember correctly 
the fermion $++$ correlators  are given by 
$$
 {\rm Bdet}\left| \frac{d}{dz}\ln E(z_i-w_j)\right|
$$ where 
$$
E(z)=\frac
{ \theta\left[ \begin{matrix}{\textstyle{\frac 12}\\ \textstyle{\frac 12}}\end{matrix}\right] (z|\tau)} { \theta'\left[ \begin{matrix}{\textstyle{\frac 12}\\ \textstyle{\frac 12}}\end{matrix}\right] (0|\tau)}
$$
is the prime form and 
$$
{\rm Bdet}|a_{ij}|= \left|\matrix{ 0 &1         & 1         & \ldots    &1          \cr   
                                        1 & a_{11} & a_{12} & \ldots   & a_{iM} \cr
                                         1& a_{21} & a_{22}  &\ldots   & a_{2M}\cr
                                 \vdots & \vdots & \vdots   & \ddots &\vdots &\cr
                                         1 & a_{M1}&a_{M2} & \ldots  & a_{MM} \cr}
                                                     \right|.
$$
I worked this out myself for some reason by taking the $(a,b)\to (1/2,1/2)$ limit in the theta fuctions.
I don't know any reference except that my notes cite  G. Mason, M. P.Tuite, A. Zuevsky, Torus $n$-Point Functions for ${\mathbb R}$-graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds, Communications in Mathematical Physics,
 vol 283.  pp305-342 (2008); arXiv:0708.0640; for the idea of the bordered determinant.
This is topic I had meant to return to, so I'll be interested to see what other people have to say.
A: Check out D’Hoker & Phong’s 1988 review, ‘geometry of string perturbation theory’ (where you can also find numerous additional references); it is very pedagogical and uses the path integral throughout. Their paper  'loop amplitudes for the fermionic string, Sec 4' is very explicit and directly addresses your question. I seem to also remember Kiritsis' book containing a nice discussion of one-loop determinants. 
Some important foundational references are the Verlinde brothers' paper NPB288(1987) (although the path integral derivation is less explicit), and  Seiberg and Witten's 1986 paper, 'spin structures in string theory'.
This is by no means an exhaustive list, but it will get you started.
