I am trying to understand what the obstructions are to orienting moduli spaces of pseudoholomorphic curves with totally real boundary condition. 


I believe that Fukaya-Oh-Ohta-Ono have shown that if a Lagrangian is relatively spin, the moduli spaces of disks with boundary in it can be oriented. 


My question has 3 related parts:


1. is there any sense in which the FOOO relative spin condition is also necessary?
2. if I consider curves of higher genus and/or more boundary components, do I need to impose additional conditions on the Lagrangian to guarantee orientability of the moduli spaces?
3. There has been a fair bit of recent work in the case in which the Lagrangian is the fixed point set of an anti-symplectic involution (Crétois, Georgieva-Zinger). Are the orientation difficulties in this case the same as in the general case, or do some special features appear here?


EDIT: Penka Georgieva pointed out that I was mistaken. Her paper (arxiv/1207.5471) deals with the general case of curves with boundary on a Lagrangian.



Orientations for pseudoholomorphic curves with totally real boundary condition
1) The problem of orienting moduli spaces of pseudo-holomorphic discs with totally real boundary conditions is really a problem in index theory. It was solved Vin de Silva in his (unpublished) D. Phil. thesis, using Atiyah's Real K-theory, and independently by FOOO. There's an excellent account in Seidel's book (section 11, especially Lemma 11.7).


A totally real Cauchy-Riemann problem is by definition a loop in the totally real Grassmannian $\mathrm{Gr}(V)$ of a complex vector space $V$ with a given real structure. The resulting space $L\mathrm{Gr}(V)$ parametrizes a family of Fredholm operators (the Cauchy-Riemann operator for functions on the closed disc, valued in $V$, with boundary conditions specified by the loop). Hence there is a determinant index bundle $$\underline{det} \to L \mathrm{Gr}(V),$$ and the basic orientation problem is to describe $w\_1(\underline{det}) \in H^1(L \mathrm{Gr}(V);\mathbb{Z}/2)$. For the component $L\_k\mathrm{Gr}(V)$ of Maslov index $k$ loops, $H^1(L\_k \mathrm{Gr}(V);\mathbb{Z}/2)$ is 2 dimensional, as one calculates using a homotopy equivalence $\mathrm{Gr}(\mathbb{C}^n) \simeq U(n)/O(n)$, so there are just four possible answers to the orientation question for each $k$.


The answer is simpler to state assuming $k$ is even. It is then as follows (I learned this from de Silva's thesis): Take a loop $\gamma\colon S^1\to L\_k\mathrm{Gr}(V)$. Then $\langle w\_1(\underline{det}), \gamma\rangle = \langle w\_2, T\_\gamma \rangle$, where $w\_2$ is the second SW class of the universal totally real bundle on $\mathrm{Gr}(V)$, and $T\_\gamma\colon S^1\times S^1\to \mathrm{Gr}(V)$ is the torus of boundary values swept out by $\gamma$.


So, in a space of pseudo-holomorphic discs in a symplectic manifold $X$ attached to an orientable Lagrangian $\Lambda$, $w\_1$ of the determinant bundle evaluates on a loop $\gamma$ by evaluating the torus of boundary values $T\_\gamma$ on $w\_2(T\Lambda)$. Essentially for this reason, it's natural to trivialize the determinant bundle by trivializing $w\_2(\Lambda)$, i.e. specifying a Pin structure. More generally, one can trivialize $w\_2(\Lambda)$ relative to a fixed background class $b\in H^2(\Lambda;\mathbb{Z}/2)$ which restricts to $w\_2(\Lambda)$, i.e. specify a relative (or "twisted") Pin structure. That suffices essentially because the torus of boundary values is the boundary of a 3-chain in $X$, and so vanishes if $w\_2$ is the restriction of $b$. Choosing Pin structures relative to a fixed background class $b$ gives a uniform way to orient moduli spaces of pseudo-holomorphic discs for relatively Pin Lagrangians.


2) For Cauchy-Riemann operators on other curves, one can degenerate to a nodal union of discs and closed Riemann surfaces, combining the orientations for the space of discs with the complex orientation of the determinant line bundle over the moduli of closed curves. Thus no additional obstructions to orientation appear.


To be precise, what goes into this is a gluing theorem for the index bundle, which is part of the linear analysis that underpins Floer theory and Gromov-Witten theory. It implies that the determinant index line over a connected sum is canonically isomorphic to the tensor product of the determinant index lines on the summands. Again, see Seidel's book, section 11 for the argument. For the underlying analysis, I'd recommend Donaldson's Floer homology book, chapter 3.


3) I don't know, but there are further concrete calculations for real loci in the work of Welschinger and also Solomon.


