Orientations for pseudoholomorphic curves with totally real boundary condition

Question

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.

Answer 1

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 It was solved Vin de Silva in his (unpublished) D. Phil.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.

Answer 2

(0) I do not know what is contained in a thesis which is not published and is not even available online (12 years after the defense).

(1) By Proposition 8.1.4 in the 1000-page FOOO book, a relatively spin structure on a (necessarily orientable) Lagrangian submanifold $L$ of a symplectic manifold $M$ determines an orientation on the moduli spaces of $J$-holomorphic disks $(D^2,S^1)\longrightarrow (M,L)$. The same reasoning applies to a family of real Cauchy-Riemann operators induced (as in Remark 1.3 of 1207.5471) by a bundle pair $(E,F)\longrightarrow(M,L)$, where $L$ is any submanifold of any manifold $M$. Proposition 8.1.7 gives an example of a non-orientable family of real Cauchy-Riemann operators, crediting it to Vin de Silva, and including a proof. I do not believe this book contains other, substantially different, statements on orientability in open GW-theory. Thus, this book discusses orienting moduli spaces of disks in some cases, but says fairly little about their orientability in general.

(2) By Theorem 1.1 in 0606429, a relatively Pin structure on a non-orientable Langrangian induces an isomorphism between the orientation line bundle of the moduli space of open $J$-holomorphic maps from Riemann surfaces with a fixed complex structure and a product of pull-backs of the orientation line bundle of the Lagrangian by evaluation maps. I do not believe this paper contains other, substantially different, statements on orientability in open GW-theory. Thus, this paper contains a number of results on both orienting and orientability of moduli spaces.

(3) Lemma 11.7 in Seidel's book does what Tim says in (1). Unfortunately, it requires more than a quick look to understand and see that it implies Proposition 8.1.4 in FOOO and the disk case of Theorem 1.1 in 0606429.

(4) Theorem 1.1 in 1207.5471 describes the holonomy of the orientation bundle of a family of real Cauchy-Riemann operators over bordered Riemann with varying complex structures. Its statement is absolutely clear from looking at the preceding half a page, at the beginning of the introduction. In particular, it is almost immediately clear that this theorem implies Proposition 8.1.4 in FOOO and the full statement of Theorem 1.1 in 0606429. The proof, contained in Section 3, is beautifully simple and uses no K-theory or even homotopy exact sequences.

(5) In the case of anti-symplectic involutions, one often wants to orient moduli spaces of real maps, not their halves, even if they are halvable. In the case of maps from $S^2$ with the standard conjugation, if the corresponding moduli space of disk maps is orientable, the space of real maps is orientable if the flip map on the disk space is orientation-preserving. So, this becomes a problem about computing the sign of this flip map. Results on this are Proposition 5.1 in 0606429 and Theorems 1.1 and 1.3 in 0912.2646. If the involution on $S^2$ has no fixed points, the orientation problem has nothing to do with any Lagrangians. Results on orientability in this case are Theorem 1.3 and Example 2.5 in 1205.1809 and Theorem 1.1 and Corollary 1.8 in 1301.1074.

(6) 1301.1074 says less about the Lagrangian case than 1207.5471. The point of 1301.1074 is to study the orientability problem for $J$-holomorphic maps that commute with involutions on the domain and the target. These maps need not be halvable to a map from a bordered Riemann surface with Lagrangian boundary conditions. An application is Corollary 1.8.