[Paper Review] Real bundle automorphisms, Cauchy Riemann operators and orientability of moduli spaces

Rémi Crétois|arXiv (Cornell University)|Sep 12, 2013

Geometric and Algebraic Topology7 references3 citations

TL;DR

This paper computes the sign of automorphism actions on orientations of determinant line bundles over moduli spaces of real Cauchy-Riemann operators on complex vector bundles with real structures over real curves. It establishes that the orientability of moduli spaces of real pseudoholomorphic curves is governed by topological invariants, including the signature of permutation actions on $\mathrm{Pin}^{\pm}$ structures and bordism classes of real $\mathrm{Spin}$ structures, leading to explicit formulas for the first Stiefel-Whitney class of such moduli spaces in terms of canonical line bundles and real cohomology classes.

ABSTRACT

This paper consists in a very brief English summary of the results appearing in French in two previous articles. We omit the proofs and focus on explaining our approach and theorems. This paper is not intended to be published. Questions are welcome. In our work, we considered a complex vector bundle $N$ equipped with a real structure $c_N$ over a real curve of arbitrary genus. We computed the sign of the action of an automorphism of $(N,c_N)$ on the orientations of the determinant line bundle over the space of Cauchy-Riemann operators on $(N,c_N)$. We first considered the automorphisms lifting the identity on the curve (in the first article). In this case, we obtained the sign as a product of two terms. The first one computes the signature of the permutations induced by the automorphisms acting in the $Pin^\pm$ structures on the real part of $(N,c_N)$. The second one comes from the action of the automorphisms of $(N,c_N)$ on the bordism classes of real $Spin$ structures on the curve. We then studied the general case in the second article. As an application of these results, we computed the first Stiefel-Whitney class of the moduli space of real pseudoholomorphic curves in many cases.

Motivation & Objective

• To determine the sign of the action of real bundle automorphisms on orientations of determinant line bundles over spaces of real Cauchy-Riemann operators.
• To understand the orientability of moduli spaces of real pseudoholomorphic curves in real symplectic manifolds.
• To compute the first Stiefel-Whitney class of these moduli spaces using topological invariants such as $\mathrm{Pin}^{\pm}$ structures and $\mathrm{Spin}$ bordism classes.
• To establish conditions under which canonical orientations exist via polarizations and polarizing sections.

Proposed method

• Analyzes automorphisms of real vector bundles $(N, c_N)$ over real curves $(\Sigma_g, c_\Sigma)$, focusing on those lifting the identity on $\Sigma_g$.
• Decomposes the orientation sign into two components: the signature of permutations on $\mathrm{Pin}^{\pm}$ structures over the real part $\mathbb{R}N$, and the action on bordism classes of real $\mathrm{Spin}$ structures on $\Sigma_g$.
• Extends results to general automorphisms via a generalization of the action computation, incorporating diffeomorphisms $\varphi$ on $\Sigma_g$ and their lifts $\Phi$ on $N$.
• Applies the results to moduli spaces of real pseudoholomorphic curves by relating the determinant bundle over the moduli space to canonical line bundles and the tautological bundle $T_D$ associated with transverse intersections with a polarizing divisor.
• Uses polarizing sections to trivialize certain bundles ($\mathfrak{p}^+_X$, $O_X^{\mathrm{Spin}}$, $H^1_w(\mathbb{R}\Sigma_g, \mathbb{R})$), simplifying the determinant line bundle to $\mathfrak{p}^+_X \otimes \det(H^1(\Sigma_g, \mathbb{R})_{-1})^{\otimes n-1} \otimes T_D$.

Experimental results

Research questions

• RQ1 How does the action of a real bundle automorphism affect the orientation of the determinant line bundle over the space of real Cauchy-Riemann operators?
• RQ2 What topological invariants govern the orientability of moduli spaces of real pseudoholomorphic curves?
• RQ3 Under what conditions does a real symplectic manifold admit a polarization that induces a canonical orientation on its moduli space?
• RQ4 How can the first Stiefel-Whitney class of the moduli space be computed in terms of geometric and topological data?

Key findings

• The sign of the automorphism action on determinant line bundle orientations is given by a product of the signature of permutations on $\mathrm{Pin}^{\pm}$ structures and the bordism class of real $\mathrm{Spin}$ structures.
• For real pseudoholomorphic curves in smooth hypersurfaces $X_\delta \subset \mathbb{C}P^N$ with $\delta \equiv N+1 \mod 4$, the first Stiefel-Whitney class of the moduli space is $w_1(\mathbb{R}_\tau \mathcal{M}^d_{g,r}(X_\delta,J)) = w_1(L_r) + (\delta-1)w_1(\det(H^1(\Sigma_g, \mathbb{R})_{-1}))$.
• The determinant bundle over the moduli space of curves transverse to a polarizing divisor $D$ is canonically isomorphic to $\mathfrak{p}^+_X \otimes \det(H^1(\Sigma_g, \mathbb{R})_{-1})^{\otimes n-1} \otimes T_D$.
• Any real symplectic manifold admits a polarization via a polarizing section, ensuring the existence of a canonical orientation on the moduli space of transverse curves.
• The canonical isomorphism of the determinant bundle eliminates dependence on $\mathfrak{D}_{(\underline{\mathbf{d}})}(N)$, simplifying the orientation theory.