Connected components of real loci in moduli spaces of vector and Higgs bundles over a Klein surface

Technical Summary: Connected Components of Real Loci in Moduli Spaces of Vector and Higgs Bundles over a Klein Surface

Problem Statement
Let $X$ be a compact, connected Riemann surface of genus $g \geqslant 2$, equipped with an antiholomorphic involution $\sigma: X \to X$ (a real structure). The authors investigate the topology of the real locus $RN(r, d)$, defined as the fixed-point set of the induced antiholomorphic involution on the moduli space $N(r, d)$ of semistable vector bundles of rank $r$ and degree $d$ over $X$.

While the connected components of the geometrically stable locus $RN^{st}(r, d)$ were previously understood via the modular interpretation of real and quaternionic vector bundles (specifically when $\gcd(r, d) = 1$), the general case where $\gcd(r, d) \neq 1$ presents significant challenges. In the non-coprime case, strictly semistable points do not admit a straightforward modular description as real or quaternionic bundles, and the relationship between the real locus of the vector bundle moduli space and the Picard variety $R\text{Pic}_d$ becomes more complex. The paper aims to determine the number of connected components of $RN(r, d)$ for all $r \geqslant 1$ and $d \in \mathbb{Z}$, and subsequently extend these results to the moduli spaces of Higgs bundles.

Methodology
The authors employ a gauge-theoretic approach, building upon the framework established in previous works (e.g., [5], [16]). Key methodological elements include:

1. Lagrangian Quotients: Instead of relying solely on algebraic geometry, the authors construct topological spaces representing moduli of real and quaternionic bundles. They utilize the space of unitary connections $\mathcal{A}_h$ on a smooth Hermitian vector bundle $(E, \tau)$, where $\tau$ is a real or quaternionic structure. The set of $\tau$-compatible holomorphic structures corresponds to the fixed points of an anti-symplectic involution $\beta_\tau$.
2. Yang-Mills Connections: By restricting to the space of Yang-Mills connections $\mathcal{A}_{YM}$, which correspond to polystable bundles via the Donaldson-Narasimhan-Seshadri theorem, they define Lagrangian quotients $\mathcal{A}^\tau_{YM} / \mathcal{G}^\tau_h$. Theorem 3.2 establishes that these quotients are non-empty and connected.
3. Inductive Decomposition: To handle the case where $\gcd(r, d) = m > 1$, the authors use an inductive strategy on $m$. They decompose the real locus $RN(r, d)$ into unions of images of continuous maps from products of lower-rank moduli spaces. Specifically, they analyze the fibers of the determinant map $c_{r,d}: RN(r, d) \to \pi_0(R\text{Pic}_d)$.
4. Intersection Arguments: The core of the proof involves showing that these smaller connected subspaces (images of Lagrangian quotients and direct sum constructions) have non-empty intersections, thereby proving the connectedness of the fibers $RN(r, d, s)$.

Key Contributions and Results

1. Connectedness of Fibers (Theorem 1.1 / Theorem 4.1):
   The primary result is that for any $s \in \pi_0(R\text{Pic}_d)$, the fiber $RN(r, d, s) := c_{r,d}^{-1}(s)$ is connected. Consequently, the decomposition of $RN(r, d)$ into connected components is determined entirely by the image of the map $c_{r,d}$.
   $$RN(r, d) = \bigsqcup_{s \in \text{Im } c_{r,d}} RN(r, d, s)$$

2. Counting Connected Components (Theorem 4.5 and Table 1):
   The authors provide a complete count of the connected components of $RN(r, d)$ based on the parity of $g$, $r$, and $d$, and the existence of real points on the curve ($X^\sigma \neq \emptyset$ vs. $X^\sigma = \emptyset$):

   • Case $X^\sigma \neq \emptyset$: The real locus $RN(r, d)$ always has $2^{n-1}$ connected components (where $n$ is the number of connected components of $X^\sigma$), regardless of whether $\gcd(r, d) = 1$. The determinant map induces a bijection between the components of $RN(r, d)$ and $R\text{Pic}_d$.
   • Case $X^\sigma = \emptyset$: The number of components can be strictly smaller than that of $R\text{Pic}_d$ when $\gcd(r, d) \neq 1$.
     • If $g$ is even, $d$ is odd, and $r$ is even, $RN(r, d)$ is empty, whereas $R\text{Pic}_d$ may have components.
     • If $g$ is odd, $d$ is even, and $r$ is even, $RN(r, d)$ has 1 connected component (containing both real and quaternionic bundles), whereas $R\text{Pic}_d$ has 2 components.
   • Quaternionic Bundles: The paper clarifies that when $X^\sigma \neq \emptyset$ and $\gcd(r, d) \neq 1$, quaternionic vector bundles (which require even rank and degree) exist but are contained entirely within the fiber corresponding to the trivial Stiefel-Whitney class of the determinant line bundle.

3. Extension to Higgs Bundles (Theorem 1.2 / Theorem 5.1):
   The results are generalized to the moduli space of semistable Higgs bundles, $R\mathcal{M}_{Dol}(r, d)$. The authors prove that the subspace $R\mathcal{M}_{Dol}(r, d, s)$ is the connected component of the real Higgs moduli space containing $RN(r, d, s)$. This confirms that the number of connected components for Higgs bundles matches that of the underlying vector bundles.

4. Topological Implications for Branes (Theorem 1.3 / Theorem 5.3):
   Applying these results to the non-abelian Hodge correspondence, the authors study the topology of fixed-point sets of specific involutions on the character variety $M_B(r, 0)$. They prove that the $(A, A, B)$-brane (fixed by $\beta$) and the $(A, B, A)$-brane (fixed by $\tilde{\beta}$) are homeomorphic. The number of their connected components is given by the same counts derived for $RN(r, d)$.

Significance
The paper resolves the topological classification of real loci in moduli spaces of vector bundles for arbitrary rank and degree, removing the restrictive assumption that $\gcd(r, d) = 1$. It demonstrates that while the modular interpretation of stable points fails in the strictly semistable case, the gauge-theoretic construction of connected Lagrangian quotients provides a robust framework for determining connectivity.

The authors highlight that the behavior of the real locus depends critically on the arithmetic properties of $r$ and $d$ relative to the genus and the existence of real points on the curve. In particular, the paper corrects the naive expectation that the number of components of $RN(r, d)$ always equals that of $R\text{Pic}_d$ when real points are absent, showing that the image of the determinant map can be a proper subset of the components of the Picard variety. These findings are applied to refine the understanding of the topology of branes in hyperkähler quotients, specifically establishing the homeomorphism between distinct types of branes defined by different real structures.