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Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure

Marco Bertola, Chaya Norton, Giulio Ruzza

Abstract

We consider the moduli space of vector bundles of rank $n$ and degree $ng$ over a fixed Riemann surface of genus $g\geq 2$. We use the explicit parametrization in terms of the Tyurin data. In the moduli space there is a "non-abelian" Theta divisor, consisting of bundles with $h^1\geq 1$. On the complement of this divisor we construct a non-abelian Cauchy kernel explicitly in terms of the Tyurin data. With the additional datum of a non-special divisor, we can construct a reference flat holomorphic connection which is also dependent holomorphically on the moduli of the bundle. This allows us to identify the bundle of Higgs fields, i.e. the cotangent bundle of the moduli space, with the affine bundle of holomorphic connections and provide a monodromy map into the ${\rm GL}_n$ character variety. We show that the Goldman symplectic structure on the character variety pulls back along this map to the complex canonical symplectic structure on the cotangent bundle and hence also on the space of affine connections. The pull-back of the Liouville one-form to the affine bundle of connections is then shown to be a logarithmic form with poles along the non-abelian theta divisor and residue given by $h^1$.

Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure

Abstract

We consider the moduli space of vector bundles of rank and degree over a fixed Riemann surface of genus . We use the explicit parametrization in terms of the Tyurin data. In the moduli space there is a "non-abelian" Theta divisor, consisting of bundles with . On the complement of this divisor we construct a non-abelian Cauchy kernel explicitly in terms of the Tyurin data. With the additional datum of a non-special divisor, we can construct a reference flat holomorphic connection which is also dependent holomorphically on the moduli of the bundle. This allows us to identify the bundle of Higgs fields, i.e. the cotangent bundle of the moduli space, with the affine bundle of holomorphic connections and provide a monodromy map into the character variety. We show that the Goldman symplectic structure on the character variety pulls back along this map to the complex canonical symplectic structure on the cotangent bundle and hence also on the space of affine connections. The pull-back of the Liouville one-form to the affine bundle of connections is then shown to be a logarithmic form with poles along the non-abelian theta divisor and residue given by .

Paper Structure

This paper contains 25 sections, 23 theorems, 141 equations, 4 figures.

Table of Contents

  1. Introduction and results
  2. Outline of results.
  3. Example: line bundles.
  4. Common notations
  5. Note added.
  6. Acknowledgements.
  7. Tyurin parametrization of vector bundles
  8. Moduli of framed vector bundles
  9. Tyurin space
  10. Non-abelian theta divisor
  11. Non-abelian Cauchy kernel
  12. Framed Higgs fields and T*V0
  13. From framed to unframed vector bundles.
  14. Flat connections and the Cauchy kernel
  15. Affine bundles of connections
  16. ...and 10 more sections

Key Result

Lemma 2.1

Let $G(z)\in\mathrm{Mat}_n(\mathcal{R})$ be such that $\det G(z)\in\mathcal{R}$ does not vanish identically. There exist unique matrices $H\in\mathrm{GL}_n(\mathcal{R})$ and $P\in\mathrm{Mat}_n(\mathbb{C}[z])$ such that with $P$ (termed polynomial normal form of $G$) a polynomial matrix of the form where $p_j(z)$ are monic polynomials of degree $d_j$ with all their zeros in $\mathbb D$, and the

Figures (4)

  • Figure 1: Illustration of the main spaces and maps.
  • Figure 2: The fundamental flat section $\Psi$ satisfying (\ref{['dpsi']}) is analytic in the interior of the fundamental polygon. Points on the boundaries are identified in pairs $p_\pm$, related by the indicated deck transformations ($\gamma_k:=\alpha_k\beta_k\alpha_k^{-1}\beta_k^{-1}$). The corresponding boundary values $\Psi_\pm(p)$ are then related by the associated monodromy transformation.
  • Figure 3: Half edges of $\Sigma$ incident at $v=\infty$ for $g=2$.
  • Figure 4: The Krichever graph $\Sigma_{\text{Kr}}$ with indicated the jump matrices as in kricheveriso for comparison, so that $J_k = B_k^{-1} A_{k}^{-1} B_k A_k$. The two-form associated to this graph following Def. \ref{['defOmega']} is equivalent, after graph contractions, to the form in AlMal and hence to the Goldman symplectic form. The matrices that Krichever denotes by $A_k, B_k$ coincide with what we denote here by $J_{\alpha_k}, J_{\beta_k}$.

Theorems & Definitions (40)

  • Lemma 2.1
  • Remark 2.2: Construction of a framed bundle from its moduli.
  • Example 2.3
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • Remark 2.7: Relationship with Tyurin vectors
  • Remark 2.8: Cohomological interpretation of the Tyurin space
  • Proposition 3.1
  • Corollary 3.2
  • ...and 30 more