Covering spaces of character varieties. (English) Zbl 1339.57002
Consider a finitely presented group \(\Gamma\) and a reductive affine algebraic group \(G\) over \(\mathbb C\). This article studies normal coverings of the \(G\)-character variety of \(\Gamma\), \(\chi_\Gamma(G):=\text{hom}(\Gamma,G)//G\) (algebraic geometric quotient under conjugation), when the fundamental group \(\pi_1(G)\) has no torsion.
The first main result proves that any covering homomorphism of such groups \(p:H\mapsto G\) induces a covering between corresponding character varieties \(p_*:\chi_\Gamma(G)\mapsto \chi_\Gamma(H)\), with fiber \(\chi_\Gamma(\text{ker }p)\). The same statement is valid in the compact Lie group category, i.e., replacing \(G\) by a maximal compact subgroup.
Let now \(\Sigma\) be a (real) surface of genus \(g\) with \(p\) punctures. The second main result is the formula
\[ \pi_1(\chi_{\pi_1(\Sigma)}(G))=\pi_1(G)^{b_1(\Sigma)} \] where \(b_1(\Sigma)\) is the first Betti number of \(X\), so it equals \(2g\) in the closed surface case (no punctures) and \(2g+p-1\) in the open (punctured) surface case. The formula above also works, when \(\Sigma\) is replaced by an \(n\)-dimensional (real) torus (replacing \(b_1(\Sigma)\) by \(n\)), in the cases when the derived group of \(G\) is a product of simple groups of type \(A_n\) or \(C_n\). It has also been recently extended to the case where \(\pi_1(G)\) has torsion in [I. Biswas, S. Lawton and D. A. Ramras, Math. Z. 281, No. 1-2, 415–425 (2015; Zbl 1349.14042)].
Let now \(\Gamma^g\) be the fundamental group of a closed surface \(\Sigma_g\) of genus \(g\geq 1\). Note that, under the non-Abelian Hodge correspondence, \(\chi_{\Gamma^g}(G)\) is homeomorphic to a certain moduli space of (semistable) \(G\)-Higgs bundles over \(\Sigma_g\). Thus, the above formula can be seen as an extension of the formula, obtained in [S. B. Bradlow, O. García-Prada and P. B. Gothen, Topology 47, No. 4, 203–224 (2008; Zbl 1165.14028)], for the moduli space of \(GL_n(\mathbb C)\)-Higgs bundles with coprime degree and rank (the smooth case) over \(\Sigma_g\).
The last part of the article presents an interesting application of these results to the stable character varieties of surface groups, defined as a natural colimit \(\chi_{\Gamma^g}(SU):=\text{colim}_{n\mapsto\infty}\chi_{\Gamma^g}(SU(n))\). Note that by the Narasimhan-Seshadri Theorem, there is a homeomorphism between the character variety \(\chi_{\Gamma^g}(SU(n))\) (the compact version of \(\chi_{\Gamma^g}(SL_n(\mathbb C))\)) and the moduli space of (semistable) trivial determinant vector bundles over \(\Sigma_g\). The last main result is the proof of the homotopy equivalence between the stable character varieties \(\chi_{\Gamma^g}(SU)\) and \(\mathbb CP^\infty\) (proving that their homotopy type is independent of the genus).
This result has a very nice connection with geometric quantization of the moduli space of bundles: it is well known that the moduli spaces of bundles \(\chi_{\Gamma^g}(SU(n))\) carry a natural so-called determinant line bundle (which coincides with the pre-quantum line bundle for \(\chi_{\Gamma^g}(SU(n))\) in the context of geometric quantization) whose holomorphic sections form the vector space of conformal blocks of a conformal field theory, the dimension of which was computed by the celebrated Verlinde formula. Since \(BU(1)\), the classifying space for \(\mathbb C\)-line bundles, is just \(\mathbb CP^\infty\), a map \(\chi_{\Gamma^g}(SU)\mapsto\mathbb CP^\infty\) defines a line bundle by pull-back of the universal line bundle over \(\mathbb CP^\infty\). Following a question stated in this article, it has been shown in the recent preprint [L. C. Jeffrey, D. A. Ramras and J. Weitsman, “The prequantum line bundle on the moduli space of flat \(SU(n)\) connections on a Riemann surface and the homotopy of the large \(n\) limit”, arXiv:1411.4360] that the map \(\chi_{\Gamma^g}(SU)\mapsto \mathbb CP^\infty\) corresponding to the determinant line bundle over \(\chi_{\Gamma^g}(SU)\) (which behaves well under stabilization in the large \(n\) limit) is indeed a homotopy equivalence.
Finally, in the Appendix, written by N.-K. Ho and C.-C. M. Liu, it is shown that, for \(\Sigma_g\) a hyperbolic surface \((g\geq 2)\) and a complex reductive group \(G\), \(\pi_0(\text{Hom}(\pi_1(\Sigma_g)))\cong \pi_1(DG)\), where \(DG\) is the derived group of \(G\), generalizing the same statement previously established in the complex semisimple case.
The first main result proves that any covering homomorphism of such groups \(p:H\mapsto G\) induces a covering between corresponding character varieties \(p_*:\chi_\Gamma(G)\mapsto \chi_\Gamma(H)\), with fiber \(\chi_\Gamma(\text{ker }p)\). The same statement is valid in the compact Lie group category, i.e., replacing \(G\) by a maximal compact subgroup.
Let now \(\Sigma\) be a (real) surface of genus \(g\) with \(p\) punctures. The second main result is the formula
\[ \pi_1(\chi_{\pi_1(\Sigma)}(G))=\pi_1(G)^{b_1(\Sigma)} \] where \(b_1(\Sigma)\) is the first Betti number of \(X\), so it equals \(2g\) in the closed surface case (no punctures) and \(2g+p-1\) in the open (punctured) surface case. The formula above also works, when \(\Sigma\) is replaced by an \(n\)-dimensional (real) torus (replacing \(b_1(\Sigma)\) by \(n\)), in the cases when the derived group of \(G\) is a product of simple groups of type \(A_n\) or \(C_n\). It has also been recently extended to the case where \(\pi_1(G)\) has torsion in [I. Biswas, S. Lawton and D. A. Ramras, Math. Z. 281, No. 1-2, 415–425 (2015; Zbl 1349.14042)].
Let now \(\Gamma^g\) be the fundamental group of a closed surface \(\Sigma_g\) of genus \(g\geq 1\). Note that, under the non-Abelian Hodge correspondence, \(\chi_{\Gamma^g}(G)\) is homeomorphic to a certain moduli space of (semistable) \(G\)-Higgs bundles over \(\Sigma_g\). Thus, the above formula can be seen as an extension of the formula, obtained in [S. B. Bradlow, O. García-Prada and P. B. Gothen, Topology 47, No. 4, 203–224 (2008; Zbl 1165.14028)], for the moduli space of \(GL_n(\mathbb C)\)-Higgs bundles with coprime degree and rank (the smooth case) over \(\Sigma_g\).
The last part of the article presents an interesting application of these results to the stable character varieties of surface groups, defined as a natural colimit \(\chi_{\Gamma^g}(SU):=\text{colim}_{n\mapsto\infty}\chi_{\Gamma^g}(SU(n))\). Note that by the Narasimhan-Seshadri Theorem, there is a homeomorphism between the character variety \(\chi_{\Gamma^g}(SU(n))\) (the compact version of \(\chi_{\Gamma^g}(SL_n(\mathbb C))\)) and the moduli space of (semistable) trivial determinant vector bundles over \(\Sigma_g\). The last main result is the proof of the homotopy equivalence between the stable character varieties \(\chi_{\Gamma^g}(SU)\) and \(\mathbb CP^\infty\) (proving that their homotopy type is independent of the genus).
This result has a very nice connection with geometric quantization of the moduli space of bundles: it is well known that the moduli spaces of bundles \(\chi_{\Gamma^g}(SU(n))\) carry a natural so-called determinant line bundle (which coincides with the pre-quantum line bundle for \(\chi_{\Gamma^g}(SU(n))\) in the context of geometric quantization) whose holomorphic sections form the vector space of conformal blocks of a conformal field theory, the dimension of which was computed by the celebrated Verlinde formula. Since \(BU(1)\), the classifying space for \(\mathbb C\)-line bundles, is just \(\mathbb CP^\infty\), a map \(\chi_{\Gamma^g}(SU)\mapsto\mathbb CP^\infty\) defines a line bundle by pull-back of the universal line bundle over \(\mathbb CP^\infty\). Following a question stated in this article, it has been shown in the recent preprint [L. C. Jeffrey, D. A. Ramras and J. Weitsman, “The prequantum line bundle on the moduli space of flat \(SU(n)\) connections on a Riemann surface and the homotopy of the large \(n\) limit”, arXiv:1411.4360] that the map \(\chi_{\Gamma^g}(SU)\mapsto \mathbb CP^\infty\) corresponding to the determinant line bundle over \(\chi_{\Gamma^g}(SU)\) (which behaves well under stabilization in the large \(n\) limit) is indeed a homotopy equivalence.
Finally, in the Appendix, written by N.-K. Ho and C.-C. M. Liu, it is shown that, for \(\Sigma_g\) a hyperbolic surface \((g\geq 2)\) and a complex reductive group \(G\), \(\pi_0(\text{Hom}(\pi_1(\Sigma_g)))\cong \pi_1(DG)\), where \(DG\) is the derived group of \(G\), generalizing the same statement previously established in the complex semisimple case.
MSC:
57M10 | Covering spaces and low-dimensional topology |
14D20 | Algebraic moduli problems, moduli of vector bundles |
32G13 | Complex-analytic moduli problems |
14L30 | Group actions on varieties or schemes (quotients) |
53D50 | Geometric quantization |