Integral cohomology of moduli spaces of principal G-bundles on a Riemann surface: torsion and integral relations beyond the unitary case

Atiyah and Bott's 1983 equivariant Morse-theoretic study of the Yang-Mills functional over a compact Riemann surface computed the rational cohomology of moduli spaces of stable bundles and proved torsion-freeness of the integral cohomology for the unitary groups U(n) in the coprime rank-degree case, while leaving open the integral structure for general structure groups. This problem asks for a determination of the integral cohomology ring of the moduli space $M_G$ of semistable principal $G$-bundles over a compact Riemann surface of genus at least 2, for an arbitrary compact connected structure group $G$: a presentation by the tautological Atiyah-Bott classes with a complete set of relations over the integers, together with the torsion subgroup, especially for groups $G$ whose classifying spaces have torsion in their integral cohomology, such as $SO(3)$ and $PU(n)$. Of the issues left open in the source, the flow-convergence question was resolved by Daskalopoulos and Rade, the complete rational relations and the Newstead conjectures were resolved by Thaddeus, Earl-Kirwan, Jeffrey-Weitsman, Teleman-Woodward, and Lim-Moreira-Pi, and the Tamagawa-counting correspondence was conceptually explained by Heinloth-Schmitt and the Gaitsgory-Lurie program; the integral structure for general $G$, and the integral relations even in the unitary case, form the surviving open core.

Background

Let $X$ be a compact Riemann surface of genus $g$ and $G$ a compact connected Lie group. A principal $G$-bundle $P\to X$ carries a space of connections $\mathscr{A}$ on which the gauge group $\mathscr{G}=\mathrm{Aut}(P)$ acts, and the Yang-Mills functional $\mathrm{YM}(A)=\int_X\lvert F_A\rvert^2$, the squared $L^2$-norm of the curvature $F_A$, is $\mathscr{G}$-invariant. Atiyah and Bott applied equivariant Morse theory to this infinite-dimensional setting. Because $\mathscr{A}$ is contractible, the equivariant cohomology $H^*_{\mathscr{G}}(\mathscr{A})$, defined as the ordinary cohomology of the Borel construction $\mathscr{A}\times_{\mathscr{G}}E\mathscr{G}$ for a contractible space $E\mathscr{G}$ on which $\mathscr{G}$ acts freely, equals the ordinary cohomology $H^*(B\mathscr{G})$ of the classifying space of the gauge group, which is a union of connected components of the mapping space $\mathrm{Map}(X,BG)$ indexed by the topological types of the bundles, and coincides with the cohomology of the moduli stack $\mathrm{Bun}_G$ of principal $G$-bundles on $X$. Over a surface the critical points of $\mathrm{YM}$ are the Yang-Mills connections by definition, and the minimum set modulo $\mathscr{G}$ consists of the central Yang-Mills connections (connections of constant central curvature), reducing to the flat connections exactly in the topologically trivial (more generally torsion) types; by the Narasimhan-Seshadri correspondence between unitary connections and polystable holomorphic structures (direct sums of stable bundles of equal slope), it corresponds to the moduli of polystable bundles of the given topological type. Here a holomorphic vector bundle $E$ over $X$ is semistable if every proper subbundle $F\subset E$ satisfies $\mu(F)\leq\mu(E)$ for the slope $\mu=\deg/\mathrm{rank}$, stable if the inequality is strict, and polystable if it is a direct sum of stable bundles of equal slope; semistability of principal $G$-bundles is defined, following Ramanathan, by testing reductions to parabolic subgroups of $G$. The unstable part of $\mathscr{A}/\mathscr{G}$ is stratified by Harder-Narasimhan types (the canonical decomposition of an unstable bundle into successive semistable pieces of strictly decreasing slopes), the same combinatorial stratification that governs the counting of bundles over finite fields. Atiyah and Bott proved this stratification equivariantly perfect, meaning the associated equivariant spectral sequences degenerate (collapse, with no further differentials), so that $H^*(B\mathscr{G};\mathbb{Z})$ is a freely generated ring of tautological classes when $G$ is unitary, and freely generated over $\mathbb{Q}$ for every compact connected $G$; integrally the statement fails once $H^*(BG;\mathbb{Z})$ has torsion, since the evaluation fibration $\mathrm{Map}(X,BG)\to BG$, $f\mapsto f(x_0)$, admits the constant maps as a section, so that $H^*(BG;\mathbb{Z})$ injects into the cohomology of the component of the trivial bundle. The tautological (Atiyah-Bott or Mumford) classes are constructed as follows: for a representation $\rho:G\to U(n)$ one forms the associated bundle $\mathscr{E}_{\rho}$ over $X\times\mathrm{Bun}_G$ and pushes its Chern classes along the projection to $X$. In the coprime case, where the rank $n$ and degree $k$ are coprime, the moduli space $N(n,k)$ of stable holomorphic vector bundles is a smooth projective variety whose rational cohomology is generated by the restrictions of these tautological classes. Newstead's conjectures concern two vanishing phenomena in this ring: the degree-four tautological class $\beta$ (a normalization of $\mathrm{ch}_2$, the second graded component of the Chern character, of the tautological rank-2 bundle) satisfies $\beta^g=0$, and the Chern classes of the tangent bundle vanish beyond degree $r(r-1)(g-1)$ in rank $r$. The arithmetic strand: the number of $\mathbb{F}_q$-points of moduli stacks of bundles, computed by Harder-Narasimhan counting (Harder-Narasimhan; Desale-Ramanan), coincides with the equivariant Poincare series (the generating series of the equivariant Betti numbers of $\mathscr{A}$) after the substitutions $q\mapsto t^{-2}$ and $\omega_i\mapsto -t^{-1}$, where the $\omega_i$ are the Frobenius eigenvalues on $H^1$ of the curve over $\mathbb{F}_q$; the counting is governed by the Tamagawa number $\tau(G)$, the volume of $G(k(X))\backslash G(\mathbb{A})/K$, where $k(X)$ is the function field of the curve, $\mathbb{A}$ its adele ring, and $K$ a maximal compact subgroup, with respect to the canonical (Tamagawa) measure. The torsion issue at the heart of this problem: for a compact connected $G$, the classifying-space cohomology $H^*(BG;\mathbb{Z})$ is a torsion-free polynomial ring for the unitary-type groups ($U(n)$, $SU(n)$, $Sp(n)$), but has $p$-torsion for many others, for example $H^3(BPU(n);\mathbb{Z})\cong\mathbb{Z}/n$ and $H^*(B\mathrm{SO}(3);\mathbb{Z})$ has 2-torsion, with analogous torsion for the exceptional groups, and this torsion persists in $H^*(B\mathscr{G};\mathbb{Z})$ by the section argument above. The semistable objects form the coarse moduli space $M_G$ of semistable principal $G$-bundles, a normal projective variety constructed by Ramanathan, whose connected components are indexed by the topological types of the bundles, that is by the set $[X,BG]\cong H^2(X;\pi_1(G))\cong\pi_1(G)$. The Atiyah-Bott torsion-freeness proof for $N(n,k)$ is an induction over the Harder-Narasimhan strata using integral Gysin sequences, and it breaks precisely when such torsion is present in the strata or their boundary fibrations, which is why the general integral statement is open.

Problem Statement

Determine the integral cohomology ring of the moduli space $M_G(X)$ of semistable principal $G$-bundles over a compact Riemann surface $X$ of genus $g\geq2$ (the cases $g<2$ being degenerate for this question: for $g=0$ the semistable locus collapses to a point, and $g=1$ is governed by the explicit low-genus classification of bundles, outside the Harder-Narasimhan regime at issue; the partial results cited below all treat $g\geq2$), for arbitrary compact connected structure groups $G$ (equivalently, for the corresponding connected complex reductive groups). Here $M_G(X)$ is Ramanathan's coarse moduli space, a normal projective variety whose connected components are indexed by the topological types of the bundles, that is by the set $[X,BG]\cong H^2(X;\pi_1(G))\cong\pi_1(G)$; the problem asks for the answer on every component, for every $g\geq2$. The problem has two coupled deliverables. (a) Presentation: prove that $H^*(M_G;\mathbb{Z})$ is generated, modulo its torsion subgroup, by the tautological Atiyah-Bott classes, that is, the restrictions to the semistable locus of the classes arising from the Künneth components of the characteristic classes of the universal bundle over $\mathrm{Bun}_G\times X$, equivalently obtained from representations $\rho:G\to U(n)$ by pushing the Chern classes of the associated bundle down along $X$, and determine a complete set of relations over $\mathbb{Z}$ among them; this includes the unitary coprime case $N(n,k)$, where torsion-freeness and the complete rational relations are known but the integral lattice of relations is not. (b) Torsion: compute the torsion subgroup of $H^*(M_G;\mathbb{Z})$, in particular for groups $G$ whose classifying spaces have $p$-torsion in their integral cohomology, such as $G=SO(3)$ or $PU(n)$, and thereby decide whether torsion in the structure group induces torsion in the moduli cohomology. $SO(3)$ is the smallest genuinely torsion-containing case; the full goal is the general theorem for all compact connected $G$, all $g\geq2$, and all components. The known rational presentations (Atiyah-Bott; Earl-Kirwan), the stack-level rational computation of Heinloth-Schmitt, and the known low-degree integral results (Picard group, fundamental group, Brauer groups) constrain the possible answers.

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