nLab moduli space of bundles

Idea

Moduli spaces and moduli stacks of vector bundles and of $G$-principal bundles for a complex algebraic group $G$ have been widely studied in geometry, with many deep results, especially for the case that the base space is a complex curve or algebraic curve. In a classical work of Grothendieck presented in FGA (see FGA explained), he studied moduli schemes of coherent sheaves with certain parameters fixed, so called Quot schemes. Later, geometric invariant theory defined other class of moduli spaces of bundles. Especially important is the case of the moduli space of stable bundles on a Riemann surface, which is deeply relevant for mathematical physics, as the Narasimhan–Seshadri theorem relates it to the moduli space of flat connections (and to self-dual solutions of Yang-Mills equations; study of spaces of conformal blocks; representation theory of affine Lie algebras and loop groups; integrable systems, esp. Hitchin systems, etc.).

Examples

Bundles over curves and the Langlands correspondence

For $G$ some complex Lie group and $\Sigma$ some complex curve, then the moduli stack of $G$-principal bundles on $\Sigma$ (which are equivalently holomorphic vector bundles when $G = \coprod_{n} GL(n,\mathbb{C})$) has a standard description as a double coset space quotient stack of the collection of formal power series around finitely many points in $\Sigma$ – the Weil uniformization theorem. We frist disucss an easy toplogical version of this statement in

• Over smooth real surfaces

and then we discuss the complex-analytic version

• Over complex curves

Notice here that the sub-moduli space of stable $GL$-principal bundles is related via the Narasimhan-Seshadri theorem to that of flat $GL$-principal connections which is the phase space of $G$-Chern-Simons theory and via the holographic relation of that to the WZW model an ingredient of the modular functor and of equivariant elliptic cohomology etc. This relation serves to explain to some extent why this object is of such interest.

Now the double quotient description is noteworthy because in this incarnation the moduli stack has, via the function field analogy, an immediate analog in algebraic geometry and in fact in arithmetic geometry over any number field. This we discuss below in the section

• Over algebraic curves.

This parallel or analogy between the moduli stack of $G$-bundles over curves in analytic geometry and in arithmetic geometry is the underlying reason for the parallel between the number theoretic Langlands correspondence and the geometric Langlands correspondence (review includes Frenkel 05, section 3.2). It is also at the heart of the Weil conjecture on Tamagawa numbers.

In summary/preview, the analogy is this:

function field analogy

	number fields (“function fields of curves over F1”)	function fields of curves over finite fields $\mathbb{F}_q$ (arithmetic curves)	Riemann surfaces/complex curves
affine and projective line
	$\mathbb{Z}$ (integers)	$\mathbb{F}_q[z]$ (polynomials, polynomial algebra on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	$\mathcal{O}_{\mathbb{C}}$ (holomorphic functions on complex plane)
	$\mathbb{Q}$ (rational numbers)	$\mathbb{F}_q(z)$ (rational fractions/rational function on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	meromorphic functions on complex plane
	$p$ (prime number/non-archimedean place)	$x \in \mathbb{F}_p$, where $z - x \in \mathbb{F}_q[z]$ is the irreducible monic polynomial of degree one	$x \in \mathbb{C}$, where $z - x \in \mathcal{O}_{\mathbb{C}}$ is the function which subtracts the complex number $x$ from the variable $z$
	$\infty$ (place at infinity)		$\infty$
	$Spec(\mathbb{Z})$ (Spec(Z))	$\mathbb{A}^1_{\mathbb{F}_q}$ (affine line)	complex plane
	$Spec(\mathbb{Z}) \cup place_{\infty}$	$\mathbb{P}_{\mathbb{F}_q}$ (projective line)	Riemann sphere
	$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ (Fermat quotient)	$\frac{\partial}{\partial z}$ (coordinate derivation)	“
	genus of the rational numbers = 0		genus of the Riemann sphere = 0
formal neighbourhoods
	$\mathbb{Z}/(p^n \mathbb{Z})$ (prime power local ring)	$\mathbb{F}_q [z]/((z-x)^n \mathbb{F}_q [z])$ ($n$-th order univariate local Artinian $\mathbb{F}_q$-algebra)	$\mathbb{C}[z]/((z-x)^n \mathbb{C}[z])$ ($n$-th order univariate Weil $\mathbb{C}$-algebra)
	$\mathbb{Z}_p$ (p-adic integers)	$\mathbb{F}_q[ [ z -x ] ]$ (power series around $x$)	$\mathbb{C}[ [z-x] ]$ (holomorphic functions on formal disk around $x$)
	$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-arithmetic jet space” of $X$ at $p$)		formal disks in $X$
	$\mathbb{Q}_p$ (p-adic numbers)	$\mathbb{F}_q((z-x))$ (Laurent series around $x$)	$\mathbb{C}((z-x))$ (holomorphic functions on punctured formal disk around $x$)
	$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ (ring of adeles)	$\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
	$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ (group of ideles)	$\mathbb{I}_{\mathbb{F}_q((t))}$ ( ideles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$
theta functions
	Jacobi theta function
zeta functions
	Riemann zeta function	Goss zeta function
branched covering curves
	$K$ a number field ($\mathbb{Q} \hookrightarrow K$ a possibly ramified finite dimensional field extension)	$K$ a function field of an algebraic curve $\Sigma$ over $\mathbb{F}_p$	$K_\Sigma$ (sheaf of rational functions on complex curve $\Sigma$)
	$\mathcal{O}_K$ (ring of integers)		$\mathcal{O}_{\Sigma}$ (structure sheaf)
	$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ (spectrum with archimedean places)	$\Sigma$ (arithmetic curve)	$\Sigma \to \mathbb{C}P^1$ (complex curve being branched cover of Riemann sphere)
	$\frac{(-)^p - \Phi(-)}{p}$ (lift of Frobenius morphism/Lambda-ring structure)	$\frac{\partial}{\partial z}$	“
	genus of a number field	genus of an algebraic curve	genus of a surface
formal neighbourhoods
	$v$ prime ideal in ring of integers $\mathcal{O}_K$	$x \in \Sigma$	$x \in \Sigma$
	$K_v$ (formal completion at $v$)		$\mathbb{C}((z_x))$ (function algebra on punctured formal disk around $x$)
	$\mathcal{O}_{K_v}$ (ring of integers of formal completion)		$\mathbb{C}[ [ z_x ] ]$ (function algebra on formal disk around $x$)
	$\mathbb{A}_K$ (ring of adeles)		$\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))$ (restricted product of function rings on all punctured formal disks around all points in $\Sigma$)
	$\mathcal{O}$		$\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all formal disks around all points in $\Sigma$)
	$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles)		$\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$
Galois theory
	Galois group	“	$\pi_1(\Sigma)$ fundamental group
	Galois representation	“	flat connection (“local system”) on $\Sigma$
class field theory
	class field theory	“	geometric class field theory
	Hilbert reciprocity law	Artin reciprocity law	Weil reciprocity law
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ (idele class group)	“
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$	“	$Bun_{GL_1}(\Sigma)$ (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
	number field Langlands correspondence	function field Langlands correspondence	geometric Langlands correspondence
	$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ (constant sheaves on this stack form unramified automorphic representations)	“	$Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$, by Weil uniformization theorem)
	Tamagawa-Weil for number fields	Tamagawa-Weil for function fields
theta functions
	Hecke theta function		functional determinant line bundle of Dirac operator/chiral Laplace operator on $\Sigma$
zeta functions
	Dedekind zeta function	Weil zeta function	zeta function of a Riemann surface/of the Laplace operator on $\Sigma$
higher dimensional spaces
zeta functions	Hasse-Weil zeta function

Over smooth real surfaces

Let

• $\Sigma$ be a smooth closed manifold of dimension 2 – a real surface;

• $G$ a connected topological group.

• $x \in X$ any point;

• $X^\ast \coloneqq X - \{x\}$

• $D \subset X$ a neighbourhood of $x$ homeomorphic to a disk;

• $D^\ast \coloneqq D - \{x\}$ the corresponding punctured disk around $x$.

Proposition

There is a bijection between

$$[X^\ast,G] \backslash [D^\ast,G] / [D,G] \simeq G Bund(X)_{\sim}$$

between the double coset space of topological groups as shown on the left and the set of equivalence classes of topological $G$-principal bundles on $X$.

e.g. (Sorger 99, prop. 4.1.1)

Proof

The key observation is that in $X^\ast$ every $G$-bundle trivializes. Therefore

$$X^\ast \coprod D \longrightarrow X$$

is a cover of $X$ which is good enough in that degree-1 nonabelian Cech cohomology on this cover with coefficients in $G$ classifies $G$-principal bundles.

For this cover the group $[D^\ast, G]$ is precisely that of Cech cocycles, and $[D \coprod X^\ast, G]$ that of Cech coboundaries.

Over complex curves

(…)

e.g. Frenkel 05, section 3.2

(…)

Over algebraic curves

Let

• $k$ an algebraically closed field;

• $G$ an affine algebraic group;

• $x\in X$ a closed point;

• $X^\ast = X- \{x\}$;

• $D \coloneqq Spf(k[ [t_x] ])$;

• $D^\ast = \coloneqq Spf(k( (t_x)) )$;

Proposition

(Weil uniformization theorem)

There is an Equivalence of stacks

$$[X^\ast] \backslash [D^\ast,G]/[D,G] \simeq Bun_G(X)$$

between the double quotient stack as shown on the left and the stack of algebraic $G$-principal bundles on $X$.

e.g. (Sorger 99, theorem 5.1.1)

Related concepts

• Narasimhan–Seshadri theorem, Donaldson-Uhlenbeck-Yau theorem, Kobayashi-Hitchin correspondence

moduli spaces

• moduli space of complex structures

• moduli space of conformal structures

• moduli space of curves

• moduli space of Riemannian metrics

• moduli stack of elliptic curves

• moduli space of Calabi-Yau spaces

• moduli stack of tori

• moduli stack of formal groups

• moduli space of connections, moduli space of flat connections

• Hitchin moduli space

• moduli space of monopoles

• moduli space of framed manifolds

• moduli space of bundles

• moduli space of (higher) line bundles

• quiver variety

References

General

Over complex curves

• Mudumbai Narasimhan, C. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. Math. 82, No. 3 (Nov., 1965), pp. 540-567, jstor, doi

• A. Ramanathan, Stable principal bundles on a compact Riemann surface, Math Ann 213, 129-152 (1975).

• Michael Atiyah, Raoul Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London 308 (1983), 523–615.

• A. Beauville, Y. Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), 385–419.

Over algebraic curves

• Gerd Faltings, Vector bundles on curves, 1995 Bonn lectures, write up by M. Stoll, pdf

• Gerd Faltings], Moduli-stacks for bundles on semistable curves, Math. Ann. 304, 3 (1996) 489-515; Stable $G$-bundles and projective connections, J. Algebraic Geom. 2, 3 (1993) 507-568, doi, Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc. 5 (2003), 41-68.

• Gerd Faltings, Line-bundles on the moduli-space of G-torsors, lecture at MSRI 2002, video and pdf

• Günter Harder, Mudumbai Narasimhan, On the cohomology groups of moduli spaces of vector bundle on curves, Math Ann. 212, 215-248 (1975).

• Nigel Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987) pp. 91–114

• V. B. Mehta, C. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), 205–239.

• P. E. Newstead, Characteristic classes of stable bundles of rank 2 over an algebraic curve, Trans. Amer. Math. Soc. 169 (1972), 337–345.

• S. Ramanan, The moduli spaces of vector bundles over an algebraic curve, Math. Ann. 200 (1973), 69–84.

• Carlos Simpson, Higgs bundles and local systems, Publ. Mathématiques de l’IHÉS 75 (1992), p. 5-95, numdam

• Constantin Teleman, C. T. Woodward, The index formula for the moduli

  of G-bundles on a curve_, Ann. Math. 170, 2, 495–527 (2009) pdf

• Christoph Sorger, Lectures on moduli of principal $G$-bundles over algebraic curves, 1999 (pdf)

• Jochen Heinloth, Uniformization of $\mathcal{G}$-bundles (pdf)

• Jonathan Wang?, The moduli stack of $G$-bundles, arXiv:1104.4828.

• References for moduli spaces of bundles over singular curves are discussed at MathOverflow here

Over arithmetic curves

• Norbert Hoffmann, On vector bundles over algebraic and arithmetic curves, 2002 (pdf)

Review in the context of geometric Langlands duality is in

• Edward Frenkel, Lectures on the Langlands Program and Conformal Field Theory, in Frontiers in number theory, physics, and geometry II, Springer Berlin Heidelberg, 2007. 387-533. (arXiv:hep-th/0512172)