(see also Chern-Weil theory, parameterized homotopy theory)
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
analytic geometry (complex, rigid, global)
geometry+analysis/analytic number theory
analytic space, analytic variety, Berkovich space
transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
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), moduli schemes of coherent sheaves with certain parameters fixed, so called Quot schemes. Later the geometric invariant theory defined other class of moduli spaces of bundles. Especially important is the case of moduli space of stable bundles on a Riemann surface which is extremely important for mathematical physics, as the Narasimhan–Seshadri theorem related it to the moduli space of flat connection (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 system etc.).
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
and then we discuss the complex-analytic version
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
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:
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, function 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 functions) | meromorphic functions on complex plane | |
$p$ (prime number/non-archimedean place) | $x \in \mathbb{F}_p$ | $x \in \mathbb{C}$ | |
$\infty$ (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 | |||
$\mathbb{Z}_p$ (p-adic integers) | $\mathbb{F}_q[ [ t -x ] ]$ (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 |
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 $x$.
There is a bijection between
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)
The key observation is that in $X^\ast$ every $G$-bundle trivializes. Therefore
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.
(…)
(…)
Let
$k$ an algebraically closed field;
$G$ an affine algebraic group;
$x\in X$ a closed point;
$X^\ast = X- \{x\}$;
$D \coloneqq Spf(k[ [t_x] ])$;
$D^\ast = \coloneqq Spf(k( (t_x)) )$;
(Weil uniformization theorem)
There is an Equivalence of stacks
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)
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.
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
Review in the context of geometric Langlands duality is in