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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:
(“ of over ”) | of over $\mathbb{F}_q$ () | / | |
---|---|---|---|
and | |||
$\mathbb{Z}$ () | $\mathbb{F}_q[z]$ (, on $\mathbb{A}^1_{\mathbb{F}_q}$) | $\mathcal{O}_{\mathbb{C}}$ ( on ) | |
$\mathbb{Q}$ () | $\mathbb{F}_q(z)$ () | on | |
$p$ (/non-archimedean ) | $x \in \mathbb{F}_p$ | $x \in \mathbb{C}$ | |
$\infty$ () | $\infty$ | ||
$Spec(\mathbb{Z})$ () | $\mathbb{A}^1_{\mathbb{F}_q}$ () | ||
$Spec(\mathbb{Z}) \cup place_{\infty}$ | $\mathbb{P}_{\mathbb{F}_q}$ () | ||
$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ () | $\frac{\partial}{\partial z}$ ( ) | “ | |
= 0 | = 0 | ||
$\mathbb{Z}_p$ () | $\mathbb{F}_q[ [ t -x ] ]$ ( around $x$) | $\mathbb{C}[ [z-x] ]$ ( on around $x$) | |
$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-” of $X$ at $p$) | in $X$ | ||
$\mathbb{Q}_p$ () | $\mathbb{F}_q((z-x))$ ( around $x$) | $\mathbb{C}((z-x))$ ( on punctured around $x$) | |
$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ () | $\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field ) | $\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ ( of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks) | |
$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ () | $\mathbb{I}_{\mathbb{F}_q((t))}$ ( ) | $\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$ | |
curves | |||
$K$ a ($\mathbb{Q} \hookrightarrow K$ a possibly ) | $K$ a of an $\Sigma$ over $\mathbb{F}_p$ | $K_\Sigma$ ( on $\Sigma$) | |
$\mathcal{O}_K$ () | $\mathcal{O}_{\Sigma}$ () | ||
$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ ( with archimedean ) | $\Sigma$ () | $\Sigma \to \mathbb{C}P^1$ ( being ) | |
$\frac{(-)^p - \Phi(-)}{p}$ (lift of / structure) | $\frac{\partial}{\partial z}$ | “ | |
$v$ prime ideal in $\mathcal{O}_K$ | $x \in \Sigma$ | $x \in \Sigma$ | |
$K_v$ ( at $v$) | $\mathbb{C}((z_x))$ ( on punctured around $x$) | ||
$\mathcal{O}_{K_v}$ ( of ) | $\mathbb{C}[ [ z_x ] ]$ ( on around $x$) | ||
$\mathbb{A}_K$ () | $\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))$ ( of on all punctured around all points in $\Sigma$) | ||
$\mathcal{O}$ | $\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all around all points in $\Sigma$) | ||
$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ () | $\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$ | ||
“ | $\pi_1(\Sigma)$ | ||
“ | (“”) on $\Sigma$ | ||
“ | |||
$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ () | “ | ||
$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$ | “ | $Bun_{GL_1}(\Sigma)$ (, by ) | |
non-abelian class field theory and automorphy | |||
number field | function field | ||
$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ ( on this form ) | “ | $Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$, by ) | |
Tamagawa-Weil for number fields | Tamagawa-Weil for function fields | ||
of /chiral on $\Sigma$ | |||
/ on $\Sigma$ | |||
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
Last revised on August 2, 2017 at 14:04:34. See the history of this page for a list of all contributions to it.