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516
The monodromy groups of Schwarzian equations on closed Riemann surfaces
 ANN. OF MATH
, 2000
"... Let θ: π1(R) → PSL(2, C) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. Theorem. Necessary and sufficient for θ to be the monodromy representation associated with a complex projective stucture on R, either ..."
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Cited by 56 (1 self)
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Let θ: π1(R) → PSL(2, C) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. Theorem. Necessary and sufficient for θ to be the monodromy representation associated with a complex projective stucture on R, either unbranched or with a single branch point of order 2, is that θ(π1(R)) be nonelementary. A branch point is required if and only if the representation θ does not lift to
Exact results for perturbative ChernSimons theory with complex gauge group
 Commun. Number Theory Phys
"... Abstract: We develop several methods that allow us to compute allloop partition functions in perturbative ChernSimons theory with complex gauge group GC, sometimes in multiple ways. In the background of a nonabelian irreducible flat connection, perturbative GC invariants turn out to be interestin ..."
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Cited by 54 (15 self)
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Abstract: We develop several methods that allow us to compute allloop partition functions in perturbative ChernSimons theory with complex gauge group GC, sometimes in multiple ways. In the background of a nonabelian irreducible flat connection, perturbative GC invariants turn out to be interesting topological invariants, which are very different from finite type (Vassiliev) invariants obtained in a theory with compact gauge group G. We explore various aspects of these invariants and present an example where we compute them explicitly to high loop order. We also introduce a notion of “arithmetic TQFT ” and conjecture (with supporting numerical evidence) that SL(2, C) ChernSimons theory is an example of such a theory. CALT682716 Contents
Duality and equivalence of module categories in noncommutative geometry II: Mukai . . .
, 2006
"... This is the second in a series of papers intended to set up a framework to study categories of modules in the context of noncommutative geometries. In [3] we introduced ..."
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Cited by 46 (6 self)
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This is the second in a series of papers intended to set up a framework to study categories of modules in the context of noncommutative geometries. In [3] we introduced
An analog of a modular functor from quantized Teichmüller theory
"... The program of the quantization of the Teichmüller spaces T (Σ) of Riemann surfaces Σ which was started in [Fo, CF] and independently in [Ka1] 1 is motivated by certain problems and conjectures from mathematical physics. One of the main aims of this program is to construct a ..."
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Cited by 46 (6 self)
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The program of the quantization of the Teichmüller spaces T (Σ) of Riemann surfaces Σ which was started in [Fo, CF] and independently in [Ka1] 1 is motivated by certain problems and conjectures from mathematical physics. One of the main aims of this program is to construct a
A Hitchin–Kobayashi correspondence for Kaehler fibrations
 J. REINE ANGEW. MATH
, 1999
"... Let X be a compact Kaehler manifold and E → X a principal K bundle, where K is a compact connected Lie group. Let A 1,1 be the set of connections on E whose curvature lies in Ω 1,1 (E ×Ad k). Let k = Lie(K), and fix on k a nondegenerate biinvariant bilinear pairing. This allows to identify k ≃ k ∗ ..."
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Cited by 42 (4 self)
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Let X be a compact Kaehler manifold and E → X a principal K bundle, where K is a compact connected Lie group. Let A 1,1 be the set of connections on E whose curvature lies in Ω 1,1 (E ×Ad k). Let k = Lie(K), and fix on k a nondegenerate biinvariant bilinear pairing. This allows to identify k ≃ k ∗. Let F be a Kaehler left Kmanifold and suppose that there exists a moment map µ: F → k ∗ for the action of K on F. Let S = Γ(E ×K F). In this paper we study the equation ΛFA + µ(Φ) = c for A ∈ A 1,1 on E and a section Φ ∈ S, where FA is the curvature of A and c ∈ k is a fixed central element. We study which orbits of the action of the complex gauge group on A 1,1 ×S contain solutions of the equation and we define a positive functional on A 1,1 ×S which generalises the YangMillsHiggs functional and whose local minima coincide with the solutions of the equation.
Surface group representations and U(p, q)Higgs bundles
, 2002
"... Using the L² norm of the Higgs field as a Morse function, we study the moduli spaces of U(p, q)Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on (p, q). A key step is the identification of the function’s local minima as modul ..."
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Cited by 41 (9 self)
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Using the L² norm of the Higgs field as a Morse function, we study the moduli spaces of U(p, q)Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on (p, q). A key step is the identification of the function’s local minima as moduli spaces of holomorphic triples. In a companion paper [7] we prove that these moduli spaces of triples are nonempty and irreducible. Because of the relation between flat bundles and fundamental group representations, we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple PU(p, q)representations. The topological invariants of the flat bundles are used to label subspaces. These invariants are bounded by a Milnor–Wood type inequality. For each allowed value of the invariants satisfying a certain coprimality condition, we prove that the corresponding subspace is nonempty and connected. If the coprimality condition does not hold, our results apply to the closure of the moduli space of irreducible representations
Moduli spaces of parabolic Higgs bundles and parabolic K(D) pairs over smooth curves: I
, 1998
"... This paper concerns the moduli spaces of rank two parabolic Higgs bundles and parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles occur in stable K(D) pairs and stable Higgs bundles is determined. Using Morse theory, the moduli space of parabolic Higgs bundles is shown to b ..."
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Cited by 39 (2 self)
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This paper concerns the moduli spaces of rank two parabolic Higgs bundles and parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles occur in stable K(D) pairs and stable Higgs bundles is determined. Using Morse theory, the moduli space of parabolic Higgs bundles is shown to be a noncompact, connected, simply connected manifold, and a computation of its Poincaré polynomial is given.
Mixed Hodge polynomials of character varieties
"... We calculate the Epolynomials of certain twisted GL(n,C)character varietiesMn of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lietype GL(n,Fq) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geomet ..."
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Cited by 37 (9 self)
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We calculate the Epolynomials of certain twisted GL(n,C)character varietiesMn of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lietype GL(n,Fq) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,C)character variety. The calculation also leads to several conjectures about the cohomology of Mn: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious Hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain