Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces / William M. Goldman, Eugene Z. Xia.

Format:

Book

Author/Creator:

Goldman, William Mark, author.

Xia, Eugene Zhu, 1963- author.

Series:

Memoirs of the American Mathematical Society ; no. 904.

Memoirs of the American Mathematical Society, 0065-9266 ; number 904

Language:

English

Subjects (All):

Surfaces, Deformation of.

Riemann surfaces.

Geometry, Differential.

Geometry, Algebraic.

Physical Description:

1 online resource (86 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2008]

Language Note:

English

Summary:

Details the theory of rank one Higgs bundles over a closed Riemann surface $X$ and their relation to representations of the fundamental group of $X$. This work constructs an equivalence between the deformation theories of flat connections and Higgs pairs, providing an identification of moduli spaces arising in different contexts.

Contents:

""Contents""; ""Introduction""; ""1. Equivalences of deformation theories""; ""2. The Betti and de Rham deformation theories and their moduli spaces""; ""2.1. The Betti groupoid""; ""2.2. The de Rham groupoid""; ""2.3. Equivalence of de Rham and Betti groupoids""; ""3. The Dolbeault groupoid""; ""3.1. Holomorphic line bundles""; ""3.2. The moduli spaces""; ""3.3. Geometric structure of the Dolbeault moduli space""; ""4. Equivalence of de Rham and Dolbeault groupoids""; ""4.1. Construction of the equivalence""; ""4.2. Higgs coordinates""; ""4.3. Involutions""

""5. Hyperkahler geometry on the moduli space""""5.1. The quaternionic structure""; ""5.2. The Riemannian metric""; ""5.3. Complex-symplectic structure""; ""5.4. Quaternionization""; ""6. The twistor space""; ""6.1. The complex projective line""; ""6.2. The twistor space as a smooth vector bundle""; ""6.3. A holomorphic atlas for the twistor space""; ""6.4. The twistor lines""; ""6.5. The real structure on the twistor space""; ""6.6. Symplectic geometry of the twistor space""; ""6.7. The lattice quotient""; ""6.8. Functions and flows""; ""7. The moduli space and the Riemann period matrix""

""7.1. Coordinates for the Betti moduli space""""7.2. Abelian differentials and their periods""; ""7.3. Flat connections""; ""7.4. Higgs fields""; ""7.5. The C*-action in terms of the period matrix""; ""7.6. The C*-action and the real points""; ""Bibliography""

Notes:

"May 2008, volume 193, number 904 (fourth of 5 numbers)."

Includes bibliographical references (pages 67-69).

Description based on print version record.

ISBN:

1-4704-0510-5