Weil–Petersson form

Weil–Petersson form
Name	Weil–Petersson form
Field	Complex analysis; Differential geometry; Algebraic geometry
Named after	André Weil; Robert Petersson

Weil–Petersson form The Weil–Petersson form is a canonical closed two-form arising on spaces parameterizing complex structures, notably on Teichmüller space and moduli spaces of curves. It intertwines contributions from harmonic theory, hyperbolic geometry, and Hodge theory, and has played a central role in developments related to Riemann surfaces, mapping class groups, and string theory.

Definition and construction

The form is defined by pairing tangent vectors via the Petersson inner product on spaces of quadratic differentials associated to a Riemann surface; this uses constructions appearing in the work of André Weil, Robert Petersson, Bernhard Riemann, Henri Poincaré, and Oswald Teichmüller. Analytically one identifies tangent vectors to Teichmüller space with Beltrami differentials and pairs them using L^2 integrals against hyperbolic area forms introduced by Ludwig Schläfli and studied by Georg Cantor and Felix Klein. Geometric constructions employ the curvature forms of the Petersson metric on bundles studied by David Mumford, Pierre Deligne, Armand Borel, and Jean-Pierre Serre.

Properties

The Weil–Petersson form is closed and invariant under the action of the mapping class group; foundational results were proved by William Thurston, Cornelia Druţu, and Yair Minsky in the context of geometric group actions. It is Kähler where nondegenerate, linking to the work of Shing-Tung Yau, S.-T. Yau, and Simon Donaldson on Kähler–Einstein metrics and to Hodge theoretic insights by Phillip Griffiths and Joe Harris. Degeneracy phenomena at the boundary of moduli space were analyzed by John H. Hubbard, Curt McMullen, and Ken'ichi Ohshika with contributions from Robert Penner and Dennis Sullivan. Symplectic invariance appears in work of Maxim Kontsevich, Edward Witten, and Alexander Grothendieck via intersection theory on compactified moduli spaces.

Weil–Petersson metric and geometry

The Weil–Petersson form induces a (possibly incomplete) Riemannian metric on Teichmüller space studied extensively by Howard Masur, Yair Minsky, Curt McMullen, Maryam Mirzakhani, and Scott Wolpert. Geodesic behavior and negative curvature properties connect to results of M. Gromov, Grigori Perelman, and Richard Hamilton in geometric analysis. Volume computations for moduli spaces using the Weil–Petersson metric were pioneered by Maryam Mirzakhani and build on techniques from Edward Witten and Maxim Kontsevich involving intersection numbers on Deligne–Mumford compactification as developed by Pierre Deligne and David Mumford.

Relation to Teichmüller and moduli spaces

On Teichmüller space the form descends to the orbifold moduli space of curves after quotienting by the mapping class group, with compactification issues treated in the Deligne–Mumford compactification framework. Connections to algebraic geometry were expanded by Alexander Grothendieck, David Mumford, Joe Harris, and Carel Faber through tautological classes and intersection theory. Links to string theory and to the Virasoro algebra arise in work by Edward Witten, Maxim Kontsevich, and Nathan Seiberg, while relationships with quantum invariants connect to Witten–Reshetikhin–Turaev constructions and to researchers like Vladimir Fock and Leon Takhtajan.

Cohomology and symplectic aspects

Cohomological interpretations of the Weil–Petersson form involve tautological classes such as ψ-classes and κ-classes studied by Carel Faber, Ravi Vakil, Dima Arinkin, and William Fulton. Its role as a symplectic form on smooth loci ties to the work of Aleksandr Kirillov, Alan Weinstein, and Jean-Michel Bismut on symplectic geometry and analytic torsion. Localization techniques and equivariant cohomology applied to Weil–Petersson volumes draw on methods developed by Atiyah–Bott, Nigel Hitchin, and E. Witten, while relations to Hodge theory reference Phillip Griffiths and Wilfried Schmid.

Applications and examples

Applications range across counting problems in geometry and physics: Mirzakhani’s volume recursion for moduli spaces used Weil–Petersson geometry, impacting enumerative geometry influenced by Maxim Kontsevich and Edward Witten. Explicit examples include computations on low-genus moduli spaces (g=0,1,2) treated by Maryam Mirzakhani, Carel Faber, and Joe Harris, and connections to hyperbolic 3-manifold theory via William Thurston and Brian Bowditch. Further applications appear in quantum field theory and string theory contexts explored by Edward Witten, Nathan Seiberg, and Anton Alekseev, and in dynamics on character varieties studied by William Goldman and Nigel Hitchin.

Category:Complex geometry