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| Gross–Taylor expansion | |
|---|---|
| Name | Gross–Taylor expansion |
| Field | Mathematical physics |
| Introduced | 1993 |
| Contributors | David Gross; Washington Taylor |
| Related | Large N expansion; string theory; two-dimensional Yang–Mills theory |
Gross–Taylor expansion The Gross–Taylor expansion is a formulation in Mathematical physics connecting large N limits of Yang–Mills theory on two-dimensional manifolds to sums over branched covering maps reminiscent of String theory worldsheet expansions. Developed in the early 1990s by David Gross and Washington Taylor, the expansion links gauge theory amplitudes on surfaces such as the sphere and torus with combinatorial and topological data from symmetric groups and Hurwitz theory, offering a bridge between Quantum field theory and enumerative geometry in contexts related to Matrix model techniques and dualities like the AdS/CFT correspondence.
The Gross–Taylor expansion arose from efforts to understand the large N behavior of SU(N) and U(N) Yang–Mills theory on compact Riemann surfaces such as the sphere, torus, and higher genus Riemann surfaces, motivated by connections to String theory proposals by figures including Edward Witten and Alexander Polyakov. The original works by David Gross and Washington Taylor proposed that the 1/N expansion organizes gauge theory partition functions into sums over maps from string worldsheets to target surfaces, invoking combinatorial structures studied by Adrian Hurwitz and modernized by researchers like Rahul Pandharipande and Maxim Kontsevich. Subsequent developments engaged communities around institutions such as Princeton University, Harvard University, and the Institute for Advanced Study.
Mathematically, the Gross–Taylor expansion expresses the partition function Z_{YM} of two-dimensional Yang–Mills theory on a surface Σ as an asymptotic series in 1/N whose coefficients are weighted counts of branched covers of Σ by source surfaces of varying genus. The formulation uses representation theory of symmetric and unitary groups, notably characters of Symmetric groups and dimensions of Irreducible representations classified by Young diagrams, linking to Schur function expansions employed in Random matrix theory and the Gross–Witten model. The expansion can be written in terms of Hurwitz numbers studied in algebraic geometry by authors such as Hurwitz, Dijkgraaf, and Okounkov, with corrections governed by combinatorial factors akin to those in the Weingarten calculus applied in Free probability contexts explored by Dan Voiculescu.
Physically, the Gross–Taylor expansion provides an explicit realization of gauge/string duality in a solvable setting, paralleling ideas from the AdS/CFT correspondence formulated by Juan Maldacena and relating to topological strings studied by Cumrun Vafa and Hirosi Ooguri. Applications include exact computations of Wilson loop observables reminiscent of constructions by Kenneth Wilson and comparison with lattice gauge theory results from groups such as Brookhaven National Laboratory collaborations. The picture supports interpretations in terms of closed string sectors, D-brane contributions analogous to descriptions by Joseph Polchinski, and connections to matrix string proposals by Robbert Dijkgraaf and René Donagi in enumerative problems tied to moduli spaces investigated by Pierre Deligne and David Mumford.
Derivations employ saddle-point analysis in the large N limit, group-theoretic expansions using Schur–Weyl duality connected to work by Hermann Weyl and Issai Schur, and exact evaluation of lattice partition functions following approaches by Miguel A. Virasoro-inspired methods and developments in Integrable systems associated with Baxter and Ludwig Faddeev. Techniques draw on enumerative geometry tools involving Hurwitz theory elaborated by Alexei Zorich and modular geometry linked to Pierre Deligne and André Weil, while diagrammatic expansions echo constructions in perturbative String perturbation theory pioneered by Michael Green and John Schwarz.
Classic examples include the expansion on the sphere and torus, where explicit Hurwitz number expressions and character sums are computable; these calculations relate to partitions studied by Srinivasa Ramanujan and combinatorial identities investigated by George Andrews. Special cases involve target surfaces with punctures or boundaries connecting to open string sectors and boundary states in Boundary conformal field theory developed by John Cardy and Gabriel Watts. Limits reducing to matrix models connect to the Gross–Witten phase transition and to double-scaling limits analyzed by David Gross and Igor Klebanov.
The Gross–Taylor expansion is closely related to large N expansions in Matrix model literature, the topological string expansions of Gopakumar–Vafa type studied by Rajesh Gopakumar and Cumrun Vafa, and field/string correspondences exemplified by the AdS/CFT correspondence of Juan Maldacena. It connects to the 1/N expansion techniques of Gerard 't Hooft and to dualities explored in Seiberg–Witten theory by Nathan Seiberg and Edward Witten, and to modern enumerative dualities in the work of Maxim Kontsevich and Yuri Manin on moduli of curves.
Open problems include rigorous control of nonperturbative corrections analogous to instanton contributions studied by Alexander Belavin and Vladimir Zakharov, extensions to higher-dimensional gauge theories beyond the two-dimensional solvable case addressed by Edward Witten and Michael Atiyah, and precise matching with holographic duals in settings influenced by Juan Maldacena and Andrew Strominger. Active research groups at institutions such as MIT, Caltech, University of Cambridge, and the Perimeter Institute pursue refinements connecting Gross–Taylor structures to modern topics in enumerative geometry, modularity studied by Don Zagier, and categorical approaches influenced by Maxim Kontsevich and Jacob Lurie.