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| Kodaira–Spencer theory of gravity | |
|---|---|
| Name | Kodaira–Spencer theory of gravity |
| Authors | Kiyoshi Kodaira, Donald C. Spencer |
| Introduced | 1990s |
| Field | Mathematical physics, String theory, Algebraic geometry |
Kodaira–Spencer theory of gravity is a quantum field theory that encodes deformation theory of complex structures on Calabi–Yau manifolds and provides an effective description of the B-model of topological string theory. Developed in the context of attempts to connect mirror symmetry with physical observables, it builds on techniques from complex manifold theory, Hodge theory, and perturbative methods from quantum field theory. The theory plays a central role in relating holomorphic anomaly equations, enumerative invariants, and the geometry of moduli spaces studied in string theory, algebraic geometry, and symplectic geometry.
Kodaira–Spencer theory of gravity arose from efforts to formalize the mathematical structure underlying the B-model on Calabi–Yau manifolds and to link results from mirror symmetry conjectures to calculable field-theoretic quantities. Influenced by foundational work of Kiyoshi Kodaira and Donald C. Spencer on deformation theory, the subject integrates ideas from Hodge decomposition, Dolbeault cohomology, Beltrami differential techniques, and perturbative expansions familiar from Feynman diagrammatics. Key contributors to its development include researchers connected to the Institute for Advanced Study, Princeton University, and groups working on Seiberg–Witten theory and Gromov–Witten invariants.
The mathematical foundations rest on the deformation theory of complex structures of compact Calabi–Yau manifolds using the machinery of Dolbeault operators, Beltrami differentials, and the Maurer–Cartan formalism familiar from Lie algebra and differential graded contexts. Central tools include Hodge theory, the Tian–Todorov theorem, the theory of Moduli spaces of complex structures, and obstructions governed by appropriate cohomology groups. The formalism leverages structures analogous to those in BRST quantization, BV formalism, and uses ideas from Kodaira–Spencer cohomology and the Calabi conjecture as clarified by work related to Shing-Tung Yau.
The action functional of Kodaira–Spencer theory is written in terms of a field that is a (0,1)-form with values in the holomorphic tangent bundle, capturing infinitesimal complex structure deformations represented by Beltrami differentials and elements of H^{0,1}(T^{1,0}). The cubic interaction arises from the wedge product combined with contraction by the holomorphic volume form, echoing structures in Chern–Simons theory and Holomorphic Chern–Simons theory. The theory can be placed in the Batalin–Vilkovisky formalism and compared with actions in topological quantum field theory, with gauge symmetries related to holomorphic vector fields and anomalies tied to the holomorphic anomaly equation discovered in the context of Bershadsky-Cecotti-Ooguri-Vafa work.
Kodaira–Spencer theory provides the effective spacetime description of the closed-string sector of the B-model on a Calabi–Yau manifold. Its perturbative amplitudes compute generating functions for higher-genus free energies that match predictions from mirror symmetry and calculations of Gromov–Witten invariants on mirror manifolds. The relation is formalized via the identification of the Kodaira–Spencer partition function with the all-genus B-model free energy subject to the holomorphic anomaly equation and fits into broader dualities such as mirror symmetry and conjectural correspondences with Matrix model techniques and large N duality instances studied in connection with Chern–Simons theory and Gopakumar–Vafa invariants.
Moduli spaces in the theory are moduli of complex structures on Calabi–Yau manifolds, stratified by Hodge-theoretic data captured by variations of Hodge structure and period maps into Griffiths transversality-constrained domains. Deformations are encoded by solutions to the Maurer–Cartan equation in a suitable differential graded Lie algebra whose obstruction theory is governed by higher Massey product–like structures familiar from homological algebra and Derived category considerations. The interplay with moduli of stable maps and compactifications appearing in Deligne–Mumford stacks informs enumerative predictions and wall-crossing phenomena connected to broader programs involving Donaldson–Thomas invariants and Pandharipande–Thomas theory.
Quantization proceeds via BV quantization and produces a perturbative expansion organized by genus, with Feynman diagrams built from propagators derived from Green’s operators on forms and vertices from cubic couplings. Regularization employs techniques parallel to those in renormalization group analyses and BPHZ-like subtraction adapted to the holomorphic context; anomalies manifest as failures of gauge invariance corresponding to the holomorphic anomaly equation of Bershadsky–Cecotti–Ooguri–Vafa. The perturbative series connects to asymptotic expansions studied in resurgence theory and to exact results accessible through localization and connections with matrix models and Topological recursion frameworks developed by researchers linked to Princeton University and IHES.
Physically, Kodaira–Spencer theory serves as the target-space effective theory capturing closed-string B-model dynamics, influencing computations of superpotential terms in compactifications of Type II string theory and contributing to understanding of N=2 supersymmetry structures and special geometry. Applications extend to cross-checks of mirror symmetry predictions for enumerative invariants, insights into dualities such as open-closed string duality and large N duality, and mathematical outputs including relationships between Gromov–Witten invariants and Donaldson–Thomas invariants. The framework has inspired further developments linking derived algebraic geometry, homological mirror symmetry conjectures advanced by Maxim Kontsevich, and categorical approaches pursued at institutions like Harvard University and University of Cambridge.
Category:Mathematical physics Category:String theory Category:Algebraic geometry