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| Wall-crossing | |
|---|---|
| Name | Wall-crossing |
| Field | Mathematics, Theoretical physics |
| Introduced | 1990s |
| Contributors | Kontsevich; Seiberg; Witten; Donaldson; Thomas; Joyce; Gaiotto; Moore; Neitzke |
Wall-crossing
Wall-crossing describes changes in mathematical or physical invariants when parameters cross loci (walls) in moduli spaces associated with structures such as Calabi–Yau manifold, Seiberg–Witten theory, Donaldson theory, Gromov–Witten theory or Stability condition chambers. It connects techniques from Algebraic geometry, Symplectic geometry, Quantum field theory, String theory and Representation theory and has influenced work by researchers linked to Maxim Kontsevich, Edward Witten, Nathan Seiberg, Richard Thomas and D. Joyce.
The concept arose from studying discontinuities in invariants attached to families parameterized by moduli spaces, such as moduli of sheafs on K3 surfaces, moduli of Higgs bundles on Riemann surfaces, and BPS spectra in N=2 supersymmetry theories studied by Seiberg–Witten theory. Early interactions occurred between the communities around Donaldson invariants, Gromov–Witten invariants, Mirror symmetry and the study of stability for objects in Derived categorys, leading to frameworks proposed by Kontsevich–Soibelman and others.
Wall-crossing is formulated using moduli spaces such as the moduli of stable sheafs on a projective variety or the moduli of stable objects in a derived category of a Calabi–Yau manifold. Stability is often defined in terms of slope functions like those in Mumford–Takemoto stability or Bridgeland stability conditions on derived categories introduced by Tom Bridgeland. The walls are real codimension-one loci in spaces like the space of Bridgeland stability conditions, the ample cone of a Kähler manifold, or the parameter space of Fayet–Iliopoulos terms in gauge theories studied in the context of Seiberg–Witten theory and Supersymmetric gauge theory.
Key invariant types include enumerative invariants such as Donaldson–Thomas invariants, Gromov–Witten invariants, instanton counting invariants related to Nekrasov partition function, and BPS indices in String theory compactifications on Calabi–Yau threefolds. Algebraic structures controlling changes include Hall algebras of abelian categorys, cluster algebras linked to Fomin–Zelevinsky theory, and Kontsevich–Soibelman motivic wall-crossing structures connected to motivic integration.
In geometry, wall-crossing describes how moduli spaces of stable objects jump when crossing walls in the space of polarizations on a projective surface or the space of Bridgeland stability conditions on a derived category of a Calabi–Yau manifold. Classic examples involve changes in moduli of sheaves on P^2 or K3 surfaces, with comparisons to work by Donaldson on four-manifolds and later developments by Thomas.
In physics, wall-crossing governs BPS spectrum changes in N=2 supersymmetric gauge theory and string theory compactifications on Calabi–Yau threefolds, with influential analyses by Gaiotto, Moore, and Neitzke who connected wall-crossing to spectral networks and cluster coordinates. It appears in contexts including line operator algebras in Chern–Simons theory, spectral curve analyses in Hitchin systems, and soliton counts in Landau–Ginzburg models studied in relation to Mirror symmetry.
Prominent formulae include the Kontsevich–Soibelman wall-crossing formula formulated in terms of automorphisms of motivic quantum tori and products of quantum dilogarithm series, and the primitive wall-crossing formulae used in Seiberg–Witten theory and the study of BPS indices by Kontsevich and Soibelman. Joyce and Song developed alternative descriptions in the context of Calabi–Yau threefold enumerative theories, relating generalized Donaldson–Thomas invariants to stable pair invariants as in the Pandharipande–Thomas framework.
These formulas link to algebraic tools like the Ringel–Hall algebra of an abelian category, cluster transformations in cluster algebras, and to wall-crossing identities in Lie algebraic and quantum groups contexts such as Yokonuma–Hecke algebra analogues. Invariants affected include Donaldson invariants, Euler characteristics of moduli spaces, and protected spin characters in BPS state counts.
Concrete geometric examples include computations for moduli of sheaves on P^2, moduli of Higgs bundles related to the Hitchin fibration on Riemann surfaces, and derived-category wall-crossing for K3 surfaces and abelian surfaces. Physical applications include analyses of BPS spectra in Seiberg–Witten theory for gauge groups like SU(2), wall-crossing of line defects in N=2 theories classified by Gaiotto–Moore–Neitzke and implications for dualities such as S-duality and Mirror symmetry dual pairs including examples from Type IIA string theory/Type IIB string theory compactifications on Calabi–Yau threefolds.
Interdisciplinary applications tie to enumerative predictions in the MNOP conjecture, calculations in topological string theory, and the study of stability for brane configurations in works involving Douglas and others.
Computational approaches employ virtual fundamental class techniques from Behrend and Fulton intersection theory, localization in toric settings using Atiyah–Bott localization and the Nekrasov partition function, and wall-crossing via scattering diagrams following Gross–Siebert methods. Algorithms exploit combinatorics of quiver representations, mutation sequences in cluster algebras, and computations in Hall algebras often implemented with symbolic algebra systems used by researchers linked to SAGE and Mathematica in collaboration with groups around Nakajima and Reineke.
Numerical and experimental methods include spectral network computations for BPS counts by Gaiotto–Moore–Neitzke approaches, exact WKB analysis in problems related to the Hitchin system, and stability condition sampling in spaces studied by Bridgeland.
Active research topics include categorification of Kontsevich–Soibelman identities pursued by groups working with Khovanov-type theories, deeper relations to the Langlands program via the geometric Langlands correspondence for Hitchin moduli spaces, refinements of motivic wall-crossing invariants, and extensions to higher-dimensional Calabi–Yau categories. Other directions involve precise enumeration of BPS spectra in theories with exceptional gauge groups like E8 and understanding wall-crossing in noncompact settings such as local del Pezzo geometries, with ongoing interaction among researchers affiliated with institutions like Institute for Advanced Study, Princeton University, Harvard University and international collaborations including Simons Foundation programs.