This entry is about discontinuities in parameter dependence of (often asymptotic) solutions of differential equations and similar phenomena (notably in the context of BPS states formalized via Bridgeland stability conditions) with stability parameters (and their stability slopes) in algebraic geometry which are often interpreted as crossing the walls of marginal stability in physics. For the different notions of the same name in Morse theory see at Cerf wall crossing and for the (Weyl chamber wall) crossing functors in representation theory see wall crossing functor.
In the study of solitons, one may try a WKB-style approximation to a nonlinear wave equation (see also eikonal equation, Maslov index, etc.). Stokes has observed that when trying to connect the local solutions, one has discontinuities along certain lines, now called Stokes lines. This is called the Stokes phenomenon. Similar issues appear in study of isomonodromic deformations of nonlinear ODEs in the complex plane, which is also relevant in soliton theory, and integrable systems, and special functions like Painlevé transcendents. This has especially been studied by the Kyoto school (Jimbo, Miwa, Sato, Kashiwara etc.), including the use of D-modules and microlocal analysis. The Kyoto school found a connection of isomonodromic theory to what is called holonomic quantum fields.
The solutions of meromorphic differential equations can be expressed in terms of meromorphic connections. Then the slopes related to the solutions can be viewed as features of particular objects in a category of -modules. More generally, slope filtrations are structures which appear in many other additive categories, e.g. in Hodge theory, theory of Dieudonné modules and so on. Many of those are related to the stability of the objects, which is important in the construction of moduli spaces.
In algebraic geometry, Grothendieck has shown how to correctly define and construct some fundamental moduli spaces, like Hilbert schemes and Quot schemes for coherent sheaves. The work has been continued by David Mumford who geometrized classical invariant theory into geometric invariant theory. To keep moduli under control, one needs to impose stability conditions on objects and also look at classes with some fixed data: those involve slopes or equivalently phase factors. This is thus similar to the phases of eikonal in the case of Stokes phenomenon. Cf. also Harder-Narasimhan filtration, Castelnuovo-Mumford regularity? (cf. wikipedia) etc.
In super Yang-Mills theory the number of BPS states is locally constant as a function of the parameters of the theory, but it may jump at certain “walls” in the moduli spaces of parameters. The precise behaviour of the BPS states as one crosses these walls is studied as “wall crossing phenomena”.
Another example are the moduli spaces of Higgs bundles, studied by Carlos Simpson and others, which have special cases with interpretations both in geometry and in the gauge theory (instantons). It appears that sometimes they can be linked to the geometric picture. Riemann-Hilbert correspondence, spectral transform and similar correspondences again play a major role.
Surely, one often works at the derived level. An adaptation of the notion of stability into the setup of triangulated categories has been introduced by Bridgeland. Bridgeland stability for the derived categories of (boundary conditions of) D-branes (B-model) are relevant for string theory.
| singularity | field theory with singularities |
|---|---|
| boundary condition/brane | boundary field theory |
| domain wall/bi-brane | QFT with defects |
Sergio Cecotti, Trieste lectures on wall-crossing invariants (2010) [pdf&rbrack
Greg Moore, PiTP Lectures on BPS states and wall-crossing in , theories (pdf)
Tudor Dimofte, Refined wall crossing (pdf), part I of Refined BPS invariants, Chern-Simons theory, and the quantum dilogarithm, 2010 (pdf, web)
Maxim Kontsevich, Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435; Motivic Donaldson-Thomas invariants: summary of results, 0910.4315; Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry, arxiv/1303.3253
Arend Bayer, Yuri I. Manin, Stability conditions, wall-crossing and weighted Gromov-Witten invariants, math.AG/0607580, Mosc. Math. J. 9 (1), 2009.
Mina Aganagic, Hirosi Ooguri, Cumrun Vafa, Masahito Yamazaki, Wall crossing and M-theory, arxiv/0908.1194
Mina Aganagic, Wall crossing, quivers and crystals, arxiv/1006.2113
Gregory Moore, PiTP lectures on BPS states and wall-crossing in d = 4, N = 2 theories pdf
S. Cecotti, Cumrun Vafa, BPS wall crossing and topological strings, arXiv/0910.2615
Davide Gaiotto, Greg Moore, Andrew Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation, Advances in Mathematics 234 (2013) 239–403 (arXiv:0907.3987 doi)
E. Diaconescu, Greg Moore, Crossing the wall: branes vs. bundles, arXiv/0706.3193
E. Andriyash, Frederik Denef, Daniel Jafferis, G. W. Moore, Wall-crossing from supersymmetric galaxies, arxiv/1008.0030
Tom Bridgeland, Valerio Toledano-Laredo, Stability conditions and Stokes factors, arxiv/0801.3974
M. C. N. Cheng, E. P. Verlinde, Wall crossing, discrete attractor flow and Borcherds algebra, SIGMA 4 (2008), 068, 33 pages, pdf
Masahito Yamazaki, Crystal melting and wall crossing phenomena, Ph.D. thesis, arxiv/1002.1709
Michele Cirafici, Annamaria Sinkovics, Richard Szabo, Instanton counting and wall-crossing for orbifold quivers, arxiv/1108.3922
H.-Y. Chen, N. Dorey, K. Petunin, Moduli space and wall-crossing formulae in higher-rank gauge theories, JHEP 11 (2011) 020, doi; Wall crossing and instantons in compactified gauge theory, JHEP 06 (2010) 024 arXiv:1004.0703
Frederik Denef, Supergravity flows and D-brane stability, JHEP 0008:050,2000 (arXiv:hep-th/0005049)
Frederik Denef, Quantum Quivers and Hall/Hole Halos, JHEP 0210:023,2002 (arXiv:hep-th/0206072)
Daniel Jafferis, Gregory Moore, Wall crossing in local Calabi Yau manifolds (arXiv:0810.4909)
(past) Kontsevich in Aarhus, August 2010, master class on wall crossing; we will keep a nlab page on it
(past) Focus Week on New Invariants and Wall Crossing, May 18-22, 2009, Kashiwa Campus of the University of Tokyo
(past) Wall-crossing in Mathematics and Physics, May 24-28, 2010, Department of Mathematics, University of Illinois at Urbana-Champaign
Description of seminar on stability conditions and Stokes factors in Bonn, pdf
Also (Gaiotto-Moore-Witten 15).
A categorification of wall crossing formulas to an (infinity,2)-category of sorts is discussed in
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