# nLab wall crossing

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) 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.

# Contents

## Overview

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 deformation?s 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 $D$-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 supersymmetric field theory

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.

## References

### Introductions and lectures

• Sergio Cecotti, Trieste lectures on wall-crossing invariants (2010) (pdf)

• Greg Moore, PiTP Lectures on BPS states and wall-crossing in $d = 4$, $\mathcal{N} = 2$ theories (pdf)

• Tudor Dimofte, Refined wall crossing (pdf), part I of Refined BPS invariants, Chern-Simons theory, and the quantum dilogarithm, 2010 (pdf, web)

### Conferences and seminars

Also (Gaiotto-Moore-Witten 15).

### Categorification

A categorification of wall crossing formulas to an (infinity,2)-category of sorts is discussed in

Revised on August 29, 2015 13:32:39 by David Corfield (87.115.222.68)