Four-Dimensional Wall-Crossing via Three-Dimensional Field Theory

By D. Gaiotto, G. Moore, Andrew Neitzke. 2008

588 citations recorded by the source corpus.

We give a physical explanation of the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum in Seiberg-Witten theories. In the process we give an exact description of the BPS instanton corrections to the hyperkähler metric of the moduli space of the theory on $${\mathbb R^3 \times S^1}$$. The wall-crossing formula reduces to the statement that this metric is continuous. Our construction of the metric uses a four-dimensional analogue of the two-dimensional tt* equations.

View Paper: https://www.semanticscholar.org/paper/0364b031eb2ddd785da071d13aea6f9841e65e33

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Citations in the corpus

Wall-crossing in coupled 2d-4d systems (2011, methodology): “This approach was developed for purely 4-dimensional theories in [1].”
$\mathcal{N} = 2$ quantum field theories and their BPS quivers (2011, background): “More recently, significant progress has been made both in mathematics and in physics, in understanding the universal rules that govern potential decay processes of BPS particles [22, 24, 45, 52, 61], and continuing progress in this subject [1–3, 15, 16, 18, 19, 21, 27, 28, 46–48, 66, 71] suggests th”
3-Manifolds and 3d Indices (2011, background): “This higher dimensional definition allows a connection to the rich subject of fourdimensional theories that can be defined in terms of the A1 6d SCFT compactified on C [32,33].”
Refined, Motivic, and Quantum (2009, methodology): “Finally, and more physically, one can try to derive the motivic wall crossing 6 See [16] and also [23] (which we follow here in the unrefined case) for a more precise description of these symplectomorphisms.”
Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (2008, methodology): “A “physical” explanation of our formulas in this context was given in [ 24 ], see also our Section 2.7.”
R-Twisting and 4d/2d Correspondences (2010, methodology): “As discussed at length in [4,35],M admits important canonically defined coordinate functions Xγ , which can be thought of as a kind of complexification of the holonomies of the Abelian gauge fields around S(1), or more precisely as the vacuum expectation values of certain supersymmetric line operato”
Framed BPS states (2010, unknown): “F (c3) is given by [−3, 2] + ρ2[−1, 2] + [1, 2] + [1, 0] (4.”
Spectral Networks (2012, unknown): “Some aspects of the WKB approximation are difficult to generalize to rank K > 2, and for a long time, this proved a stumbling block to giving nontrivial illustrations of the general statements of [17,5] in the higher rank case.”
Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces (2015, methodology): “They introduced the concept of “limiting configuration”, and they proved the convergence to the limiting configuration under the assumption, inspired by the work of Gaiotto, Moore and Neitzke [4, 5].”
D-instantons and twistors (2008, methodology): “Thus, one may “pile up” the BPS rays up in just two composite rays located on the positive and negative imaginary axis [38].”
Gauge Theories Labelled by Three-Manifolds (2011, background): “XE of [2, 16]) on the moduli space of the compactified 4d theory.”
Quadratic differentials as stability conditions (2013, background): “We learnt this idea from Gaiotto, Moore and Neitzke’s work [14, Section 6], although in retrospect, it is an immediate consequence of well-known results in the theory of quadratic differentials.”