Wall-crossing, Hitchin Systems, and the WKB Approximation

2009

575 citations recorded by the source corpus.

We consider BPS states in a large class of d = 4,N = 2 eld theories, obtained by reducing six-dimensional (2; 0) superconformal eld theories on Riemann surfaces, with defect operators inserted at points of the Riemann surface. Further dimensional reduction on S 1 yields sigma models, whose target spaces are moduli spaces of Higgs bundles on Riemann surfaces with ramication. In the case where the Higgs bundles have rank 2, we construct canonical Darboux coordinate systems on their moduli spaces. These coordinate systems are related to one another by Poisson transformations associated to BPS states, and have well-controlled asymptotic behavior, obtained from the WKB approximation. The existence of these coordinates implies the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum. This construction provides a concrete realization of a general physical explanation of the wall-crossing formula which was proposed in (1). It also yields a new method for computing the spectrum using the combinatorics of triangulations of the Riemann surface.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/3512c5cc4bc54ee36c5fedc170cd80d476bfc6bc

Citations in the corpus

• Framed BPS states (2010, unknown): “F (c3) is given by [−3, 2] + ρ2[−1, 2] + [1, 2] + [1, 0] (4.”
• Partition Functions of $${\mathcal{N}=(2,2)}$$N=(2,2) Gauge Theories on S2 and Vortices (2012, unknown): “surprising aspects of the dynamics of multiple M5-branes – whose reduction on a Riemann surface Σ can give either an N = 2 [60] or an N = 1 [61, 62, 63] four-dimensional theory depending on the twist, some aspects being discussed in [64, 65, 66] – most notably the AGT correspondence [67].”
• Instanton counting with a surface operator and the chain-saw quiver (2011, unknown): “3 of [47]) that the codimension-two defect of type L corresponding to the partition N = n1+ · · ·+nM can be realized by considering a Type IIA configuration with N D4-branes ending on M D6-branes such that nI D4-branes end on the I-th D6-brane.”
• Loop and surface operators in $ \mathcal{N} = 2 $ gauge theory and Liouville modular geometry (2009, unknown): “We already encountered this phenomenon in Liouville theory, and understood the relation to the WKB analysis of [1]: expectation values of UV line operators are linear combinations of individual contributions, each taking the form expected from an IR line operator.”
• Spectral Networks (2012, unknown): “There is a class of regular defects labeled by Young diagrams with K boxes [2, 3].”
• Tinkertoys for Gaiotto duality (2010, unknown): “With a certain twist, to preserve N = 2 supersymmetry [3], the Kaluza-Klein reduction to 4D is the N = 2 gauge theory of interest.”
• Wild quiver gauge theories (2011, unknown): “These correspond to asymptotically free gauge theories, as exemplified in [2, 29] and, in the A1 case, reduce to the D1 and the D2 type singularities studied in our paper.”
• Ordered Exchange Graphs (2013, unknown): “mas invariants [65] and in Ordered Exchange Graphs 7 calculating the complete spectrum of a BPS (Bogomol’nyi–Prasad–Sommerfield) particle in string theory [6, 22, 104] (this also appears implicitly in [37]). The main feature of an ordered exchange graph is the property (i): the arrows in the Hasse qu”
• Nilpotent orbits and codimension-two defects of 6d N=(2,0) theories (2012, unknown): “The study of these untwisted and twisted defects was initiated in [11], and further developed in various papers including [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].”