TY - JOUR

T1 - Wall-crossing, Hitchin systems, and the WKB approximation

AU - Gaiotto, Davide

AU - Moore, Gregory W.

AU - Neitzke, Andrew

N1 - Funding Information:
The work of GM is supported by the DOE under grant DE-FG02-96ER40949 . We thank the KITP at UCSB for hospitality during the course of part of this work and therefore this research was supported in part by DARPA under Grant No. HR0011-09-1-0015 and by the National Science Foundation under Grant No. PHY05-51164 . GM would like to thank the Galileo Galilei Institute and the Aspen Center for Physics for hospitality during the completion of this work. The work of AN is supported by the NSF under grant numbers PHY-0503584 and PHY-0804450 . D.G. is supported in part by the DOE grant DE-FG02-90ER40542 . D.G. is supported in part by the Roger Dashen membership in the Institute for Advanced Study .

PY - 2013/2/5

Y1 - 2013/2/5

N2 - We consider BPS states in a large class of d = 4, N=2 field theories, obtained by reducing six-dimensional (2, 0) superconformal field 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 ramification. 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 Gaiotto etal.[40]. It also yields a new method for computing the spectrum using the combinatorics of triangulations of the Riemann surface.

AB - We consider BPS states in a large class of d = 4, N=2 field theories, obtained by reducing six-dimensional (2, 0) superconformal field 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 ramification. 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 Gaiotto etal.[40]. It also yields a new method for computing the spectrum using the combinatorics of triangulations of the Riemann surface.

KW - Donaldson-Thomas invariants

KW - Fock-Goncharov coordinates

KW - Hitchin systems

KW - Hyperkähler geometry

KW - Supersymmetric gauge theory

KW - Wall-crossing

UR - http://www.scopus.com/inward/record.url?scp=84870333631&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84870333631&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2012.09.027

DO - 10.1016/j.aim.2012.09.027

M3 - Article

AN - SCOPUS:84870333631

VL - 234

SP - 239

EP - 403

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -