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 [1]. It also yields a new method for computing the spectrum using the combinatorics of triangulations of the Riemann surface. arXiv:0907.3987v2 [hep-th] 23 Sep 2011 157 13.4 The regular solution 157 13.5 The large R limit of X γ 160 13.6 The real section 160 13.7 Relation to Hitchin flows 161 14. Comparison with [1]: differential equations and the Riemann-Hilbert problem 162 -3 -A. Expressing monodromy matrices in terms of Fock-Goncharov coordinates 164 B. Computing the Hamiltonian flows 169 C. WKB error analysis 171 D. Holomorphic coordinates on multi-center Taub-NUT 172 E. Configurations of integers with nonpositive second discrete derivative 1744 Actually, we should consider all multiples of γ0, thus the correct transformation to use isIn the examples we study only a single charge will contribute to the discontinuity. 5 Another relation between four-dimensional super Yang-Mills theory and the TBA has recently been discussed by Nekrasov and Shatashvili [7]. γ is their asymptotic behavior for ζ → 0, ∞ and R → ∞. It is this property that motivates our definition of a WKB triangulation. As described in Section 6, we define WKB curves of phase ϑ to satisfy λ, ∂ t = e iϑ . Of course, we have already met this condition above, when discussing BPS states! It is equivalent to the assertion that in the local coordinate w = z z 0 λ, where z 0 is a 9 The periodicity of ϑ can be an integer multiple of 2π, or it might even live in the universal cover R.

We consider a class of line operators in d = 4, N = 2 supersymmetric field theories, which leave four supersymmetries unbroken. Such line operators support a new class of BPS states which we call "framed BPS states." These include halo bound states similar to those of d = 4, N = 2 supergravity, where (ordinary) BPS particles are loosely bound to the line operator. Using this construction, we give a new proof of the KontsevichSoibelman wall-crossing formula (WCF) for the ordinary BPS particles, by reducing it to the semiprimitive WCF. After reducing on S 1 , the expansion of the vevs of the line operators in the IR provides a new physical interpretation of the "Darboux coordinates" on the moduli space M of the theory. Moreover, we introduce a "protected spin character" (PSC) that keeps track of the spin degrees of freedom of the framed BPS states. We show that the generating functions of PSCs admit a multiplication, which defines a deformation of the algebra of holomorphic functions on M. As an illustration of these ideas, we consider the sixdimensional (2, 0) field theory of A 1 type compactified on a Riemann surface C. Here, we show (extending previous results) that line operators e-print archive: http://lanl.arXiv.org/abs/1006.0146v2 GAIOTTO ET AL.are classified by certain laminations on a suitably decorated version of C, and we compute the spectrum of framed BPS states in several explicit examples. Finally, we indicate some interesting connections to the theory of cluster algebras.

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 R 3 × 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.

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We give a new derivation of the quasinormal frequencies of Schwarzschild black holes in d 4 and Reissner-Nordstr m black holes in d = 4 , in the limit of in nite damping. For Schwarzschild in d 4 w e nd that the asymptotic real part is T Hawking log(3) for scalar perturbations and for some gravitational perturbations this con rms a result previously obtained by other means in the case d = 4 . For Reissner-Nordstr m in d = 4 w e nd a speci c generally aperiodic behavior for the quasinormal frequencies, both for scalar perturbations and for electromagneticgravitational perturbations. The formulae are obtained by studying the monodromy of the perturbation analytically continued to the complex plane the analysis depends essentially on the behavior of the potential in the \unphysical" region near the black hole singularity.e-print archive: http://lanl.arXiv.org/abs/hep-th/0301173