A q-form global symmetry is a global symmetry for which the charged operators are of space-time dimension q; e.g. Wilson lines, surface defects, etc., and the charged excitations have q spatial dimensions; e.g. strings, membranes, etc. Many of the properties of ordinary global symmetries (q = 0) apply here. They lead to Ward identities and hence to selection rules on amplitudes. Such global symmetries can be coupled to classical background fields and they can be gauged by summing over these classical fields. These generalized global symmetries can be spontaneously broken (either completely or to a subgroup). They can also have 't Hooft anomalies, which prevent us from gauging them, but lead to 't Hooft anomaly matching conditions. Such anomalies can also lead to anomaly inflow on various defects and exotic Symmetry Protected Topological phases. Our analysis of these symmetries gives a new unified perspective of many known phenomena and uncovers new results.

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We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of N = 2 SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0, 1.

By analyzing brane configurations in detail, and extracting general lessons, we develop methods for analyzing S-duality of supersymmetric boundary conditions in N = 4 super Yang-Mills theory. In the process, we find that S-duality of boundary conditions is closely related to mirror symmetry of three-dimensional gauge theories, and we analyze the IR behavior of large classes of quiver gauge theories. S-DUALITY OF BOUNDARY CONDITIONS (p, q) Fivebranes and fractional Chern-Simons couplings 891Acknowledgment 893References 893

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.