90 research outputs found

    Line Defects, Tropicalization, and Multi-Centered Quiver Quantum Mechanics

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    We study BPS line defects in N=2 supersymmetric four-dimensional field theories. We focus on theories of "quiver type," those for which the BPS particle spectrum can be computed using quiver quantum mechanics. For a wide class of models, the renormalization group flow between defects defined in the ultraviolet and in the infrared is bijective. Using this fact, we propose a way to compute the BPS Hilbert space of a defect defined in the ultraviolet, using only infrared data. In some cases our proposal reduces to studying representations of a "framed" quiver, with one extra node representing the defect. In general, though, it is different. As applications, we derive a formula for the discontinuities in the defect renormalization group map under variations of moduli, and show that the operator product algebra of line defects contains distinguished subalgebras with universal multiplication rules. We illustrate our results in several explicit examples.Comment: 76 pages, 10 figures; v2: minor revisions, correction to Coulomb branch calculation for defects in SU(2) SY

    Hyperkahler Sigma Model and Field Theory on Gibbons-Hawking Spaces

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    We describe a novel deformation of the 3-dimensional sigma model with hyperk\"ahler target, which arises naturally from the compactification of a 4-dimensional N=2\mathcal{N}=2 theory on a hyperk\"ahler circle bundle (Gibbons-Hawking space). We derive the condition for which the deformed sigma model preserves 4 out of the 8 supercharges. We also study the contribution from a NUT center to the sigma model path integral, and find that supersymmetry implies it is a holomorphic section of a certain holomorphic line bundle over the hyperk\"ahler target. We study explicitly the case where the original 4-dimensional theory is pure U(1)U(1) super Yang-Mills, and show that the contribution from a NUT center in this case is simply the Jacobi theta function.Comment: 48 page

    Spectral networks and Fenchel-Nielsen coordinates

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    We explain that spectral networks are a unifying framework that incorporates both shear (Fock-Goncharov) and length-twist (Fenchel-Nielsen) coordinate systems on moduli spaces of flat SL(2,C) connections, in the following sense. Given a spectral network W on a punctured Riemann surface C, we explain the process of "abelianization" which relates flat SL(2)-connections (with an additional structure called "W-framing") to flat C*-connections on a covering. For any W, abelianization gives a construction of a local Darboux coordinate system on the moduli space of W-framed flat connections. There are two special types of spectral network, combinatorially dual to ideal triangulations and pants decompositions; these two types of network lead to Fock-Goncharov and Fenchel-Nielsen coordinates respectively.Comment: 63 pages; v2: expository improvements, journal versio

    Asymptotics of Hitchin's metric on the Hitchin section

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    We consider Hitchin's hyperk\"ahler metric gg on the moduli space M\mathcal{M} of degree zero SL(2)\mathrm{SL}(2)-Higgs bundles over a compact Riemann surface. It has been conjectured that, when one goes to infinity along a generic ray in M\mathcal{M}, gg converges to an explicit "semiflat" metric gsfg^{\mathrm{sf}}, with an exponential rate of convergence. We show that this is indeed the case for the restriction of gg to the tangent bundle of the Hitchin section B⊂M\mathcal{B} \subset \mathcal{M}.Comment: 22 pages, 1 figure. v2: Minor revision
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