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Y. He; M. van Loon (2014)
Publisher: Springer/SISSA
Types: Article
Subjects: QA, Differential and Algebraic Geometry, Brane Dynamics in Gauge Theories, High Energy Physics - Theory, Mathematics - Algebraic Geometry, AdS-CFT Correspondence

Classified by OpenAIRE into

arxiv: High Energy Physics::Theory
We study and classify regular and semi-regular tessellations of Riemann surfaces of various genera and investigate their corresponding supersymmetric gauge theories. These tessellations are generalizations of brane tilings, or bipartite graphs on the torus as well as the Platonic and Archimedean solids on the sphere. On higher genus they give rise to intricate patterns. Special attention will be paid to the master space and the moduli space of vacua of the gauge theory and to how their geometry is determined by the tessellations.
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    • 1 Department of Mathematics, City University, London, EC1V 0HB, UK and 2 School of Physics, NanKai University, Tianjin, 300071, P.R. China and 3 Merton College, University of Oxford, OX1 4JD, UK
    • 2 Bipartite Tessellation of Riemann Surfaces 5 2.1 Dimer Models: Bipartite Graphs on Riemann Surfaces . . . . . . . . 5 2.2 Master Space and Moduli Space . . . . . . . . . . . . . . . . . . . . . 7 2.3 Classi cation of (Semi-)Regular Tessellations . . . . . . . . . . . . . . 9 2.3.1 Terminology and De nitions . . . . . . . . . . . . . . . . . . . 9 2.3.2 Classi cation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.3 Regular Tessellations . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.4 Semi-Regular Tessellations . . . . . . . . . . . . . . . . . . . . 18 2.4 Outline of Computation . . . . . . . . . . . . . . . . . . . . . . . . . 21
    • 3 Regular Tessellations 22 3.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Detailed results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.1 Genus 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.2 Genus 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2.3 Genus 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.4 Genus 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
    • 4 Semi-Regular Tessellations 67 [1] B. Feng, A. Hanany and Y. -H. He, \D-brane gauge theories from toric singularities and toric duality," Nucl. Phys. B 595, 165 (2001), arXiv:hep-th/0003085.
    • [14] Y. -H. He and J. McKay, \N=2 Gauge Theories: Congruence Subgroups, Coset Graphs and Modular Surfaces," J. Math. Phys. 54, 012301 (2013) [J. Math. Phys. 54, 012301 (2013)], arXiv:1201.3633 [hep-th].
    • [15] Y. -H. He, \Bipartita: Physics, Geometry & Number Theory," arXiv:1210.4388 [hep-th].
    • [16] Y. -H. He, J. McKay and J. Read, \Modular Subgroups, Dessins d'Enfants and Elliptic K3 Surfaces," LMS J. Comp. Math. 16, 271 (2013), arXiv:1211.1931 [math.AG].
    • [17] N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov and J. Trnka, \Scattering Amplitudes and the Positive Grassmannian," arXiv:1212.5605 [hep-th].
    • [18] J. Golden, A. B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, \Motivic Amplitudes and Cluster Coordinates," arXiv:1305.1617 [hep-th].
    • [28] A. Hanany and R. -K. Seong, \Brane Tilings and Specular Duality," JHEP 1208, 107 (2012), arXiv:1206.2386 [hep-th].
    • [29] S. Cremonesi, A. Hanany and R. -K. Seong, \Double Handled Brane Tilings," JHEP 1310, 001 (2013), arXiv:1305.3607 [hep-th].
    • [30] Y. -H. He and J. Read, \Hecke Groups, Dessins d'Enfants and the Archimedean Solids," arXiv:1309.2326 [math.AG].
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