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Quantum mirror symmetry and twistors
, 2009
"... Using the twistor approach to hypermultiplet moduli spaces, we derive the worldsheet, D(−1), and D1instanton contributions to the generalized mirror map, relating Type IIA and Type IIB string theory compactified on generic mirror CalabiYau threefolds. For this purpose, we provide a novel descript ..."
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Using the twistor approach to hypermultiplet moduli spaces, we derive the worldsheet, D(−1), and D1instanton contributions to the generalized mirror map, relating Type IIA and Type IIB string theory compactified on generic mirror CalabiYau threefolds. For this purpose, we provide a novel description of the twistor space underlying the Type IIB hypermultiplet moduli space where the SL(2, Z)action is found to be free from quantum corrections. The extent to which instanton effects may resolve the perturbative singularities of the moduli space metric is discussed.
Fourier expansions of Kac–Moody Eisenstein series and degenerate Whittaker vectors,” 1312.3643 [hepth
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Small automorphic representations and degenerate Whittaker vectors,” 1412.5625 [math.NT
"... We investigate Fourier coefficients of automorphic forms on split simplylaced Lie groups G. We show that for automorphic representations of small GelfandKirillov dimension the Fourier coefficients are completely determined by certain degenerate Whittaker vectors on G. Although we expect our result ..."
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We investigate Fourier coefficients of automorphic forms on split simplylaced Lie groups G. We show that for automorphic representations of small GelfandKirillov dimension the Fourier coefficients are completely determined by certain degenerate Whittaker vectors on G. Although we expect our results to hold for arbitrary simplylaced groups, we give complete proofs only for G = SL(3) and G = SL(4). This is based on a method of Ginzburg that associates Fourier coefficients of automorphic forms with nilpotent orbits of G. Our results complement and extend recent results of Miller and Sahi. We also use our formalism to calculate various local (real and padic) spherical vectors of minimal representations of the exceptional groups E6, E7, E8 using global (adelic) degenerate Whittaker vectors, correctly reproducing existing results for such spherical vectors obtained by very different methods. ar X iv
Arithmetic and Hyperbolic Structures in String Theory
, 2010
"... This monograph is an updated and extended version of the author’s PhD thesis. It consists of an introductory text followed by two separate parts which are loosely related but may be read independently of each other. In Part I we analyze certain hyperbolic structures arising when studying gravity in ..."
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This monograph is an updated and extended version of the author’s PhD thesis. It consists of an introductory text followed by two separate parts which are loosely related but may be read independently of each other. In Part I we analyze certain hyperbolic structures arising when studying gravity in the vicinity of a spacelike singularity (the “BKLlimit”). In this limit, spatial points decouple and the dynamics exhibits ultralocal behaviour which may be mapped to an auxiliary problem given in terms of a (possibly chaotic) hyperbolic billiard. In all supergravities arising as lowenergy limits of string theory or Mtheory, the billiard dynamics takes place within the fundamental Weyl chambers of certain hyperbolic KacMoody algebras, suggesting that these algebras generate hidden infinitedimensional symmetries of the theory. We develop in detail the relevant mathematics of Lorentzian KacMoody algebras and hyperbolic Coxeter groups, and explain with many examples how these structures are intimately connected with gravity. We also construct a geodesic sigma model invariant under the hyperbolic KacMoody group E10, and analyze to what extent its dynamics reproduces the dynamics of type II and elevendimensional supergravity.