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The Riemann Zeros and Eigenvalue Asymptotics
 SIAM Rev
, 1999
"... Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many feat ..."
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Cited by 40 (5 self)
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Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display randommatrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The RiemannSiegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.
Dimensional mutation and spacelike singularities,” Phys
 Rev. D
, 2006
"... I argue that string theory compactified on a Riemann surface crosses over at small volume to a higher dimensional background of supercritical string theory. Several concrete measures of the count of degrees of freedom of the theory yield the consistent result that at finite volume, the effective dim ..."
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Cited by 18 (2 self)
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I argue that string theory compactified on a Riemann surface crosses over at small volume to a higher dimensional background of supercritical string theory. Several concrete measures of the count of degrees of freedom of the theory yield the consistent result that at finite volume, the effective dimensionality is increased by an amount of order 2h/V for a surface of genus h and volume V in string units. This arises in part from an exponentially growing density of states of winding modes supported by the fundamental group, and passes an interesting test of modular invariance. Further evidence for a plethora of examples with the spacelike singularity replaced by a higher dimensional phase arises from the fact that the sigma model on a Riemann surface can be naturally completed by many gauged linear sigma models, whose RG flows approximate time evolution in the full string backgrounds arising from this in the limit of large dimensionality. In recent examples of spacelike singularity resolution by tachyon condensation, the singularity is ultimately replaced by a phase with all modes becoming heavy and decoupling. In the present case, the opposite behavior ensues: more light degrees of freedom arise in the small radius regime. We comment on the emerging zoology of cosmological singularities that results. October 20051. Dimensionality and Singularities Many timelike singularities are resolved in a way that involves new light degrees of freedom appearing at the singularity. In spacelike singularities studied recently [1,2], ordinary spacetime ends where a tachyon background becomes important. The tachyon at first constitutes a new light mode in the system, which goes beyond the spectrum evident in general relativity. However its condensation then replaces the wouldbe shortdistance singularity with a phase where degrees of freedom ultimately become heavy [3,1]. In this note, we will find strong indications that there is a whole zoo of possible behaviors at cosmological spacelike singularities, including examples in which the general relativistic singularity is replaced by a phase with more light degrees of freedom, in the form of a larger effective number of dimensions. 1
New dimensions for wound strings: The modular transformation of geometry to topology,” Phys
 Rev. D
, 2007
"... We show, using a theorem of Milnor and Margulis, that string theory on compact negatively curved spaces grows new effective dimensions as the space shrinks, generalizing and contextualizing the results in [1]. Milnor’s theorem relates negative sectional curvature on a compact Riemannian manifold to ..."
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Cited by 7 (0 self)
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We show, using a theorem of Milnor and Margulis, that string theory on compact negatively curved spaces grows new effective dimensions as the space shrinks, generalizing and contextualizing the results in [1]. Milnor’s theorem relates negative sectional curvature on a compact Riemannian manifold to exponential growth of its fundamental group, which translates in string theory to a higher effective central charge arising from winding strings. This exponential density of winding modes is related by modular invariance to the infrared small perturbation spectrum. Using selfconsistent approximations valid at large radius, we analyze this correspondence explicitly in a broad set of timedependent solutions, finding precise agreement between the effective central charge and the corresponding infrared small perturbation spectrum. This indicates a basic relation between geometry, topology, and dimensionality in string theory. December
UTHET090102 Exact Gravity Dual of a Gapless Superconductor
, 902
"... A model of an exact gravity dual of a gapless superconductor is presented in which the condensate is provided by a charged scalar field coupled to a bulk black hole of hyperbolic horizon in asymptotically AdS spacetime. Below a critical temperature, the black hole acquires its hair through a second ..."
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A model of an exact gravity dual of a gapless superconductor is presented in which the condensate is provided by a charged scalar field coupled to a bulk black hole of hyperbolic horizon in asymptotically AdS spacetime. Below a critical temperature, the black hole acquires its hair through a second order phase transition while an electromagnetic perturbation of the background Maxwell field determines the conductivity of the boundary theory.