Random Complex Geometry and Vacua, Or: How to Count Universes in String/m Theory
نویسنده
چکیده
This lecture is concerned with counting and equidistribution problems for critical points of random holomorphic functions and their applications to statistics of vacua of certain string/M theories. To get oriented, we begin with the distribution of complex zeros of Gaussian random holomorphic polynomials of one variable. We then consider zeros and critical points of Gaussian random holomorphic sections of line bundles over complex manifolds. The main focus is on critical points ∇s(z) = 0 of a holomorphic sections relative to a smooth metric connection ∇ on a holomorphic line bundle. In string/M theory compactified on a Calabi-Yau manifold, the possible vacuum states of the universe (vacua) are critical points of a holomorphic section (the ‘superpotential’) of a line bundle over the moduli space of Calabi-Yau manifolds. Physicists sometimes estimate the number of possible vacua to be around 10. We describe some rigorous results from [DSZ3, DD] on the number and distribution of vacua in such string theories. Finally, we discuss some results from [DSZ1, DSZ2] on the pure geometry of critical points: how the average number of critical points is asymptotically minimized by Calabi extremal metrics and some hints on the correlations between critical points on small scales.
منابع مشابه
2 0 Fe b 20 04 CRITICAL POINTS AND SUPERSYMMETRIC VACUA
Supersymmetric vacua ('universes') of string/M theory may be identified with certain critical points of a holomorphic section (the 'superpotential') of a Hermitian holomor-phic line bundle over a complex manifold. An important physical problem is to determine how many vacua there are and how they are distributed. The present paper initiates the study of the statistics of critical points ∇s = 0 ...
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Supersymmetric vacua (‘universes’) of string/M theory may be identified with certain critical points of a holomorphic section (the ‘superpotential’) of a Hermitian holomorphic line bundle over a complex manifold. An important physical problem is to determine how many vacua there are and how they are distributed. The present paper initiates the study of the statistics of critical points ∇s = 0 o...
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