"Modular invariance in math and physics"의 두 판 사이의 차이
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==path integral in string theory== | ==path integral in string theory== | ||
| − | * [[path integral and moduli space of Riemann surfaces]] :<math>Z=\sum_{g=0}^{\infty} g_{s}^{-\chi(\Sigma_{g})}Z_{g}=\sum_{g=0}^{\infty} g_{s}^{2g-2}Z_{g}=\frac{1}{g_{s}^2}Z_{0}+g_{s}^{0}Z_{1}+g_{s}^2Z_{2}+\cdots</math | + | * [[path integral and moduli space of Riemann surfaces]] :<math>Z=\sum_{g=0}^{\infty} g_{s}^{-\chi(\Sigma_{g})}Z_{g}=\sum_{g=0}^{\infty} g_{s}^{2g-2}Z_{g}=\frac{1}{g_{s}^2}Z_{0}+g_{s}^{0}Z_{1}+g_{s}^2Z_{2}+\cdots</math> |
* <math>Z_{1}</math> is an integral over <math>M_1 = \mathbb{H}/SL(2,\mathbb{Z})</math> i.e. the fundamental domain. | * <math>Z_{1}</math> is an integral over <math>M_1 = \mathbb{H}/SL(2,\mathbb{Z})</math> i.e. the fundamental domain. | ||
| − | * string theory (symmetries, modular group) has a natural covariant UV cutoff! | + | * string theory (symmetries, modular group) has a natural covariant UV cutoff! |
** compare with http://www.physics.thetangentbundle.net/wiki/Quantum_field_theory/Schwinger_proper_time_formalism | ** compare with http://www.physics.thetangentbundle.net/wiki/Quantum_field_theory/Schwinger_proper_time_formalism | ||
2020년 11월 16일 (월) 07:47 판
introduction
- Is it useful?
- Why is it important?
- Kac http://www.ams.org/publications/online-books/hmbrowder-hmbrowder-kac.pdf
- Modular invariance in lattice statistical mechanics http://aflb.ensmp.fr/AFLB-26j/aflb26jp287.pdf
path integral in string theory
- path integral and moduli space of Riemann surfaces \[Z=\sum_{g=0}^{\infty} g_{s}^{-\chi(\Sigma_{g})}Z_{g}=\sum_{g=0}^{\infty} g_{s}^{2g-2}Z_{g}=\frac{1}{g_{s}^2}Z_{0}+g_{s}^{0}Z_{1}+g_{s}^2Z_{2}+\cdots\]
- \(Z_{1}\) is an integral over \(M_1 = \mathbb{H}/SL(2,\mathbb{Z})\) i.e. the fundamental domain.
- string theory (symmetries, modular group) has a natural covariant UV cutoff!
circle method
- modular invariant partition functions
- Kac-Peterson modular S-matrix
- Mock theta and physics
- Blackhole theory
- Hardy-Ramanujan tauberian theorem
questions
expositions
- Nikolov, Nikolay M., and Ivan T. Todorov. 2004. “Lectures on Elliptic Functions and Modular Forms in Conformal Field Theory”. ArXiv e-print math-ph/0412039. http://arxiv.org/abs/math-ph/0412039.
- Goddard, Peter. ‘Modular Invariance and Infinite-Dimensional Algebras’. In Superstrings, edited by Peter G. O. Freund and K. T. Mahanthappa, 3–16. NATO ASI Series 175. Springer US, 1988. http://link.springer.com/chapter/10.1007/978-1-4613-1015-0_1.
- Lepowsky, J. “Affine Lie Algebras and Combinatorial Identities.” In Lie Algebras and Related Topics, edited by David Winter, 130–56. Lecture Notes in Mathematics 933. Springer Berlin Heidelberg, 1982. http://link.springer.com/chapter/10.1007/BFb0093358.
- Lepowsky, J. "Lie algebras and combinatorics." Proc. Internat. Congr. Math.(Helsinki, 1978)(to appear) (1978). http://www.mathunion.org/ICM/ICM1978.2/Main/icm1978.2.0579.0584.ocr.pdf