"Modular invariance in math and physics"의 두 판 사이의 차이
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* [[path integral and moduli space of Riemann surfaces]]<br><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}+\cdtos</math><br> | * [[path integral and moduli space of Riemann surfaces]]<br><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}+\cdtos</math><br> | ||
* <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 | + | * string theory (symmetries, modular group) has a natural covariant UV cutoff!<br> |
| + | ** compare with http://www.physics.thetangentbundle.net/wiki/Quantum_field_theory/Schwinger_proper_time_formalism | ||
2012년 8월 26일 (일) 03:28 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
\(Z=\sum_{g=0}^{\infty} g_{s}^{-\chi(\Sigma_{g})}Z_{g}=\sum_{g=0}^{\infty} g_{s}^{2g-2}Z_{g}=\frac{1}{g_{s}^2}Z_{0}+g_{s}^{0}Z_{1}+g_{s}^2Z_{2}+\cdtos\) - \(Z_{1}\) 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