BRST quantization and cohomology
imported>Pythagoras0님의 2012년 10월 29일 (월) 09:50 판
introduction
- gauge theory = principal G-bundle
- we require a quantization of gauge theory
- BRST quantization is one way to quantize the theory and is a part of path integral
- gauge theory allows 'local symmetry' which should be ignored to be physical
- this ignoring process leads to the cohomoloy theory.
- gauge theory allows 'local symmetry' which should be ignored to be physical
- BRST = quantization procedure of a classical system with constraints by introducing odd variables (“ghosts”)
- re-packaging of Faddeev-Popov quantization
- the conditions D = 26 and α0 = 1 for the space-time dimension D and the zero-intercept α0 of leading trajectory are required by the nilpotency QB2 = 0 of the BRS charge
gauge fixing
ghost variables
Faddeev-Ghost determinant
- Faddeev-Popov ghosts, Hitoshi Murayama
path integral and ghost sector
- \(Z = \int\!\mathcal{D}X\,\mathcal{D}c \mathcal{D}b \mathcal{D}\bar{c} \mathcal{D}\bar{b} \,e^{-\int\left(\partial X \partial \bar{X} -b_{zz}\partial_{\bar{z}}c^{z}+b_{\bar{z}\bar{z}}\partial_{z}c^{\bar{z}}\right)}\)
- \(e^{S_1(X)+S_2(b,c,\bar{b},\bar{c},\cdots,X)\)
- DX : matter and Db : ghost Dc : antighost
- bc system of \epsilon=+1 (in Faddeev–Popov ghost fields)
- \lambda=2
- c_{b,c}=-26
- [c]=-1,[b]=2
- global issues
- discrepancies in conformal gauge
- moduli spaces
- CKV
- path integral and moduli space of Riemann surfaces
nilpotency of BRST operator
- new, global symmetry (BRST)
- Q is fermionic
- Q_{BRST}^2=0
- http://bolvan.ph.utexas.edu/~vadim/Classes/2008f.homeworks/brst.pdf
- [1]http://www.nuclecu.unam.mx/~chryss/papers/brst_final.pdf
construction of Hilbert space of states
- BRST charge acts on a huge space
- Q.v =0 <=> physical condition
- if the total central charge is not 0 but c, Q_{BRST}^2=c
BRST cohomology
- \(\Lambda_{\infty}\) semi-infinite form
- \(\mathfrak{g}\) \[\mathbb{Z}\]-graded Lie algebra
- \(\sigma\) : anti-linear automorphism sending \(\mathfrak{g}_{n}\) to \(\mathfrak{g}_{-n}\)
- \(H^2(\mathfrak{g})=0\) (i.e. no non-trivial central extension)
applications
- BRST approach to minimal models BRST approach to minimal models http://dx.doi.org/10.1016/0550-3213(89)90568-3
- BRST approach to no-ghost theorem
- BRST approach to coset constructions
books
- Polchinski, vol. I. $3.1-3.4, 4.2-4.3
- GSW, I. 3.1-3.2
encyclopedia
- http://en.wikipedia.org/wiki/BRST_quantization
- http://www.scholarpedia.org/article/Becchi-Rouet-Stora-Tyutin_symmetry
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
expositions
- Introduction to Lie algebra cohomology with a view towards BRST cohomology ,Friedrich Wagemann, 2010-8
- PG minicourse: BRST cohomology (http://empg.maths.ed.ac.uk/Activities/BRST/Notes.pdf José Figueroa-O’Farrill 2006
- D'Hooker, E., Phong, D.H.: The geometry of string perturbation theory. Rev. Mod. Phys. 60,. 917-1065 (1988)
articles
- Igor B. Frenkel, Anton M. Zeitlin, Quantum Group as Semi-infinite Cohomology
- J.W., van Holten. 1990. “The BRST complex and the cohomology of compact lie algebras”. Nuclear Physics B 339 (1) (7월 23): 158-176. doi:10.1016/0550-3213(90)90537-N
- D. Bernard and G. Felder, 1990, Fock representations and BRST cohomology inSL(2) current algebra
- BRST cohomology in classical mechanics
- Symplectic Reduction, BRS Cohomology, and Infinite-Dimensional Clifford algebras, B. Kostant, S. Sternberg, Ann. Physics 176 (1987) 49–113
- I. B. Frenkel,. H. Garland, and. G. J. Zuckerman, PNAS November 1, 1986 vol. 83 no. 22 8442-8446, Semi-infinite cohomology and string theory
- http://dx.doi.org/10.1103/RevModPhys.60.917
blogs
- http://www.math.columbia.edu/~woit/notesonbrst.pdf
- http://www.math.columbia.edu/~woit/wordpress/?cat=12
- Notes on BRST I: Representation Theory and Quantum Mechanics
- Notes on BRST II: Lie Algebra Cohomology, Physicist’s Version
- Notes on BRST III: Lie Algebra Cohomology
- Notes on BRST IV: Lie Algebra Cohomology for Semi-simple Lie Algebras
- Notes on BRST V: Highest Weight Theory
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