"BRST quantization and cohomology"의 두 판 사이의 차이

2020년 11월 16일 (월) 07:48 판

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.
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 \(\alpha_0=1\) for the space-time dimension \(D\) and the zero-intercept \(\alpha_0\) of leading trajectory are required by the nilpotency \(Q_B^2 = 0\) of the BRS charge

\(Z = \int\!\mathcal{D}X\,\mathcal{D}c \mathcal{D}b \mathcal{D}\bar{c} \mathcal{D}\bar{b} \,e^{-\int\left(\partial X \partial \bar{X} -b_{zz}\partial_{\bar{z}}c^{z}+b_{\bar{z}\bar{z}}\partial_{z}c^{\bar{z}}\right)}\)
\(e^{S_1(X)+S_2(b,c,\bar{b},\bar{c},\cdots,X)}\)
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

\(\Lambda_{\infty}\) 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)

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

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

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