BRST quantization and cohomology

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.
• 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

ghost variables

• ghost fields

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

related items

• 물리학과 cohomology
• Homological algebra
• Lie algebra cohomology

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

expositions

• http://www.mathcs.emory.edu/~rudolf/BRST.pdf
• 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