4 BRST quantization of the bosonic string 4.1 BRST quantization The BRST formalism is a consistent quantization prescription for a gauge theory at the level of path integral, assuming that the gauge algebra is not field dependent. To begin with, consider an action S[φ], where φ collectively denotes all fields in the theory, that is invariant with respect to gauge symmetries of the form δφ = dµ ✏↵δ φ ✏↵δ φ. (4.1) ↵ ↵ ⌘ ↵ Z Here ↵ is a continuous label that includes the spacetime coordinates. dµ↵ is a suitable measure on ↵-space, whereas ✏↵ is a function in ↵. When there is no room for confusion, we will omit the integration measure in our notation. The gauge variation δ↵ acts on composite fields by Leibniz’s rule as a di↵erentiation, and obey an (infinite dimensional) Lie algebra γ [δ↵,δβ]=f↵ δγ, (4.2) γ 7 where the structure constants f↵ by assumption are field independent. Formally, the partition function can be expressed as the Euclidean path integral S[φ] Z = [Dφ] e− , (4.6) Z up to some unspecified normalization of the integration measure [Dφ]. To define the latter properly, we can fix the gauge by replacing (4.6) with A A S[φ] Z = [Dφ] δ(F [φ]) det(δ↵F [φ])e− , (4.7) Z 7As an example, suppose the gauge symmetry of interest is di↵eomorphism. In this case, ↵ will be a ↵ ⇢ d spacetime coordinate y together with a vector index ⇢, ✏ will be a vector field " (y), with dµ↵ = d y.A scalar field ' transforms as δ '(x)= δd(x y)@ '(x). (4.3) y,⇢ − − ⇢ (4.1) then reproduces the usual infinitesimal di↵eomorphism '(x)= "⇢(x)@ '(x). From the action of the − ⇢ commutator of a pair of di↵eomorphisms on ', d d [δ ,δ ]'(x)=δ (x z)@ σ δ (x y)@ '(x) (y, ⇢ z,σ) y,⇢ z,σ − x − ⇢ − $ (4.4) = ddw δ⌫ δ⇥d(w z)@ δd(x ⇤y)+δ⌫ δd(w y)@ δd(x z) δ '(x), − ⇢ − σ − σ − ⇢ − w,⌫ Z ⇥ ⇤ we deduce the structure constant of di↵eomorphism, f (w,⌫) = δ⌫ δd(w z)@ δd(x y)+δ⌫ δd(w y)@ δd(x z). (4.5) (y,⇢)(z,σ) − ⇢ − σ − σ − ⇢ − 58 where A is an index that labels components of the gauge condition F A[φ]=0,suchthat A ↵ the Jacobian δ↵F [φ] is non-degenerate. We can then introduce the ghosts bA, c ,anda Lagrangian multiplier field BA,sothat(4.7)canbewrittenas Z = [DφDB Db Dc↵]exp S[φ]+iB F A[φ] b c↵δ F A[φ] . (4.8) A A − A − A ↵ Z In the full gauge fixed action with the ghosts included, the gauge symmetry is replaced by afermionicglobalsymmetryδB,knownasBRSTsymmetry,thatactsonthefieldsas δ φ = ic↵δ φ, B − ↵ δBBA =0, (4.9) δBbA = BA, i δ c↵ = f ↵cβcγ. B 2 βγ 2 The definition is such that δB =0,and iB F A[φ] b c↵δ F A[φ]=δ ib F A[φ] . (4.10) A − A ↵ B A It follows that δB is a symmetry of the action appearing in (4.8). One may also construct a Noether current jB associated with BRST symmetry, and define a corresponding BRST charge Q ,suchthatδ = i Q , ,where , stands for B B { B ·}cl {· ·}cl the Poisson bracket. If the path integral measure is also BRST invariant, then QB can be 2 promoted to a Hermitian quantum operator, such that QB =0. Aphysicalstate will be defined to be one that is BRST-invariant, or BRST-closed, | i Q =0.Intermsofthewavefunctional [φ], this condition is Q [φ] iδ [φ]= B| i B ⌘ B 0. It follows that the transition amplitude between two physical states is invariant under deformations of the gauge condition F A[φ]=0.Toseethis,considerachangeF A A A ! F + δ0F .Wehave A ↵ φf S[φ]+δB (bAF ) A δ0 f Ufi i = [DφDBADbADc ] ⇤ [φf ]e− δB(bAδ0F ) i[φi]=0, (4.11) h | | i φi f Z where Ufi is the time evolution operator defined by the path integral with boundary condi- tions φ = φi at the initial time and φ = φf at the final time. Furthermore, shifting a physical state by a BRST-exact state Q χ for some χ | i B| i | i does not a↵ect transition amplitudes between and any other physical states. Thus we | i may view as being equivalent to + Q χ , and identify the Hilbert space of physical | i | i B| i states with the cohomology of QB. 59 4.2 BRST on the worldsheet As already seen in section (2.4), the gauge fixing of the worldsheet theory of the critical bosonic string with the gauge condition gab =ˆgab leads to the action µ a µ a S[X ,bab,c ]=SP [ˆgab,X ]+Sgh[ˆgab,bab,c ], (4.12) In the language of BRST quantization, we have already integrated out the Lagrangian mul- tiplier field Bab as well as the dynamical metric gab.TheBRSTtransformationofbab, δBbab = Bab,isnowreplacedanoperatorthatisequivalenttoBab by the equation of motion, µ a which is the stress-energy tensor of the (X ,bab,c )system,namely X gh Tab = Tab + Tab . (4.13) The classical BRST transformation of the worldsheet fields, up to equations of motion, are µ a µ δBX = ic @aX , δBbab = iTab, (4.14) δ ca = icb ˆ ca. B rb B In the quantum theory, it is more precise to express this in terms of the BRST current ja as a local operator. In the conformal gaugeg ˆab = δab,workingincomplexcoordinates(z,z¯), the ghost fields can be written as b b , b b , and c cz, c cz¯,withthefollowing ⌘ zz ⌘ z¯z¯ ⌘ ⌘ singular OPE among holomorphic ghost fields e e 1 1 b(z)c(0) ,c(z)b(0) . (4.15) ⇠ z ⇠ z The bc system by itself is a CFT with holomorphic stress-energy tensor T gh = (@b)c 2b@c. (4.16) − − As usual, an implicit normal ordering is defined by subtracting o↵singular terms in the coincidence limit, and this form of the ghost stress-energy tensor is unambiguously fixed by the requirement that b is a primary of conformal weight (2, 0) and whereas c is one of weight ( 1, 0). One can indeed verify that the OPE of T gh with itself takes the form of the Virasoro − algebra with central charge cgh = 26. − Now we will denote the holomorphic and anti-holomorphic components of the BRST current by j j , j j ,andwritetheBRSTchargeas Bz ⌘ B Bz¯ ⌘ B dz dz¯ eQ = j (z) j (¯z). (4.17) B 2⇡i B − 2⇡i B I I e 60 Note that the holomorphic and anti-holomorphic BRST currents are independently con- served. The holomorphic BRST current is 3 j = cT X + bc@c + @2c, (4.18) B 2 X 1 µ µ where T = @X @Xµ is the stress-energy tensor of the X CFT, of central charge ↵0 − 2 B D.Thelasttermproportionalto@ c does not a↵ect QB,andisincludedsothatj can transform as a conformal primary. In fact, jB(z)isaconformalprimaryonlywhenD =26. QB obeys Q ,b = T = T X + T gh. (4.19) { B } The nilpotency of QB implies that T (z)T (0) = T (z) Q ,b(0) = Q ,T(z)b(0) (4.20) { B } { B } has only 1 singularity, and thus the total central charge c = D 26 must be zero. We z2 − will see later that this is also the condition for the cancelation of Weyl anomaly on a curved worldsheet. For computing BRST cohomology, it is useful to work with the oscillator representation of the BRST charge, X m n QB = cnL n + − : cmcnb m n : c0 +(anti holomorphic), (4.21) − 2 − − − − n Z m,n Z X2 X2 where the normal ordering prescription is defined by moving all the positively graded os- cillators to the right of the negatively graded ones, including the appropriate signs when a pair of Grassmannian fields are exchanged.8 One can also directly verify from (4.21) that Q ,b = LX + Lgh. { B n} n n Let us comment on the space of states in the bc ghost system. The space of local vertex operators are generated by the OPE of b, c, and their derivatives. However, the lowest weight operators are not the identity operator, but c and c@c, both of which has conformal weight 1. In the Fock space description, they correspond to two degenerate ground states and − |#i ,thatobey |"i b0 = c0 =0, |#i |"i (4.22) c = ,b = . 0|#i |"i 0|"i |#i Indeed, under the state/operator mapping, corresponds to the operator c(0), whereas |#i b 1 is mapped to the identity operator. Combing left and right movers, we will write e.g. − |#i , for the state that corresponds to the operator cc(0). |# #i 8Note that this normal ordering on the Fourier modes is a priori unrelated to the normal ordering prescription on the products of local field operators. e 61 The bc system has a U(1) ghost number symmetry that assigns (left-moving) ghost num- ber 1 to c and 1tob. The corresponding Noether current is − j = bc. (4.23) gh − While jgh is a conserved current of weight (1, 0), it is not a Virasoro primary. Indeed, the gh OPE of T with jgh takes the form 3 1 1 T gh(z)j (0) + j (0) + @j (0). (4.24) gh ⇠z3 z2 gh z gh This leads to an anomaly in the ghost number symmetry on a curved worldsheet, namely the operator jgh has a nonzero divergence 3 ja = R(g), (4.25) ra gh 4 where R(g)istheRicciscalaroftheworldsheetmetricgab.Consequently,thecorrelation functions of the bc CFT on a genus g Riemann surface must violate ghost number by 3 − 3g.Thereisasimilarviolationoftheghostnumberintheanti-holomorphicbc CFT.