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Vertex Operators :::::::::::::::::::::::::: :::::::::::::::::::::::::: :::::::::::::::::::::::::: Vertex operators :::::::::::::::::::::::::: Open-string operator products For later calculations, we'll need a short list of operator products. But first we need to emphasize some differences and similarities for open and closed strings. For the closed string we have left- and right-handed modes XbL and XbR, while for the open string XbL = XbR: 8 q α0 < [XbL(z) + XbR(¯z)] for closed X(z; z¯) = 2 q α0 : 2 [Xb(z) + Xb(¯z)] for open All these Xb's have conveniently normalized propagators 0 0 0 0 hXb(z) Xb(z )i = hXbL(z) XbL(z )i = hXbR(z) XbR(z )i = −ln(z − z ) from which follows directly those for X itself: 2 −ln(jz − z0j2) for closed hX(z; z¯) X(z0; z¯0)i = α0 −ln(jz − z0j2) − ln(jz − z¯0j2) for open where the open string has extra contributions from crossterms, now involving the same Xb. For open-string amplitudes involving only open-string external states, all the ver- tex operators will be on the boundary, p z =z ¯ ) X(z; z¯) = 2α0Xb(z) Therefore, when Fourier transforming wave functions we use the exponentials ( p ik^·X(z) 2α0 for open ik·X(z;z¯) e b for open ^ e = ) k = k × q α0 ik^·X (z) ik^·X (¯z) for closed e bL e bR for closed 2 Closed-string vertex operators are the product of left- and right-handed ones, which are functions of z andz ¯, respectively, and thus take the form of the product of 2 independent open-string vertex operators. Working directly in terms of Xb, we then have the \operator products" ^ 1 ^ (i@Xb)(z0) eik·Xb(z) ≈ k^ eik·Xb(z) z0 − z ( p 1 2α0 for open 0 q or (@Xb)(z ) f(X(z; z¯)) ≈ − 0 (@f)(X(z; z¯)) × α0 z − z 2 for closed 2 1 (i@Xb)(z0)(i@Xb)(z) ≈ (z0 − z)2 (Note the context: @Xb is a z derivative, @f is an x derivative. The \i" associated with @Xb is from Wick rotation.) For example, we can use these results to determine the proper normalization of massless vertex operators, by comparison with that of tachyons: For the tachyon, 1 0 W (z) = eik^·Xb(z); k^2 = 2 ) W (z0) W (z) ≈ eik^·[Xb(z )−Xb(z)] k^ k^ −k^ (z0 − z)2 (For the closed string, we have the product of left and right versions of the above. Note this correctly gives k2 = 1/α0 for the open string tachyon and 4/α0 for the closed, where α0 is the slope of the open-string Regge trajectory, and the parameter that appears in the action that describes both open- and closed-string states.) The z factors are canceled in string field theory by considering the gauge-fixed kinetic term h0jV (c0 )V j0i, where V = cW . Gauge-independent vertex operators When ghosts are included, vertex operators can be generalized to arbitrary gauges for the external gauge fields. (This result follows from the same method applied to relate integrated and unintegrated vertices in subsection XIIB8 of Fields. We'll do a better job of that here.) The main point is the existence of integrated and unintegrated vertex operators: Integrated ones are natural from adding backgrounds to the gauge-invariant action; unintegrated ones from adding backgrounds to the BRST operator. We'll relate the two by going in both directions. The following discussion will be for general quantum mechanics (except in the relativistic case we use τ in place of t), but we'll add some special comments for open strings at the end. The action can be written as Z S ∼ dτ HI plus the usual terms for converting Hamiltonian to (first-order) Lagrangian, where the interacting Hamiltonian consists of the free part plus linearized vertex HI = H0 + W BRST invariance with respect to the free BRST operator then implies [Q0;S] ≈ 0 R ) [Q0; dτ W ] ≈ 0 ) [Q0;W ] ≈ @τ V ) fQ0;V g ≈ 0 Vertex operators 3 for some V , where \≈" means \at the linearized level". The BRST invariants R W and V are thus our integrated and unintegrated vertex operators, respectively. Going in the other direction, we start with interacting BRST QI = Q0 + V where fully interacting BRST invariance implies at the linearized level 2 QI = 0 ) fQ0;V g ≈ 0 The full gauge-fixed action is then defined (in relativistic quantum mechanics, or otherwise in the ZJBV formalism) by HI = fQI ; bg ≈ H0 + W It then follows that 0 = [QI ;HI ] ≈ [Q0;H0] + ([Q0;W ] + [V; H0]) which agrees with the above, since H0 gives the (free) time development: [H0;V ] = @τ V The only modifications for the open string are eliminating σ dependence: Z dσ Z dσ Z dσ Q = J; H = T; b ! b (0 − mode) 0 2π 0 2π 2π V ! V jσ=0;W ! W jσ=0 After combining the left and right-handed modes into functions of just z over the whole plane, as usual, we can then replace σ and τ with z in our definitions in an appropriate way. Vector vertex The simplest case is the massless vector. The choice for integrated vertex was obvious from the gauge transformation of the external field: . W = X · A(X); δA(x) = −@λ(x) Z Z Z Z ) dτ W = dX · A(X); δ dτ W = − dλ(X) = 0 As usual, the τ integral gets converted into a z integral over the boundary (real axis). 4 Besides this \background" gauge invariance, we also need the \quantum" BRST invariance. The unintegrated vertex V and the BRST invariance of R W then follow from the same calculation: Z [Q; W ] = @V ) Q W = QV = 0 We use the BRST operator Z I 0 dz 1 2 dz 0 Q = J; J = cT + c(@c)b; T = 2 (i@Xb) ; [Q; W (z)] = J(z ) W (z) 2πi z 2πi (For the open string, this is all of Q; the closed string has Q = QL + QR, with QL and QR given by the above, with \L" or \R" subscripts on everything. For now, we stick to the open string. There is a sign convention change from Fields for Q and T .) For T (z0)W (z), we get \single-contraction" (tree/classical) terms from the singu- lar part of either @X with W (one propagator), and nonsingular (ordinary) product of the other @X (no propagator). So we evaluate p a 0 1 a 0 1 b a (@Xb )(z ) [(@Xb)·A(X)](z) ≈ − A (X(z))− 2α [(@Xb) @ Ab(X)](z) (z0 − z)2 z0 − z We also get \double-contraction" (1-loop) terms from the singular part of the product of the second @X with the above: (@Xb)(z0) · (right-hand side of above) ≈ p 1 1 2 2α0 @ · A(X(z)) + 2α0 (@Xb · A)(z) (z0 − z)3 (z0 − z)2 We then need to integrate, using I 0 dz 1 0 1 n 0 n+1 f(z ) = @ f(z) z 2πi (z − z) n! Putting it all together, 0 a b W = (i@Xb) · A ) [Q; W ] = @V − α (i@c)(@Xb )@ Fba rα0 V = c(i@Xb) · A − (i@c)@ · A 2 p 0 (We have repeatedly used the identity @zf(X) = (@X) · @f = 2α (@Xb) · @f.) Thus BRST invariance of R W and V requires the background satisfy only the b (free) gauge-covariant field equations @ Fba = 0. This was to be expected, since quantum BRST invariance of Yang-Mills in a Yang-Mills background requires the same in field theory. We also find an order α0 correction to the vertex operator V : This can be explained by noting that, while c@X creates a Yang-Mills state from the vacuum, @c creates its Nakanishi-Lautrup field plus @ · A, in a combination that vanishes by that field’s equation of motion. Zero-modes 5 RNS trees Here are the method and some simple examples for tree graphs in the Ramond- Neveu-Schwarz formalism. We'll restrict ourselves to just external vectors: Tachyons are easier, but irrelevant for the superstring; fermions are much easier in manifestly supersymmetric formalisms. :::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::: Zero-modes :::::::::::::::::::::::::::::: Calculations are pretty straightforward, as for the bosonic string. However, while ghosts for the bosonic string were almost trivial, and could almost be ignored, ghosts in RNS affect the non-ghost factors of vertex operators. Such complications (\pic- tures") can be discussed without ghosts, but the ghosts make the existence of pictures clearer. As for the bosonic string, for tree amplitudes with external bosons, ghosts contribute only their zero-modes. Oscillators We first give a general definition of zero-modes for tree graphs using conformal field theory. To describe the ground-state wave function for a string, we begin by considering ground states in a more general context. Define a general set of variables in terms of coordinates and momenta: [pi; xjg = −iηij; xiy = xi; piy = pi Z Y dxi I = p jxihxj 2π for some arbitrary indices i; j and arbitrary (constant) metric ηij. Then define general harmonic oscillators: ai = p1 (xi + ipi); aiy = p1 (xi − ipi) 2 2 ) [ai; ajyg = ηij The ground state has the usual Gaussian wave function: aij0i = 0; aiyj0i 6= 0 i j ) hxj0i = e−ηij x x =2; h0j0i = 1 6 RNS trees But the string also has non-oscillator modes, i.e., zero-modes. A general set of zero-modes satisfies instead pij0i = 0 D E Y i ) hxj0i = 1; 0 δ(x ) 0 = 1 p in terms of the same kind of coordinates and momenta as above (with some 2π's defined into our δ functions), since the vacuum expectation value is now just an integral over the x's (without Gaussians).
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