A geometric approach to BRST quantization of the bosonic string Valdo Tatitscheff valdo.tatitscheff@ens.fr Abstract BRST quantization find its roots in the years 1974-1976 in papers by Becchi, Rouet, and Stora [1], and independently, by Tyutin [4]. At first, it was a very heuristic method of quantizing gauge theories, but since then, a great work has been done to understand geometrically this procedure. This is what we want to describe in this paper. The first section constructs the BRST cohomology in a simplified framework, and in the second, links are proposed between the simplified mathematical approach and the physical approach in the bosonic string theory. In particular, the BRST symmetry is closely related to the Fadeev-Popov gauge fixing. 1 Geometric BRST quantization 1.2 Lie algebra cohomology 1.1 Symplectic reduction Any smooth function on B induces a function on C which is constant on the fibers, and conversely, Let (X; O) be a symplectic manifold on which any function on C which is constant on the fibers a connected Lie group G acts by symplectomor- induces a function on B. phisms, such that there is a moment map: Let's highlight that the hamiltonian Killing vec- tor fields Ξi generated by the Lie algebra g of G φ : X ! g∗ form a global basis of the tangent spaces to the fibers. Let's chose a dual basis !i of the cotan- that is, a function satisfying that 8x 2 X and i i gent space to the fibers, that is, ! (Ξj) = δj, and 8ξ 2 g: define: d(φ(x)(ξ)) = ι Ω d : C1(C) ! Ω1 (C) ξ~x V V ~ by: where ξ is the fundamental vector field generated X d (f) = (L f)!i by ξ. We can define a map δ : g ! F un(X) by V Ξi setting: i where L f denotes the Lie derivative of f in δ(ξ)(x) = φ(x)(ξ) Ξi the direction Ξi. One extends this function to a for all ξ 2 g and x 2 X. Provided 0 2 g∗ is a derivation: regular value of the moment map, we can define: p p+1 dV :ΩV (C) ! ΩV C −1 C = φ (0) by defining: i 1 X i j k which is then a closed embedded coisotropic sub- dV ! = − fjk! ^ ! ? 2 manifold. Moreover, 8x 2 X, TxM0 is precisely j;k the subspace of T M spanned by the Killing vec- x 0 This function satisfies the relation d ◦ d = 0 (it tor fields induced by the action of G. V V is proved by the following proposition) and thus Now define B = C=G. B is the Marsden- (V g ⊗ C1(C); d ) form a cochain complex from Weinstein symplectic reduction of X. Note that V which one can compute the Lie algebra cohomol- there are two steps in this procedure : one step ogy. of restriction from X to C, and one step where one quotients by the group's action. For more de- Proposition 1.1. The differential dV coincide tails on symplectic reduction and the rest of this with the Chevalley-Eilenberg's differential corre- section, see [2], [5]. sponding to the trivial representation of g on R. 1 Proof. Let xa and xb be in g. One computes: Let's add the Lie algebra cohomology to obtain: 1 X j d !i(x ; x ) = − f i (δj δk − δ δk) Vp;q 1 δ Vp;q−1 1 V a b 2 jk a b b a g ⊗ C (X) / g ⊗ C (X) j;k dV i i dV ! (xa; xb) = −f ab Vp+1;q g ⊗ C1(X) i 1+2 i dV ! (xa; xb) = (−1) ! ([xa; xb]) where we wrote: Vp;q g = Vp g∗ ⊗ Vq g. Since δ ◦ d = d ◦ δ, it is a double complex. Let: There are interpretations of the low-degree co- homology groups of the Chevalley-Eilenberg com- p q M ^ ∗ ^ 1 0 V 1 Pm = g ⊗ g ⊗ C (X) plex. For example, HV ( g ⊗ C (C)) is the sub- space of C1(C) constituted by all functions which p−q=m are constant on the fibers. and: 1.3 BRST cohomology Dm : Pm ! Pm+1 We found a cohomology describing the property be given by: of being a scalar function under a gauge change, m but unfortunately it is defined on V g ⊗ C1(C), Dm(:) = d(:) + (−1) δ(:) whereas one want to work on globally defined ob- jects, that is, with C1(X) instead of C1(C). C Eventually we have the following cochain complex: is defined defined by the constraints C = φ−1(0) Dm−1 Dm Dm+1 which corresponds to the gauge fixing conditions. ··· / Pm / Pm+1 / ··· Despite these constraints, we'd like to work on globally defined objects. which computes the classical BRST cohomology. n Suppose that: It can be proved ([2]) that HD = 0 if n < 0 and Hn = Hn (C) if n ≥ 0. 1 1 D V C (C) = C (X)=I That was exactly the purpose of the BRST pro- cedure : this cohomology allows to work on glob- where I is some ideal in C1(X). By the very def- ally defined objects without loosing the profound inition of C, 8ξ 2 g, δx 2 I. Suppose that I is sense of gauge. generated by the δ(ξ), that is: In physics literature, elements of V g∗ are called V I = C1(X)δ(g) ghosts, those of g are called antighosts, while m = p − q is the ghost number. Now one can extend δ as a linear anti-derivation Now let's recall Fadeev-Popov gauge fixing for V 1 on g ⊗ C (X) by letting δ(ξ ⊗ 1) = 1 ⊗ δ(ξ) and the bosonic string action, and try to sketch the δ(1 ⊗ f) = 0. We thus have a complex: links between the geometrical approach and the path-integral approach, and in particular the na- k ^ ! g ⊗ C1(X) !···! g ⊗ C1(X) !C1(X) ture and role of ghost fields. with differential δ. It is called the Koszul complex of V g ⊗ C1(X). The Koszul construction aim to 2 BRST string quantization algebraically represent the quotient: In the framework of field theory (especially for C1(C) = C1(X)=I string theory), the symplectic manifold is a space of fields and is not finite-dimensional anymore. Its k-th homology space is indeed zero for k > 0, Yet, we won't bother with such considerations and and: will make direct links, even if it implies some loss δ ^ 1 1 H0 ( g ⊗ C (X)) = C (X)=I of mathematical rigor. 2 2.1 Fadeev-Popov gauge fixing the fiducial metric to g. This step corresponds to the gauge fixing, or, to stick to the notations of Let (M; g) be the two-dimensional world-sheet the previous section, to restrict oneself from X to Riemannian manifold, and Xµ(σ1; σ2), µ = C. 0; ··· ; (d − 1), be d embeddings of M into ( d; η) R One can insert the former expression into Z: where ηµν is Minkowski's metric in d dimensions. Starting from Polyakov's action: Z [dXdgdζ] Z[^g] = ∆ (g)δ(g − g^ζ )e−S[X;g] Z V FP 1 2 p ab µ gauge S[X; g] = 0 d σ gg @aX @bXµ 4πα M One may then integrate over g, and shift the inte- one defines the following path-integral: gration variable X ! Xζ to get: Z [dX][dg] Z ζ Z = exp(−S[X; g]) [dζdX ] ζ −S[Xζ ;g^ζ ] Vgauge Z[^g] = ∆FP (^g )e Vgauge Here, the gauge group G is the group of By definition of the gauge group: reparametrizations of the world-sheet, as well as Weyl's transformations of the metric. G acts on S[Xζ ; g^ζ ] = S[X; g^] the space of metrics on M by taking the pull-back by any reparametrization of M and combining it and because: with a Weyl transformation of parameter !. Let Z 0 Z −1 0 ζ 2 G, then: [dζ0]δ(gζ − g^ζ ) = [dζ0]δ(g − g^ζ ζ ) c d 0 ζ 0 2!(σ) @σ @σ (ζ · g) (σ ) = g (σ ) = e g (σ) ζ αβ ab @σa @σb cd one also has: ∆FP (^g ) = ∆FP (^g) for every ζ. Thus, one can integrate over the fields ζ. In the Z path integral, one has to divide by This integration is trivial and only gives the vol- the gauge group's volume, in order to take into ume of the gauge group, because one had "three" account the fact that the space of fields is redun- gauge functions, namely the reparametrization of dant, in the sense that two different configurations the world-sheet by changing the two coordinates of fields on M can correspond to the same physical on M and the local rescaling of the metric, and the system. This is the sense of the gauge group's ac- gauge-fixing functions fix the three coordinates of tion on this space of fields: all orbits correspond to the metric on M. Unrigorously, it shows that the the same physical system. We can use the gauge measure of the remaining subgroup of G which freedom to fix some fields, for example, the metric, leaves the metric invariant is a measure zero sub- to a canonical form. Unfortunately, there may re- group. main some freedom, that is some non trivial gauge Eventually, ons is led to: transformation which leave the metric invariant, thus one has to take the quotient of the subman- Z ifold of the space of fields defined by the gauge- Z[^g] = [dX]∆FP (^g) exp(−S[X; g^]) fixing conditions.