A Geometric Approach to BRST Quantization of the Bosonic String

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 vector 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 ωi of the cotan- that is, a function satisfying that ∀x ∈ X and i i gent space to the fibers, that is, ω (Ξj) = δj, and ∀ξ ∈ g: define: d(φ(x)(ξ)) = ι Ω d : C∞(C) → Ω1 (C) ξ˜x V V ˜ by: where ξ is the fundamental vector field generated X d (f) = (L f)ωi 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 ∈ g and x ∈ X. Provided 0 ∈ 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 ω = − fjkω ∧ ω ⊥ 2 manifold. Moreover, ∀x ∈ 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 ⊗ C∞(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 ωi(x , x ) = − f i (δj δk − δ δk) Vp,q ∞ δ Vp,q−1 ∞ V a b 2 jk a b b a g ⊗ C (X) / g ⊗ C (X) j,k dV i i dV ω (xa, xb) = −f ab Vp+1,q g ⊗ C∞(X) i 1+2 i dV ω (xa, xb) = (−1) ω ([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 cohomology groups of the Chevalley-Eilenberg com- p q M ^ ∗ ^ ∞ 0 V ∞ Pm = g ⊗ g ⊗ C (X) plex. For example, HV ( g ⊗ C (C)) is the subspace of C∞(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 ⊗ C∞(C), Dm(.) = d(.) + (−1) δ(.) whereas one want to work on globally defined objects, that is, with C∞(X) instead of C∞(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. ∞ ∞ D V C (C) = C (X)/I That was exactly the purpose of the BRST procedure : this cohomology allows to work on glob- where I is some ideal in C∞(X). By the very def- ally defined objects without loosing the profound inition of C, ∀ξ ∈ g, δx ∈ 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 = C∞(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 ∞ 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 ⊗ C∞(X) → · · · → g ⊗ C∞(X) → C∞(X) ture and role of ghost fields. with differential δ. It is called the Koszul complex of V g ⊗ C∞(X). The Koszul construction aim to 2 BRST string quantization algebraically represent the quotient: In the framework of field theory (especially for C∞(C) = C∞(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 ^ ∞ ∞ 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[ˆg] = ∆ (g)δ(g − gˆζ )e−S[X,g] Z V FP 1 2 √ 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 ,gˆζ ] Vgauge Z[ˆg] = ∆FP (ˆg )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 , gˆζ ] = S[X, gˆ] 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 ω. Let Z 0 Z −1 0 ζ ∈ G, then: [dζ0]δ(gζ − gˆζ ) = [dζ0]δ(g − gˆζ ζ ) c d 0 ζ 0 2ω(σ) ∂σ ∂σ (ζ · g) (σ ) = g (σ ) = e g (σ) ζ αβ ab ∂σa ∂σb cd one also has: ∆FP (ˆg ) = ∆FP (ˆg) 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[ˆg] = [dX]∆FP (ˆg) exp(−S[X, gˆ]) fixing conditions. To put it in a nutshell, in order to have a well-defined path-integral, one has which is a better-defined path-integral, because to take a symplectic reduction of the symplectic the gauge has already been fixed. Now, the last manifold of all fields on M. Such a space is in thing to do is to obtain the expression of ∆FP (ˆg). a one-to-one correspondance with the space of all One could say that this function is an induced physically-distinguishable configurations. measure on the submanifold C of X. Because of −1 For g some metric on M, one writes: the expression of ∆FP (ˆg) , one is going to con- Z sider a path-integral on all gauge transformation ζ 1 = ∆FP (g) [dζ]δ(g − gˆ ) which leave the metric invariant. It’s equivalent to look at all Killing vectors, that is, fundamental whereg ˆ is the trivial, or fiducial, metricg âb = δab. vector fields induced by vectors in the Lie algebra The delta function ensures that the only transfor- g of G or infinitesimal transformations, under the mations which will be kept are the ζ which bring flow of whose the form metric does not change.

3 One writes: and is related to the Lie algebra cohomology, and the quotient of the space of ﬁelds by the gauge δgab = 2δωgab − ∇aδσb − ∇bδσa group action, while

−1 we could also write ∆FP (ˆg) under the form: δBBab = 0 Z δBbab = Bab [dδωdδσ]δ(−(2δω − ∇ˆ · δσ)ˆgab + 2Pˆ1δσ) are related to the Koszul complex, and thus to where: the restriction to the submanifold defined by the constraints on the form of the metric. 1 (Pˆ δσ) = (∇ δσ + ∇ δσ − gˆ ∇ δσc) To conclude our geometric insight of BRST 1 ab 2 a b b a ab c symmetry in the bosonic string theory, let’s make After expressing the delta functional as an expo- a two remarks. 0 0 nential, and integrating over δω, one is left with: First, since HB = HV (C), the initial action Z Z functional S, which is gauge invariant and thus in 0 2 p 0ab 0 [dβ dδσ]exp{4πi d σ (ˆg)β (Pˆ1δσ)ab} HV (C), is identified in this cohomology space with all functional which differ only by exact terms. For −1 example, adding the S2 + S3 term doesn’t change for the expression of ∆FP (ˆg) , where the β’s are traceless symmetric tensors. the equivalence class one is looking at, since: This determinant can be inverted be replacing −1 ab S2 + S3 = i δB(babF (g)) all bosonic fields by a corresponding ghost field: a a 0 δσ → c and βab → bab. One is left with the The second important idea we’d like to put in ex- gauge-fixed path integral : ergue is the notion of physical state. These are Z special states, those which are meaningful from an Z = [dXdcdbdB]exp{−S − Sgh − SG.F } experimental point of view. In particular, different choices of gauges should not affect the scattering Ghost fields c and b have a very natural interpre- amplitudes of the theory. In the quantized theory, tation in the geometrical framework: on the one fields are operators acting on the states of the the- hand, c-fields (or ghosts) are related to a global ory. The BRST symmetry is a transformation of basis of the dual of the Lie algebra g. If one the operators, but since we can express it in terms BRST-quantizes an abstract theory, at the end of an anti-commutator with a BRST operator QB, there should be one c-field for each vector of a which acts on the states as well. This operator basis of g, that is, one c-field for each infinites- QB has the same properties as the BRST differ- imal transformation of the gauge group G. On ential δB, and in particular, is nilpotent of order 2 the other, b-fields (or anti-ghosts) are related to and induces a cochain complex of Hilbert spaces, the gauge-fixing conditions, there is one b-field for where the k-th cochain space is the set of all states each constraint relation. of ghost number k. As done in Polchinski’s book [3], it can be shown that physical states are closed 2.2 Bosonic string BRST symmetry in the sense that |Ψi is physical if:

So one can class the BRST transformations of the QB |ψi = 0 bosonic string theory into two distinct subgroups: On the other hand, states which are exact for Q the ﬁrst one contains the transformations: B are called null states. Null states are orthogonal to every physical state. If one wants to avoid redun- δBhab = −i(2cωhab − ∇acb − ∇bca) dancies, one can quotient the closed states by the µ a µ δBX = −(c ∂aX ) exact states, and thus the spectrum of the theory a b a is obtained as the direct sum of the cohomology δBc = −ic ∂bc groups of the QB cohomology on the states. The ω b ω δBc = −ic ∂bc spectrum obtained this way is in fact equivalent to

4 the spectra one has after quantizing the classical theory by means of the Old Covariant Quantiza- tion, or Light-cone Quantization [3], but BRST quantization corresponds to this great philosophi- cal idea of directly relating the BRST cohomology as seen as symplectic reduction of the phase space, and BRST cohomology as seen as the algebraic way of understanding physical states.

References

[1] C. Becci, A. Rouet, and R. Stora. The abelian higgs kibble model, unitarity of the s-operator. Physics Letters B, 52(3):344–346, 1974.

[2] T. Kimura and J.M. Figueroa-O’Farrill. Ge- ometric brst quantization. i. prequantization. Comm. Math. Phys., 136(2):209–229, 1991.

[3] Joseph Polchinski. String Theory, volume 1 of Cambridge Monographs on Mathematical Physics. Cambridge University Press, second edition, 1998.

[4] I.V. Tyutin. Gauge invariance in ﬁeld theory and statistical physics in operator formal- ism. Preprint of P.N. Lebedev Physical Insti- tute, 39, 1975.

[5] F. Wagemann. Introduction to lie algebra cohomology with a view towards brst cohomology. 2010.