
Week 7 Reading material from the books Polchinski, Chapter 3,4 • Becker, Becker, Schwartz, Chapter 3 • Green, Schwartz, Witten, chapter 3 • 1 Polyakov path integral and BRST coho- mology We need to discuss now the details of the Polyakov path integral. The main idea is to do a “gauge fixed” version of the path integral over Riemann surfaces that we have discussed so far. The main problem is to try to understand the precise inner workings of this path integral from a global point of view, because we have an “infinite symmetry” to contend with. Locally, as we have discussed so far, we can always bring the string met- ric to lightsheet coordinates (for the Lorentzia metric), or equivalently, to complex analytic coordinates where only gzz¯ = 0. However, this is not global enough for our purposes. 6 To understand a little better how things behave globally, we need to step back a little bit and discuss the symmetries of the action that we want to consider. We have Weyl invariance, and diffeomorphism invariance. Weyl invari- annce is easy, because all we need to do is change the value of the metric keeping the coordinate points fixed: g(σ) exp(2w(σ))g(σ), or in a coor- dinate independent formulation g exp(→w)g (1 + w)g for w a global (bounded function) on the manifold.→ ∼ The one that is more tricky is diffeomorphism invariance. How do we express the statement that we are allowed to make general coordinate trans- formations in a coordinate independent way? Well, we want to think about it for infinitesimal coordinate transforma- tions. This in general moves points just a little bit. We can describe this by an infinitesimal global vector field on the Riemann surface V . In this language, 1 any coordinate function σa varies locally as follows σa σa + V (σa)= σa + V a (1) → a where V = V ∂a. Remember that vector fields are derivations: they take functions into functions, and at least locally, one can describe them in terms of a local basis for derivations ∂a. This is the content expressed above. Similarly, we can see that g will be affected by these infinitesimal changes of coordinates in the following form gab gab + aVb + bVa (2) → ∇ ∇ This involves lowering the indices of the vector field, so it depends explicitly on V . Also, derivatives are covariantized. This can be proved by explicitly noticing that what stays invariant is a b a′ b′ gabdσ dσ = ga′b′ dσ dσ (3) This final result is not immediately obvious. We use the fact that there is a local coordinate system where g δab and the local Christoffel symbols vanish Γ = 0 (this just states that ∂g∼= 0 to first order). For this coordinate system we get that the metric varies as ∂bVa + ∂aVb, just from the general infinitesimal coordinate transformation. Going to a general coordinate frame just covariantizes all the derivatives, and that way we get a general result. Also notice that we have lowered the indices of V with respect to the metric g, so the variation of the metric depends very explicitly on the metric. We find it convenient to absorb part of the action of V as a Weyl trans- formation, so that the variation of the metric is traceless: c gab gab + aVb + bVa gab cV Diff’ (4) → ∇ ∇ − ∇ gab gab(1 + w) Weyl (5) → This is just a ”linear change of variables” in the V,w plane, and it’s functional determinant is one (w w + V ). → This is most easily visualized in a local complex coordinate system, where gzz =0,gz¯z¯ = 0 and gzz¯ 1. The total variations are ∼ z¯ δgzz ∂V (6) ∼ δgz¯z¯ ∂V¯ z (7) ∼ z z¯ δgzz¯ wgzz¯ + ∂V + ∂V¯ (8) ∼ 2 but we can make a linear change of variables in w to absorb ∂V + ∂¯V¯ The process of gauge fixing the path integral over Riemann surfaces, is that we choose a complex coordinate system on the Riemann surface, so that gzz = gz¯z¯ = 0; while we have gzz¯ = 0. Locally, we can choose gzz¯ = Const, but this is not allowed globally. 6 However, in two dimensions, we have Riemann’s uniformization theorem that states that we can always choose gzz¯ to have constant curvature. For different genus, we get different answers. For g = 0, the metric is a round sphere. For g = 1, the metric is flat. For g > 1 the metric has constant negative curvature. This means that for g > 1 all of the metrics result from isometries to the upper half plane with the Poincare metric and with identifications by the group of isometries of the upper half plane SL(2, R). This is the covering space of the Riemann surface. Selecting a metric amounts to selecting a finite subgroup of SL(2, R). This is described by some finite number of generators and hence, a finite number of functions mapping to SL(2, R). This means that the problem of classifying all Riemann surfaces up to Weyl invariance results into a finite dimensional space of equivalence classes (the so-called moduli space of metrics). This is fortunate for us. This means that up to a finite number of integrals, we are left over with a gauge fixing that eliminates all degrees of freedom from the metric. The metric with constant curvature is important for us. We will also normalize it so that the volume of the metric is one. In complex coordinates, making V traceless makes our life easier. That ∇ means that gzz¯ does not vary under V transformations and the variations factorize neatly. gzz gzz + zVz (9) → ∇ gz¯z¯ gz¯z¯ + z¯Vz¯ (10) → ∇ gzz¯ gzz¯(1 + w) (11) → Our choice of gauge fixing is gzz =0= gz¯z¯, and gzz¯ =g ˆzz¯ where in the last case we are using a constant curvature metric with unit volume. So now we want to do a path integral with insertions of delta functions of this gauge condition, but we need to ensure that the result is gauge invariant. This is where the Faddeev-Popov determinant comes into place. 3 This relies on the observation that for finite integrals (or for finite group invariant integrals) one has the following identity: ′ dθδ(f(θ))=1/f (θ) f=0 (12) Z | This is proved by changing variables from θ to f, and using dyδ(y)=1. Or more generally R k k i i d θδ (f (θ))=1/ det(∂jf (θ)) f=0 (13) Z | where dkθ is a Haar-metric for the group, that near the identity is just dθi. The determinant is just the Jacobian from changing of variables fromQθ to the f i. The determinant found there is the Faddeev-Popov determinant. Clearly we have the identity i k k i det(∂jf (θ)) f=0 d θδ (f (θ))=1 (14) | Z Now, in the above, we can take path integral limit, and consider this as an identity for a path integral over the “gauge orbit” with a gauge fixing condition f = 0. What we then do, is to write the above identity in the full Polyakov path integral: we insert one. The functional integral over the dθ is just the volume of the gauge orbit (group action). So we have [g] X I δf D D θ δ(fθ) det exp( S) (15) Z Diff Weyl Z D δθ − × f=0 where f are the complete set of gauge fixing functions. Now, we integrate over [g], which is where we set our gauge conditions. Modulo a finite number of parametersD associated to global choices, which we will discuss in more details later, the local integral gives us det( z) det( z¯) (16) ∇ ∇ from the variations with respect to V . For w, we get the functional deter- minant of g over the Riemann surface. This is ultralocal (does not depend 4 on derivatives), and can be renormalized away or absorbed in the volume of the surface (which we have chosen fixed and proportional to the curvature). The cancellation of the variation of the action with respect to w, once the matter is included, also leads to the condition ctotal = 0. The other two pieces involve functional determinants of nice differential operators on the worldsheet. This determinant can be written as 2 z¯z¯ b c exp d zb z¯cz¯ (17) Z D D Z ∇ z¯z¯ z and if we lower and raise indices b bzz, cz¯ c , while the z¯ becomes ∂z. ∼ ∼ ∇ z A similar result holds for the other variation. Note here that bzz and c transform as 2 tensors and ( 1) tensors respectively, and that the action is that of a CFT of b, c type. − What we find, by matching conformal weights to the b-c theory, that cghost = 26, with a stress tensor given by − Tbc A : b∂c :+B : c∂b : (18) ∼ We also need the contractions that arise from the Greens’ function of ∂¯ and it’s give by 1 b(z)c(w)= (19) z w − Matching the weight of c gives us B = 1, and getting the full OPE for the stress tensor with c right gives us A + B− = 1. Again, this is a free field theory on the string worldsheet, and we can quantize it in a straightforward way. If we take mode oscillators on the cylinder worldsheet, we have zero modes and non-zero modes. Quantizing the non-zero modes is easy. We define a ground state by b m 0 = c m 0 = 0 for m > 0. However, we get an ambiguity at m = 0, − | i − | i where b0,c0 =1 (20) { } and we have two states 0 and 0 .
Details
-
File Typepdf
-
Upload Time-
-
Content LanguagesEnglish
-
Upload UserAnonymous/Not logged-in
-
File Pages11 Page
-
File Size-