Week 161 1 Sigma models Let us consider a target space for type II string theory, with some metric, some B field turned on, and the dilaton turned on. We want the mani- fold to have large radius of curvature, so that we can make a low energy approximation that results into a supergravity theory. How does the string couple to all of these spacetime fields? We need that the string action have the following properties 1. The worldsheet action must be supersymmetric. 2. We would expect the action to couple linearly to the spacetime fields, because we are expanding around weakly coupled flat space. 3. Consistency of worldsheet gravity requires that the theory be confor- maly invariant in order to satisfy the Virasoro constraints. Thus, in the large radius of curvature limit we expect that the theory is close to flat 10D space. Putting all of this ingredients together, we get what is called a non-linear σ-model. The action will look as follows, in order to get a classically conformally (scale) invariant theory Z 1p Z 1 Z p d2σ −ηηαβg @ Xµ@ Xν + B dXµ^dXν + −ηR Φ+ fermions 2 µν α β 2 µν η (1) The form of the action given above contains the coupling of the worldsheet metric η to the fields. R is the Ricci curvature tensor associated to η. As written, the action is classically scale invariant (we can rescale η by a constant factor and the action does not change). However, we need more than classical scale invariance. We need the worldsheet theory to be conformally invariant at the quantum level. To understand wether a sigma model is conformally invariant or not, we need to check the β-functions of the couplings and require that they vanish. 1 c David Berenstein, 2006 1 The fields gµν(X), etc, encode such couplings. Indeed, if we think of gµν as a polynomial (power series) in X, each coefficient of g acts as a coupling constant. Similarly for B and Φ. Thus the β functions are functions of "functions", as we have an infinite number of coupling constants. Thus we usually call them β-functionals to indicate this. What we want to do now is to find a coordinate system where these expansions in power series simplify. Remember we have reparametrization invariance in target space for all couplings above. These are just field redefinitions. We also have to remember that B is a gauge field, so we can choose the best gauge as well. These special coordinates are called Gaussian normal coordinates. i They satisfy, around a given point in the target space, that Γjkj0 = 0, so that ρ σ gµν = δµν + aRµρνσ(g)j0X X + ::: (2) with a a universal coefficient that depends on conventions of signs and nor- malizations for curvatures. Indeed, in these coordinates, only R and it's covariant derivatives appear in the expansion to the right. Similarly, Bµν has a gauge invariance B ! B + dΛ, so we can make B vanish at a point, and it's first derivative can be made proportional to dB = H. ρ Bµν ∼ HµνρX + @HXX + ::: (3) i generically, we would expect covariant derivatives, but since Γjkj0 = 0, the covariant derivatives are ordinary ones to first order. Also, Φ can be expanded up to second order and further. The recipe for perturbation theory is that we want to expand around a background metric γ such that Rγ = 0, and we keep all quadratic terms in the action to define propagators. The β functionals determine the quantum and classical violations of con- formal invariance. The beta functionals just pick the logarithmic divergences of loop integrals. So we need to calculate those. First, let us analyze the case where Φ is constant. Then, for H, we get one from the contractions of X in @HXX. This is proportional to µ @µHρσ = 0 (4) 2 These are the equations of motion of the field B in the supergravity limit. Similarly, for g, we get a logarithmic divergence from the contraction of the two XX term in G, to second order. This is one loop in the field theory, and it is proportional to β(gµν) ∼ Rµν(g) (5) We also get a log divergence from a contraction of HH, this gives an integral proportional to Z 1 H(z)H(z0) @ X@¯ Xd2z0 (6) jz − z0j2 z z (There is no fermionic contribution). Indeed, the worlsheet fermions act to cancel the quadratic divergences in contracting @X@X¯ . These are cancellations due to worldsheet supersymme- try. To get these to cancel, we need to use the superfield formalism, so that the auxiliary fields also contribute. These type of terms are proportional to δ2(0), and only happen for coinciding operators. The total β function of g contains all contributions to this logarithmic divergence that are symmetric in the µν indices. We get this way that 2 Rµν + Hµν = 0 (7) and these are Einstein's equations for the background field. They get cor- rected at higher (worldsheet QFT) loop orders. These corrections depend on a dimensionful parameter. In our case, this is exactly α0, which acts as the ~ parameter in the string action. These corrections are called α0 corrections, and they are part of the tree level amplitudes in string theory. No higher genus worldsheet is required. The equations of motion obtained from the β functions set to zero are compatible with a Lagrangian in spacetime. The Lagrangian (up to coeffi- cients related to normalizations) is Z p p µνρ −gR + −gHµνρH (8) In the bosonic string case, the quadratic divergences are not cancelled automatically. Instead, they have to be cancelled by an off-shell tachyon contribution, which is just a potential of X generated as a power series in ~. 3 If we consider a case of varying Φ, there are also classical violations of con- formal invariance, proportional to r2Φ. These have to be cancelled against the quantum variations of the action. Cancelling different orders in perturbation theory against each other to guarantee a local symmetry at the quantum level is called the Green-Schwarz mechanism of anomaly cancellation. Thus the spacetime equations of motion of the dilaton are of the form 2 X r Φ + β(ci)ciOi = Tzz¯ = 0 (9) where ci is the coupling constant associated to Oi, and the term ciOi is the contribution to Tzz¯ from the operator Oi. 2 Exponentials of vertex operators Why does the above relations between beta functions and spacetime equa- tions of motion work? This is what we will try to answer here. To understand how this works, we need to be able to turn on a non-trivial background in a field theory. How do we get a classical background field in a quantum theory? We need to turn on a coherent state that satisfies the equations of motion, order by order in perturbation theory. A coherent state (for a free field) is of the form X y exp( smam)j0i (10) for a general system made of harmonic oscillators m. The coefficients sm are usually the Fourier transforms of the wave packets. In any case, to turn on a wave packet in the string formalism, we need to use a vertex operator. The on-shell conditions mean that the vertex op- erator satisfies the linearized equations of motion for perturbations of the background. After all, we found that massless strings correspond to a spin 2 massless particle, etc. If we want to compute scattering with respect to such a wave form, we need to insert the state in the path integral, and write correlations with respect to the state X y exp( smam)j0i (11) 4 so that we end up computing X ∗ Y X y h0j exp( smam) Oi exp( smam)j0i (12) In our case, we would insert a vertex operator with some wave form y dependence to get the wave packet associated to smam. Namely, we get P y R 2 smam ∼ d zV (X), where V has only positive frequency. Notice that by using a sufficiently complicated amplitude we can always use the background fields as integrated vertex operators: this is actually what we want. The V with negative frequency is in the am lowering operators. Putting both together, we get that we need to compute correlations with the string action modified to first order by Z 2 S ∼ S0 + d zV + ::: (13) where V is a solution of the linearized equation of motion, and the terms with ::: include the non-linearities from interactions of V with itself. Thus, string coherent states result in modifications of the string action as one would expect from a sigma model calculation in the new background. The consistency of the deformation by physical states is what gives the equation of motion of the background. The formalism of coherent states will then force us to compute exactly the same type of problems associated to σ- models, if we only turn on the massless fields of the string. For type II strings, this is just the corresponding type II supergravity. For the heterotic string, this also includes Vertex operators associated to the massless vector particles. Thus the σ-model expressions correspond to special backgrounds of string theory where only the supergravity modes have been turned on. Similarly, in the presence of D-branes, we need to consider boundary coherent state in- sertions. These will in general correspond to non-trivial gauge configurations for the YM fields on the D-brane. 3 Anomaly cancellation and the Green-Schwarz mechanism We have been discussing how to get the full spacetime action of the super- string theory.