Low energy supergravity

Superstring interactions

In this chapter we will examine superstring interactions from two com-plementary points of view. First we study the interactions of the massless degrees of freedom, which are highly constrained by supersymmetry. The first section discusses the tree-level interactions, while the second discusses an important one-loop effect: the anomalies in local spacetime symme-tries. We then develop superstring perturbation theory. We introduce superfields and super-Riemann surfaces to give superconformal symme-try a geometric interpretation, and calculate a variety of tree-level and one-loop amplitudes.

12.1 Low energy supergravity

The ten-dimensional supersymmetric string theories all have 32 or 16 supersymmetry generators. This high degree of supersymmetry completely determines the low energy action.

Type IIA superstring

We begin by discussing the field theory that has the largest possible space-time supersymmetry and Poincar´e invariance, namely eleven-dimensional supergravity. As explained in the appendix, the upper limit on the di-mension arises because nontrivial consistent field theories cannot have massless particles with spins greater than two.

This theory would seem to have no direct connection to superstring theory, which requires ten dimensions. Our immediate interest in it is that, as discussed in section B.5, its supersymmetry algebra is the same as that of the IIA theory. The action of the latter can therefore be obtained by dimensional reduction, toroidal compactification keeping only fields that are independent of the compact directions. For now this is just a trick to

84

12.1 Low energy supergravity 85 take advantage of the high degree of supersymmetry, but in chapter 14 we will see that there is much more going on.

The eleven-dimensional supergravity theory has two bosonic fields, the metric GMN and a 3-form potential AMNP ≡ A3 with field strength F4. Higher-dimensional supergravities contain many different p-form fields; to distinguish these from one another we will denote the rank by an italicized subscript, as opposed to numerical tensor indices which are written in roman font. In terms of the S O(9) spin of massless states, the metric gives a traceless symmetric tensor with 44 states, and the 3-form gives a rank 3 antisymmetric tensor with 84 states. The total number of bosonic states is then 128, equal to the dimension of the S O(9) vector-spinor gravitino.

The bosonic part of the action is given by 2κ²₁₁S₁₁= The form action, written out fully, is proportional to

The p! cancels the sum over permutations of the indices, so that each independent component appears with coefficient 1. Forms are written as tensors with lower indices in order that their gauge transformations do not involve the metric.

We will take such results from the literature without derivation. Our interest is only in certain general features of the various actions, and we will not write out the full fermionic terms or supersymmetry transformations.

For the supergravities arising from string theories, one can verify the action by comparison with the low energy limits of string amplitudes; a few such calculations are given later in the chapter and in the exercises.

Also, many important features, such as the coupling of the dilaton, will be understood from general reasoning.

Now dimensionally reduce as in section 8.1. The general metric that is invariant under translations in the 10-direction is

ds² = G¹¹_MN(x^µ)dx^Mdx^N

= G¹⁰_µν(x^µ)dx^µdx^ν + exp(2σ(x^µ))[dx¹⁰+ Aν(x^µ)dx^ν]² . (12.1.3) Here M, N run from 0 to 10 and µ, ν from 0 to 9. We have added a su-perscript 11 to the metric appearing in the earlier supergravity action and introduced a new ten-dimensional metric G¹⁰_µν = G¹¹µν. The ten-dimensional metric will appear henceforth, so the superscript 10 will be omitted.

The eleven-dimensional metric (12.1.3) reduces to a ten-dimensional metric, a gauge field A1, and a scalar σ. The potential A3 reduces to two

potentials A3 and A2, the latter coming from components where one index is along the compact 10-direction. The three terms (12.1.1) become

S1 = 1 We have compactified the theory on a circle of coordinate period 2πR and defined κ²₁₀ = κ²₁₁/2πR. The normalization of the kinetic terms is canonical for 2κ²₁₀= 1.

In the action (12.1.4) we have defined

F4 = dA3 − A1 ∧ F3 , (12.1.5) the second term arising from the components G^{µ 10} in the 4-form ac-tion (12.1.2). We will use Fp+1 = dAp to denote the simple exterior deriva-tive of a potential, while field strengths with added terms are distinguished by a tilde as in eq. (12.1.5). Note that the action contains several terms where p-form potentials appear, rather than their exterior derivatives, but which are still gauge invariant. These are known as Chern–Simons terms, and we see that they are of two types. One involves the wedge product of one potential with any number of field strengths, and it is gauge invariant as a consequence of the Bianchi identities for the field strengths. The other appears in the kinetic term for the modified field strength (12.1.5). The second term inF^4 has a gauge variation

− dλ0 ∧ F3 =−d(λ0 ∧ F3). (12.1.6) It is canceled by a transformation

δ^A3 = λ0 ∧ F3 , (12.1.7) which is in addition to the usual δA3 = dλ2. In the present case, the Kaluza–Klein gauge transformation λ0 originates from reparameterization of x¹⁰, and the transformation (12.1.7) is simply part of the eleven-dimensional tensor transformation. Since the combinationF^4 is invariant under both λ0 and λ2 transformations we should regard it as the physical field strength, but with a nonstandard Bianchi identity

dF^4 =−F2 ∧ F3 . (12.1.8) Poincar´e duality of the form theory, developed in section B.4 for forms without Chern–Simons terms, interchanges these two kinds of Chern–

Simons term.

12.1 Low energy supergravity 87 The fields of the reduced theory are the same as the bosonic fields of the IIA string, as they must be. In particular the scalar σ must be the dilaton Φ, up to some field redefinition. The terms in the action have a variety of σ-dependences. Recall that the string coupling constant is determined by the value of the dilaton. As discussed in section 3.7, this means that after appropriate field redefinitions the tree-level spacetime action is multiplied by an overall factor e^−2Φ, and otherwise depends on Φ only through its derivatives. ‘Appropriate redefinitions’ means that the fields are the same as those appearing in the string world-sheet sigma model action.

Since we have arrived at the action (12.1.4) without reference to string theory, we have no idea as yet how these fields are related to those in the world-sheet action. We will proceed by guesswork, and then explain the result in world-sheet terms. First redefine

G_µν = e^−σG_µν(new), σ = 2Φ

3 . (12.1.9)

The original metric will no longer appear, so to avoid cluttering the equations we do not put a prime on the new metric. Then

SIIA = SNS+SR+SCS , (12.1.10a)

We have regrouped terms according to whether the fields are in the NS–NS or R–R sector of the string theory; the Chern–Simons action contains both. It will be useful to distinguish R–R from NS–NS forms, so for the R–R fields we henceforth use Cp and Fp+1 for the potential and field strength, and for the NS–NS fields B2 and H3. Also, we will use A1

and F2 for the open string and heterotic gauge fields, and B2 and H3 for the heterotic antisymmetric tensor.

The NS action now involves the dilaton in standard form. Eq. (12.1.9) is the unique redefinition that does this. The R action does not have the expected factor of e^−2Φ, but can be brought to this form by the further redefinition

C₁ = e^−ΦC₁^ , (12.1.11)

d¹⁰x (−G)^1/2|F²|² =

d¹⁰x (−G)^1/2e^−2Φ|F2^|² , (12.1.12a) F₂^ ≡ dC1^ − dΦ ∧ C1^ , (12.1.12b) and similarly for F and C . The action (12.1.12) makes explicit the dilaton dependence of the loop expansion, but at the cost of complicating the Bianchi identity and gauge transformation,

dF₂^ = dΦ∧ F2^ , δC₁^ = dλ^₀ − λ^0dΦ . (12.1.13) For this reason the form (12.1.10) is usually used. For example, in a time-dependent dilaton field, it is the charge to which the unprimed fields couple that will be conserved.

Let us now make contact with string theory and see why the background R–R fields appearing in the world-sheet action have the more complicated properties (12.1.13). We work at the linearized level, in terms of the vertex operators

VαV˜β(CΓ^µ1...µp)_αβeµ₁...µp(X) . (12.1.14) Here Vα is the R ground state vertex operator (10.4.25) and Γ^µ1...µp = Γ^[µ1. . . Γ^µp]. The nontrivial physical state conditions are from G0 ∼ pµψ^µ₀ and ˜G0 ∼ pµψ˜₀^µ, and amount to two Dirac equations, one acting on the left spinor index and one on the right:

Γ^νΓ^µ1...µp∂νeµ1...µp(X) = Γ^µ1...µpΓ^ν∂νeµ1...µp(X) = 0 . (12.1.15) By antisymmetrizing all p + 1 gamma matrices and keeping anticommu-tators one obtains

Γ^νΓ^µ1...µp = Γ^νµ1...µp+ pη^ν[µ1Γ^µ2...µp] , (12.1.16a) Γ^µ1...µpΓ^ν = (−1)^pΓ^νµ1...µp + (−1)^p+1pη^ν[µ1Γ^µ2...µp] . (12.1.16b) The Dirac equations (12.1.15) are then equivalent to

de_p= d∗ep= 0 . (12.1.17) These are first order equations, unlike the second order equations encoun-tered previously for bosonic fields. In fact, they have the same form as the field equation and Bianchi identity for a p-form field strength. Thus we identify the function eµ1...µp(X) appearing in the vertex operator as the R–R field strength rather than potential. To confirm this, observe that in the IIA theory the spinors in the R–R vertex operator (12.1.14) have opposite chirality and so their product in table 10.1 contains forms of even rank, the same as the IIA R–R field strengths.

This has one consequence that will be important later on. Amplitudes for R–R forms will always contain a power of the momentum and so

4 3

after which

12.1 Low energy supergravity 89 vanish at zero momentum. The zero-momentum coupling of a gauge field measures the charge, so this means that strings are neutral under all R–R gauge fields.

The derivation of the field equations (12.1.17) was for a flat background.

Now let us consider the effect of a dilaton gradient. It is convenient that the linear dilaton background gives rise to the free CFT (10.1.22),

T_F = i(2/α^)^1/2ψ^µ∂X_µ− 2i(α^/2)^1/2Φ_,µ∂ψ^µ , (12.1.18a) G₀ ∼ (α^/2)^1/2ψ^µ₀(pµ+ iΦ,µ) , (12.1.18b) and similarly for ˜T_F and ˜G₀. The field equations are modified to

(d− dΦ∧)ep= (d− dΦ∧) ∗ ep = 0 . (12.1.19) Thus the Bianchi identity and field equation for the string background fields are modified in the fashion deduced from the action. There is no such modification for the NS–NS tensor. It couples to the world-sheet through its potential,

1 2πα^

M

B₂ . (12.1.20)

This is invariant under δB2 = dλ1 independent of the dilaton, and so H3 = dB2 is invariant and dH3 = 0.

Massive IIA supergravity

There is a generalization of the IIA supergravity theory which has no simple connection with eleven-dimensional supergravity but which plays a role in string theory. The IIA theory has a 2-form and a 4-form field strength, and by Poincar´e duality a 6-form and an 8-form as well,

F6 =∗F^4 , F^8 =∗F2 ; (12.1.21) again, a tilde denotes a field strength with a nonstandard Bianchi identity.

The pattern suggests we also consider a 10-form F10 = dC9. The free field equation would be

d∗F10 = 0 , (12.1.22)

and since ∗F10 is a scalar this means that

∗ F10 = constant . (12.1.23) Thus there are no propagating degrees of freedom. Nevertheless, such a field would have a physical effect, since it would carry energy density. This is closely analogous to an electric field F2 in two space-time dimensions, where there are no propagating photons but there is an energy density and a linear potential that confines charges.

Such a field can indeed be included in IIA supergravity. The action is Here ˜SIIAis the earlier IIA action (12.1.10) with the substitutions

F2 → F2 + MB2 , F4 → F4 + 1

2MB2 ∧ B2 , F^4 →F^4 + 1

2MB2 ∧ B2 . (12.1.25) The scalar M is an auxiliary field, meaning that it appears in the action without derivatives (and in this case only quadratically). Thus it can be integrated out, at the cost of introducing a rather nonlinear dependence on B2.

We will see in the next chapter that this massive supergravity does arise in the IIA string. To put the 9-form potential in perspective, observe that the maximum-rank potential that gives rise to a propagating field in ten dimensions is an 8-form, whose 9-form field strength is dual to a 1-form.

The latter is just the gradient of the R–R scalar field C0. A 10-form potential also fits in ten dimensions but does not give rise to propagating states. We saw in section 10.8 that this does exist in the type I string, so we should not be surprised that the 9-form will appear in string theory as well.

Type IIB superstring

For low energy IIB supergravity there is a problem due to the self-dual field strength F5=∗F5. As discussed in section B.4 there is no covariant action for such a field, but the following comes close:

SIIB = SNS+SR+SCS , (12.1.26a) The NS–NS action is the same as in IIA supergravity, while the R–R and Chern–Simons actions are closely parallel in form. The equation of

12.1 Low energy supergravity 91 motion and Bianchi identity for F^5 are

d∗F^5 = dF^5 = H3 ∧ F3 . (12.1.28) Recall that the spectrum of the IIB string includes the degrees of free-dom of a self-dual 5-form field strength. The field equations from the action (12.1.26) are consistent with

∗F^5 =F^5 (12.1.29)

but they do not imply it. This must be imposed as an added constraint on the solutions; it cannot be imposed on the action or else the wrong equations of motion result.

This formulation is satisfactory for a classical treatment but it is not simple to impose the constraint in the quantum theory. This will not be important for our purposes, and we leave further discussion to the references. Our main interest in this action is a certain S L(2, R) symmetry.

Let the Einstein metric (12.1.30a) being used everywhere. This is invariant under the following S L(2, R) symmetry:

τ^ = aτ + b invariance of the τ kinetic term is familiar, and that of the F3 kinetic term follows from

M^ = (Λ⁻¹)^TMΛ⁻¹ . (12.1.33) This S L(2, R) invariance is as claimed in the second line of table B.3. Any given value τ is invariant under an S O(2, R) subgroup so the moduli space is the coset S L(2, R)/S O(2, R). If we now compactify on tori, the moduli

and other fields fall into multiplets of the larger symmetries indicated in the table and the low energy action has the larger symmetry.

Observe that this S L(2, R) mixes the two 2-form potentials. We know that the NS–NS form couples to the string and the R–R form does not.

The S L(2, R) might thus seem to be an accidental symmetry of the low energy theory, not relevant to the full string theory. Indeed, this was assumed for some time, but now we know better. As we will explain in chapter 14, the discrete subgroup S L(2, Z) is an exact symmetry.

Type I superstring

To obtain the type I supergravity action requires three steps: set to zero the IIB fields C0, B2, and C4 that are removed by the Ω projection; add the gauge fields, with appropriate dilaton dependence for an open string field; and, modify the F3 field strength. This gives

SI =Sc+So , (12.1.34a) The open string S O(32) potential and field strength are written as matrix-valued forms A1 and F2, which are in the vector representation as indicated by the subscript on the trace. Here

F3 = dC2 −κ²₁₀

g₁₀² ω3 , (12.1.35) and ω3 is the Chern–Simons 3-form

ω3 = Tr_v

Again the modification of the field strength implies a modification of the gauge transformation. Under an ordinary gauge transformation δA1 = dλ− i[A1, λ], the Chern-Simons form transforms as

δω3 = dTrv(λdA1). (12.1.37) Thus it must be that

δC2 = κ²₁₀

g₁₀² Trv(λdA1) . (12.1.38) The 2-form transformation δC2 = dλ1 is unaffected.

The action appears to contain two parameters, κ10with units of L⁴and g10 with units of L³. We can think of κ10 as setting the scale because

12.1 Low energy supergravity 93 it is dimensionful, but there is one dimensionless combination κ₁₀g₁₀^−4/3. However, under an additive shift Φ→ Φ + ζ, the couplings change κ10→ e^ζκ₁₀ and g→ e^ζ/2g and so this ratio can be set to any value by a change of the background. Thus the low energy theory reflects the familiar string property that the coupling is not a fixed parameter but depends on the dilaton. The form of the action (12.1.34) is fixed by supersymmetry, but when we consider this as the low energy limit of string theory there is a relation between the closed string coupling κ10, the open string coupling g₁₀, and the type I α^. We will derive this in the next chapter, from a D-brane calculation, as we did for the corresponding relation in the bosonic string.

Heterotic strings

The heterotic strings have the same supersymmetry as the type I string and so we expect the same action. However, in the absence of open strings or R–R fields the dilaton dependence should be e^−2Φ throughout:

Shet= 1 2κ²₁₀

d¹⁰x (−G)^1/2e^−2Φ



R + 4∂_µΦ∂^µΦ− 1

2|H^3|²−κ²₁₀

g²₁₀Tr_v(|F2|²)

 . (12.1.39) Here

H3 = dB2 −κ²₁₀

g²₁₀ω3 , δB2 = κ²₁₀

g²₁₀Tr_v(λdA1) (12.1.40) are the same as in the type I string, with the form renamed to reflect the fact that it is from the NS sector.

Because of the high degree of supersymmetry, the type I and heterotic actions can differ only by a field redefinition. Indeed the reader can check that with the type I and heterotic fields related by

G_Iµν = e^−ΦhG_hµν , Φ_I=−Φh , (12.1.41a) FI3 =H^h3 , AI1 = Ah1 , (12.1.41b) the action (12.1.34) becomes the action (12.1.39). For the heterotic string, the relation among κ10, g10, and α^ will be obtained later in this chapter, by two different methods; it is, of course, different from the relation in the type I theory.

For E8× E8 there is no vector representation, but it is convenient to use a normalization that is uniform with S O(32). In place of Trv(t^at^b) in the action use ₃₀¹Tr_a(t^at^b). This has the property that for fields in any S O(16)× SO(16) subgroup it reduces to Trv(t^at^b).

12.2 Anomalies

It is an important phenomenon that some classical symmetries are anoma-lous, meaning that they are not preserved by quantization. We encountered this for the Weyl anomaly in chapter 3. We also saw there that if the left-and right-moving central charges were not equal there was an anomaly in two-dimensional coordinate invariance.

In general, anomalies in local symmetries make a theory inconsistent, as unphysical degrees of freedom no longer decouple. Anomalies in global symmetries are not harmful but imply that the symmetry is no longer exact.

Both kinds of anomalies play a role in the Standard Model. Potential local anomalies in gauge and coordinate invariance cancel among the quarks and leptons of each generation. Anomalies in global chiral symmetries of the strong interaction are important in accounting for the π⁰ decay rate and the η^ mass.

In this section we consider potential anomalies in the spacetime gauge and coordinate invariances in the various string theories. If the theories we have constructed are consistent these anomalies must be absent, and in fact they are. Although this can be understood in purely string theoretic terms it can also be understood from analysis of the low energy field theory, and it is useful to take both points of view.

We can analyze anomalies from the purely field theoretic point of view because of the odd property that they are both short-distance and long-distance effects. They are short-long-distance in the sense that they arise because the measure cannot be defined — the theory cannot be regulated — in an invariant way. They are long-distance in the sense that this impossibility follows entirely from the nature of the massless spectrum.

Let us illustrate this with another two-dimensional example, which is also of interest in its own right. Suppose we have left- and right-moving current algebras with the same algebra g, with the coefficients of the Schwinger terms being ˆkL,Rδ^ab. Couple a gauge field to the current,

Sint=

d²z (j_z^aA^a_¯z + j_¯z^aA^a_z) . (12.2.1) The OPE determines the jj expectation value, so to second order the path integral is

Z [A] = 1 2

d²z₁d²z₂

ˆkL

z²₁₂A^a_¯z(z1, ¯z₁)A^a_¯z(z2, ¯z₂) + ˆkR

¯

z²₁₂A^a_z(z1, ¯z₁)A^a_z(z2, ¯z₂)

 . (12.2.2) Now make a gauge transformation, which to leading order is δA^a₁ = dλ^a.

12.2 Anomalies 95

Let us run through the logic here. The path integral (12.2.2) is nonlocal, but its gauge variation is local. The latter is necessarily true because the variation can be thought of as arising from the regulator if we actually evaluate the path integral by brute force. Although the variation is local, it is not in general the variation of a local operator. When it is so, as is the case for ˆkL = ˆkR here, one can subtract that local operator from

Let us run through the logic here. The path integral (12.2.2) is nonlocal, but its gauge variation is local. The latter is necessarily true because the variation can be thought of as arising from the regulator if we actually evaluate the path integral by brute force. Although the variation is local, it is not in general the variation of a local operator. When it is so, as is the case for ˆkL = ˆkR here, one can subtract that local operator from

In document Polchinski J. String Theory. Vol. 2. Superstring Theory and Beyond (Page 106-125)