— DRAFT — IASSNS-HEP-97/72

Lectures 2, 3, 4

II. Spectrum of Free Bosonic Strings

Eric D’Hoker

We begin by constructing the Hilbert space of physical states and the spectrum of a single free bosonic closed oriented string, whose worldsheet has the topology of a cylinder (or equivalently an annulus) and of a single free open oriented string, whose worldsheet has the topology of half a cylinder (or equivalently half an annulus).

A free string should not couple to any other strings, or to any non-trivial background fields in the space-time manifold M. Thus we take M = ℝ^{D}, with constant G_{μν} and B_{μν} = Ψ = T = 0 to start. We wish to examine the properties of physical strings in physical space-time, so it is essential that the space-time metric be Minkowskian, with G_{μν} = η_{μν} = diag(− + + ⋯ +) and μ, ν = 0, 1, …, D − 1. In quantum field theory, the problem of constructing the Hilbert space of states and the spectrum of a single particle amounts to finding the 1-particle unitary representations of the Poincaré group — a problem for which the Minkowski nature of space-time is again essential.

A single string has an infinite number of excitation states, each of which corresponds to a representation of the Poincaré group. In this lecture, we wish to identify which representations of the Poincaré group occur (i.e. find the spectrum) and determine the conditions under which all these representations are unitary. We will find that conformal invariance plays an essential rôle.

Before we discuss the details of this fundamental analysis of string theory, let us outline the main steps of our strategy.

1) In the first lecture, we postulated a prescription for the transition amplitudes of N external string states — represented by vertex operators V_{i}, i = 1, …, N — in a general Riemannian space-time M with metric G (as well as possibly other fields B_{μν}, Φ, T)

A = ∑_{h=0}^{∞} ∫_{Met(Σ)} Dg \frac{1}{𝒩(g)} ∫_{Diff(Σ,M)} Dx V_{1} … V_{N} e^{−S[x,g]}  (2.1)

II.1 2) To investigate the spectrum of the string in Minkowski ℝ_{M}^{D} we shall start with the above amplitudes in Euclidean ℝ_{E}^{D} with G_{μν} = δ_{μν}, B_{μν} = Φ = T = 0 and action

S[x, g] = \frac{1}{8π} ∫_{Σ} dμ_{g}g^{mn}∂_{m}x · ∂_{n}x .  (2.2)

(For flat Euclidean or Minkowskian metric, we have x ∈ ℝ_{E}^{D} or ℝ_{M}^{D} respectively and we use the notation (v, v′) = v · v′.) The worldsheet metric g is always kept Riemannian. The string tension κ introduced in §I has been set to κ = 1/8π. In the physics literature one often finds the string tension expressed in terms of α′ (the Regge slope parameter) by κ = \frac{1}{4πα′}; so in our conventions α′ = 2.

3) At fixed g, the quantum field theory defined by S[x, g] provides an example of a conformal field theory (see also the lectures by Gawedzki). All we shall need of the conformal field theory to solve for the free string spectrum is contained in the operator product expansions.

4) Once we have derived the OPE’s for Euclidean ℝ_{E}^{D}, we shall continue the time component of the map x, let x^{0} → ix^{0} and obtain well-defined OPE’s for the operators in Minkowskian ℝ_{M}^{D}. In particular, we obtain two Virasoro algebras Vir_{L} ⊗ Vir_{R} for the closed string, and a single Virasoro algebra Vir_{L} for the open string.

5) From the Minkowskian OPE’s and associated algebra, we construct Fock spaces ℱ_{open} and ℱ_{closed} for open and closed strings respectively along the same lines that Gawedzki constructed a Hilbert space. Due to the Minkowski signature of the space-time metric, however, the Fock space we obtain does not carry a positive definite inner product. Vir_{L} ⊗ Vir_{R} acts on ℱ_{closed} and Vir_{L} acts on ℱ_{open}.

6) The remaining integration over g — which is not present in conformal field theory, but characteristic to string theory — effectively selects out of ℱ (either ℱ_{closed} or ℱ_{open}) a subspace ℱ^{+} on which we have a positive definite inner product, under certain conditions which we shall determine. In particular, this is how the critical dimension D = 26 arises.

(The occurrence of negative norm states in a standard Fock space construction is not unique to string theory. It will take place for gauge theories and for gravity as well, or more generally in any theory with particles of spin ≥ 1.)

II.2 A) Basics of Conformal Field Theory

• An oriented surface Σ, with a (fixed) Riemannian metric g automatically carries a complex structure, and we choose local complex coordinates z and z̄ in which g is conformally flat, and given by

g = g_{zz̄}(dz ⊗ dz̄ + dz̄ ⊗ dz) = 2g_{zz̄}|dz|^{2}  (2.3)

This metric is left invariant under analytic or conformal diffeomorphisms z → z′ = f(z), with f complex analytic, combined with Weyl transformations on g. A quantum theory invariant under this transformation is a conformal field theory (abbreviated CFT).

• Fundamental operators in CFT are conformal tensors or primary fields ϕ of conformal weight (h, h̄) satisfying the transformation law

ϕ = ϕ(z, z̄)(dz)^{h}(dz̄)h̄ = ϕ′(z′, z̄′)(dz′)^{h}(dz̄)h̄ .  (2.4)

For integer weights h and h̄, these are operator-valued sections of K^{h} ⊗ K̄h̄, where K is the holomorphic cotangent (or canonical) bundle of Σ. In general, h and h̄ don’t have to be integers, but their difference is the spin of the field ϕ and must satisfy h − h̄ = spin ∈ \frac{1}{2}ℤ.

• In the conformal field theories we shall deal with in these lectures, the matrix elements of products of primary fields (and other fields) may be expressed in terms of functional integrals:

⟨ϕ_{1}(ξ_{1}) … ϕ_{N}(ξ_{N})⟩_{g} = ∫ Dx ϕ_{1}(ξ_{1}) … ϕ_{N}(ξ_{N})e^{−S[x,g]}  (2.5)

where it is understood that ϕ_{i} can be expressed in terms of the canonical fields x. Notice that here we do not normalize correlation functions by dividing out by the partition function ⟨1⟩_{g}.

• The stress tensor is defined as the variation of the action S with respect to the inverse metric g^{−1} = g^{mn}∂_{m} ⊗ ∂_{n}, where the metric g is viewed as a background source field.

δ_{g}S = −\frac{1}{4π} ∫_{Σ} dμ_{g}T_{mn}δg^{mn}  (2.6)

II.3 Assuming that the variation g has support away from the operator positions ξ_{1}, …, ξ_{N}, the variation of correlation functions is given by

δ_{g} ⟨ϕ_{1}(ξ_{1}) … ϕ_{N}(ξ_{N})⟩ = \frac{1}{4π} ∫_{Σ} dμ_{g}δg^{mn} ⟨T_{mn}ϕ_{1}(ξ_{1}) … ϕ_{N}(ξ_{N})⟩  (2.7)

The latter equation may be used as a definition of T_{mn}, even in conformal field theories where no Lagrangian formulation is available.

• Applying Noether’s theorem to conformal symmetry yields the following conserved currents:

j_{z} = v^{z}T_{zz}  jz̄ = vz̄T_{z̄z̄}  (2.8)

where v^{z} (resp. vz̄) is any (locally) holomorphic (resp. anti-holomorphic) vector field. Conformal symmetry requires ∂z̄j_{z} = 0, ∂_{z}jz̄ = 0, so that we must also have

∂z̄T_{zz} = 0  ∂_{z}T_{z̄z̄} = 0  (2.9)

Thus, T (z) = T_{zz} and T̃(z̄) = T_{z̄z̄} are complex analytic and anti-analytic respectively. In correlation functions, however, a product of T (z) and another field ϕ(w) may produce singularities, as z → w, so the above analyticity condition is to be understood as valid when T is away from other operators.

• It was shown in Gawedzki’s lectures how to extract these OPE singularities from conformal Ward identities for locally flat metrics g; for the case of general metric g, the derivation is the subject of Problem Set #2. One finds

\begin{aligned}T (z)ϕ(w, w̄) & ∼ (\frac{h}{(z − w)^{2}} + \frac{1}{z − w} \frac{∂}{∂w} ) ϕ(w, w̄) \\ T (z)T (w) & ∼ \frac{c/2}{(z − w)^{4}} + \frac{2}{(z − w)^{2}} T_{ww} + \frac{1}{z − w} ∂_{w}T_{ww} \end{aligned}  (2.10)

where c is the central charge of the CFT and h is the holomorphic conformal weight of ϕ. (Analogous relations hold for T̃.) As always, these relations are to be understood as valid when inserted into any correlation function. The ∼ sign used in (2.10) stands for the fact

II.4 that we have retained only the singular part on the r.h.s. When c = 0, T (z) is a conformal tensor of weight (2, 0), but for c ≠ 0, T transforms as a projective connection instead, with

T′(z′)(dz′)^{2} = T (z)(dz)^{2} − \frac{c}{12}\{z′; z\}(dz)^{2}  (2.11)

where the Schwarzian derivative { ; } is defined by

{f; z} ≡ f′′′/f′ − (3/2)(f′′/f′)^{2},  f′ = ∂f/∂z.

• Analytic currents j_{α} = j_{αz}dz, α labeling different species of currents, give rise to timeindependent or conserved charges:

Q_{α}(C) = \frac{1}{2πi} ∮_{C} j_{α}

Here Q_{α}(C) is the charge enclosed in the region bounded by C and equals Q_{α}(C′) provided no sources for Q_{α} occur between C and C′. The action of charges on fields ϕ may be evaluated using only the residues in the OPE of j_{α} with ϕ:

Q_{α}(C)ϕ(w, w̄) = \frac{1}{2πi} ∮_{C} j_{α}(z)ϕ(w, w̄) = Res_{z→w} j_{α}(z)ϕ(w, w̄)  (2.12)

provided no other sources for j_{α} lie within C. The commutator between two charges is recovered as follows:

\begin{matrix}Q_{α}( & C_{1})Q_{β}(C_{2}) − Q_{α}(C_{3})Q_{β}(C_{2}) \\ = [Q_{α}, Q_{β}](C_{2}) \\ = \frac{1}{2πi} ∮_{C_{2}} Res_{z→z_{2}} j_{α}(z)j_{β}(z_{2}) \end{matrix}  (2.13)

• The conserved charges associated with conformal transformations are obtained from

vector fields v_{(m)}^{z} = z^{m+1} and vz̄ ; (m) = z̄^{m+1}:

\begin{aligned}L_{m} & ≡ ∮ \frac{dz}{2πi}z^{m+1}T (z) \\ L̃_{m} & ≡ ∮ \frac{dz̄}{2πi}z̄^{m+1}T̃(z̄). \end{aligned}  (2.14)

II.5 From the OPE of two T ’s (and two T̃’s), we recover two Virasoro algebras

\begin{aligned}[L_{m}, L_{n}] & = (m − n)L_{m+n} + \frac{c}{12}m(m^{2} − 1)δ_{m+n,0} \\ [L̃_{m}, L̃_{n}] & = (m − n)L̃_{m+n} + \frac{c}{12}m(m^{2} − 1)δ_{m+n,0}. \end{aligned}  (2.15)

From the OPE of T (and T̃) with a primary field ϕ of conformal weight (h, h̄), we obtain

\begin{aligned}[L_{m}, ϕ(w, w̄)] & = w^{m+1}∂_{w}ϕ(w, w̄) + h(m + 1)w^{m}ϕ(w, w̄) \\ [L̃_{m}, ϕ(w, w̄)] & = w̄^{m+1}∂w̄ϕ(w, w̄) + h̄(m + 1)w̄^{m}ϕ(w, w̄) \end{aligned}  (2.16)

• An important example of a CFT is the free scalar field x: Σ → ℝ (or ℝ/λℤ as discussed in Witten’s lectures), with action

S[x] = \frac{1}{4π} ∫_{Σ} d^{2}z ∂_{z}x∂z̄x  (2.17)

The OPE is derived from the Green function for x:

x(z, z̄)x(w, w̄) ∼ − ln |z − w|^{2}  (2.18)

It is customary to define normal ordering by

: x(z, z̄)x(w, w̄) :≡ x(z, z̄)x(w, w̄) + ln |z − w|^{2}  (2.19)

The normal ordered product admits a smooth limit as w → z. The renormalized (also called the quantum) stress tensor may now be defined by

T (z) = −\frac{1}{2} : (∂_{z}x)^{2} :≡ −\frac{1}{2} lim_{w→z} (∂_{z}x∂_{w}x + \frac{1}{(z − w)^{2}} ) .  (2.20)

Elementary primary fields in this CFT are listed below, with their corresponding conformal weight.

\begin{aligned}& ∂_{z}x \\ & ∂z̄x \\ : & exp ikx : \end{aligned}  \begin{matrix}(1, 0) \\ (0, 1) \\ (\frac{1}{2} k^{2}, \frac{1}{2} k^{2}) \end{matrix}  (2.21)

The central charge is c = 1. Notice that while the derivative of x and the exponential are primary fields, x itself is not a well-defined quantum field (cf. Witten’s lecture).

II.6 B) The free closed bosonic string conformal field theory

The worldsheet topology for the free closed bosonic string is that of the cylinder or equivalently the annulus.

The metric g determines global complex coordinates z, z̄ in which g = 2g_{zz̄}|dz|^{2}.

Recall from 2) that our starting point is the flat Euclidean ℝ_{E}^{D} worldsheet action

S = \frac{1}{4π} ∫_{Σ} d^{2}z ∂_{z}x · ∂z̄x  (2.22)

which for fixed metric g defines a CFT of D scalar fields x^{μ}, which are the coordinates of the map x: Σ → ℝ_{E}^{D}. The quantum stress tensor is

T (z) = −\frac{1}{2} : ∂_{z}x · ∂_{z}x :  (2.23)

and the central charge is c = D.

Following point 3), we obtain the OPE’s and associated algebraic structure. To do this, we take advantage of the fact that ∂_{z}x(z) and T (z) are holomorphic, and thus admit a Laurent expansion inside the annulus:

∂_{z}x(z) = −i ∑_{m∈ℤ} x_{m}z^{−m−1}  T (z) = ∑_{m∈ℤ} L_{m}z^{−m−2}
∂z̄x(z̄) = −i ∑_{m∈ℤ} x̃_{m}z̄^{−m−1}  T̃(z̄) = ∑_{m∈ℤ} L̃_{m}z̄^{−m−2}  (2.24)

From the definition of T ≡ −\frac{1}{2}: ∂_{z}x · ∂_{z}x:, we have

\begin{aligned}L_{m} & = \frac{1}{2} ∑_{n} x_{m−n} · x_{n}  m ≠ 0 \\ L_{0} & = \frac{1}{2}x_{0}^{2} + ∑_{n=1}^{∞} x_{−n} · x_{n} \end{aligned}  (2.25)

II.7 From the OPE of ∂_{z}x with itself and with T (z), we get

[x_{m}^{μ}, x_{n}^{ν}] = mδ_{m+n,0}δ^{μν}  [L_{m}, x_{n}] = −nx_{m+n} ,  (2.26a)

also

[x̃_{m}^{μ}, x̃_{n}^{ν}] = mδ_{m+n,0}δ^{μν}  [L̃_{m}, x̃_{n}] = −nx̃_{m+n} ,  (2.26b)

while tilded and untilded operators mutually commute. L_{m} and L̃_{m} satisfy Virasoro algebras with central charge c = D.

Following point 4), we find that the above OPE’s and their algebra contents can safely be continued to x^{0} → ix^{0} where x^{0} is the time component of the map x: Σ → ℝ_{M}^{D}. It is clear that this means

\begin{matrix}x_{m}^{0} → ix_{m}^{0} \\ x̃_{m}^{0} → ix̃_{m}^{0} \end{matrix} \}  all m ∈ ℤ.  (2.27)

The stress tensor is now

T (z) = −\frac{1}{2}: ∂_{z}x^{μ}∂_{z}x^{ν}: η_{μν} ≡ −\frac{1}{2}: ∂_{z}x · ∂_{z}x:  (2.28)

and the algebraic relations are

\begin{aligned}[x_{m}^{μ}, x_{n}^{ν}] & = mδ_{m+n,0}η^{μν} [L_{m}, x_{n}^{μ}] = −nx_{m+n}^{μ} \\ L_{m} & = \frac{1}{2} ∑_{n}: x_{m−n}^{μ}x_{n}^{ν}: η_{μν} \end{aligned}  (2.29)

and analogously for x̃ and L̃. The central charge is unchanged, and we still have Vir_{L}⊗Vir_{R}.

The field x itself is not a conformal tensor, but may be obtained by integrating ∂_{z}x and ∂z̄x:

x(z, z̄) = x_{L}(z) + x_{R}(z̄)  (2.30)

with

\begin{aligned}x_{L}(z) & = q_{L} − ix_{0} ln z + i ∑_{m≠0} x_{m} \frac{z^{−m}}{m} \\ x_{R}(z̄) & = q_{R} − ix̃_{0} ln z̄ + i ∑_{m≠0} x̃_{m} \frac{z̄^{−m}}{m} . \end{aligned}  (2.31)

II.8 Since x : Σ → ℝ_{M}^{D}, x must be a single-valued map as a function of z. This will require that x_{0} = x̃_{0} = p, and p is to be identified with the momentum operator of the string, while q = q_{L} + q_{R} is the overall position operator of the string.

It will turn out to be useful to record the explicit form of the operators L_{0} and L̃_{0}, expressed in terms of p.

\begin{aligned}L_{0} & = \frac{1}{2}p^{2} + ∑_{n=1}^{∞} x_{−n} · x_{n} \\ L̃_{0} & = \frac{1}{2}p^{2} + ∑_{n=1}^{∞} x̃_{−n} · x̃_{n} \end{aligned}  (2.32)

For non-simply connected space-time manifolds M, the map x need not be single-valued, since the string can wrap around non-trivial 1-cycles in M. As a result, x_{0} need not be equal to x̃_{0}, but the difference x_{0} − x̃_{0} belongs to a lattice, dual to H^{1}(M, ℤ).

C) The free open bosonic string conformal field theory The worldsheet of a free open string has the topology of a strip or a half annulus.

Fundamental open strings should have freely moving boundary points. Thus Neumann conditions ∂_{Im z}x = 0 at the boundary B are appropriate. (Dirichlet b.c. would fix the end points of the string on some submanifold 𝒟 of M. Such submanifolds are D-branes, or depending upon their space dimension p: p-branes. In certain string theories, D-branes become dynamical (extended) objects, perhaps to be discussed later on.)

It is standard to double Σ by reflecting Σ onto the lower half plane into Σ^{∗} and then considering the union Σ′ = Σ∪Σ^{∗} which is the full annulus. Functions on Σ with Neumann

II.9 boundary conditions on B are then even functions under z → z̄. Thus, the mode expansion of the open string is obtained from this restriction applied to x(z, z̄)

x(z, z̄) = x_{L}(z) + x_{L}(z̄)  (2.33)

and only involves the oscillators x_{m} but there are no oscillators x̃_{m}. Similarly, we only have Virasoro generators L_{n} but there are no L̃_{n}. As a result, the energy-momentum tensor satisfies

T (z) = T̃(z̄) = 0  on  Im(z) = 0.  (2.34)

This condition guarantees conservation of T across B, and results from the fact that Diff(Σ) leaves B invariant. We have the mode expansion

x(z, z̄) = x + 2ip ln |z|^{2} − i ∑_{m≠0} x_{m} (\frac{z^{−m}}{m} + \frac{z̄^{−m}}{m} )  (2.35)

Notice that x_{0} = 2p, in contrast with the closed string. It is useful to also record the Virasoro generators

\begin{aligned}L_{0} & = 2p^{2} + ∑_{n=1}^{∞} x_{−n} · x_{n} \\ L_{m} & = ∑_{n∈ℤ} x_{m−n} · x_{n}  m ≠ 0 \end{aligned}  (2.36)

Thus the open string only has a single Virasoro algebra Vir_{L}.

D) Fock Space, Negative Norm States

We now follow point 5) and construct the Fock space of the conformal field theory of x, continued to Minkowski space-time. (This parallels Gawedzki’s construction).

To do so, we must choose a polarization and split the operators {x_{m}^{μ}} and {x̃_{m}^{μ}} into a group of raising, lowering and central generators. This choice must be consistent with Poincaré invariance. (Also, here we must use a property inherited from the Minkowski nature of the worldsheet, namely that x_{m} and x̃_{m} oscillators are independent, except at m = 0. We shall not expand upon this issue now, but come back to this part later on.)

II.10 The correct choice is (x_{0}^{μ})^{†} = x_{0}^{μ}, while x_{m}^{μ} and x̃_{m}^{μ} are raising (resp. lowering) for m < 0 (resp. m > 0). Also

(x_{m}^{μ})^{†} = x_{−m}^{μ}  (x̃_{m}^{μ})^{†} = x̃_{−m}^{μ}  (2.37)

and this implies

L_{m}^{†} = L_{−m}  L̃_{m}^{†} = L̃_{−m}.  (2.38)

The Fock space ℱ_{k} is constructed from x_{m}^{μ} oscillators only. The ground state | 0, k⟩_{L} (also called the highest weight state) is labeled by the momentum k (the eigenvalue of the momentum operator p)

\{ \begin{matrix}p| 0, k⟩_{L} = k| 0, k⟩_{L} \\ x_{m}^{μ}| 0, k⟩_{L} = 0 & m > 0. \end{matrix}  (2.39)

We may choose its normalization as follows:

L⟨0, k|0, k′⟩_{L} = δ(k − k′) .  (2.40)

Arbitrary vectors in ℱ_{k} are obtained by applying x_{−m}^{μ}’s for m > 0 to the ground state

| ε, k⟩_{L} = ε_{μ_{1}...μ_{n}}(k; m_{1}, …, m_{n})x_{−m_{1}}^{μ_{1}}, … x_{−m_{n}}^{μ_{n}}| 0, k⟩_{L}  (2.41)

for all possible Lorentz polarization tensors ε_{μ_{1}...μ_{n}} (k; m_{1}, …, m_{n}); n ∈ ℕ, with all possible m_{i} ∈ ℕ.

• The open string Fock space is a sum over all momenta k of ℱ_{k}:

ℱ_{open} = ⨁_{k∈ℝ_{M}^{D}} ℱ_{k}

• The closed string Fock space is a sum over all momenta — which must be the same for left and right movers, in view of the constraint x_{0} = x̃_{0} — of the tensor product of ℱ_{k} and ℱ̃_{k}, where ℱ̃_{k} is built from x̃_{m}^{μ}:

ℱ_{closed} = ⨁_{k∈ℝ_{M}^{D}} ℱ_{k} ⊗ ℱ̃_{k}.

II.11 • Using the norm on | 0, k⟩_{L} introduced in (2.40), ℱ_{k} naturally inherits a quadratic form. However, ℱ_{k} automatically contains negative norm states. To see this, consider states | ε, k⟩_{L} = ε_{μ}x_{−m}^{μ} |0, k⟩_{L} and with m > 0 evaluate their norm:

L ⟨ε, k|ε, k′⟩_{L} = ⟨0, k |ε · x_{−m}^{†}ε · x_{−m} |0, k′⟩ = mε^{2}δ(k − k′)  (2.42)

For ε^{2} > 0, the norm of the state | ε, k⟩_{L} is positive. But, for ε^{2} < 0, which is allowed with Minkowski metric, the norm of | ε, k⟩_{L} is negative. Such states are called ghosts and their presence violates quantum mechanics: they cannot be present in a physically sensible theory.

E) Integration over Met(Σ) — Virasoro Constraints

We now deal with point 6): the integration over Met(Σ), and the elimination of ghosts from the spectrum. We shall perform this integration with great care in forthcoming lectures, but we can already here extract its key effects.

The metric g enters in the action S as a non-dynamical field, i.e. without derivatives. The rôle of the metric g here is completely analogous to that of the time-like component A_{0} of the gauge field in Yang-Mills theory (see also Faddeev’s lecture). The effect of such non-dynamical fields is to supply a constraint on the dynamics of the remaining degrees of freedom, and to reduce their phase space. (You may loosely view the effect in the functional integral as generating a delta function ∫ dA_{0}e^{iA_{0}x} = δ(x).) The constraints also generate a group action on the phase space, and the reduced phase space may be viewed as the quotient by this group action.

Let us analyze the effect of the constraints at the classical level first. The variational equations are

\{ \begin{matrix}∂z̄∂_{z}x = 0 & by varying x \\ T (z) = T̄(z̄) = 0 & by varying g \end{matrix}  (2.43)

The second equation is a constraint on x since T (z) = −\frac{1}{2}∂_{z}x · ∂_{z}x (similarly for T̃(z̄)). Since ∂z̄T (z) = 0, the first equation shows that T will vanish throughout Σ as long as T = 0 has been imposed on the boundary. The action of the constraints at the classical level is

II.12 something we have already analyzed, namely the Virasoro algebra but with zero central charge. Geometrically, the constraint reflects the fact that vibrations of x(Σ) tangent to the surface x(Σ) are eliminated, leaving only transverse direction:

\begin{matrix}(D − 2) transverse \\ directions \end{matrix} \}  ^{\} \begin{matrix}2 longitudinal directions \\ eliminated by T̄ = T = 0 \end{matrix}}

We now analyze the effects of the constraints at the quantum level. There are basically three ways of dealing with the constraints:

1) Eliminate the longitudinal degrees of freedom at the classical level by solving for them in terms of the transverse degrees of freedom, and then quantize the transverse degrees of freedom. This procedure goes under the name of light-cone gauge quantization. It cannot be achieved in a manifestly Lorentz invariant parametrization. We shall discuss it very briefly later on in this lecture.

2) Ultimately, in lecture VII, we shall reformulate this constraint problem in terms of the powerful BRST quantization method. This is a more complex scheme based on introducing extra, unphysical degrees of freedom, called Faddeev-Popov ghosts, and enforcing BRST (Becchi-Rouet-Stora-Tyutin) invariance. This method is manifestly Lorentz invariant. The physical Hilbert space will in fact arise as a cohomology group of a semi-infinite cohomology complex, as developed by Feigin and later on by Freeman and Olive in physics and Frenkel, Garland and Zuckerman in mathematics.

3) Before launching into BRST, however, we shall analyze the quantum effects of the constraints in a more direct fashion, which will also yield immediate results on the spectrum. In this third method, we impose the constraint as an invariance condition and use it to

II.13 select an invariant subspace out of the full Fock space ℱ (either ℱ_{open} for open strings or ℱ_{closed} for closed strings). In Yang-Mills theory, the corresponding constraint was just Gauss’ law, enforcing gauge invariance on physical states. In string theory, the constraints are the vanishing of the stress tensor, enforcing Diff(Σ) invariance on physical states. Due to the central extension in Vir, the reduced Fock space ℱ^{+} = ℱ/Vir will have a positive definite norm only when certain extra conditions are satisfied, which we now determine.

F) Physical Spectrum, No-Ghost Theorem We use the Virasoro constraints to select an invariant subspace out of the full Fock space ℱ. It is tempting to try and impose L_{n}| ϕ⟩ = L̃_{n}| ϕ⟩ = 0 for all n ∈ ℤ. However, unless c = 0, these conditions imply that | ϕ⟩ = 0. Instead, we impose invariance under half of Vir. The correct physical state conditions are therefore

| ϕ⟩_{L} ∈ ℱ_{k}^{phys}  if  (L_{m} − aδ_{m,0})| ϕ⟩_{L} = 0  m ∈ ℕ
| ϕ⟩_{R} ∈ ℱ̃_{k}^{phys}  if  (L̃_{m} − aδ_{m,0})| ϕ⟩_{R} = 0  m ∈ ℕ.

(2.44)

and the full physical Fock spaces are obtained by

ℱ_{open}^{phys} = ⊕_{k} ℱ_{k}^{phys} ,
ℱ_{closed}^{phys} = ⊕_{k} ℱ_{k}^{phys} ⊗ ℱ̃_{k}^{phys} .

The constant a is undetermined at this point, and is allowed for since L_{0} and L̃_{0} were naturally defined only up to an additive constant in view of the ordering choice we made in their definition. For closed strings, the value of a is taken the same for L_{0} and L̃_{0} so that L_{0} − L̃_{0} = \frac{∂}{∂σ} enforces rotation invariance on the annulus. Notice that states | ϕ⟩ , | ψ⟩ ∈ ℱ^{phys} automatically yield zero matrix elements of Virasoro generators L_{m} − aδ_{m,0} for all m ∈ ℤ (similarly for L̃_{m} − aδ_{m,0} in the case of the closed string). Hence, we have now realized the physical state condition ⟨ϕ|T |ψ⟩ = ⟨ϕ|T̃|ψ⟩ = 0!

States of the form | ϕ⟩ = L_{−m}| ψ⟩, for m > 0, are called spurious, and we define

ℱ_{k}^{spurious} = {| ϕ⟩ = ⨁_{m=1}^{∞} L_{−m}| χ_{m}⟩}

II.14 It is easy to see that ℱ_{k}^{spurious} is generated by states of the form L_{−1}| χ_{1}⟩ ⊕ L_{−2}| χ_{2}⟩, using the commutation relations of the Virasoro algebra which imply that [L_{−1}, L_{−n}] is a non-zero multiple of L_{−1−n} for n ≥ 2. Notice that spurious states are orthogonal to all physical states. States that are both physical and spurious are null,

ℱ_{k}^{spurious} ∩ ℱ_{k}^{phys} ⊂ ℱ_{k}^{null} ,

where ℱ_{k}^{null} stands for the space of null states of momentum k.

The special Virasoro conditions (L_{0} − a)| ϕ⟩ = (L̃_{0} − a)| ϕ⟩ = 0 determine the value of k^{2} for a given level of operators, and thus determines the mass of the state. To see this, introduce the number (also called level) operators

N ≡ ∑_{n=1}^{∞} x_{−n} · x_{n}  Ñ ≡ ∑_{n−1}^{∞} x̃_{−n} · x̃_{n} .  (2.45)

The operators N and Ñ count the number of operators x_{−n} and x̃_{−n}, n ≥ 1 with weight n, applied to the ground state | 0, k⟩. The special Virasoro conditions thus become

closed  (k^{2} + M^{2})| ϕ⟩ = 0  M^{2} = 2N − 2a  and N = Ñ| ϕ⟩ = 0
open  (k^{2} + M^{2})| ϕ⟩ = 0  M^{2} = \frac{1}{2}N − \frac{1}{2}a . (2.46)

Thus, the mass squared of various string states are spread by 2× integers for closed strings and \frac{1}{2}× integers for open strings.

The mass squared of a state is one of the Casimir values that label the representation of the Poincaré group under which the state transforms. The remaining data that fully specify the Poincaré representation are those given by the representation of the stabilizer group (i.e. the subgroup that leaves the momentum invariant). This information is contained in the expression of the Lorentz generators of the string.

J^{μν} = q^{u}p^{ν} − q^{ν}p^{μ} − i ∑_{n=1}^{∞} \frac{1}{n}(x_{−n}^{μ}x_{n}^{ν} − x_{−n}^{ν}x_{n}^{μ})

(−i ∑_{n=1}^{∞} \frac{1}{n}(x̃_{−n}^{μ}x̃_{n}^{ν} − x̃_{−n}^{ν}x̃_{−n}^{μ}) for closed strings)

(2.47)

It follows that the string ground state | 0, k⟩ transforms as a scalar (i.e. trivial representation under the stabilizer).

II.15

We observed in the first lecture that, at long distances compared to the Planck length ℓ_{P}, the size of strings effectively becomes unobservable. We can then approximate the dynamics of the string by that of point particles, and string theory by quantum field theory. Thus, we shall seek to identify string states with states in quantum field theory. To do so, we require that string and quantum field states transform under the same representations of the Poincaré group, as well as of other symmetry groups that are available. Also their interactions should agree in this long distance limit.

We examine under what conditions the Virasoro constraints eliminate all negative norm states, so that ℱ_{k}^{phys} has positive norm. Before stating the general result, let’s look at the lowest level cases (for the closed string; for the open string it suffices to let M^{2} → M^{2}/4).

(1) Ground state | 0, k⟩_{L} with mass square M^{2} = −2a (actually | 0, k⟩_{L} ⊗ | 0, k⟩_{R} for the closed string). The Virasoro conditions are automatically satisfied.

(2) First excited state | ε, k⟩_{L} = ε · x_{−1}| 0, k⟩_{L} with mass square M^{2} = −2a + 2:

L_{1}| ε, k⟩_{L} = 0 ⟹ ε · k = 0 .

The remaining Virasoro conditions L_{n}| ε, k⟩_{L} = 0, n ≥ 2, are automatically satisfied. There are three possibilities:

(i) a > 1, so that k^{2} is space-like. If we pick k = (0, k^{1}, 0 … 0) then we are allowed to choose ε = (ε^{0}, 0 … 0), and we see that ghosts are still present.

(ii) a < 1, so that k^{2} time-like. If we pick k = (k^{0}, 0 … 0) we see that we always have ε^{2} > 0 and there are D − 1 independent states with positive norm.

(iii) a = 1, so that k^{2} is light-like. If we pick k = (k^{1}, k^{1}, 0 … 0), we see that there are (D −2) massless states of positive norm, and one of zero norm.

Case (i) is clearly eliminated: Virasoro constraints do not suffice to eliminate all negative norm states. Case (ii) has massive M^{2} > 0 vector states (open string), or massive rank 2

II.16 tensor states (closed string). Case (iii) has massless M^{2} = 0 vector states (open strings), which suggests the presence of massless Yang-Mills states, and has massless M^{2} = 0 rank 2 tensor states (closed strings) which suggests the presence of gravity (for symmetric tensors).

3) At the next level | ε, k⟩_{L} = {ε_{μν}x_{−1}^{μ}x_{−1}^{ν} + η_{μ}x_{−2}^{μ}}| 0, k⟩_{L}, with mass squared M^{2} = −2a + 4 and

\begin{matrix}L_{1} & | ε, k⟩_{L} = 0 \\ L_{2} & | ε, k⟩_{L} = 0 \end{matrix}  \{ \begin{matrix}ε_{μν}k^{ν} + η_{μ} = 0 \\ ε_{μ}^{μ} + 2k_{μ}η^{μ} = 0 . \end{matrix}

The norm of the state is

L⟨ε, k|ε, k′⟩_{L} = 2𝒩_{L}⟨0, k|0, k′⟩_{L}

with

\begin{aligned}𝒩 & = ε_{μν}ε^{μν} + η_{μ}η^{μ} \\ & = (ε_{00})^{2} − 2(ε_{0i})^{2} + (ε_{ij})^{2} − (η_{0})^{2} + (η_{i})^{2}. \end{aligned}

Choosing a rest frame where k^{0} = M, k^{i} = 0, the Virasoro conditions become

\{ \begin{matrix}ε_{00}M + η_{0} = 0 \\ ε_{i0}M + η_{i} = 0 \\ −ε_{00} + ε_{ii} + 2Mη_{0} = 0 . \end{matrix}

The Virasoro conditions impose no constraints on the traceless part of ε_{ij}, and its contribution to 𝒩 is ≥ 0. Hence, we may retain only the trace part: ε_{ij} = εδ_{ij}. Eliminating η_{i} and η_{0} and ε_{00} in favor of ε_{0i} and ε, we have

𝒩 = \frac{ε^{2}(D − 1)}{(2M^{2} + 1)^{2}} \{(2M^{2} + 1)^{2} − (M^{2} − 1)(D − 1)\} + (M^{2} − 2)(ε_{0i})^{2}.

Assuming D > 1, we find the conditions

\{ \begin{matrix}M^{2} ≥ 2 \\ (2M^{2} + 1)^{2} − (M^{2} − 1)(D − 1) ≥ 0 . \end{matrix}

the first of which is already satisfied since M^{2} = −2a + 4 and we have seen before that we must have a ≤ 1. For M^{2} = 2 (corresponding to massless states at level 1), we find from the second inequality that D ≤ 26. In fact, for D = 26, the second inequality implies M^{2} ≤ 2, so that we must have M^{2} = 2 for D = 26.

II.17

For D ≤ 25, the quadratic form (2M^{2} + 1)^{2} − (M^{2} − 1)(D − 1) is non-negative for all M^{2} ≥ 2, so the state | ε, k⟩ always has non-negative norm.

For D ≥ 27, the quadratic form is negative for a region in which M^{2} ≥ 2. In fact, by analyzing higher level states it can be shown that for D ≥ 27 no value of a can be chosen so as to eliminate all ghosts. The complete result can be summarized in the following

No Ghost Theorem (R. Brower, P. Goddard and C. Thorn).

• For D > 26 or a > 1 ℱ^{phys} contains negative norm states;

• For D = 26 and a < 1 ℱ^{phys} contains negative norm states;

• For D = 26 and a = 1 ℱ^{phys} contains only non-negative norm states, in particular, two towers of null states and ℱ^{+} ≡ ℱ^{phys}/ℱ^{null} has positive definite norm;

ℱ^{null} = \{(L_{−2} + \frac{3}{2}L_{−1}^{2})| χ_{2}⟩ ⊕ L_{−1}| χ_{1}⟩\}

• For D < 26 and a ≤ 1 ℱ^{phys} but contains only positive norm states for a < 1 (contains one tower of null states L_{−1}| χ_{1}⟩ for a = 1).

We shall not prove the no ghost theorem here, since later on, using BRST quantization, a simple proof can be given.

The case D ≤ 25 and a ≤ 1 is contained in D = 26, a = 1 by setting 26 − D space-like operators to zero, and adjusting the 26−D positive squared momenta p_{i}^{2}, i = 1, …, 26−D, so that

\frac{1}{2} ∑_{i} p_{i}^{2} − a + 1 = 0 .

II.18 G) Spectrum of the critical bosonic string with D = 26, a = 1

Based on the masses and Lorentz tensor structure of the lowest lying states of the spectrum, we may identify which particles they naturally correspond to in field theory.

It is easy to see that the spectrum consists entirely of bosons.

Open strings

(1) States corresponding to | 0, k⟩ have M^{2} = −\frac{1}{2}, hence they are tachyons (particles traveling faster than light), and Lorentz scalars;

(2) States of the form ε · x| 0, k⟩ with M^{2} = 0 are vectors under SO(1, D − 1). There are (D − 2) positive norm states and one null state, which constitute a massless gauge boson in ℝ^{D}. Gauge transformations have the form ε_{μ} → ε_{μ} + k_{μ}γ (invariance under these transformations is clear since they amount to shifting the state by a null state γL_{−1}| 0, k⟩ = γk · x_{−1}| 0, k⟩);

(3) M^{2} > 0 states, with M^{2} = \frac{1}{2}n, n ∈ ℕ, transform under various tensor representations of SO(1, D − 1).

Closed strings

(1) States corresponding to | 0, k⟩ = | 0, k⟩_{L}⊗| 0, k⟩_{R} have M^{2} = −2, so they are tachyons, and Lorentz scalars;

(2) States of the form ε_{μν}x_{−1}^{μ}x̃_{−1}^{ν}| 0, k⟩ have M^{2} = 0. Corresponding to the trace-part of ε there is a Lorentz scalar, the dilaton, with positive norm. The symmetric traceless part of ε gives a Lorentz rank 2 tensor, the graviton. There are \frac{D(D+1)}{2} −1 such states, \frac{D(D−1)}{2} −1 of them having positive norm. As for the antisymmetric part of ε, it gives a Lorentz rank 2 tensor called the anti-symmetric tensor with \frac{D(D−1)}{2} states, out of which \frac{(D−1)(D−2)}{2} have positive norm. These states exhibit gauge invariance under

ε_{μν} → ε_{μν} + γ_{μ}k_{ν} + γ′_{ν}k_{μ}  (2.48)

(which amounts to shifting by null states);

II.19 (3) M^{2} > 0 states, with M^{2} = −2n, n ∈ ℕ, transform under various tensor representations of SO(1, D − 1).

H) Lightcone gauge, density of states

For D = 26 and a = 1, the null states in ℱ^{null} may be ignored completely. We may retain positive norm states only at the cost of giving up manifest Lorentz invariance, SO(1, 25), by retaining D − 2 = 24 transverse oscillators only. Let M′ be a space-like subvector space of M, with dim M′ = 24, and let ε, ε̃ be in M′; then ε · x_{−n} and ε̃· x̃_{−n} for n > 0 create only positive norm states. This construction has manifest SO(24) invariance. It can be checked that the full Lorentz algebra SO(1, 25) closes in this realization. Also, it can be shown in this case that there is a one-to-one correspondence between those states constructed above and the states in ℱ^{+} = ℱ^{phys}/ℱ^{null} in the covariant approach.

If the same construction were attempted for D ≠ 26 or a ≠ 1, one would find that the Lorentz algebra in this realization does not close.

Using the above correspondence, it is very easy to estimate the total number of states at each mass level. Introducing a generating function

\begin{aligned}G(w) & = Tr_{ℱ^{+}} (w^{2M^{2}+2}) = ∏_{n=1}^{∞}(1 − w^{n})^{−24} \\ & ≡ ∑_{n=1}^{∞} d_{n}w^{2n} . \end{aligned}  (2.49)

We find the number of states asymptotically for large M^{2}: d_{M^{2}} = M^{−25/2}e^{M/4π\sqrt{2}}.

I) Primary fields and Vertex Operators for Physical States

The construction given above for the physical Hilbert space has the disadvantage that the operators x_{−n}^{μ} with n > 0, create states in Fock space that are not in general physical. We now construct vertex operators, which have the property that they send physical states into physical states, and spurious states into other spurious states.

Since physical states obey (L_{n} − δ_{n,0})| ψ⟩ = (L̃_{n} − δ_{n,0})| ψ⟩ = 0, n ≥ 0, it is clear that any operator commuting with the entire Virasoro algebra will yield such a vertex operator.

II.20 Also, since [L_{n}, V] = [L̃_{n}, V] = 0, V automatically sends spurious states into spurious states.

Since L_{n} generates conformal reparametrizations, V should be conformal invariant. To construct V ’s, we consider general tensors, then construct invariants. We use the action of L_{n} on any primary field:

\begin{aligned}[L_{n}, ϕ(z, z̄)] & = z^{n+1}∂_{z}ϕ(z, z̄) + h(n + 1)z^{n}ϕ(z, z̄) \\ & = ∂_{z}(z^{n+1}ϕ(z, z̄)) + (h − 1)(n + 1)z^{n}ϕ(z, z̄) \\ [L̃_{n}, ϕ(z, z̄)] & = ∂z̄(z̄^{n+1}ϕ(z, z̄)) + (h̄ − 1)(n + 1)z̄^{n}ϕ(z, z̄) . \end{aligned}  (2.50)

It is clear that amongst primary fields, the only operator that commutes with all of L_{n} and L̃_{n} is the unit operator ϕ = 1.

However, integrals of ϕ may be invariant. For any primary field of conformal weight (h, h̄) = (1, 1), its commutators with L_{n}, L̃_{n} are total derivatives, and so its integral will be Virasoro invariant. In addition, primary fields ϕ of weight (1, 1) form natural volume forms on Σ. Hence the integral of ϕ

V = ∫ ϕ(z, z̄)  satisfies  [L_{n}, V] = [L̃_{n}, V] = 0 .  (2.51)

This is not the only way to construct vertex operators though: if ϕ is a holomorphic primary field (e.g. ∂_{z}x^{μ}), then it will commute with L̃_{n} provided h̄ = 0 and its commutator with L_{n} is a total derivative if h = 1. In addition (1, 0) and (0, 1) forms are naturally integrated along contours. Thus, the operators

∮ dz ϕ_{(1,0)}

and

∮ dz̄ϕ_{(0,1)}

are also vertex operators provided that ∂z̄ϕ_{(1,0)} = 0 and ∂_{z}ϕ_{(0,1)} = 0. The operators ϕ_{(1,0)} and ϕ_{(0,1)} are precisely what we defined previously as conformal currents; their charges are conserved. Such operators do not in fact create new physical states, as will become clear by considering examples below.

II.21

Vertex operators for the closed bosonic string are now easily reconstructed from the basic field x^{μ} associated with D copies of the c = 1 CFT. We have the following primary fields

∂_{z}x^{μ} holomorphic, weight  (1, 0)
∂z̄x^{μ} anti-holomorphic, weight (0, 1)
: e^{ik·x} : weight (\frac{k^{2}}{2} , \frac{k^{2}}{2} )

From the holomorphic current ∂_{x}x^{μ} we get the conserved charge of momentum:

p^{μ} = \frac{1}{2π} ∮ dz ∂_{z}x^{μ} .  (2.52)

Physical states in ℱ_{k}^{phys} will just be eigenstates of p^{μ}, so p^{μ} does not really create any new states, but simply measures their momentum k.

A single exponential yields a vertex operator for k^{2} = 2, i.e. when the conformal weight is (1, 1):

V_{T} (k) = ∫_{Σ} d^{2}z : e^{ikx} :  (2.53)

This scalar operator creates a state of mass M^{2} = −2, which is the tachyon discovered previously. But the exponential may also be combined with derivatives for k^{2} = 0

V_{G,B,Φ}(k) = ∫_{Σ} d^{2}z ε_{μμ̄}(k) ∂_{z}x^{μ}∂z̄xμ̄ : e^{ik·x} :  (2.54)

which reproduces the states for the graviton, the antisymmetric tensor, and the dilaton as symmetric traceless, anti-symmetric, and trace part respectively, with k_{μ}ε^{μμ̄}(k) = kμ̄ε^{μμ̄}(k) = 0. For higher powers of ∂_{z}x^{μ} and ∂z̄xμ̄, we will have masses of the form M^{2} = 2n − 2 in general. However, higher derivatives ∂_{z}^{2}x^{μ} may also appear. While these are not primary fields (they mix with ∂_{z}x under conformal reparametrizations), they may combine with other operators and still yield primary fields.

When we obtained the Virasoro conditions on the polarization tensors ε_{μμ̄}, we established an invariance of the states in ℱ^{phys} under the transformations

ε_{μμ̄} → ε_{μμ̄} + k_{μ}γ̄μ̄ + kμ̄γ_{μ}  (2.55)

II.22 (We shall shortly interpret this invariance as a manifestation of Diff(M)-invariance for the closed string.) This invariance is also easily established for the vertex operators: for example a shift by γ̄ yields

\begin{aligned}∫_{Σ} d^{2}z k_{μ}γ̄μ̄∂_{z}x^{μ}∂z̄xμ̄ : e^{ikx} : & = −i ∫_{Σ} d^{2}zγ̄μ̄∂z̄xμ̄(∂_{z}e^{ikx}) \\ & = i ∫_{Σ} d^{2}zγ̄μ̄(∂_{z}∂z̄xμ̄)e^{ik·x} \end{aligned}  (3.56)

But we have ∂_{z}∂z̄x^{μ} = 0 by the field equations — away from any other singularities — and thus the integral vanishes. (The effects of the singularities may also be taken into account, as we shall show later on and do not modify the conclusion above.)

J) Identifying the graviton: vertex operators from background fields

One of the most important discoveries in string theory is that the k^{2} = 0, ε_{μμ̄}(k)- symmetric traceless physical state, found above, should be identified with the graviton. This is the particle in quantum field theory that represents excitations of the space-time metric and that mediates the force of gravity. Here, we provide further evidence for this identification. Along the same lines, we provide evidence for the identification of the open string k^{2} = 0, ε_{μ}(k)-state with the Yang-Mills particle in this section and in §K.

Consider a space-time M which is almost flat with the following metric

G_{μν}(x) = η_{μν} + ε_{μν}(x)  (2.57)

where ε_{μν}(x) is everywhere small compared to η_{μν}. The tensor ε_{μν}(x) may be viewed as a superposition of plane waves by Fourier transform:

ε_{μν}(x) = ∫ d^{26}k ε_{μν}(k)e^{ik·x}  (2.58)

Now, consider string theory in this background: the action on the worldsheet is given by

S_{G} = \frac{1}{8π} ∫_{Σ} dμ_{g} g^{mn}∂_{m}x^{μ}∂_{n}x^{ν}G_{μν}(x)  (2.59)

II.23 and the transition amplitude involves the exponential of S_{G}. Since ε_{μν} is small, we may expand in powers of ε_{μν}:

e^{−S_{G}} = e^{−S_{η}} ∑_{n=0}^{∞} \frac{(−1)^{n}}{n!} S_{n} \frac{1}{(8π)^{n}}  (2.60)

with S_{η} the action corresponding to the flat metric and S_{n} defined by

S_{n} = ∫ d^{26}k_{1} … d^{26}k_{n}V (k_{1}) … V (k_{n})  (2.61)

where

V (k) = ε_{μν}(k) ∫_{Σ} dμ_{g}g^{mn}∂_{m}x^{μ}∂_{n}x^{ν}e^{ikx}

The operator V (k) is precisely the vertex operator we constructed for the graviton — written now in general metric g — for a definite momentum k.

Thus, we see that strings couple to the fluctuations in the background metric precisely by the graviton vertex operator. The plane waves may be regarded as creating excitations in the background space-time metric, which make it deviate from the flat metric. (A completely analogous correspondence holds for the other massless states, the dilaton and the anti-symmetric tensor.)

We have however established more precise information: in order to maintain conformal invariance and eliminate negative norm states, the momentum k and the polarization tensor ε_{μν} had to satisfy on-shell conditions

\begin{aligned}k^{2}ε_{μν}(k) = 0 & ⟹ □ε_{μν}(x) = □G_{μν}(x) = 0 \\ k^{μ}ε_{μν}(k) = 0 & ⟹ ∂^{μ}ε_{μν}(x) = ∂^{μ}G_{μν}(x) = 0 \\ ε_{μ}^{μ}(k) = 0 & ⟹ ε_{μ}^{μ}(x) = 0 , \end{aligned}  (2.62)

where we have listed to the right of the double arrows the restatement of the conditions on the metric.

Now, since G_{μν} is almost flat (ε being small), corrections quadratic and higher in ε may be neglected. Considering the Ricci tensor in this approximation, we have

\begin{aligned}R_{μν} & = R_{μλν}^{λ} = ∂_{ν}Γ_{μλ}^{λ} − ∂_{λ}Γ_{μν}^{λ} + O(ε^{2}) \\ 2R_{μν} & = ∂_{μ}∂_{ν}ε_{λ}^{λ} − ∂_{λ}∂_{μ}ε_{ν}^{λ} − ∂_{λ}∂_{ν}ε_{μ}^{λ} + □ε_{μν} \end{aligned}  (2.63)

II.24 The conditions on G_{μν} obtained from the on-shell conditions of ε_{μν}(k) now imply that

R_{μν} = 0 ,  (2.64)

within the almost flat (or linearized) approximation.

In addition, the fact that shifts by null states are physically unobservable implies the gauge invariance of the physical states in ℱ^{+} under the shifts

ε_{μν}(k) ⟶ ε_{μν}(k) + k_{μ}γ_{ν} + k_{ν}γ_{μ}  (2.65)

(with k^{μ}γ_{μ} = 0 for on-shell states). In terms of the metric, this gauge invariance translates into

ε_{μν}(x) → ε_{μν}(x) + ∂_{μ}γ_{ν}(x) + ∂_{ν}γ_{μ}(x).  (2.66)

We recognize this transformation as an infinitesimal diffeomorphism generated by the vector field γ^{μ}(x), in the approximation where O(ε^{2}) terms are neglected. R_{μν} = 0 is invariant under these transformations, as can be checked explicitly.

In fact R_{μν} = 0 is the only Diff(M)-invariant equation that reduces to □G_{μν}(x) = 0 in the linearized limit. Thus, we find that — to leading order in metric fluctuations – the conformal invariance constraints translate into Ricci flatness, i.e. Einstein’s equations in the absence of matter. This discovery supports our proposed identification of ε_{μν}(k), k^{2} = 0 string states with gravitons.

Later on (Lecture VIII), we shall further confirm this identification in another important limit, that of small variations in the derivatives of the metric, the so-called low energy limit.

K) Internal degrees of freedom of open strings: Chan-Paton rules

The end points of the open string are distinguished points. It is consistent with Poincaré and 2-d conformal invariance to put degrees of freedom or charges at the end points. During interactions of several strings, the charges simply flow along the boundary, and should thus

II.25 be conserved.

The simplest case corresponds to adding electric charge to the end points of oriented strings:

Consistency with the interactions requires that the charges be opposite (i.e. corresonding to conjugate representatives) at the two ends. An external U(1) gauge field A_{μ} couples to these charges just in the same way as it couples to point particles with electric charge ±e, moving on worldlines:

S_{A} = ie ∫_{B} dτ ẋ^{μ}(τ)A_{μ}(x(τ)).  (2.67)

Following the derivation of vertex operators from the coupling to background fields that we used for the closed string, we easily obtain the vertex operators for the open string gauge states, by decomposing A_{μ} into Fourier modes

A_{μ}(x) = ∫ d^{D}k ε_{μ}(k)e^{ik·x} ,  (2.68)

where ε_{μ}(k) is the polarization tensor. Thus, the vertex operators are

V (ε, k) = ε_{μ}(k) ∫_{B} dτ ẋ^{μ}(τ)e^{ik·x(τ)}  (2.69)

II.26 with conformal invariance for k^{2} = 0.

We may add further degrees of freedom to the end points of the string in the following way. Let λ^{i}, i = 1, …, N be Grassmann valued fields that live only on the boundary B of the string worldsheet (recall that Neumann b.c. were imposed on B for the map x: Σ → M). A natural candidate for the action for such degrees of freedom is

S_{0}(λ) = ∫_{B} dτ λ̄^{i} \frac{d}{dτ} λ_{i} .  (2.70)

This action is Diff(B) invariant, and does not depend upon the Σ metric. The equation of motion for λ_{i} simply requires that λ_{i} be τ-indepenent, so that λ_{i} may indeed be interpreted as conserved charges. The action S_{0} automatically has SO(N) symmetry when λ_{i} are real (and λ̄^{i} = δ^{ij}λ_{j}) and SO(2N) symmetry when λ_{i} is complex. As a result, quantization automatically produces a Hilbert space of states on which a spinor representation of SO(N) (for λ_{i} real) or SO(2N) (for λ_{i} complex) acts.

To restrict to a subgroup G of SO(N) or SO(2N), and/or a subrepresentation Λ of the corresponding spinor representations, it is necessary to restrict the Hilbert space of states. This is conveniently achieved by introducing a Lagrange multiplier, σ: B → ℝ. For G = U(N), for example, the restriction to the rank p (0 ≤ p ≤ N) totally anti-symmetric tensor representation of U(N), we use the following modification of S_{0}:

S(λ) = ∫_{B} dτ (λ̄^{i} \frac{d}{dτ} λ_{i} + σ(τ)(λ̄^{i}λ_{i} − p))  (2.71)

Analogous modifications may be used for G = SO(N) and Sp(N). Let us now restrict to G = U(N), and p = 1, so that Λ equals the defining representation. Quantization of λ_{i} generates N conserved charges attached to the boundaries of the string

II.27 The string for U(N) gauge group should be oriented, since the charges at both ends of the string transform under inequivalent representations of U(N). The string state now transforms under the representation Λ̄ ⊗ Λ of U(N), which is the adjoint of U(N). The representation matrices T^{a}, with a = 1, …, dim U(N) = N^{2} describe the string state belonging to the representation Λ̄ ⊗ Λ, by the index a. The matrix elements (T^{a})_{i}^{j} of T^{a} specify which λ_{i} and λ̄^{j} are created at the boundary B of the open string. The full Fock space of the open string is given by ℱ_{open} ⊗ Λ̄ ⊗ Λ, and the physical state space is ℱ_{open}^{phys} ⊗ Λ̄ ⊗ Λ.

We observe that for G = U(N) and Λ the defining representation, Λ̄ ⊗ Λ transforms in the adjoint representation and the open string states are thus natural candidates for Yang-Mills particles with gauge group G = U(N). Had we considered any other representation for Λ (other than the defining or its complex conjugate), the string states would not have tranformed precisely under the adjoint representation of U(N), and could not have been interpreted as Yang-Mills particles. Thus the defining representation is singled out by its relevance to describing Yang-Mills particles.

The strings for SO(N) and Sp(N) should be unoriented, since the charges at both ends transform under equivalent representations of SO(N) and Sp(N), respectively. Considering Λ to be the defining representation of SO(N), we obtain string states in the adjoint of SO(N) by the anti-symmetrized tensor product of Λ by itself (Λ ⊗ Λ)_{A}. On the other hand, for Λ the defining representation of Sp(N), we obtain the string states in the adjoint of Sp(N) by the symmetrized tensor product of Λ by itself (Λ ⊗ Λ)_{S}. In each case we recover precisely the string states suitable for Yang-Mills particles.

For exceptional groups, no construction is known that produces string states in the adjoint representation only, and it is widely believed that it is impossible to obtain Yang-Mills particles from open strings with these groups.

A non-Abelian gauge field can couple to the boundaries of the open string, much in the same way as this happens for Abelian gauge fields. Here, however, we must include the coupling of the charges of the gauge field A_{μ}^{a} to those of the string. Let us restrict again

II.28 to the defining representation of U(N). The coupling of A_{μ}^{a} is given as follows:

S_{A}(λ) = ∫_{B} dτλ̄^{i} (δ_{i}^{j} \frac{d}{dτ} + iA_{μ}^{a}ẋ^{μ}(T^{a})_{i}^{j}) λ_{j} + σ(τ)(λ̄^{i}λ_{i} − 1) .  (2.72)

The coupling to the gauge field A_{μ}^{a} may again be used to construct the vertex operators for the creation and annihilation of gauge states of the string. One finds upon quantization that the vertex opertor for the U(1) case is simply to be multiplied by the matrices (T^{a})_{i}^{j}:

V^{a}(ε, k) = T^{a}V (ε, k) .  (2.73)

As the internal degree of freedom and the matrices T^{a} commute with Virasoro, the physical state conditions on V^{a} are satisfied whenever they are on V (ε, k).

It is instructive to examine the effects of these degrees of freedom on interactions as well. For simplicity, we restrict to A_{μ}^{a} = 0. Charges then simply continue to be translated along the boundary B in a conserved way, as shown schematically in the figure below.

Interactions are only allowed if charges are conserved.

In a full string diagram, the effect will be to include a trace over the matrices T, just as in Yang-Mills Feynman rules, as shown schematically below.

∼ tr T^{a} T^{b} T^{c} T^{d}

II.29 With the help of the Yang-Mills vertex operators V^{a} of (2.73) and (2.69), one recovers Yang-Mills scattering amplitudes in the limit where momenta are small compared to the Planch length.

II.30