— DRAFT — IASSNS-HEP-97/72

Lectures 5, 6

III. String Amplitudes and Moduli Space of Curves

Eric D’Hoker

In the previous Chapter, we have quantized free bosonic strings, determined their spectrum and Hilbert space of states under certain conditions: D = 26 and the presence of massless gravitons and Yang-Mills particles. (The case D ≤ 25, allowed by the no-ghost theorem, was shown to be a subcase.)

Now, we shall quantize interacting strings, and investigate quantum mechanical transition amplitudes for the scattering of string states. While operator methods were convenient for determining the spectrum of free strings, the functional integral formulation in terms of a summation over surfaces is better suited for dealing with interactions. Thus, our starting point is the transition amplitude

A = ∑_{topologies Σ (h=0,...,∞)} ∫_{Met(Σ)} Dg \frac{1}{𝒩(g)} ∫_{Diff(Σ,M)} V_{1} … V_{N}e^{−S[x,g]}  (3.1)

with Riemannian space-time M, and external string states represented by vertex operators V_{i}, i = 1, …, N. This was precisely our original starting point with Euclidean signature metric on M. Minkowski amplitudes (for example, for flat M = ℝ_{M}^{D}) are obtained by analytic continuation in external momenta and polarization tensors. The vertex operators were determined in §II in terms of conformal primary fields of weight (1, 1). The summation in the transition amplitude A is over topologies of compact oriented surfaces, since we shall limit ourselves here to the study of the closed oriented string.

The strategy is as follows:

1) At fixed topology of Σ, we use the product structure of Met(Σ) = Weyl(Σ)× conformal classes (Σ) to decompose the measure Dg on Met(Σ) into a product of measures on the orbits of Weyl, on the orbits of Diff(Σ) and on the moduli space ℳ_{h} = Met(Σ)/Diff(Σ) ⋉

III.1 Weyl(Σ). A Jacobian determinant arises, which is essentially the Faddeev-Popov determinant. This decomposition holds for any matter conformal field theory, in particular for any space-time M, defining a conformal field theory.

2) For flat Euclidean space-time M, the combined measure — including the Faddeev-Popov determinant — and the correlation functions, are found to be Weyl-invariant (i.e. Weyl(Σ) is a symmetry group) if and only if D = 26 and gravitons are massless on surfaces of any topology (equivalently, the a = 1 condition of §II holds).

3) Under the assumptions of 2) the string theory is usually called critical. In this case, the formal measure Dg/𝒩(g) on Met(Σ), divided by the volume of the orbits of the symmetry group Diff(Σ) ⋉ Weyl(Σ), naturally projects down to an intrinsically defined measure on ℳ_{h}. Thus, the transition amplitudes for critical strings reduce to well-defined integrals over ℳ_{h} of the matter conformal field theory correlation functions.

4) More generally, the assumptions of 2) may be relaxed, and the matter part may be any conformal field theory of central charge c = 26 and with correlation functions satisfying certain Weyl invariance conditions.

5) In particular, in Lecture 9, we shall generalize the Weyl invariance conditions to the case of curved space-time manifolds M — in certain limits.

6) Under the assumptions of 2), we shall obtain in this chapter 0- and 1-loop transition amplitudes for critical bosonic string theory.

A. Finite-Dimensional Case

Let P be a finite-dimensional manifold, with Riemannian metric G, and associated measure dμ_{G}. Let H be a Lie group acting on P, and assume that dμ_{G} is H-invariant. We wish to reduce the integrals of H-invariant functions on P to integrals on the quotient P̄ = P/H. Let f(x): P → ℂ be H-invariant,

f(xh) = f(x)  for all  x ∈ P, h ∈ H.

III.2 Then for any section s: P̄ → P, we have

∫_{P} dμ_{G}f = ∫P̄ dμ̄ s^{∗}(f)J ∫_{H} dν_{H} ,

where dμ̄ is the measure on P̄ induced from dμ_{G}, dν_{H} is the left-invariant measure on H, and J is a Jacobian (or Faddeev-Popov) determinant which we define below, as follows. First, let us present a figure:

The tangent space at x to P may be decomposed into the part tangent to the orbit of H, generated by the Lie algebra ℋ,

i: ℋ → T_{x}P

and a part from the push-forward of the tangent space to Tx̄P̄:

s_{∗}: Tx̄P̄ → T_{x}(P) ,

as follows:

s_{∗} ⊕ i: Tx̄P̄ ⊕ ℋ → T_{x}(P).

The Jacobian J is now obtained as

J = det(Tx̄P̄ ⊕ ℋ → T_{x}P).

B. Basic Notation: Tensors, Derivatives

Fix a positive definite metric g on Σ and choose local complex coordinates z, z̄ in which

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

Tensors decompose into one-dimensional components of weight (m, n), which are sections of a line bundle K^{(m,n)}:

ϕ = ϕ_{\underbrace{z … z}_{m} \underbrace{z̄… z̄}_{n}}(dz)^{m}(dz̄)^{n} ∈ K^{(m,n)} .

Using the metric g, we identify tensors of weight (m, n) with tensors of weight (m − n, 0). The space of tensors of weight (m, 0) is denoted K^{m}, with K^{1} the space of sections of the canonical line bundle T_{(1,0)}^{∗}(Σ). (Generally, we do not distinguish between a bundle and the space of its sections.)

On sections of K^{m} we have the L^{2} inner product

(ϕ_{1}, ϕ_{2})_{g} = ∫_{Σ} dμ_{g}(g_{zz̄})^{−m}ϕ_{1}^{∗}ϕ_{2}  (3.2)

as well as the covariant derivatives

∇ϕ = ∇(m) ; z̄ϕ + ∇_{z}^{(m)}ϕ

where ∇(m) ; z̄ = ¯∂_{(m)} is the Cauchy Riemannn operator

∇(m) ; z̄: K^{m} → K^{(m,1)} ∇(m) ; z̄ϕ = \frac{∂}{∂z̄} ϕ ⊗ dz̄ .  (3.3)

Equivalently, identifying (m, 1) ∼ (m − 1, 0), we have

\{ \begin{matrix}∇_{(m)}^{z}: K^{m} → K^{m−1} & ∇_{(m)}^{z}ϕ = (g_{zz̄})^{−1} \frac{∂}{∂z̄}ϕ ⊗ (dz)^{−1} \\ ∇_{z}^{(m)}: K^{m} → K^{m+1} & ∇_{z}^{(m)}ϕ = (g_{zz̄})^{m} \frac{∂}{∂z} (g_{zz̄})^{−m}ϕ ⊗ dz . \end{matrix}  (3.4)

(In general, we shall not exhibit the differentials dz and dz̄ in ∇.) The adjoints of these operators with respect to ( , )_{g} are

(∇_{(m)}^{z})^{†} = −∇_{z}^{(m−1)}; (∇_{z}^{(m)})^{†} = −∇_{(m+1)}^{z}  (3.5)

III.4 and the Laplace operators are

\begin{aligned}Δ(m)^{+} & = −2∇_{(m+1)}^{z}∇_{z}^{(m)} \\ Δ_{(m)}^{−} & = −2∇_{z}^{(m−1)}∇_{(m)}^{z} \end{aligned}  (3.6)

We also make use of the Riemann-Roch-Atiyah-Singer theorem

dim Ker∇_{z}^{(m)} − dim Ker∇_{(m+1)}^{z} = (m + \frac{1}{2}) χ(Σ) = (2m + 1)(1 − h) ,  (3.7)

as well as the vanishing theorems:

• h = 0, m ≥ 1,  Ker ∇(m) ; z̄ = 0

(no holomorphic forms on the sphere)

• h ≥ 2, m ≤ − 1,  Ker ∇(m) ; z̄ = 0 .

(no holomorphic vector fields for h ≥ 2)

(3.8)

C. Space of Metrics – Moduli Space of Riemann Surfaces

Let Met(Σ) denote the space of positive definite metrics on Σ, and T_{g}Met(Σ) its tangent space at g, consisting of infinitesimal deformations δg ∈ K^{(1,1)} ⊕ K^{(2,0)} ⊕ K^{(0,2)}. Following the results of §B, we have an L^{2} norm ∥δg∥_{g}^{2} = (δg, δg)_{g} on T_{g}Met(Σ), which makes Met(Σ) into an infinite dimensional Riemannian manifold. The measure Dg is defined with respect to ∥δg∥.

Diff(Σ) acts isometrically on Met(Σ), but Weyl(Σ) does not. The space of orbits of Diff(Σ) ⋉ Weyl(Σ) in Met(Σ) is the moduli space of Riemann surfaces (or complex curves) of genus h:

ℳ_{h} ≡ Met(Σ)/Diff(Σ) ⋉ Weyl(Σ) .  (3.9)

Actually, the group Diff(Σ) is not connected. The group of connected components is the mapping class group G_{Σ}:

G_{Σ} = π_{0}(Diff(Σ)) ,

which is an infinite discrete group for h ≥ 1. The component connected to the identity in Diff(Σ) is denoted by Diff_{0}(Σ). Thus, there arises another natural space: J_{h}, which is Teichmüller space

𝒯_{h} ≡ Met(Σ)/Diff_{0}(Σ) ⋉ Weyl(Σ) ,

III.5 which is the simply connected covering space of ℳ_{h}:

ℳ_{h} = 𝒯_{n}/G_{Σ} .

Teichmüller space is a complex finite dimensional manifold, while moduli space is a complex finite dimensional orbifold (the action of G_{Σ} on 𝒯_{h} produces singularities), both of dimension

dim_{ℂ} 𝒯_{h} = dim_{ℂ} ℳ_{h} = \{ \begin{matrix}0 & h = 0 \\ 1 & h = 1 \\ 3h − 3 & h ≥ 2 . \end{matrix}  (3.10)

We have explained that string transition amplitudes can be reduced to an integral over moduli space. We wish to factorize the measure Dg into its components along Diff(Σ), Weyl(Σ) and ℳ_{h}, i.e. we want to fix the gauge. To examine this factorization concretely, we decompose T_{g}Met(Σ) into the tangent spaces to the orbits of Diff_{0}(Σ) and Weyl(Σ), as explained in the finite dimensional example. We perform this factorization at an arbitrary metric g ∈ Met(Σ), and shall choose a section later on.

We use the fact that Diff_{0}(Σ) is generated by vector fields v = v^{z} ⊕ vz̄ ∈ K^{(−1,0)} ⊕ K^{(0,−1)}. A Weyl transformation will in general be denoted by e^{2σ}, where σ takes real values.

Let the reference metric be g = 2g_{zz̄}|dz|^{2}; an arbitrary metric g + δg in the neighborhood of g may then be conveniently parametrized by a change in g_{zz̄} as well as by Beltrami

differentials δηz ; z̄ ∈ K^{(−1,1)} and δη̄z̄ ; z ∈ K^{(1,−1)}.

g + δg = 2 (g_{zz̄} + δg_{zz̄}) |dz + δηz ; z̄dz̄|^{2} .  (3.11)

The space T_{g}Met(Σ) admits the orthogonal decomposition

T_{g}Met(Σ) = K^{(1,1)} ⊕ K^{(2,0)} ⊕ K^{(0,2)}
δg = 2δg_{zz̄}|dz|^{2} + 2g_{zz̄}δηz ; z̄(dz̄)^{2} + 2g_{zz̄}δη̄z̄ ; z(dz)^{2} .  (3.12)

The infinitesimal action of (δv^{z}, δvz̄, 2δσ) in the Lie aglebra of Diff_{0}(Σ) ⋉ Weyl(Σ) is then simply

\{ \begin{matrix}δg_{zz̄} & = (2δσ + ∇_{z}δv^{z} + ∇z̄δvz̄)g_{zz̄} \\ δηz ; z̄ & = ∇z̄δv^{z} \\ δη̄z̄ ; z & = ∇_{z}δvz̄ . \end{matrix}  (3.13)

While the range of Weyl(Σ) is the full K^{(1,1)}, in general the range of Diff_{0}(Σ) need not be all of K^{(2,0)} ⊕ K^{(0,2)}. Instead, we have the orthogonal decomposition

K^{(2,0)} = Range ∇_{z}^{(1)} ⊕ Ker(∇_{z}^{(1)})^{†} .  (3.14)

The space Ker (∇_{z}^{(1)})^{†} = Ker ∇(2) ; z̄ consists of holomorphic quadratic differentials on Σ, and can be identified with the holomorphic cotangent space of ℳ_{h}:

T_{(1,0)}^{∗}(ℳ_{h}) = Ker ∇(2) ; z̄  (3.15)

(Analogously, we have T_{(0,1)}^{∗}(ℳ_{h}) = Ker ∇_{z}^{(−2)}.) To see this, we notice that holomorphic differentials ϕ_{j} ∈ Ker ∇(2) ; z̄ provide linear forms on the space of differentials δη:

(δη, ϕ_{j}) = ∫_{Σ} dz dz̄ δηz ; z̄ϕ_{jzz}  (3.16)

The above pairing is Weyl(Σ)-invariant and the kernel of ( · , ϕ_{j}) (and ( · , ϕ_{j}^{∗})) is precisely the tangent space to the orbits of Diff_{0}(Σ) and Weyl(Σ); hence the identification. By the Riemann-Roch theorem, and the vanishing theorems listed above, it is easy to see that the dimensions of ℳ_{h} and of Ker ∇(2) ; z̄ ⊕ Ker ∇_{z}^{(−2)} coincide.

More precisely, the holomorphic quadratic differentials ϕ_{j} ∈ Ker ∇(2) ; z̄ form a complex vector space, thus providing a natural almost complex structure on the tangent space of ℳ_{h}. In fact, this almost complex structure is integrable and results in a complex structure on ℳ_{h}; for a given choice of ϕ, the above linear forms define local complex coordinates m_{j}, m̄_{j} on ℳ_{h} given by

δm_{j} = (δη, ϕ_{j})  δm̄_{j} = (δη̄, ϕ̄_{j}) .  (3.17)

We may represent this projection onto ℳ_{h} by a picture analogous to the one used ot discuss

III.7 the finite dimensional example at the beginning of this lecture.

Vice-versa, for a given set of complex coordinates (m_{j}, m̄_{j}) on ℳ_{h}, we may choose a section s of ℳ_{h} in Met(Σ) — also called a slice — of dimension dim ℳ_{h}, and parametrized by (m_{j}, m̄_{j}):

The tangent space at g(m_{j}, m̄_{j}) then decomposes into the tangent directions to the action of Weyl(Σ), the action of Diff_{0}(Σ) on the differentials δη, δη̄, and moduli deformation, given by a basis of Beltrami differentials μ_{j} and μ̄_{j}:

\{ \begin{matrix}δηz ; z̄ = ∇(−1) ; z̄δv^{z} + ∑_{j=1}^{dim ℳ_{h}} δm_{j}μ_{jz̄}^{z} \\ δη̄z̄ ; z = ∇_{z}^{(1)}δvz̄ + ∑_{j=1}^{dim ℳ_{h}} δm̄_{j}μ̄z̄ ; jz . \end{matrix}  (3.18)

We may choose the Beltrami differentials as follows

III.8

\{ \begin{matrix}μ_{jz̄}^{z} = g^{zz̄} \frac{∂g_{z̄z̄}}{∂m_{j}} (m, m̄) \\ μ̄z̄ ; jz = g^{zz̄} \frac{∂g_{zz}}{∂m̄_{j}} (m, m̄) , \end{matrix}  (3.19)

so that μ_{j} and μ̄_{j} are unchanged under Weyl transformations g → e^{2σ}g: μ_{j} → μ_{j}; μ̄_{j} → μ̄_{j}. Thus, under a change of section s, μ_{j} and μ̄_{j} will be shifted only by the action of a vector field in the Lie algebra of Diff_{0}(Σ).

Finally, for Σ with the topology of either the sphere or the torus, we also have holomorphic vector fields in Ker ∇(−1) ; z̄. These generate conformal automorphisms, and do not act on η at all. We have

\{ \begin{matrix}h = 0 & dim Ker∇(−1) ; z̄ = 3 & SL(2, ℂ) \\ h = 1 & dim Ker∇(−1) ; z̄ = 1 & T^{2} \\ h ≥ 2 & dim Ker∇(−1) ; z̄ = 0 & possibly finite automorphism group . \end{matrix}  (3.20)

D. Factorizing the Integration Measure

We decompose the integration measure Dg on Met(Σ) into measures on the orbits of Diff(Σ) and Weyl(Σ) and on ℳ_{h} by using the parametrization developed above for δg in terms of Weyl changes δσ and differentials δη, δη̄:

δg = 2δσg + 2g_{zz̄}δηz ; z̄(dz̄)^{2} + 2g_{zz̄}δη̄z̄ ; z(dz)^{2} .  (3.21)

The relation

∥δg∥_{g}^{2} = 4∥δσ∥_{g}^{2} + 2∥δη∥_{g}^{2} + 2∥δη̄∥_{g}^{2}  (3.22)

implies that Dg = Dσ Dη Dη̄, in obvious notation.

Next, we make use of the decomposition of δη and δη̄ into orbits of Diff_{0}(Σ) and moduli deformations, which we parametrize by coordinates (δm_{j}, δm̄_{j}) on ℳ_{h} and a choice of Beltrami differentials μ_{jz̄}^{z} and μ̄z̄ ; jz, as in (3.18). Using the orthogonal decomposition of the differentials δηz̄ ; z g_{zz̄} according to

K^{(2,0)} = Range∇_{z}^{(1)} ⊕ Ker ∇(2) ; z̄ ,  (3.23)

we evaluate ∥δη∥_{g}^{2} and find

2∥δη∥_{g}^{2} = 2∥∇(−1) ; z̄δv^{z}∥_{g}^{2} + ∥δη⊥ projection of δηz̄ ; z g_{zz̄} onto Ker ∇(2) ; z̄∥_{g}^{2} .  (3.24)

III.9 The second term on the right hand side is easily worked out by choosing an orthogonal basis for Ker ∇(2) ; z̄, with basis vectors ϕ_{j}, j = 1, …, dim ℳ_{h}:

∥δη⊥ projection of δηz̄ ; z g_{zz̄} onto Ker ∇(2) ; z̄∥_{g}^{2}

= ∑_{j}(δη, ϕ_{j})_{g}(ϕ_{j}, ϕ_{j})_{g}^{−1}(ϕ_{j}, δη)_{g}

= ∑_{j,k,ℓ} δm_{k}δm̄_{ℓ}(μ_{k}, ϕ_{j})_{g}(ϕ_{j}, ϕ_{j})_{g}^{−1}(ϕ_{j}, μ_{ℓ})_{g}

(3.25)

The first term on the right hand side of (3.24) is conveniently rewritten in terms of the Laplace operator Δ_{(−1)}^{−}:

2∥∇(−1) ; z̄δv^{z}∥_{g}^{2} = (δv^{z}, Δ_{(−1)}^{−}δv^{z})_{g}

A last point must be clarified before evaluating the measure Dη Dη̄. For the special cases of the sphere (h = 0) and the torus (h = 1), Ker ∇(−1) ; z̄ ≠ 0, and there are (continuous families of) conformal automorphisms of Σ: SL(2, ℂ) for h = 0 and T^{2} for h = 1. The corresponding vector fields leave η and η̄ invariant. Thus, in the change of variables, δη → (δv, δm_{j}), we must restrict to vector fields δv that are orthogonal to Ker ∇(−1) ; z̄. The corresponding restricted volume element on vector fields v is denoted by D′v. The associated determinant of Δ_{(−1)}^{−} must be similarly restricted to non-zero modes.

Putting all together, we find

Dg = det′Δ_{(−1)}^{−} \frac{| det(μ_{j}, ϕ_{k})_{g}|^{2}}{det(ϕ_{j}, ϕ_{k})_{g}} ∏_{j} dm_{j} dm̄_{j} D′v · Dσ .  (3.26)

While originally we chose ϕ_{j} ∈ Ker ∇(2) ; z̄ to form an orthogonal basis, it is now clear that the above formula holds for any basis Ker ∇(2) ; z̄. (The finite dimensional determinants are of dim ℳ_{h} × dim ℳ_{h} dimensional matrices.)

It is standard practice — and extremely useful — to introduce the following combination of determinants

Z_{(n)}^{±}(g) ≡ \frac{Det′Δ_{(n)}^{±}}{det(ϕ_{j}, ϕ_{k})_{g} det(ψ_{a}, ψ_{b})_{g}}  (3.27)

where

• ϕ_{j} ∈ Ker ∇_{(n+1)}^{z} for + ;  ϕ_{j} ∈ ∇_{z}^{(n−1)} for −
• ψ_{a} ∈ Ker ∇_{z}^{(n)}  for + ;  ψ_{a} ∈ ∇_{(n)}^{z}  for − .

III.10 While Z_{(n)}^{±} have non-trivial Weyl transformation properties, we shall show in the next section that this particular combination has simple transformation properties.

Putting all together, we have the following expression for the measure Dg:

Theorem:

Dg = Dσ D̃v Z_{(−1)}^{−}(g) | det(μ_{j}, ϕ_{k})|_{g}^{2} ∏_{j=1}^{dim ℳ_{h}} dm_{j} dm̄_{j} ,  (3.28)

where we have

1) D̃v ≡ D′v det(ψ_{a}, ψ_{b})_{g}. For h ≥ 2, we have D̃v = Dv, i.e. the measure over all vector fields of Diff_{0}(Σ). For h = 0, 1, we shall postpone the explicit construction of D̃v until §G.

2) The measure

| det(μ_{j}, ϕ_{k})_{g}|^{2} ∏_{j} dm_{j} dm̄_{j}  (3.29)

is intrinsically defined on ℳ_{h}. To see this, notice that we chose μ_{j} to be Weyl independent, that ϕ_{j} can be chosen Weyl independent, and that the pairing (μ_{j}, ϕ_{k})_{g} is Weyl invariant. Furthermore, we defined Beltrami differentials by

μ_{jz̄}^{z} = g^{zz̄} \frac{∂g_{z̄z̄}}{∂m_{j}}  (3.30)

so that the above combination is manifestly unchanged under reparametrizations of m_{j} and m̄_{j}.

3) Z_{(−1)}^{−}(g) will be Weyl dependent, as we shall establish in the next section. This object is called the Faddeev-Popov determinant (up to the finite dimensional determinants), expressing the fact that we gauge fixed the Diff_{0}(Σ)-invariance.

4) To complete the construction outlined in the finite-dimensional case, for the projection down onto ℳ_{h} of the measure Dg on Met(Σ), we will have to understand the Weyl dependence of the combined Z_{(−1)}^{−}(g) and matter contributions, which we do in the next section.

E. Weyl Rescalings of Functional Determinants

The determinant combinations Z_{(n)}^{±} (including those with n ∈ \frac{1}{2} + ℤ, which will enter when dealing with the superstring, as we shall see later on) will be considered repeatedly, and we shall need their behavior under Weyl(Σ) rescalings of the metric. Throughout, we shall adopt a regularization and renormalization scheme that preserves Diff(Σ) invariance.

Theorem.

Z_{(n)}^{±}(ge^{2σ}) = Z_{(n)}^{±}(g)e^{−2c_{(n)}^{±}S_{L}(σ,g)}  (3.31)

where c_{(n)}^{±} is given by

c_{(n)}^{±} ≡ 6n^{2} ± 6n + 1  (3.32)

and the Liouville action is independent of n and given by

S_{L}(σ, g) = \frac{1}{12π} ∫_{Σ} dμ_{g} [\frac{1}{2}g^{mn}∂_{m}σ∂_{n}σ + R_{g}σ + β^{2}(e^{2σ} − 1)] .  (3.33)

The constant β depends upon the renormalization scheme used, and will vanish when determinants are defined with ζ-function regularization, or with dimensional regularization.

Since special care must be paid to the effects of the zero modes, we outline the proof. (Various intermediate steps will be proven in Problem Set IV.) First, it suffices to prove the formula for the Weyl transformation of Z_{(n)}^{±} by infinitesimal σ; the result for general σ follows by straightforward integration (Gawȩdzki, prob. 8). Thus, we must prove

δ_{σ} ln Z_{(n)}^{±}(g) = −\frac{c_{(n)}^{±}}{6π} ∫_{Σ} dμ_{g}[R_{g}δσ + 2β^{2}δσ]

(Notice that from the form of this infinitesimal variation, it is clear that the determinant Z_{(n)}^{±} defines a conformal field theory with central charge related to ±c_{(n)}^{±}.)

We define determinants by introducing the ζ-function associated with the eigenvalues λ_{α} of Δ_{(n)}^{±}:

ζ(s) ≡ \begin{matrix}∑ & ′ \\ α \end{matrix} λ_{α}^{−s}

Here, ∑′ denotes the sum over all non-zero eigenvalues λ_{α} of Δ_{(n)}^{±}. It is standard to represent this sum in terms of the heat-kernel of Δ_{(n)}^{±}:

ζ(s) = \frac{1}{Γ(s)} ∫_{0}^{∞} dt t^{s−1}(Tr e^{−tΔ_{(n)}^{±}} − N_{(n)}^{±})  (3.35)

where N_{(n)}^{±} = dim Ker Δ_{(n)}^{±}.

The ζ(s) function series and the heat-kernel integral representation are absolutely convergent for Re(s) > 1. ζ(s) is holomorphic for Re(s) > 1, but may be analytically continued throughout ℂ, with a simple pole at s = 1. The determinants are then defined by

ln Det′Δ_{(n)}^{±} = −ζ′(0)  (3.36)

The non-zero spectra of Δ(n)^{+} and Δ_{(n+1)}^{−} coincide and thus their determinants are equal, so we shall henceforth focus on δ_{(n)}^{−}.

To evaluate the Weyl dependence of determinants, we compute that of ζ(s) and analytically continue to s = 0. We use the following intermediate results (proven in Problem Set IV).

1) N_{(n)}^{±} are Weyl independent

2) δ_{σ}Tr e^{−tΔ_{(n)}^{−}} = −t\frac{∂}{∂t}Tr [2nδσe^{−tΔ_{(n−1)}^{+}} − (2n − 2)δσe^{−tΔ_{(n)}^{−}}]

3) Tr δσe^{−tΔ_{(n)}^{±}} = \frac{1}{4πt} ∫_{Σ} dμ_{g}δσ + \frac{1±3n}{12π} ∫_{Σ} dμ_{g} R_{g} δσ + O(t) (3.37)

4) Tr δσe^{−tΔ_{(n)}^{−}} = −\frac{1}{2n−2}δ_{σ} ln det(ϕ_{j}, ϕ_{k})_{g} + O(e^{−λ_{1}t})

5) Tr δσe^{−tΔ_{(n−1)}^{+}} = \frac{1}{2n}δ_{σ} ln det(ψ_{a}, ψ_{b})_{g} + O(e^{−λ_{1}t})

where λ_{1} is the smallest non-zero eigenvalue of Δ_{(n)}^{−}.

We make use of the above results and carry out the analytic continuation of δ_{σ} ζ(s) to s = 0, and then consider δσζ′(0). For Re(s) > 1, using 1) and 2), we have

δ_{σ}ζ(s) = −\frac{1}{Γ(s)} ∫_{0}^{∞} dt t^{s} \frac{∂}{∂t}Tr [ · ](t)  (3.38)

where we introduced the abbreviation

Tr [ · ](t) ≡ tr[2nδσe^{−tΔ_{(n−1)}^{+}} − (2n − 2)δσe^{−tΔ_{(n)}^{−}}]  (3.39)

III.13 We shall make use of the asymptotic behaviors of Tr [ · ](t) as t → 0 and t → ∞, which are obtained from 3), 4), 5):

\{ \begin{matrix}Tr [ · ](t) = \frac{1}{2πt} ∫_{Σ} dμ_{g}δσ + ^{\frac{c_{(n)}^{−}}{6π}} ∫_{Σ} dμ_{g}δσR_{g} + O(t) \\ Tr [ · ](t) = δ_{σ} ln det(ϕ_{j}, ϕ_{k})_{g} det(ψ_{a}, ψ_{b})_{g} + O(e^{−λ_{1}t}) . \end{matrix}  (3.40)

Here, c_{(n)}^{−} was defined in (3.32).

To separate the asymptotic behaviors as t → 0, t → ∞, we cut [0, ∞] at t = t_{0} > 0, and define δ_{σ}ζ(s) = δ_{σ}ζ_{0}(s) + δ_{σ}ζ_{∞}(s)

\{ \begin{matrix}δ_{σ}ζ_{0}(s) = −\frac{1}{Γ(s)} ∫_{0}^{t_{0}} dt t^{s} \frac{∂}{∂t}Tr [ · ](t) \\ δ_{σ}ζ_{∞}(s) = −\frac{1}{Γ(s)} ∫_{t_{0}}^{∞} dt t^{s} \frac{∂}{∂t}Tr [ · ](t) . \end{matrix}  (3.41)

The first term fails to converge at s = 0, so we subtract the asymptotic behavior of Tr [ · ](t) in 1/t as t → 0:

\begin{aligned}δ_{σ}ζ_{0}(s) = & − \frac{1}{Γ(s)} ∫_{0}^{t_{0}} dt t^{s} \frac{∂}{∂t} (Tr [ · ](t) − \frac{1}{2πt} ∫_{Σ} dμ_{g}δσ) \\ & + \frac{1}{Γ(s)} \frac{1}{2π} ∫_{Σ} dμ_{g}δσ ∫_{0}^{t_{0}} dt t^{s−2} . \end{aligned}

The first integral on the right hand side is convergent at s = 0, while the second may be explicitly analytically continued to s = 0. Taking the derivative at s = 0:

δ_{σ}ζ′_{0}(0) = − ∫_{0}^{t_{0}} dt\frac{∂}{∂t} (Tr [ · ](t) − \frac{1}{2πt} ∫_{Σ} dμ_{g}δσ) − \frac{1}{2πt_{0}} ∫_{Σ} dμ_{g}δσ

= −Tr [ · ](t_{0}) + (Tr [ · ](t) − \frac{1}{2πt} ∫_{Σ} dμ_{g}δσ)|_{t=0}

= −Tr [ · ](t_{0}) + \frac{c_{(n)}^{−}}{6π} ∫_{Σ} dμ_{g}R_{g}δσ .

The term δ_{σ}ζ_{∞}(s) is given by

δ_{σ}ζ_{∞}(s) = −\frac{1}{Γ(s)} ∫_{t_{0}}^{∞} dt t^{s} \frac{∂}{∂t}Tr [ · ](t)

As t → ∞, Tr [ ] → constant + O(e^{−λ_{1}t}), λ_{1} > 0, so that the integral is exponentially convergent, and analytic around s = 0. Hence, using the asymptotics of Tr [ · ](t) as t → ∞:

δ_{σ}ζ′_{∞}(0) = − ∫_{t_{0}}^{∞} dt\frac{∂}{∂t}T r[ ](t)
= −δ_{σ} ln det(ϕ_{j}, ϕ_{k})_{g} det(ψ_{a}, ψ_{b})_{g} + Tr [ · ](t_{0}) .

Putting all together, we have

δ_{σ} ln Det′Δ_{(n)}^{−} = −δσζ′(0)

= δ_{σ} ln det(ϕ_{j}, ϕ_{k})_{g} det(ψ_{a}, ψ_{b})_{g} + \frac{c_{(n)}^{−}}{6π} ∫_{Σ} dμ_{g}R_{g}δσ ,

as announced previously. Notice that in ζ-function definitions of the determinants, no “area term” involving the additional constant β arises.

F. Critical Central Charge – Critical Dimension

We complete the reduction of the measure Dg to a measure on ℳ_{h}, making use of the Weyl transformation properties of functional determinants, derived in §E.

Just as we did in the case of the finite-dimensional example, we choose a section s of Met(Σ) and denote the corresponding metric ĝ(m_{j}, m̄_{j}). A general metric g ∈ Met(Σ) may then be parametrized in terms of the action of Diff(Σ) and Weyl(Σ) on ĝ(m_{j}, m̄_{j}): let f ∈ Diff(Σ) and e^{2σ} ∈ Weyl(Σ), then The action of Diff(Σ) is by isometries, and our definition of functional determinants was invariant with respect to Diff(Σ). Thus, the effect of f^{∗} is immaterial and may be dropped. The effect of Weyl transformations is obtained by combining the results of §D, E, and we have

Dg = D̃vDσZ_{(−1)}^{−}(ĝ)e^{−26S_{L}(ĝ,σ)}| det(μ̂_{j}, ϕ_{k})ĝ|^{2} dm_{j} dm̄_{j} .  (3.42)

We now consider a full transition amplitude A_{h}, evaluated on surfaces Σ of genus h. (To avoid certain complications due to conformal automorphisms, we postpone the cases h = 0, 1 and restrict to h ≥ 2. The cases h = 0, 1 are completely analogous and will be dealt with separately in §I.) Recall that

A_{h} = ∫_{Met(Σ)} Dg \frac{1}{𝒩(g)} ⟨V_{1} … V_{N}⟩_{g} ,  (3.43)

where

⟨V_{1} … V_{N}⟩_{g} = ∫_{Diff(Σ,M)} Dx V_{1} … V_{N}e^{−S[x,g]} .  (3.44)

Given the Weyl transformation properties of Dg established above, it is clear that a matter theory (i.e. the quantum field theory of x fields for given metric g) is singled out that makes the Weyl dependence of the amplitude trivial. We define a critical (bosonic) string theory as one governed by a matter conformal field theory with central charge c = 26, and with physical vertex operators V_{i}, i = 1, …, N such that

⟨V_{1} … V_{N}⟩_{e^{2σ} ĝ} = e^{26S_{L}(ĝ,σ)} ⟨V_{1} … V_{N}⟩ĝ .  (3.45)

(This last condition puts restrictions on the allowed vertex operators in the theory.) For critical string theories, the amplitudes at fixed topology (h ≥ 2) reduce to

A_{h} = ∫_{ℳ_{h}} ∏_{j} dm_{j} dm̄_{j}| det(μ̂_{j}, ϕ_{k})ĝ|^{2}Z_{(−1)}^{−}(ĝ) ⟨V_{1} … V_{N}⟩ĝ
· ∫_{Weyl(Σ)} Dσ ∫_{Diff_{0}(Σ)} Dv/𝒩_{0}(g)

(3.46)

(Recall that for h ≥ 2, D̃v = Dv.) Now, recall that the formal measure Dg/𝒩(g) was the measure on Met(Σ), divided by the volume of the orbits of the group Weyl(Σ) × Diff(Σ).

III.16 Since we restricted our integration to one over ℳ_{h} (and not Teichmüller space T_{h}), we effectively quotiented by the mapping class group. The remaining normalization factor is 𝒩_{0}(g), formally the volume of Weyl × Diff_{0}(Σ). Thus, the formal integrals over Weyl(Σ) and Diff_{0}(Σ) cancel 𝒩_{0}(g):

∫_{Weyl(Σ)} Dσ ∫_{Diff_{0}(Σ)} Dv/𝒩_{0}(g) = 1 .  (3.47)

Thus, we finally obtain an ultimate expression for amplitudes in critical string theory at fixed topology (h ≥ 2); the expression reduces to integrals over moduli of Riemann surfaces:

A_{h} = ∫_{ℳ_{h}} ∏_{ℳ_{h}}dm_{j} dm̄_{j}| det(μ̂_{j}, ϕ_{k})ĝ|^{2}Z_{(−1)}^{−}(ĝ) ⟨V_{1} … V_{N}⟩ĝ  (3.48)

The cases for h = 0, 1 will be discussed in §I, J.

G. Flat Space-Time Manifold M

For flat Euclidean space-time M = ℝ^{D}, we have already established that the central charge of the conformal field theory defined by maps x ∈ Diff(Σ, M) is c = D. Thus, the flat space-time string theory is a critical string theory precisely when D = 26. (We now notice that this condition is the same as the one found for the free string: this is rather fortunate!) Let us work out in detail under what conditions the vertex operators also satisfy the criteria of a critical string.

To do so, it is convenient to start with a study of the Weyl transformation properties of correlation functions of unintegrated exponents:

⟨∏_{i=1}^{N} e^{ik_{i}·x(z_{i})}⟩_{g} = ∫_{Diff(Σ,M)} Dx ∏_{i=1}^{N} e^{ik_{i}·x(z_{i})}e^{−S_{0}[x,g]}  (3.49)

where S_{0} is the Gaussian action, including the constant dilaton Φ_{0}: (We ignore the constant tachyon.)

S_{0}[x, g] = \frac{1}{8π} ∫_{Σ} dμ_{g} x · Δ_{(0)}x + \frac{Φ_{0}}{4π} ∫_{Σ} dμ_{g}R_{g}  (3.50)

The Gaussian integral is carried out by separating the constant mode in x:

x = x_{0} + x′ , ∫_{Σ} dμ_{g} x′ = 0

and making use of the Green functions for x′:

\{ \begin{matrix}Δ_{(0)}G_{g}(z, w) = 4πδ_{g}(z, w) − \frac{4π}{∫_{Σ} dμ_{g}} . \\ ∫_{Σ} dμ_{g}G_{g}(z, w) = 0 \end{matrix}  (3.51)

The Green function at coincident points must be regularized and renormalized in a Diff(Σ)-invariant way. This may be carried out with the help of heat-kernel methods, or simply by renormalizing by the Diff(Σ)-invariant distance:

G_{g}^{R}(z, z) ≡ lim_{w→z}(G_{g}(z, w) + 2 ln dist_{g}(z, w)) .  (3.52)

The Weyl scaling properties of the Green function are then

\{ \begin{matrix}G_{e^{2σ} ĝ}(z, w) & = Gĝ(z, w) + f(z) + f(w) \\ G_{e^{2σ} ĝ}^{R}(z, w) & = GR ; ĝ(z, z) + 2f(z) + 2σ(z) , \end{matrix}  (3.53)

where f(z) is a function required to maintain ∫_{Σ} dμ_{g}G_{g} = 0 under Weyl rescalings. We do not need its explicit form here. One finds

⟨∏_{i} e^{k_{i}·x(z_{i})}⟩_{g} = δ(k)e^{− \frac{1}{2} χΦ_{0}} (\frac{Det′Δ_{(0)}}{∫_{Σ} dμ_{g}} )^{−D/2} ∏_{i,j} e^{\frac{1}{2} k_{i}·k_{j}G_{g}(z_{i},z_{j})} ,  (3.54)

where k ≡ ∑_{i} k_{i}, δ(k) is the D-dimensional δ-function, and the factor ∫_{Σ} dμ_{g} under the determinant arises from the normalization of the zero mode x_{0}. As a result, under g = e^{2σ}ĝ, we have

∏_{i,j} e^{− \frac{1}{2} k_{i}·k_{j}G_{g}(z_{i},z_{j})} = ∏_{i,j} e^{− \frac{1}{2} k_{i}·k_{j}Gĝ(z_{i},z_{j})}e^{−k_{i}^{2}σ(z_{i})} .  (3.55)

All dependence on f(z) cancels in view of momentum conservation k = ∑_{i} k_{i} = 0, as guaranteed by the overall δ(k) factor in (3.54). The Weyl scaling property of the determinant factor is precisely that of Z_{(0)}(g)^{−D/2}. (Z_{(0)}(g) contains an additional factor involving holomorphic Abelian differentials, but this factor is manifestly Weyl invariant.)

Combining all of the above, we find for D = 26

⟨∏_{i}^{N} e^{ik_{i}·x(z_{i})}⟩_{ĝe^{2σ}} = ∏_{i=1}^{N} e^{−k_{i}^{2}σ(z_{i})}e^{26S_{L}(ĝ,σ)}⟨∏_{i=1}^{N} e^{ik_{i}·x(z_{i})}⟩ĝ .  (3.56)

It is now straightforward to examine the Weyl covariance conditions of vertex operators for low mass cases.

1) Tachyon vertex operator

V_{T} (k) = ε(k) ∫_{Σ} dμ_{g}e^{ik·x} .

In order to cancel the explicit Weyl dependence of the measure dμ_{g} on Σ:

dμ_{e^{2σ} ĝ} = dμĝe^{2σ}

against the Weyl dependence of the correlation function, we recover the familiar condition k^{2} = 2 for the tachyon. (This condition also corresponded to a = 1 in §III.)

2) The massless vertex operators

V_{ε}(k) = ε_{μν}(k) ∫_{Σ} dμ_{g}g^{mn}∂_{m}x^{μ}∂_{n}x^{ν}e^{ik·x} ;

the Weyl dependence of the part dμ_{g}g^{mn} is trivial, so we need the familiar condition k^{2} = 0. In addition, Weyl dependence appears through the contraction of ∂_{m}x^{μ} with the exponential, and is cancelled by the transversality conditions

k^{μ}ε_{μν}(k) = k^{ν}ε_{μν}(k) = 0 .

Finally, from the contraction of ∂_{m}x^{μ} and ∂_{n}x^{ν}, (actually only the trace-part contributes), a curvature R_{g} term appears, and Weyl invariance for the trace-part is achieved through the addition of a (classically non-Weyl invariant) part involving the curvature:

See Prob. Set #5.

V_{ε}(k) = ε_{μν}(k) ∫_{Σ} dμ_{g}g^{mn}∂_{m}x^{μ}∂_{n}x^{ν}e^{ikx} + αε_{μ}^{μ} ∫_{Σ} dμ_{g}R_{g}e^{ik·x} .

The physical particle corresponding to this vertex operator is the dilaton. Recall that the dilaton field Φ entered the generalized sigma model action with a Gaussian curvature term R_{g} as well.

3) Higher mass states: the general structure of vertex operators is given by

V_{ε}(k) = ε_{μ_{1}...μ_{2p}} ∫_{Σ} dμ_{g}g^{m_{1}n_{1}} … g^{m_{p}n_{p}}∂_{m_{1}}x^{μ_{1}} … ∂_{n_{p}}x^{μ_{2p}}e^{ik·x}

plus terms involving higher derivatives on x. To cancel the overall exponential Weyl dependence, we must require k^{2} = −2p+2, yielding the same mass spectrum as in the free string case. In complete analogy with the case 2), there will be further conditions of transversality on ε and and further terms involving R_{g} will have to be added to render the trace parts Weyl invariant.

H. Non-Critical Strings

Conformal matter field theories with c ≠ 26 are called non-critical string theories. The amplitudes for these theories retain a non-trivial Weyl dependence through the Liouville action, and through the vertex operators. Its direct quantization requires understanding the dynamics of Liouville theory. Despite much effort, there is no complete picture to date of how this can be achieved consistently, and without the presence of a tachyon.

A tremendous number of results was obtained indirectly (by matrix model techniques) when c is less than 1 and rational.

I. Tree Level Amplitudes

Tree level amplitudes correspond to the worldsheet topology of a sphere S^{2}, which we stereographically project onto the complex plane ℂ. The moduli space ℳ_{0} of the sphere consists of just a single point, which we may choose to represent by the round metric, of curvature Rĝ = 1, and area 4π:

ĝ = \frac{4|dz|^{2}}{(1 + |z|^{2})^{2}}  z ∈ ℂ .  (3.57)

The group of conformal automorphisms is PSL(2, ℂ) and acts by Möbius transformations

ζ: z → ζ(z) = \frac{az + b}{cz + d} ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ PSL(2, ℂ) .

A convenient basis for the associated vector fields (called conformal Killing vectors) is

ψ_{1}^{z} = 1  ψ_{2}^{z} = z  ψ_{3}^{z} = z^{2} ,

and similarly for ψz̄ ; a, a = 1, 2, 3.

III.20

To obtain the general expression for tree level transition amplitudes, we must first complete the treatments of conformal automorphisms, which we had postponed in §F, and of Weyl transformation of the measure. The starting point is our expression (3.28) for the measure Dg, considered now for h = 0

Dg = D_{σ}D̃vZ_{(−1)}^{−}(ĝ)e^{−26S_{L}(ĝ,σ)}  (3.58)

where D̃v = D′v det(ψ_{a}, ψ_{b})_{g}, ψ_{a} ∈ Ker∇(−1) ; z̄, and the measure D′v is over Diff_{0}(Σ) vector fields orthogonal to Ker∇(−1) ; z̄ ⊕ Ker∇_{z}^{(1)}. Tree level amplitudes are then given by

A_{0} = ∫_{Weyl(Σ)} Dσ ∫_{Diff_{0}(Σ)/PSL(2,ℂ)} D̃v/𝒩(g)Z_{(−1)}^{−}(ĝ) ⟨V_{1} … V_{N}⟩ĝ  (3.59)

using the same notations as in III.19.

While the group PSL(2, ℂ) leaves the metric g invariant (up to Weyl rescaling), it does transform the positions of vertex operator insertions non-trivially. It is natural and — as we shall establish later on — required for unitarity of the transition amplitudes, to divide by the (formal) volume of the full Diff_{0}(Σ) group, and not just of Diff_{0}(Σ)/PSL(2, ℂ). To achieve this, we proceed as follows.

1) For N ≥ 3 cases Let the last three vertex operators be given by (of course, we may choose any three

vertex operators)

V_{i} = ∫_{ℂ} d^{2}z_{i} W_{i}(z_{i}) , i = N − 2, N − 1, N .

We propose to make a change of variables from the vertex insertion points z_{i}, z̄_{i} i = N − 2, N − 1, N to the group elements of PSL(2, ℂ), thus completing the measure D′v into Dv. This is possible since, given any three points z_{i} ∈ ℂ, there is a unique element ζ ∈ PSL(2, ℂ), such that ζ maps z_{i}^{0} onto any triplet z_{i} ∈ ℂ, with

z_{i} = ζ(z_{i}^{0}) .

III.21 to carry this out in detail, let ψ_{a}^{z} be a basis of the Ker∇(−1) ; z̄, a = 1, 2, 3, chosen to be Weyl invariant. We parametrize general vector fields v in psl(2, ℂ) by complex parameters λ_{a}, λ̄_{a} with

v^{z} = ∑_{a=1}^{3} λ_{a}ψ_{a}^{z} vz̄ = ∑_{a=1}^{3} λ̄_{a}ψz̄ ; a .

The measure on PSL(2, ℂ) vector fields is then

dμ_{v}^{(6)} = Dv/D′v = ∏_{a=1}^{3} dλ_{a} dλ̄_{a} det(ψ_{a}, ψ_{b})_{g}

where the determinant det(ψ_{a}, ψ_{b})_{g} arises as the Jacobian of the coordinate change from v to λ. On the other hand, from the action of PSL(2, ℂ) on the points z_{i}^{0}: z_{i} = ζ(z_{i}^{0}) = e^{v}z_{i}^{0}, we have dz_{i} = ∑_{a} dλ_{a} ψ_{a}(z_{i}), so that the measure on d^{2}z_{i} becomes:

∏_{i=N−2}^{N} d^{2}z_{i} = ∏_{a=1}^{3} dλ_{a} dλ̄_{a} | det ψ_{a}(z_{i})|^{2} .

Clearly, all factors in this identity are Weyl-invariant. Combining all the results obtained, we have

D̃v ∏_{i=N−2}^{N} d^{2}z_{i} = D′v det(ψ_{a}, ψ_{b})_{g} ∏_{a=1}^{3} dλ_{a} dλ̄_{a} | det ψ_{a}(z_{i})|^{2}
= Dv | det ψ_{a}(Z_{i})|^{2}

(3.60)

We may now divide by the formal volume of Weyl(Σ) ⋉ Diff(Σ) just as in the cases h ≥ 2, and we obtain the final expression for tree level amplitudes with N ≥ 3:

A_{0} = z_{(−1)}^{−}(ĝ)| det ψ_{a}(z_{i})|^{2}⟨V_{1} … V_{N−3} ∏_{i=N−2}^{N} W (Z_{i})⟩ĝ ,  (3.61)

valid for any 3-points z_{i}. We shall evaluate some of these amplitudes momentarily for flat space-time.

2) N = 0, 1, 2 cases Clearly, here, no three vertex operators are available to complete the integration D′v

into Dv. In fact, these amplitudes vanish! To justify this, notice that D′v can at most be

III.22 completed by the vector fields in PSL(2, ℂ)/S_{N} where S_{N} is the stabilizer of N points on the sphere. Thus, we are left to divide out by the volume of S_{N}

∫_{Weyl(Σ)} Dσ ∫_{Diff_{0}/(PSL(2,ℂ)/S_{N})} D′v/𝒩(g) ∼ \frac{1}{Vol(S_{N})} = 0 .  (3.62)

For N = 2, fix the two points at 0 and ∞, and S_{2} is the group of dilations, whose volume is ∞.

An alternative argument that the amplitudes should vanish is that no conformal invariant amplitudes can be constructed, except with value 0. The physical interpretations of these vanishing amplitudes are as follows:

1)  N = 0  tree-level vacuum energy is zero

“vanishing of the cosmological constant”.

2) N = 1 the “tadpole” amplitudes vanish,

i.e. conformal field theory provides solutions (no linear terms) to string theory, at tree level.

3)  N = 2  no mass corrections to tree level.

3) Green function and determinants The scalar Green function on the sphere, with metric ĝ, is given by

G(z, z′) = − ln \frac{|z − z′|^{2}}{(1 + |z|^{2})(1 + |z′|^{2})} .  (3.63)

The determinants involving Δ_{(0)} and Δ_{(−1)}^{−} are just constants. They may be computed explicitly, but we shall not do so here. Instead, we abbreviate

𝔷 ≡ Z_{(−1)}^{−}(ĝ) (\frac{Det′Δ_{(0)}}{∫_{Σ} dμĝ} )^{−13} .  (3.64)

Furthermore, the only remaining finite-dimensional determinant is

\begin{aligned}| det ψ_{a}(z_{i})|^{2} & = | det ( \begin{matrix}1 & 1 & 1 \\ z_{N−2} & z_{N−1} & z_{N} \\ z_{N−2}^{2} & z_{N−1}^{2} & z_{N}^{2} \end{matrix} ) |^{2} \\ & = |(z_{N−2} − z_{N−1})(z_{N−1} − z_{N})(z_{N} − z_{N−2})|^{2} . \end{aligned}  (3.65)

4) Tachyon Amplitudes The tachyon vertex operator is particularly simple:

V_{T} (k) = ∫_{Σ} d^{2}z W_{T} (z, k)  W_{T} (z, k) = ε\sqrt{ĝ} e^{ik·x} .  (3.66)

Here, ε is a constant, independent of k, and plays the same rôle as the field normalization Z-factors in the Lehmann Symanzik Zcmmerman (LSZ) formalism in quantum field theory: ⟨k|φ(x)|0⟩ = Z e^{ik·x} for a scalar field φ(x) and its associated 1-particle state |k⟩ momentum k, and |0⟩ the ground state.

We now make use of the results in p. II.23 on the correlation function of exponential operators. We introduce the renormalized ε_{R} by absorbing a multiplicative renormalization factor of the exponential

ε_{R} = ε e^{−G_{R}(z_{i},z_{i})ĝ}  (3.67)

The amplitude now takes the form

A_{0} = δ(k)e^{−2Φ_{0}}ε_{R}^{N} 𝔷 ∏_{i=1}^{N−3} ∫_{ℂ} d^{2}z_{i} ∏_{i=1}^{N} \sqrt{ĝ(z_{i})} e^{− \frac{1}{2} ∑_{i≠j} k_{i}·k_{j}G(z_{i},z_{j})} · | det(ψ_{a}(z_{i}))|^{2} . (3.68)

The contributions to G of the form ln(1+|z_{i}|^{2}) sum up and, using momentum conservation and k_{i}^{2}, precisely cancel the \sqrt{ĝ} prefactors. We are left with

A_{0}(k_{1} … k_{N}) = δ(k)e^{−2Φ_{0}}ε_{R}^{N} 𝔷| det ψ_{a}(z_{i})|^{2} ∏_{i=1}^{N−3} d^{2}z_{i} ∏_{i<j}^{N} |z_{i} − z_{j}|^{2k_{i}·k_{j}}  (3.69)

Now, z_{N−2}, z_{N−1}, z_{N} were any 3 points on the space, so we may let them equal 0, 1, ∞, respectively, and we find

A_{0}(k_{1}, …, k_{N}) = δ(k)e^{−2Φ_{0}}ε_{R}^{N} 𝔷 ∏_{i=1}^{N−3} ∫_{ℂ} d^{2}z_{i} ∏_{i<j}^{N−1} |z_{i} − z_{j}|^{2k_{i}·k_{j}} .  (3.70)

It is very instructive to examine amplitudes for small N:

(1) \underline{three point function}

A_{0}(k_{1}, k_{2}, k_{3}) = 𝔷 e^{−2Φ_{0}}ε_{R}^{3}δ(k) .  (3.71)

III.24 This 3-point function provides the 3-tachyon on shell coupling.

(2) \underline{four point function}

A_{0}(k_{1}, …, k_{4}) = 𝔷 e^{−2Φ_{0}}ε_{R}^{4}δ(k) ∫_{ℂ} d^{2}z |z|^{2k_{1}·k_{2}}|1 − z|^{2k_{1}·k_{3}} .  (3.72)

It is standard to introduce the 3 Lorentz invariants s, t, u that characterize the kinematics of the 4-point function:

\{ \begin{matrix}s ≡ −(k_{1} + k_{2})^{2} \\ t ≡ −(k_{2} + k_{3})^{2} \\ u ≡ −(k_{1} + k_{3})^{2} . \end{matrix}  (3.73)

(Note that s + t + u = −8.) In terms of s, t, u the four point function is

A_{0}(k_{1}, …, k_{u}) = 𝔷 e^{−2Φ_{0}}ε_{R}^{4}δ(k) ∫_{ℂ} d^{2}z|z|^{−s−4}|z − 1|^{−u−4} .  (3.74)

This integral expression is absolutely convergent in the following region 𝒟, defined by

\{ \begin{matrix}Re(s) < −2 \\ Re(u) < −2 \\ Re(t) < −2 \end{matrix}  or  \{ \begin{matrix}Re(s) < −2 \\ Re(t) < −2 \\ Re(s + t) > −6 . \end{matrix}  (3.75)

Taking just the real parts:

Let s, t ∈ 𝒟, then we may evaluate the amplitude by elementary methods, and we find the Virasoro-Shapiro amplitude:

A_{0}(s, t) = 𝔷e^{−2Φ_{0}}ε_{R}^{4}δ(k)π \frac{Γ(−1 − s/2)Γ(−1 − t/2)Γ(−1 − u/2)}{Γ(2 + s/2)(Γ(2 + t/2)Γ(2 + u/2)} .  (3.76)

Since we know the analytic continuation of Γ(z) throughout ℂ, with poles only at z ∈ −ℕ, we automatically have an analytic continuation of the full amplitude A_{0}(s, t), throughout (s, t) ∈ ℂ × ℂ. As a byproduct, we also immediately have the amplitude in Minkowski space-time momenta, by letting

k_{i}^{0} → −ik_{i}^{0}

\{ \begin{matrix}s → s + iϵ \\ t → t + iϵ & δ(k) → iδ(k) \\ u → u + iϵ \end{matrix}

where it is understood that ϵ > 0 and that we only consider the limit in which ϵ → 0. Thus, the 4-point amplitude in Minkowski space-time is given by

A_{0}(s, t) = 𝔷e^{−2Φ_{0}}ϵ_{R}^{4}δ(k)πi\frac{Γ(−1 − 2/2 − iϵ)Γ(−1 − t/2 − iϵ)Γ(−1 − u/2 − iϵ)}{Γ(2 + s/2 + iϵ)Γ(2 + t/2 + iϵ)Γ(2 + u/2 + iϵ)} . (3.77)

The amplitude has only pole singularities — as expected at tree level — and their location is easily identified. The Γ-function contributions from the denominator are entire. From the numerator, we have poles at

\{ \begin{matrix}s = −2 + 2n_{s} & n_{s} ∈ ℕ \\ t = −2_{2}n_{t} & n_{t} ∈ ℕ \\ u = −2 + 2n_{u} & n_{u} ∈ ℕ . \end{matrix}  (3.78)

These poles precisely correspond to the creation of intermediate string states whose mass

III.26 is the above value for s, t or u. The states may occur either in the s, t or u channel:

It is instructive to identify where these singularities arise from in the integral representation for the amplitude:

s poles: z → 0
t poles: z → ∞
u poles: z → 1 .

The fact that three QFT diagrams arise from a single string diagram is the simplest example of a general phenomenon mentioned in the introduction.

Let us concentrate on the s = −2 pole, which corresponds to a tachyon intermediate state. The diagram there factorizes into a propagator, and 2 three point functions for external tachyon:

Insisting upon this factorization, we obtain a non-linear equation between the 3-point function and 4-point function, expressing factorization — or unitarity — of the transition amplitudes (or so-called S-matrix elements). This relation fixes ε_{R}:

ε_{R}^{2} = \frac{2π}{𝔷} e^{2Φ_{0}} .  (3.79)

J. One loop amplitudes

At one loop, the worksheet has the topology of a torus, and there is a single complex modulus τ ∈ ℂ. We represent Σ by

Σ = ℂ/(ℤ + τZ) Im τ > 0

with the flat metric ĝ = 2|dz|^{2}. Teichmüller space is 𝒯_{1} = {τ ∈ ℂ, Im τ > 0}, the mapping class group is G_{Σ} = SL(2, ℤ), acting on τ as PSL(2, ℤ) by

τ → τ′ = \frac{aτ + b}{cτ + d}  ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℤ).  (3.80)

Moduli space is conveniently taken as 𝒯_{1}/PSL(2, ℤ) = ℳ_{1}, but we shall have to recall that the full mapping class group also divides out by {±I} in SL(2, ℤ). It is standard to take the fundamental domain

ℳ_{1} = \{τ = τ_{1} + iτ_{2}, τ_{1,2} ∈ ℝ, τ_{2} > 0, |τ_{1}| ≤ \frac{1}{2}, |τ| ≥ 1\} .  (3.81)

Determinants of Laplace operators on tensors of any integer weight are given by the case of Laplacians on functions, which was computed by Gawȩdzki:

\begin{matrix}Det′Δ_{(0)} & = τ_{2}^{2} |η(τ)|^{4} \\ η(τ) ≡ e^{iπτ/12} & ∏_{n=1}^{∞}(1 − e^{2πinτ}) . \end{matrix}  (3.82)

The Green function on the scalar field x is given by

G(z, w|τ) = − ln |\frac{ϑ_{1}(z − w|τ)}{ϑ′_{1}(0|τ)}|^{2} − \frac{π}{2τ_{2}} (z − w − z̄ + w̄)^{2} ,  (3.83)

where ϑ_{1}(z|τ) is the Jacobi theta function that has a zero at z = 0.

The N-string transition amplitude is given by the following expression

A_{1}(k_{1}, …, k_{N}) = ∫_{ℳ_{1}} \frac{d^{2}τ}{2τ_{2}^{2}} \frac{1}{(4π^{2}τ_{2})^{12}} \frac{1}{|η(τ)|^{48}} ≪ V_{1}, …, V_{N} ≫  (3.84)

where ≪ V_{1} … V_{N} ≫ stands for the normalized correlation function of the vertex operators V_{1} … V_{N}: ≪ V_{1} … V_{N} ≫= ⟨V_{1} … V_{N}⟩ / ⟨1⟩.

III.28 For example, considering only tachyon vertex operators, we have

≪ V_{1} … V_{N} ≫= δ(k)ε_{R}^{N} ∏_{i=1}^{N} ∫ d^{2}z_{i} exp (−\frac{1}{2} ∑_{i≠j} k_{i} · k_{j}G(z_{i}, z_{j}|τ))  (3.85)

There is no dependence on the dilaton constant Φ_{0} here, because the Euler number on the torus vanishes. Other correlation functions may be similarly derived.

To one loop order, we may now also have non-zero N = 0, 1, 2 point functions. Let us consider N = 0 first: ≪ 1 ≫= 1.

A_{1}( · ) = Vol(M) ∫_{ℳ_{1}} \frac{d^{2}τ}{2τ_{2}^{2}} \frac{1}{(4π^{2}τ_{2})^{12}} \frac{1}{|η(τ)|^{48}} .  (3.86)

Using the asymptotics for Im τ → ∞,

η(τ) ∼ e^{iπτ/12} = e^{iπτ_{1}/12}e^{−πτ_{2}/12}  (3.87)

we see immediately that this amplitude diverges for large τ_{2}. In fact, this η(τ) factor is common to all amplitudes, for any type of vertex operators, and produces a divergence in any amplitude, for any arrangement of external momenta. We shall not show this here in general, but leave this point to be checked by the reader. Thus, bosonic closed oriented string theory is divergent in flat space-time.

Nature of Divergence

Let us try to gain additional insight into the physical origin of this divergence. To do so, it is helpful to exhibit the dependence on internal momenta by inserting a factor:

1 = ∫ d^{26}k δ (k − \frac{1}{2πi} ∮_{A} dz ∂_{z}x) ,  (3.88)

where A is the cycle in Σ whose pre-image in ℂ runs from z = 0 to z = 1. We find

A_{1}( · ) = Vol(M) ∫_{ℝ^{26}} d^{26}k ∫_{ℳ_{1}} \frac{d^{2}τ}{2τ_{2}} \frac{1}{(4π^{2})^{12}} \frac{1}{|η(τ)|^{48}} e^{−2πτ_{2}k^{2}} .  (3.89)

Substituting the asymptotics of the η(τ)-function

A_{1}( · ) = Vol(M) ∫_{ℝ^{26}} d^{26}k ∫_{ℳ_{1}} \frac{d^{2}τ}{2τ_{2}} \frac{1}{(4π^{2})^{12}} e^{−2πτ_{2}(k^{2}−2)}(1 + #e^{−2πτ_{2}} + ⋯) , (3.90)

it is now immediately apparent why a divergence occurs, and where it occurs. For τ_{2} → ∞, the momentum range 0 ≤ k^{2} < 2 contributes an exponentially growing factor. In fact, integrating out τ_{2} for large τ_{2}:

∫ \frac{dτ_{2}}{τ_{2}} e^{−2πτ_{2}(k^{2}−2)} ∼ \frac{1}{k^{2} − 2}

This contribution arises from the tachyon! It is an I.R. problem related to the stability of flat space-time.

It is equally instructive to see where the integral is convergent: at large k^{2}, since the range of τ_{2} is cut off from below by the choice of ℳ_{2}, the integral is exponentially suppressed for large momenta k^{2}. Thus, this loop amplitude is superconvergent at high energy, in contrast with field theory.

In fact, we can use this simple example to exhibit the distinction between what this partition function would look like in quantum field theory and what it looks like in string theory. In field theory, we may imagine the partition function for the same states, given

III.30 by the same generating function. The only difference is the region ℳ_{1}:

Integration region for QFT

with the same states as string theory.

Now, since τ_{2} reaches down to τ_{2} = 0, the superconvergence disappears and the integrand is power behaved in momenta. Thus, we see here that a crucial difference between string theory and quantum field theory arises from modular invariance, producing ultraviolet finite answers in the case of strings