Structure of string amplitudes

The asymptotic states for which we want to calculate a scattering am-plitude live in string theory at the end of infinitely long worldsheets, namely strips for open strings and tubes for closed strings. In the spirit of “summing over possibilities”, we have to sum over all worldsheets allowed in the considered theory, that connect these asymptotic pieces.

Focusing for a moment on closed oriented strings, the relevant world-sheets are connected, compact, closed surfaces, which are classified by

Aclosed + + + . . .

Aopen + + + . . .

Figure 2.2: Sum over topologies for oriented closed (top) and open (bottom) string four-point amplitude.

Aclosed g⁻_s²

∫

M0,4

+

∫

M1,4

+g²_s

∫

M2,4

+ . . .

Aopen g⁻_s¹

∫

M0,4

+

∫

M1,4

+ g_s

∫

M2,4

+ . . .

Figure 2.3: Expansion of oriented four-point open and closed string amplitude in terms of integrals over moduli spaces of punctured Riemann surfaces. We have absorbed the powersgⁿ_s^candgⁿ_s^o/2into the vertex operators, cf. (2.21).

their genus (number of handles). Hence, we obtain a sum over genera, as depicted in Figure2.2. Since the dilaton contribution to the worldsheet action is justφχ, whereχ 2(1−g)is the Euler characteristic of the genusgworldsheet, in the path integral, this worldsheet is weighted by a factor e^{−2φ0(1−g)} g⁻_s^2+2g. Hence, as long as gs 1, the genus expansion is a perturbative expansion in the string coupling where the genus corresponds to the loop order.

For each genus, the string amplitude is given as a path integral over the worldsheet metricγand the embedding field Xin the Polyakov action (2.4),

1

vol(diff×Weyl)

∫

DXDγe^{−SPoly[X,γ]}, (2.19) where the integral is taken over a worldsheet with the desired asymptotic states at the boundaries and we have divided by the volume of the gauge group to account for the gauge freedom. Note that this expression does not explicitly include fermionic fields, but this and the following arguments in this section apply equally to the superstring.

In order to compute (2.19), we use the conformal symmetry of the action to pull the infinite stretches of the worldsheet back to the compact surface in the center and replace the asymptotic states by insertions of appropriate operators on the worldsheet, which now becomes a punctured Riemann surface, as depicted in Figure2.3. These operators are called vertex operators and are obtained from the CFT operator–

state correspondence. Open-string vertex operators are inserted on the boundary of the worldsheet and closed-string operators in the bulk.

The path integral overXin (2.19) then becomes aCFTcorrelator of the vertex operators and after fixing the remaining diffeomorphism× Weyl gauge freedom, the integral over the metricγbecomes an integral over the moduli spaceM_g,n of the genus g Riemann surface withn marked points. Fixing the gauge cancels the volume of the gauge group in (2.19) and introduces ghosts, as mentioned in Sections2.1.1and2.1.2.

When we are considering general string interactions involving open and closed strings on a possibly non-orientable worldsheet, we have to sum over compact, connected surfaces. These are characterized by their number of handles g, the number of boundariesband the number of cross-capscthat one has to attach to the sphere to obtain them. The Euler characteristic is given by

χ2−2g−b−c. (2.20)

Adding a handle to the worldsheet decreasesχby 2 and corresponds to emission and absorption of a closed string, hence closed string vertex operators come with a factor of g_s. Adding a boundary decreases χ by one and corresponds to emission and absorption of an open string, hence, open string vertex operators come with a factor ofg¹_s^/2.

Putting everything together, the resulting expression has the form A_g(k₁, . . . ,k_n) g^−χ+nc+

1 2no s

∫

M_g,n

dµ_g,nh Ön

i1

Vⁱ

g(k_i,z_i)ghostsi, (2.21) wherenoandncare the number of open and closed vertex operators, dµ_g,n is the measure on M_g,n, k₁, . . . ,k_n jointly denote momenta, polarizations and other data of the asymptotic states andVⁱ

g(k_i,z_i)is the genus gvertex operator of thei^thexternal state, inserted at position z_i. Note that the prefactors gⁿ_s^c andgⁿ_s^o/2are usually absorbed into the vertex operators, as in Figure2.3. In (2.21), an integral over the (unfixed) insertion positions and a sum over the ways how to distribute the open string vertex operators over the boundaries and how to order them, is included in dµg,n. The detailed structure of the ghost contribution depends on the external states and the genus of the worldsheet.

Note how peculiar (2.21) is: A spacetime amplitude in 10 or 26 dimen-sions is given in terms of a 2dCFTcorrelator. In particular, momentum conservation for the external momentak_i arises very indirectly from the zero mode integral of the worldsheet fields, as will be demonstrated in Section3.2.1.

The tree-level and one-loop open and closed, orientable and non-orientable worldsheets are collected together with their number of handles g, boundariesb, cross-capscand Euler numberχin Table2.1.

In the type-II and heterotic theories, only orientable worldsheets without boundaries are allowed and hence there are only closed,

closed/open orientability surface g b c χ

closed orientable sphere 0 0 0 2

closed orientable torus 1 0 0 0

closed non-orientable real projective plane 0 0 1 1 closed non-orientable Klein bottle 0 0 2 0

open orientable disk 0 1 0 1

open orientable cylinder 0 2 0 0

open non-orientable Möbius strip 0 1 1 0 Table 2.1: Tree-level and one-loop open and closed worldsheets of oriented

and unoriented strings withghandles,bboundaries,ccross-caps and Euler numberχ.

oriented strings, whose tree-level and one-loop contributions come from the sphere and torus, respectively.

In type-Itheory, the worldsheets can be non-orientable and can have boundaries, hence all surfaces in Table2.1contribute. For closed string scattering, we have only the sphere at tree-level (χ2), but at one-loop (χ0) not only the torus, but also the Klein bottle, cylinder and Möbius strip contribute. Additionally, we have the real projective plane and the disk at “one-half-loop” withχ1. For open strings in type-I, we have the disk at tree-level (χ1) and the cylinder and Möbius strip at one-loop (χ0).

As mentioned in Section2.1.3, theUVfiniteness of string perturbation theory is an important argument when considering string theory as a theory of quantum gravity and hence it is interesting to consider possible divergences in the expression (2.21). On the one hand, the integrand in (2.21) could have a singularity for some points of the moduli space.

For unitaryCFTs, correlation functions have no singularities and hence all possible singularities have to come from the non-unitary ghostCFT. Arguments for the absence of these singularities were given e.g. in [60–

62] and a systematic procedure to avoid them is given in [85,86]. On the other hand, the integral overM_g,nin (2.21) could diverge and although in all known examples the integral is finite, it is not known whether this is true in general. Divergences of the integral overM_g,n are due to degenerations of the Riemann surface and comparing the structure of (2.21) to field theory amplitudes shows that these degenerations correspond toIRdivergences in the field theory language and similarly singularities of the integrand correspond toUV divergences [87]. In summary, string theory appears to beUVfinite, while the question ofIR

finiteness is still open.