Oriented open theory

The open string oriented one-loop amplitudes are defined on the cylinder C2, which can be

represented as a strip of length 2πt with two boundaries at σ1 = 0, ` and the line σ2 = 0 and σ2= 2πt identified. We will set ` ≡ π in the sequel.

∼ =

• The cylinder C2 is described by one real modulus t. Unlike the torus, there is no analogue

of the modular group action P SL(2, Z). Thus the modulus can take values in the full regime 0 ≤ t ≤ ∞.

• The group of conformal Killing transformations consists of translations parallel to the σ2_-

axis, i.e. such that they preserve the boundary of the strip at σ1 _{= 0, π. Its volume is}

2πt.

The vacuum amplitude

The computation is very similar to the computation on the torus.

• For simplicity we consider the situation of a stack of N D-branes filling the entire 26- dimensional space, corresponding to (NN) boundary conditions in all dimensions. Gener- alisations to lower-dimensional branes are simple.

• As in the closed sector, the ghost contribution turns out to cancel the oscillator trace of precisely two non-transverse directions. With this in mind the amplitude is

ZC2 = Z ∞ 0 dt 2tTr e −2πt(L0−24c) _(5.118) = iV26 Z ∞ 0 dt 2t(8π 2 α0t)−13Tr0_⊗24 i=1Xiq L0−241_{. (5.119)}

For a stack of N concindent D-branes filling all of spacetime this is ZC2= iV26N 2Z ∞ 0 dt 2t(8π 2_α0_t)−13_η(it)−24_{. (5.120)}

• The IR-limit, t → ∞, works out like for the closed string: The only IR-divergent term is due to the open tachyon, which is absent in the eventual superstring theory.

• The UV-limit t → 0, on the other hand, is different from the closed string sector: Unlike on the torus, the UV-divergent region is not absent from the integral because there is no analogous modular group action serving as an intrinsic regulator.

Thus, it might seem that we do face a UV divergence, contrary to our previous claim that string theory is UV finite. To see why the divergence as t → 0 is not in contradiction with UV finiteness we need to discuss the worldsheet duality between the open and closed string channel.

Open versus closed channel

• In the UV-divergent limit t → 0, the cylinder is infinitely long.

• The remarkable insight is the following: We can either view the long cylinder as describing an open string stretching between the boundaries at σ1 _{= 0, π and running in the loop}

described by the Euclidean time σ2_{. Or, alternatively, we may interpret the annulus as a}

closed string propagating at tree-level from the left to the right. The two interpretations of the cylinder are referred to as open and closed string channel.

• Technically, the two viewpoints are related by interchanging the role of the Euclidean time and the spatial coordinate on the worldsheet. From our analysis of P SL(2, Z) transforma- tions of the torus on Assignment 10 we recall that an S-duality transformation τ → −_τ1 exchanges the coordinates σ1_{and σ}2_{. The same applies to the cylinder with it taking the}

role of τ . Including a conventional rescaling of the spatial coordinate the transition from the open to the closed string channel is accomplished by

t −→ s = π

t. (5.121)

With the help of the transformation of the Dedekind function η(it) = t−12_η i t  =s π 12 ηis π  (5.122) the annulus amplitude in closed string channel is

ZC2 = iV26N 2 1 2π(8π2_α0₎13 Z ∞ 0 ds ηis π −24 . (5.123)

• The UV limit t → 0 in the open channel has translated in the IR limit s → ∞ of the closed channel. This describes a closed string tree-level process with the string propagating over long Euclidean time. Thus we have reinterpreted the UV-divergence as an IR-divergence. This is in fact a general feature of string amplitudes:

All UV divergencies in string amplitudes can be reinterpreted as IR divergencies of dual diagrams. • In fact, we can make the propagation of the closed strings visible in the limit s → ∞ by

expanding ηis π −24 = e2s |{z} tachyon + 24 |{z} massless +O(e−2s). (5.124)

The tachyonic term is again an artifact of the bosonic theory. Of importance is the second term. It shows that the IR divergence is due to the exchange of massless closed string states at zero momentum.

Tadpoles in string theory and field theory

A diagram where a state - here a closed string state - is created from the vacuum is called a tadpole.

• The IR divergence is the 1

k2 divergence from joining two tadpoles by the propagator of a

massless state with k2_{= 0.}

• The tadpole diagram as such is computed by the 1-point function of a single closed string operator inserted in the interior of the disk. Note that such a diagram is in general non-zero even though the 1-point function on the sphere and the open 1-point function on the disk vanish - see the remark after (5.28).

• In QFT a tadpole diagram results from a term linear in the field in the lagrangian as this is what gives rise to a single field vertex. Tadpoles therefore signal an instability of the vacuum, which is defined as the locus in field space that satisfies V0(φ) = 0. For example in the presence of a tadpole for a bosonic field,

S = Z

−1 2(∂φ)

2_{+ Λφ (5.125)}

the locus φ = 0 does not correspond to the true vacuum. If we set out at φ = 0 the field configuration will change. In the presence of higher terms there may be a new vacuum at φ 6= 0 and the theory will flow to that correct vacuum. In the above action, by contrast, there is no such vacuum and the theory is entirely unstable.

• To see how to deal with the tadpole in string theory we need to include also unoriented worldsheets.