Abstract
To relate string theory to the usual description of particles and their interactions in terms of quantum field theories, it is important to have tools at hand to derive the effective point particle interactions for the massless excitation modes of the string. Such effective actions can be deduced from on-shell string scattering amplitudes which are computed as correlation functions of physical state vertex operators. We construct the vertex operators and compute various three-point functions which are needed to extract e.g. the interactions of graviton, two-form, dilaton and of gauge fields at leading order. We also compute the four-point functions of open and closed string tachyons and discuss some of their properties. Often the leading order (in α′) effective actions are already uniquely determined by symmetries, such as gauge symmetries or supersymmetry. We present the bosonic sectors of the ten-dimensional supergravity theories which are related to the ten-dimensional superstring theories. We also include a discussion of eleven-dimensional supergravity. The Dirac-Born-Infeld action, which governs the dynamics of the gauge field on a D-brane, will also be discussed.
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Notes
- 1.
Here and below we often drop normal ordering symbols.
- 2.
Given \({\epsilon }_{\mu u }\)the symmetric-traceless piece is \({ 1 \over 2} ({\epsilon }_{\mu u } + {\epsilon }_{u \mu }) -{ {\epsilon {}^{\rho }}_{\rho } \over d-2} ({\eta }_{\mu u } - {k}_{\mu }{\overline{k}}_{u } -{\overline{k}}_{u }{k}_{\mu })\).
- 3.
λ is a constant which makes λkdimensionless.
- 4.
The anti-symmetric tensor is slightly tricky, so let us give some details. An unconstrained anti-symmetric tensor \({\epsilon }_{\mu u }\)in ddimensions has \({ 1 \over 2} d(d - 1)\)independent components. The transversality condition \({k}^{\mu }{\epsilon }_{\mu u } = 0\)imposes (d − 1) constraints (because of the identity \({k}^{\mu }{\epsilon }_{\mu u }{k}^{u } \equiv 0\)not all components of \({k}^{\mu }{\epsilon }_{\mu u }\)are linearly independent). Furthermore there is the gauge freedom \({\epsilon }_{\mu u } \rightarrow {\epsilon }_{\mu u } + {k}_{\mu }{\zeta }_{u } - {k}_{u }{\zeta }_{\mu }\)where transversality imposes \(k \cdot \zeta = 0\). However there is a ‘gauge invariance for the gauge invariance’, i.e. the choice \({\zeta }_{\mu } = a{k}_{\mu }\)does not change \({\epsilon }_{\mu u }\). Taking all of this into account leaves \({ 1 \over 2} (d - 2)(d - 3)\)components.
- 5.
This is shown as follows: \({k}^{\mu } = ({k}^{0},k)\), together with \({\overline{k}}^{\mu } = (-{k}^{0},k)\)and (d − 2) transverse polarization vectors \({e}_{\mu }^{i}\)allow us to decompose
$$\begin{array}{rcl}{ F}_{{\mu }_{1}\ldots {\mu }_{p+1}}& =& {a}_{{i}_{1}\ldots {i}_{p+1}}{e}_{[{\mu }_{1}}^{{i}_{1}}\cdots {e}_{{ \mu }_{p+1}]}^{{i}_{p+1}} + \big{(}{b}_{{ i}_{1}\ldots {i}_{p}}{k}_{[{\mu }_{1}} + {c}_{{i}_{1}\ldots {i}_{p}}{\overline{k}}_{[{\mu }_{1}}^{ }\big{)}{e}_{{\mu }_{2}}^{{i}_{1}}\cdots {e}_{{ \mu }_{p+1}]}^{{i}_{p}} \\ & & \quad + {d}_{{i}_{1}\ldots {i}_{p-1}}{k}_{[{\mu }_{1}}{\overline{k}}_{{\mu }_{2}}{e}_{{\mu }_{3}}^{{i}_{1}}\cdots {e}_{{ \mu }_{p+1}]}^{{i}_{p-1}}\end{array}$$. If we impose (16.22) we find \(a = c = d = 0\)and k 2 b = 0. This also shows that the number of propagating degrees of freedom equals the number of components of \({b}_{{i}_{1}\ldots {i}_{p}}\), i.e. \(\left ({ d-2 \atop p} \right )\).
- 6.
For a general compact simple Lie algebra we can choose Hermitian generators such that \({\mathrm{tr\,}}_{R}({T}^{a}{T}^{b}) = {C}_{R}{\delta }^{ab}\)where \({C}_{R} ={ 1 \over \mathrm{rank}(G)} \sum olimits (\lambda ,\lambda )\)and the sum is over all weights of the irreducible representation R. The structure constants are defined as \([{T}^{a},{T}^{b}] = i{f}^{abc}{T}^{c}\). They depend on the normalization of the roots.
- 7.
We should caution the reader that we are not careful about overall phases of the amplitudes.
- 8.
Strictly speaking this amplitude vanishes, because for three massless particles the momenta must be co-linear. This can be avoided by allowing complex momenta.
- 9.
On the disk the conformal Killing group preserves the cyclic order of points on the boundary of the disk.
- 10.
If we work on D 2rather than \({\mathbb{H}}_{+}\), there is a nontrivial Jacobian, i.e. \(\overline{\psi }(\overline{z}) = \psi (z^{\prime}){\left ({ \partial z^{\prime} \over \partial \overline{z}} \right )}^{1/2}\). With \(z^{\prime} = 1/\overline{z}\)one finds \(\langle \psi (z)\overline{\psi }{(\overline{w})\rangle }_{{D}_{2}} ={ i \over 1-z\overline{w}}\). One verifies that this maps to the result on \({\mathbb{H}}_{+}\)under (4.140). An alternative way to get the disc propagator is to use the boundary state formalism and to compute e.g. \(\langle \mathrm{Dp}\vert \psi (z)\overline{\psi }(\overline{w})\vert 0\rangle\).
- 11.
Define \(u ={ 1 \over 2} ({x}_{1} + {x}_{2})\)and \(v ={ 1 \over 2} ({x}_{1} - {x}_{2})\). With \(z = x + iy\)we have \({d}^{2}z\,d{x}_{1}\,d{x}_{2} = 4\,dx\,dy\,du\,dv\). An infinitesimal \(SL(2, \mathbb{R})\)transformation acts as \(\delta z = \alpha + \beta z + \gamma {z}^{2}\)with \(\alpha ,\beta ,\gamma \)real. Then \(\left \vert { \partial (x,y,u) \over \partial (\alpha ,\beta ,\gamma )} \right \vert = (1 + {v}^{2})\)at \(x = 0,y = 1,u = 0\). Note that if we fix the positions of three open strings to \({x}_{1},{x}_{2},{x}_{3}\), the Jacobian is \(\vert {x}_{12}{x}_{13}{x}_{23}\vert \).
- 12.
This can be extended to spin fields but we will not do that.
- 13.
The representation \(B(\alpha ,\beta ) = {\int olimits olimits }_{-\infty }^{\infty }dx\;{ {({x}^{2})}^{\alpha -1/2} \over {(1 + {x}^{2})}^{\alpha +\beta }}\)of the Beta function is useful.
- 14.
This uses the normalization \(\mathrm{tr\,}({T}^{a}{T}^{b}) = 2{\delta }^{ab}\)for the generators in the fundamental representation. If we use group generators normalized to \(\mathrm{tr\,}({T}^{a}{T}^{b}) = {\delta }^{ab}\), we obtain \({g}_{d} ={ 2 \over \sqrt{\alpha ^{\prime}}} {\kappa }_{d}\).
- 15.
One can show that in the presence of a non-trivial dilaton background these do, in fact, correspond to the physical state conditions of the vertex operator.
- 16.
The structure of anomalies is much richer in d = 10 than in d = 4, mainly because of possible gravitational anomalies. We will not present a general discussion of anomalies in string theory, but a few aspects will be mentioned as we go along.
- 17.
We define the anti-symmetrization symbol \([\ldots \,]\)with unit weight, e.g. \({\partial }_{[M}{B}_{NP]} ={ 1 \over 3!} ({\partial }_{M}{B}_{NP} \pm \ \text{ 5 permutations})\).
- 18.
There exists a formulation which takes into account the self-duality at the level of the action, but at the expense of manifest covariance or of introducing auxiliary degrees of freedom.
- 19.
For a complex p-form we define \(\vert {F}_{p}{\vert }^{2} = \frac{1} {p!}{F}_{{M}_{1}\ldots {M}_{p}}{\overline{F}}^{{M}_{1}\ldots {M}_{p}}\).
- 20.
Note that \({\Omega }_{\mathrm{YM}}\)is first order in a derivative expansion, while \({\Omega }_{\mathrm{L}}\)is third order. For this reason, the Lorentz term is sometimes considered subleading. However, both terms are equally important for anomaly cancellation.
- 21.
C 0is also called an axion and the symmetry \({C}_{0} \rightarrow {C}_{0} + \mathrm{const}\)is called axionic shift symmetry or Peccei-Quinn symmetry. We will use this terminology also for p > 0.
- 22.
The relation between the generators associated with the two stacks of branes, i.e. Tvs. − T T, is due to the fact that the gauge field is odd under Ωand that Ωreverses the orientation of the open strings. (16.167) can be expressed in the more familiar basis where SO(2N) generators are real anti-symmetric matrices. If we write \(iT = A + iB\), A, Breal and \({A}^{\mathrm{T}} = -A\)and \({B}^{\mathrm{T}} = B\)as a consequence of T † = T, (16.167) corresponds to \(\left (\begin{array}{cc} A &B\\ - B & A \end{array} \right )\).
- 23.
The normal bundle on the submanifold \(\mathcal{W}\)of a Riemannian manifold Mconsists of the pairs (y, v), where yranges over the points of \(\mathcal{W}\)and vis an element of the tangent space of T y (M) which is orthogonal, w.r.t. the Riemann metric on M, to the tangent space \({T}_{y}(\mathcal{W})\).
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Blumenhagen, R., Lüst, D., Theisen, S. (2012). String Scattering Amplitudes and Low Energy Effective Field Theory. In: Basic Concepts of String Theory. Theoretical and Mathematical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29497-6_16
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DOI: https://doi.org/10.1007/978-3-642-29497-6_16
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