Abstract
Type II compactifications with D-branes and background fluxes are viable candidates to relate string theory to the physics we observe in four dimensions. For simple toroidal orbifold backgrounds the D-brane and orientifold sector can be described by an exact CFT, but issues such as tadpole cancellation, the Green-Schwarz mechanism, determining the massless spectrum etc. arise in a broader context and can be discussed from the low-energy-effective action perspective. String compactifications with non-vanishing NS-NS and R-R p-form field strengths provide solutions to the moduli problem, as these background fluxes modify the string equations of motion at leading order so that its solutions generically generate a potential for the would-be moduli fields. Thus they receive a vacuum expectation value and a mass. Basic knowledge of \(\mathcal{N} = 1\)supersymmetry in four dimensions is assumed.
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Notes
- 1.
These vacua can be meta-stable as long as they are long living.
- 2.
Since the r.h.s. of (17.4) is of the same order (in \(\alpha ^{\prime}\)and in \({g}_{s}\)) as the l.h.s, the flux is as much part of the background as the metric is and it cannot be viewed as a correction controlled by a small parameter.
- 3.
Additional magnetized D8-branes are only possible, if there exist non-trivial 5-cycles on the six-dimensional manifold.
- 4.
In this chapter we use \({\Omega }_{p}\)for the world-sheet parity in order to avoid confusion with the three-form \(\Omega \).
- 5.
\(\sigma \)is a reflection in the (5-7-9) directions and it acts on positive and negative chirality spinors as \(\pm i{\gamma }^{5}{\gamma }^{7}{\gamma }^{9}\). If we express the Dirac-matrices in terms of creating and annihilation operators and use that \(\vert \eta \rangle\)and \(\vert \overline{\eta }\rangle\)are highest and lowest weight states, respectively, the stated results follow.
- 6.
This is a notion from symplectic geometry where \(\omega \)is the symplectic form and Xthe phase space, i.e. the cotangent bundle of the configuration space. The latter is a Lagrangian submanifold (by Darboux’ theorem we can choose local coordinates such that \(\omega = \sum d{p}_{i} \wedge d{q}^{i}\)).
- 7.
The notion of calibrated submanifolds is more general, in particular it is defined for submanifolds of arbitrary dimensions. The supersymmetry preserving cycles of type IIB orientifolds are even cycles. They are calibrated w.r.t. \({ 1 \over p!} {\omega }^{p}\), for \(p = 0, 1, 2, 3\)for O3,O5,O7 and O9 planes, respectively. This means e.g. that \(\mathrm{vol}{\vert }_{{\Sigma }_{2}} = \omega {\vert }_{{\Sigma }_{2}}\). The calibrated cycles are precisely the complex submanifolds. In the presence of other background fields, e.g. a background Bfield, the supersymmetry conditions are modified.
- 8.
- 9.
Half of the scalars are from the R-R sector and the other half from the NS-NS sector. The invariant combinations do not mix the two sectors.
- 10.
Note that it is not sufficient that the cycle \({\Pi }_{a}\)is mapped to itself in homology under the induced action of \(\sigma \). The submanifold representative of the cycle has to be invariant. The gauge group can be determined from the Möbius strip amplitude. For tori one finds that on the world-volume of Dp-branes on top of an Op-plane, the gauge group is SO while for D(p\(-\)4) branes on an Op-plane it is Sp.
- 11.
Here we use that the anomaly coefficients \({\mathrm{tr\,}}_{R}(\{{T}^{a},{T}^{b}\}{T}^{c}) = A(R)\,{\mathrm{tr\,}}_{ }(\{{T}^{a},{T}^{b}\}{T}^{c})\)for the relevant \(SU(N)\)representations are \({A}_{ } = (N - 4){A}_{ },\,{A}_{ } = (n + 4){A}_{ },\,{A}_{\overline{ }} = -{A}_{ }\). This can be derived e.g. by looking at the decomposition w.r.t. \(SU(3)\).
- 12.
Here we define \({\mathrm{tr\,}}_{R}({Q}^{a}{T}^{b}{T}^{b}) = {A}_{abb}{\mathrm{tr\,}}_{ }({T}^{b}{T}^{b})\); there is no sum over b. We also define \({B}_{abb} = \mathrm{tr\,}({Q}^{a}{Q}^{b}{Q}^{b})\)and \({A}_{agg} = \mathrm{tr\,}({Q}^{a})\), where \({Q}^{a}\)is the \(U{(1)}_{a}\)charge.
- 13.
If \({\mathrm{tr\,}}_{R}({T}^{a}{T}^{b}) = c(R){\delta }^{ab}\), \(c( ) = (N + 2)c( ),\,c( ) = (N - 2)c( ),\,c(\overline{ }) = c( )\).
- 14.
We can regularize the triangle diagrams in such a way that \({A}_{abb}^{(1)}\)and \({B}_{abb}^{(1)}\)are proportional to the divergences of the \(U{(1)}_{a}\)current while the other two currents in the respective triangle diagrams are conserved. Bose symmetry in the three currents accounts for the factor \({ 1 \over 3}\)in \({B}_{aaa}^{(1)}\)(second line in (17.25)).
- 15.
Compared to Chap. 14we have slightly changed the notation from \({A}^{a},{B}_{a}\)to \({A}_{I},{B}^{I}\).
- 16.
The wrapping numbers \({m}_{a}^{\prime},{n}_{a}^{\prime}\)of the mirror branes are, of course, not independent of those of the branes. The precise relation depends on the action of \(\sigma \)on the homology basis, but it will not be needed.
- 17.
We know that higher order \(\alpha ^{\prime}\)corrections also destroy Ricci flatness but preserve the CY condition \({c}_{1}(M) = 0\). The corrections to \({R}_{mn}eq 0\)generated by fluxes are not of this type. The resulting manifolds might e.g. not even be Kähler.
- 18.
In the pure spinor approach it is possible to include anti-symmetric tensor backgrounds in the world-sheet action, but it has not yet been developed sufficiently far to be easily applicable to the situations we are interested in.
- 19.
The factor of \(1/2\)was introduced because \({F}_{5}\)is self-dual, D3-branes carry both electric and magnetic charges with respect to \({F}_{5}\)and the action only contains the electric interaction term.
- 20.
Taking the back-reaction into account one gets a warped CY. This will be explained below.
- 21.
For instance, if we have only one Kähler modulus Tand \(K = -3\ln (T + \overline{T}) +\tilde{ K}({\phi }_{A},{\overline{\phi }}_{A})\)and if furthermore the superpotential is independent of T, \(W = W({\phi }_{A})\), we obtain from (17.56) \({V }_{\mathrm{F}} = {e}^{{\kappa }_{4}^{2}K}{G}^{A\bar{A}}{D}_{A}W{D}_{\bar{A}}\overline{W}\). Supersymmetric minima now have vanishing potential energy. Also note that the vev of Tis not determined. This leads to a degenerate family of vacua and hence the name no-scale models.
- 22.
For ease of comparison with the relevant literature, in this section we are using the convention (14.292) for the Hodge-\(\star \). Otherwise we would either have to modify (17.64) or the duality properties of the field strengths.
- 23.
\({F}_{0} = m\)is the constant mass-parameter of Romans massive type IIA supergravity.
- 24.
The spinors \({\eta }_{+}\)(\({\eta }_{-}\)) correspond to the internal spin fields \(\Sigma (z)\)(\({\Sigma }^{\dag }(z)\)) of Chap. 15, the \(U(1)\)current \(J(z)\)to the 2-form \(\omega \), and the 3-form \(\Omega \)to the identity operator of the internal CFT.
- 25.
Elements of \({\Lambda }^{p}\)are p-form fields on M.
- 26.
Recall from Chap. 14that primitivity means \({({\mathcal{W}}_{2})}_{i\bar{\jmath}}{\omega }^{i\bar{\jmath}} = 0\). This corresponds to the decomposition \(\mathbf{3} \otimes \overline{\mathbf{3}} = \mathbf{8} + \mathbf{1}\)into traceless and trace parts with \({\mathcal{W}}_{2} \sim \mathbf{8}\)and \({\mathcal{W}}_{1} \sim \mathbf{1}\).
- 27.
A manifold is symplectic if is has a globally defined closed non-degenerate two-form, the symplectic structure. Non-degenerate means that the coefficient matrix of the two-form has an inverse.
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Blumenhagen, R., Lüst, D., Theisen, S. (2012). Compactifications of the Type II Superstring with D-branes and Fluxes. In: Basic Concepts of String Theory. Theoretical and Mathematical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29497-6_17
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DOI: https://doi.org/10.1007/978-3-642-29497-6_17
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