Abstract
An alternative to describing compactifications via a solvable conformal field theory is the perturbative approach around a geometric supergravity background at large radius. For this purpose one analyzes the string equations of motion at leading order in a typical length scale \(L/\sqrt{\alpha \prime }\). We describe this approach in detail for a class of backgrounds which preserve some amount of space-time supersymmetry in four-dimensions: compactification on Calabi-Yau manifolds. But we start with a brief discussion of the string equations of motion as the requirement of vanishing beta-functions of the non-linear sigma model for a string moving in a curved background. We then derive a generalization of T-duality to manifolds with isometries. This leads to the so-called Buscher rules. We then introduce some of the mathematical tools which are required for an adequate treatment of Calabi-Yau compactifications. With them at hand we consider compactifications of the type II and heterotic superstring on Calabi-Yau manifolds and discuss the structure of their moduli spaces. In an appendix we fix our notation and review some concepts of Riemannian geometry. The derivations of some results which are used in the main text are also relegated to the appendix.
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Notes
- 1.
There are also contributions from string loops and, of course, from both world-sheet and space-time non-perturbative effects.
- 2.
Massive ones contain more than two world-sheet derivatives and therefore lead to irrelevant operators in the two-dimensional CFT.
- 3.
In the pure spinor formulation of the superstring one uses, in addition to X μ, fermionic world-sheet scalars which transform as spinors under the space-time Lorentz group. In this formulation there is no problem to couple the R-R fields of the RNS formulation directly to the world sheet.
- 4.
We will not write down the σ-models for the superstring and the heterotic string. This is most succinctly done in (1, 1) and (0, 1) superspace, respectively. In the heterotic σ-model a background gauge field appears. For the bosonic string one should include the tachyon which couples without world-sheet derivatives.
- 5.
The Weyl anomaly functions β are not quite the renormalization group β-functions. Their relation is explained in the references at the end of this chapter.
- 6.
Note that θ is not necessarily meant to denote an angle-variable. Our units are such that the metric is dimensionless.
- 7.
We can also view this action as a gauged version of (14.10) where the isometry has been gauged, i.e. we have replaced \({\partial }_{\alpha }\theta \rightarrow {D}_{\alpha }\theta = {\partial }_{\alpha }\theta + {V }_{\alpha }\)where \(V \rightarrow V - d\epsilon \)under \(\theta \rightarrow \theta + \epsilon \). The Lagrange multiplier \(\tilde{\theta }\)enforces that the gauge field \({V }_{\alpha }\)has vanishing field strength. We obtain (14.11) after fixing the gauge θ = 0. The dual theory is obtained by integrating over the gauge field and fixing θ = 0.
- 8.
Another way to see this is to consider \(\Gamma \equiv {a}_{n}{\Gamma }^{n}\). Then \(\det ({\Gamma }^{2}) = {g}^{mn}{a}_{n}{a}_{m}\)which, on a Riemannian manifold with Euclidean signature, vanishes iff \({a}_{n} = 0,\,\forall n\). Therefore, \({a}_{n}{\Gamma }^{n}\epsilon = 0\)requires, for non-zero ε, that \({a}_{n} = 0,\,\forall n\). On Lorentzian manifolds Ricci-flatness is not a necessary condition for the existence of covariantly constant spinors.
- 9.
These conditions and also (14.41) receive α′corrections. We will comment on them later.
- 10.
The discussion of the condition of having Killing spinors is much richer than the corresponding question for Killing vectors. In the latter case it is known that Ricci-flat compact manifolds do not admit Killing vectors other than those associated with tori. Equivalently, the first Betti number b 1receives contributions only from non-trivial cycles associated to torus factors in K D .
- 11.
More precisely, there is only one family which is parametrized by a moduli space. For CY1this was discussed in Chap. 6and will be further discussed later in this section and for CY3in Sect. 14.5.
- 12.
We have \({\epsilon }^{1,2} = {\epsilon }_{R}^{1,2} \otimes {\epsilon }_{+} + {\epsilon }_{L}^{1,2} \otimes {\epsilon }_{-}\)for type IIB. For type IIA we interchange \({\epsilon }_{R}^{2} \leftrightarrow {\epsilon }_{L}^{2}\).
- 13.
n − 1-dimensional submanifolds of \({\mathbb{P}}^{n}\)or, more generally, co-dimension-one submanifolds of \({\mathbb{P}}^{n}\)are called hyperplanes.
- 14.
In this context the following result is of interest: there are no compact complex submanifolds of \({\mathbb{C}}^{n}\). This is an immediate consequence of the fact that any global holomorphic function on a compact complex manifold is constant, applied to the coordinate functions.
- 15.
\(P = {\Pi }^{1,0},\,Q = {\Pi }^{0,1}\)in terms of our previous definition.
- 16.
Given a Riemannian metric \({g}_{\mu u }\), its components in local complex coordinates are \({g}_{ij},\,{g}_{\bar{{\imath}}\bar{{\jmath}}} = {({g}_{ij})}^{{_\ast}},\,{g}_{i\bar{{\jmath}}},\,{g}_{\bar{{\imath}}j} = {({g}_{i\bar{{\jmath}}})}^{{_\ast}}\). It is Hermitian if \({g}_{ij} = 0\).
- 17.
The unitary group U(n) is the set of all complex \(n \times n\)matrices which leave invariant a Hermitian metric \(\overline{{g}_{i\bar{{\jmath}}}} = {g}_{j\bar{{\imath}}}\), i.e. \(Ug{U}^{\dag } = g\). For the choice \({g}_{i\bar{{\jmath}}} = {\delta }_{ij}\)one obtains the familiar condition \(U{U}^{\dag } = Vdash \).
- 18.
More precisely, on polydiscs \({P}_{r} =\{ z \in {\mathbb{C}}^{n} : \vert {z}^{i}\vert < r,\ \mbox{ for all}\ i = 1,\ldots ,n\}\).
- 19.
There exist other definitions in the literature; e.g. Griffiths and Harris define an operator \({{_\ast}}_{{\mathrm{GH}}^{ }} :\, {A}^{p,q} \rightarrow {A}^{n-p,n-q}\). What they call \({{_\ast}}_{{\mathrm{GH}}^{ }}\!\!\psi \)we have called \({_\ast}\overline{\psi }\).
- 20.
Proof: \(\overline{\varphi } \wedge {_\ast}\psi = \overline{\varphi \wedge {_\ast}\overline{\psi }} = \overline{\langle \varphi ,\psi \rangle }{ {\omega }^{n} \over n!} =\langle \psi ,\varphi \rangle { {\omega }^{n} \over n!} = \psi \wedge {_\ast}\overline{\varphi }\).
- 21.
The generalizations of (14.127b) to ndimensions is \({_\ast}\omega ={ 1 \over (n-1)!} {\omega }^{n-1}\); \({_\ast}\Omega = {(i)}^{n}{(-1)}^{{ 1 \over 2} n(n+1)}\)generalizes (14.128a) to (n, 0) forms. Finally, for α a primitive \(p + q = k\)-form, \({_\ast}\alpha = {(-1)}^{{ 1 \over 2} k(k+1)}{(i)}^{p-q}{ 1 \over (n-k)!} {\omega }^{n-k} \wedge \alpha \).
- 22.
Proof: Since \(\varphi \wedge {_\ast}\bar{\psi } \in {A}^{n,n-1}\), \(d(\varphi \wedge {_\ast}\bar{\psi }) =\bar{ \partial }(\varphi \wedge {_\ast}\bar{\psi })\). Integrating this over Mleads to (14.130).
- 23.
One can similarly define \({\partial }^{{_\ast}} = -{_\ast}\bar{ \partial }{_\ast}\)and \({\Delta }_{\partial }\). On a Hermitian manifold, \({d}^{{_\ast}} = {\partial }^{{_\ast}} +\bar{ {\partial }}^{{_\ast}}\).
- 24.
Proof: \((\psi ,{\Delta }_{\bar{\partial }}\psi ) = \vert \!\vert \bar{\partial }\psi \vert \!{\vert }^{2} + \vert \!\vert \bar{{\partial }}^{{_\ast}}\psi \vert \!{\vert }^{2}\)which vanishes iff \(\bar{\partial }\psi =\bar{ {\partial }}^{{_\ast}}\psi = 0\).
- 25.
From \(\bar{\partial }\varphi =\bar{ \partial }\bar{{\partial }}^{{_\ast}}\eta \)it follows that \((\bar{\partial }\varphi ,\eta ) = (\bar{\partial }\bar{{\partial }}^{{_\ast}}\eta ,\eta ) = (\bar{{\partial }}^{{_\ast}}\eta ,\bar{{\partial }}^{{_\ast}}\eta )\).
- 26.
On a Kähler manifold the three conditions \(d\omega = 0,\,abla J = 0\)and ω is harmonic, are equivalent.
- 27.
One can proof that a non-zero (p, 0) form, not necessarily holomorphic, can never be exact. Indeed, considering \(\alpha = d\beta \in {A}^{p,0}\), it is straightforward to show that \(\alpha \wedge \bar{ \alpha } \wedge {\omega }^{n-p} = f{\omega }^{n}\), where ω is the Kähler form and fa positive function. The r.h.s., when integrated over the manifold, is non-zero, while the l.h.s. integrates to zero, because it can be written as \(d(\beta \wedge \bar{ \alpha } \wedge {\omega }^{n-p})\). It then follows that any \(\alpha \in {\Omega }^{p}\)satisfies \(d\alpha = 0\): clearly, \(d\alpha = \partial \alpha \)is an exact (holomorphic) (p + 1) form which vanishes by the above argument.
- 28.
The Chern connection is obviously not the only connection with the property that the (0, 2) and (2, 0) parts of its curvature vanish, since this is a gauge invariant statement: for \(A \rightarrow UA{U}^{-1} + U\,d{U}^{-1}\)the curvature transforms as \(F \rightarrow UF{U}^{-1}\).
- 29.
This follows from \([{abla }_{i},{abla }_{\bar{{\jmath}}}]{V }_{k} = {R{}_{i\bar{{\jmath}}k}}^{l}{V }_{l}\)and \([{abla }_{i},{abla }_{\bar{{\jmath}}}]{V }^{k} = {R{}_{i\bar{{\jmath}}}{}^{k}}_{l}{V }^{l}\)and \({R{}_{i\bar{{\jmath}}k}}^{k} = -{R{}_{i\bar{{\jmath}}}{}^{k}}_{k}\).
- 30.
There exists an algorithm, due to Donaldson, for an approximate construction which converges to the exact CY metric.
- 31.
Examples of explicit non-compact Ricci-flat Kähler metrics are the Eguchi-Hanson metrics and the metric on the deformed and the resolved conifold. They play a role in the resolution of singularities (orbifold and conifold singularities, respectively) which can occur in compact CY manifolds at special points in their moduli space.
- 32.
Calabi-Yau orbifolds have discrete holonomy groups. There the condition is that it is not contained in any continuous subgroup of SU(n).
- 33.
Alternatively we could represent the torus as a homogeneous polynomial of degrees \(4\)and 6 in \({\mathbb{P}}^{2}[1, 1, 2]\)and \({\mathbb{P}}^{2}[1, 2, 3]\), respectively and the following discussion would also be valid.
- 34.
A simple example is \({\mathbb{P}}^{2}[1, 1, 2]\), i.e. \(({z}^{0},{z}^{1},{z}^{2})\)and \((\lambda {z}^{0},\lambda {z}^{1},{\lambda }^{2}{z}^{2})\)denote the same point. For \(\lambda = -1\)the point \([0 : 0 : {z}^{2}] \equiv [0 : 0 : 1]\)is fixed but λ acts non-trivially on its neighbourhood. There is a ℤ 2orbifold singularity at this point.
- 35.
The toric variety (14.194) can be considered as the \(\mathcal{O}(-1) \oplus \mathcal{O}(-1)\)bundle over a \({\mathbb{P}}^{1}\)defined by the two homogeneous coordinates z 1and z 2. Here \(\mathcal{O}(n)\)denotes the holomorphic line bundle over \({\mathbb{P}}^{1}\)with first Chern number \(\int\limits_{{\mathbb{P}}^{1}}{c}_{1}(\mathcal{O}(n)) = n\). For the size of this \({\mathbb{P}}^{1}\)going to zero, one gets the conifold singularity.
- 36.
More generally, by a theorem of Dolbeault, \({H}^{q}(M,{\wedge }^{p}{T}_{M}^{{_\ast}}\otimes V ) \simeq {H}_{\bar{\partial }}^{p,q}(M,V )\)where \({T}_{M}^{{_\ast}}\)is the holomorphic cotangent bundle and Vany holomorphic vector bundle. In the case above, we have p = 0, q = 1 and V = T M . \({H}^{q}(M,{\wedge }^{p}{T}_{M}^{{_\ast}}\otimes V )\)is a sheaf cohomology group and Vis, more accurately, any sheaf of germs of a holomorphic vector bundle. For details we refer to the cited mathematical literature.
- 37.
Another way to reach at this conclusion is to deform the almost complex structure \(\tilde{J} = J + \delta J\)and to require, to first order in \(\delta J\), \(\tilde{{J}}^{2} = -Vdash \)and \(N(\tilde{J}) = 0\)where Nis the Nijenhuis tensor. This leads to the conditions \(\delta {J}_{i}^{j} = 0\)and \({\partial }_{[\bar{{\imath}}}^{ }\delta {J}_{\bar{{\jmath}}]}^{k} = 0\). However, deformations δJare trivial, if they can be generated by a coordinate transformation. It is easy to check that under a coordinate transformation the components \({J}_{\bar{{\imath}}}^{j}\)change as \(\delta {J}_{\bar{{\imath}}}^{j} = -2i\bar{{\partial }}_{\bar{{\imath}}}{\xi }^{j}\). This leaves the closed modulo the exact forms, i.e. elements of the cohomology.
- 38.
More generally, there is an isomorphism \({H}^{q}(X,{T}_{M} \otimes V ) \simeq {H}^{q}(M,{\wedge }^{n-1}{T}_{M}^{{_\ast}}\otimes V ) \simeq {H}_{\bar{\partial }}^{n-1,q}(M,V )\).
- 39.
Our discussion of complex structure moduli is not complete. We have only considered the linearized deformation equation. It still needs to be shown that they can be integrated to finite deformations. It can be shown that this is indeed the case for Calabi-Yau manifolds. For a general complex manifold the number of complex structure deformations is less than h 2, 1.
- 40.
For n-dimensional manifolds with vanishing first Chern class, the number of geometric moduli, i.e. deformations of the Ricci-flat Kähler metric, is \({h}^{1,1} + 2{h}^{1,n-1} - 2{h}^{2,0}\). The need to subtract \(2{h}^{2,0}\)can be understood as follows: a harmonic (0, 2) form \({b}_{\overline{{\imath}}\overline{{\jmath}}}\)gives, as \({g}^{i\bar{{\jmath}}}{b}_{\bar{{\jmath}}\bar{k}}\), rise to an element of \({H}^{1}(M,T)\). It is instructive to verify the counting by comparing the number of metric components of \({T}^{2n}\), \(n(2n + 1)\), with \({h}^{1,1} + {h}^{1,n-1} = {n}^{2} + 2{n}^{2}\)and \(2{h}^{2,0} = n(n - 1)\).
- 41.
For special values of these coefficients the hypersurface is singular, i.e. there are solutions of \(p = dp = 0\).
- 42.
This might be familiar from the torus which can be realized as a degree three hypersurface in \({\mathbb{P}}^{3}\). The most general hypersurface constraint can be brought to the form \(p\,=\,{z}_{1}^{3} + {z}_{2}^{3} + {z}_{3}^{3} - 3a{z}_{1}{z}_{2}{z}_{3}\,=\,0\). The parameter acan be shown to be related to the modular parameter τ via \(j(\tau ) ={ 216{a}^{3}{(8+{a}^{3})}^{3} \over {({a}^{3}-1)}^{3}}\).
- 43.
From now on we use indices \((i,j,\ldots ,\bar{{\imath}},\bar{{\jmath}},\ldots \,)\)for the internal space and μ for the four uncompactified space-time dimensions.
- 44.
A massive B-field does not have this invariance and carries three on-shell degrees of freedom (the antisymmetric second-rank tensor of its little group SO(3)). It can be dualized to a massive vector with three degrees of freedom.
- 45.
A quaternionic manifold is a complex manifold of real dimension 4mand holonomy group \(Sp(2) \times Sp(2m)\).
- 46.
Three four-cycles \(a,b,c \in {H}^{4}\)in a six-dimensional manifold generically intersect in finitely many points. If \(\alpha ,\beta ,\gamma \in {H}^{2}\)are their Poincaré dual two-forms, the triple intersection number is \(\kappa (\alpha ,\beta ,\gamma )\).
- 47.
Here we use the isomorphisms between the anti-holomorphic tangent bundle and the holomorphic cotangent bundle, i.e. raising and lowering the index with the Hermitian metric.
- 48.
Note that the Hfield which solves (14.231) is \(\mathcal{O}(\alpha \prime )\)and therefore does not contribute to the lowest order YM equations \({\beta }^{A} = 0\), cf. (14.6).
- 49.
More precisely, \({H}^{1}(X,{V }^{{_\ast}}\otimes V ) \simeq {H}^{1}(X,\mathrm{End}V ) \oplus {H}^{1}(X)\); but \({H}^{1}(X) \simeq {H}_{\bar{\partial }}^{0,1}(X)\)is trivial on a CY manifold.
- 50.
This will be shown in Chap. 15.
- 51.
Here we use homogeneous coordinates on moduli space, \({Z}^{I} = ({Z}^{0},{Z}^{0}{t}^{a})\). The relation between the quantities used in (14.305) and (14.220) is \(\Omega {\vert }_{\mathrm{here}} = {Z}^{0}\Omega {\vert }_{\mathrm{there}}\), \({k}_{I} = \left ({ 1 \over {Z}^{0}} (1 - {t}^{a}{k}_{a}),{ 1 \over {Z}^{0}} {k}_{a}\right )\)and \({\chi }_{I} = (-{t}^{a}{\chi }_{a},{\chi }_{a})\).
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Blumenhagen, R., Lüst, D., Theisen, S. (2012). String Compactifications. In: Basic Concepts of String Theory. Theoretical and Mathematical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29497-6_14
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DOI: https://doi.org/10.1007/978-3-642-29497-6_14
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