Abstract
The five superstring theories which we have encountered so far will be argued to be different perturbative limits of one underlying theory, connected via a web of perturbative and non-perturbative dualities. We will discuss the concept of dualities and the role of BPS states in establishing non-perturbative dualities. We construct brane solutions of supergravity, discuss their BPS properties and identify those carrying R-R charge as D-branes. We discuss the place of eleven-dimensional supergravity in this duality web and the self-consistent evidence for the existence of a so far unknown eleven-dimensional quantum gravity theory, called M-theory. The known facts include that its low-energy effective field theory is given by eleven-dimensional supergravity and that it contains membranes and five-branes, which upon compactification are related to the branes of string theory. Two examples of the power of dualities will be discussed in the final two sections: F-theory, an interesting class of non-perturbative compactifications of type IIB string theory and the AdS-CFT correspondence. They derive their appeal and usefulness from the fact that non-perturbative effects can be related to simple geometric structures.
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Notes
- 1.
In field theory, solitons are non-trivial stable source-free solutions of the classical field equations with a finite action integral and non-zero topological charge. Examples are magnetic monopoles, whose energy is localized in three-dimensional space and which behave very much like point particles. In Yang-Mills theories one also has instantons, classical solutions of the equations of motion in Euclidean signature. They are localized in space and time and their non-trivial contributions to the YM path integral scale like \(\exp \left (-{S}_{\mathrm{inst}}\right )\), where \({S}_{\mathrm{inst}} \sim 1/{g}_{\mathrm{YM}}^{2}\) is the classical action evaluated on the instanton solution. The string winding modes are solitonic in the sense that they are topologically stable. The conserved topological charge is the winding number. World-sheet instantons in string theory are topologically non-trivial embeddings of the euclideanized world-sheet into the space-time manifold. We have met them in Chap. 14.
- 2.
The δ-function satisfies \(\int\limits_{M}{d}^{d}x\,{\delta }^{d}(x) =\int\limits_{M}{d}^{d}x\,\sqrt{\vert g\vert }\,{ 1 \over \sqrt{\vert g\vert }} {\delta }^{d}(x) = 1\). If \({\Sigma }_{p}\) has no boundary, \(\star {j}_{p+1}^{e}\) is essentially its Poincaré dual.
- 3.
The Dirac monopole satisfies \(\nabla \cdot B = {q}_{m}{\delta }^{3}(x)\) and it is not a soliton. Those can only occur in non-linear systems. The solitonic brane solutions are more closely related to the ’t Hooft-Polyakov monopole of the SO(3) Yang-Mills-Higgs model where the non-linearity is due to the non-Abelian gauge group and the Higgs sector.
- 4.
We could also decompose the supercharges into representations of \(SO(1,D - 1) \times SO(10 - D)\). For instance, if we compactify on T 6, we have the decomposition (14.40) and obtain the \(\mathcal{N} = 4\) SUSY algebra in D = 4.
- 5.
The solutions with \(\tilde{n} = 0\) and \(\tilde{n} = -1\) deserve special treatment. Here the Green functions are \(-\alpha \ln r\ (\tilde{n} = 0)\) and \(-\alpha \vert y\vert \ (\tilde{n} = -1)\).
- 6.
This becomes apparent in harmonic gauge \({\partial }^{M}{h}_{MN} -{ 1 \over 2} {\partial }_{N}{h}_{M}^{M} = 0\).
- 7.
With the appropriate modification this is also valid to \(\tilde{n} = 0\). For \(\tilde{n} = -1\) the integral diverges.
- 8.
We can obtain the anti-brane by changing the orientation; e.g. \({X}^{0}(\xi ) = -{\xi }^{0},\,{X}^{i}(\xi ) = {\xi }^{i}\). This is similar to saying that a positron is an electron going backwards in time: there the only way to change the orientation of the world-line is to change the direction of time. For higher dimensional world-volumes there are other ways.
- 9.
- 10.
In Einstein frame \(R \sim 1/({r}^{1/2}{\alpha }^{3/4})\).
- 11.
\({R{}^{\mu \nu }}_{\rho \sigma } = {\epsilon }^{\mu \nu \alpha }{\epsilon }_{\rho \sigma \beta }{G{}_{\alpha }}^{\beta }\) where \({G}_{\alpha \beta } = {R}_{\alpha \beta } -{ 1 \over 2} R{g}_{\alpha \beta }\) is the Einstein-tensor.
- 12.
Alternatively an anti-holomorphic function, which would lead to essentially the same analysis.
- 13.
The j-function is the third power of the partition function of eight chiral bosons compactified on the E 8 lattice or, equivalently (by bosonization) the third power of the partition function of 16 chiral fermions. It has important applications in number theory and the theory of elliptic curves. The coefficients in the power series expansion of j(q) − 744 are related to the dimensions of the irreducible representations of the so-called Monster group, the biggest of the sporadic finite groups.
- 14.
- 15.
Here we use g IIB instead of g s to emphasize that the result holds for type IIB but not for other string theories. For those we will find other relations later on.
- 16.
Going from string to Einstein frame means that we have to rescale all lengths: \({L}_{\mathrm{S}}^{2} = \sqrt{{g}_{\mathrm{IIB }}}{L}_{\mathrm{E}}^{2}\). Since L E is S-duality invariant, \({L}_{\mathrm{S}}\) transforms as \({L}_{\mathrm{S}}^{2} \rightarrow {L}_{\mathrm{S}}^{2}/{g}_{\mathrm{IIB}}\). Alternatively we may change the length scale, i.e. \(\alpha ^{\prime} \rightarrow {g}_{\mathrm{IIB}}\alpha ^{\prime}\).
- 17.
This can e.g. be seen by requiring locality of the OPE of the vertex operators of the 1–1 and the 1–9 fermions.
- 18.
The product \(\boxtimes \) is non-commutative, as the S-duality transformation also acts non-trivially on the antisymmetric tensor.
- 19.
Recall from Chap. 14 that K3 surfaces are the only compact CY manifolds in complex dimension two. Their only non-trivial Hodge number is h 1, 1 = 20 and, of course, h 2, 0 = 1. The second Betti number splits according to \({b}^{2} = {b}_{+}^{2} + {b}_{-}^{2}\) into self-dual and anti self-dual two-forms with b + 2 = 3. The Kähler form and the (2,0) and (0,2) forms are self-dual.
- 20.
With reference to the remarks at the end of Sect. 15.4, we mention that (18.100) is the moduli space of (4, 4) superconformal field theories with \((\bar{c},c) = (6, 6)\).
- 21.
It cannot be a direct product with \({\mathbb{P}}^{1}\) because this would break all supersymmetries and there are no BPS states.
- 22.
We will discuss elliptically fibered K3 surfaces in more detail in Sect. 18.8.
- 23.
Recall: \(\delta {G}_{MN} = -({\nabla }_{M}{\xi }_{N} + {\nabla }_{N}{\xi }_{M}) = -({\xi }^{P}{\partial }_{P}{G}_{MN} + {\partial }_{M}{\xi }^{P}{G}_{PN} + {\partial }_{N}{\xi }^{P}{G}_{MP})\).
- 24.
These relations are implied by the ansatz (18.105), if we write the length element in terms of dimensionless coordinates (for constant dilaton and zero 1-form)
$$d{s}^{2} = {\mathcal{l}}_{ 11}^{2}{G}_{ MN}d{x}^{M}d{x}^{N} = {l}_{ 11}^{2}({g}_{s}^{4/3}d{x}_{ 11}^{2}+{g}_{s}^{-2/3}{g}_{\mu \nu }d{x}^{\mu }d{x}^{\nu }) \equiv {R}_{ 11}^{2}d{x}_{ 11}^{2}+\alpha ^{\prime}{g}_{\mu \nu }d{x}^{\mu }d{x}^{\nu }.$$(18.109) - 25.
The compactification reduces the dimension of space-time and the M2 world-volume. This is also called double dimensional reduction.
- 26.
The metric
$$d{s}^{2} = d{x}_{ \vert \vert }^{2} + V (r)(d{r}^{2} + {r}^{2}d{\Omega }_{ 2}) + V {(r)}^{-1}{(d{x}^{11} + A(y) \cdot dy)}^{2}\,$$(18.115)with \(y \in {\mathbb{R}}^{3},\,{r}^{2} ={ y}^{2}\) and \(\nabla \times A = \nabla V\) (which implies \({\nabla }^{2}V = 0\)) solves the vacuum Einstein equations. For \(V = 1 + \alpha /r\) one finds \(F = dA = \alpha {\epsilon }_{{S}^{2}}\). This is the KK-monopole solution. It is regular at R = 0, if \({x}^{11} \sim {x}^{11} + 4\pi \alpha \). If we identify \({V }^{1/2} = {e}^{-2/3\phi } = {H}^{1/2}\) and write it in the form (18.105), we see that g μν is the D6-brane solution in string frame and dC = dA is its magnetic field strength.
- 27.
A similar analysis can be made for the IIA and IIB supersymmetry algebras.
- 28.
This modification of the supersymmetry algebra can be derived from the world-volume action of extended objects in the Green-Schwarz formulation.
- 29.
Holomorphic line bundles are completely specified by their transition functions between overlapping coordinate patches. For \({\mathbb{P}}^{1}\) there are two patches, U i with \({u}_{i}\neq 0\), with inhomogeneous coordinates \({\xi }_{1} ={ {u}_{2} \over {u}_{1}}\) and \({\xi }_{2} ={ {u}_{1} \over {u}_{2}}\). A section of the canonical bundle in U i is \({\omega }_{i}d{\xi }_{i}\) and from \(d{\xi }_{2} = -{ 1 \over {\xi }_{1}^{2}} d{\xi }_{1}\) we find the transformation \({\omega }_{2} = {c}_{21}{\omega }_{1} = -{\xi }_{1}^{2}{\omega }_{1}\). (The minus sign is irrelevant because two line bundles are isomorphic if their transitions functions are related by \({c^{\prime}}_{ij} = {h}_{i}{c}_{ij}{h}_{j}^{-1}\) where \({h}_{i}\neq 0\) on U i .) The isomorphism class of line bundles with transition functions \({c}_{ij} = {({ {u}_{j} \over {u}_{i}} )}^{n}\) is denoted by \(\mathcal{O}(n)\), i.e. \(K \simeq \mathcal{O}(-2)\). Consider a homogeneous polynomial of order n, \(f =\sum\limits_{p=0}^{n}{a}_{p}{u}_{1}^{p}{u}_{2}^{n-p} = {u}_{1}^{n}{f}_{1}^{ }({\xi }_{1}) = {u}_{2}^{n}{f}_{2}^{ }({\xi }_{2})\). They are sections of \(\mathcal{O}(n) \simeq {K}^{-n}\). The generalization to line bundles over \({\mathbb{P}}^{n}\) is immediate. For a different but equivalent characterization see Footnote 35 on page 482.
- 30.
This is somewhat analogous to viewing S 2 as a S 1 fibration over an interval. At the ends of the interval the fiber degenerates but the total space is nevertheless smooth.
- 31.
It is interesting to note that the gauge group is SU(n) rather than U(n). The explanation, which involves the Stückelberg mechanism, will not be given here.
- 32.
An example: the conditions which lead to the A n − 1 singularity are satisfied if we choose \(f = a,g = b + {u}^{n}\) with \(a = -3{({b}^{2}/4)}^{1/3}\) such that \(\Delta = 54b{u}^{n} + \mathcal{O}({u}^{2n})\). With \(x \rightarrow {({ 2 \over 27b} )}^{1/6}x + {({ b \over 2} )}^{1/3}\) the Weierstrass form is transformed to \({y}^{2} = {x}^{2} + {u}^{n} + \mathcal{O}({x}^{3})\).
- 33.
These rules seem to exclude the junction consisting of F1 ending on D1. But we must interpret it as a junction between (1, 0) (F1), (0, 1) (D1) and ( − 1, − 1) strings. The D1 string absorbs the charge carried by the F1 string and turns into a (1,1) string leaving the junction (which is equivalent to a ( − 1, − 1) string entering the junction.) In the limit of zero string coupling the tension of the strings with \(q\neq 0\) becomes infinite while the tension of F1 stays finite and we recover the picture of a straight D-string with a fundamental string ending on it at right angle.
- 34.
The Weyl tensor has the same symmetries as the Riemann tensor, is traceless on all pairs of indices and transforms homogeneously under Weyl transformations of the metric. It vanishes identically for n ≤ 3. A space is conformally flat iff its Weyl tensor vanishes.
- 35.
- 36.
The general transformation is \(\delta {e}_{\mu }^{a} = {\xi }^{\nu }{\partial }_{\nu }{e}_{\mu }^{a} + {\partial }_{\mu }{\xi }^{\nu }\,{e}_{\nu }^{a} + \sigma {e}_{\mu }^{a} + {\Omega {}^{a}}_{b}{e}_{\mu }^{b}\).
- 37.
One can construct classically Weyl-invariant actions by introducing a compensating field τ. Starting with an action \(S(g,\phi )\), the action \(S({e}^{2\tau }g,{e}^{-\Delta \tau }\phi )\) is Weyl invariant if τ transforms as \(\tau \rightarrow \tau - \sigma \) and \({g}_{\mu \nu } \rightarrow {e}^{2\sigma }{g}_{\mu \nu },\phi \rightarrow {e}^{-\Delta \sigma }\phi \). A simple example is \(S = \int \sqrt{g}R\) which leads to \(S = \int \sqrt{g}{e}^{(d-2)\tau }(R + (d - 1)(d - 2){(\partial \tau )}^{2})\). If we define \(\psi =\exp ({ d-2 \over 2} \tau )\) which has \({\Delta }_{\psi } ={ d-2 \over 2}\), we obtain the familiar action of a conformally coupled scalar \(S = \int {d}^{d}x\sqrt{g}({ 1 \over 2} {\nabla }^{i}\psi {\nabla }_{i}\psi +{ 1 \over 8} { d-2 \over d-1} R{\psi }^{2})\). In an \(Ad{S}_{d}\) background \(\psi \) has mass \({m}^{2} = -{ 1 \over 4{\mathcal{l}}^{2}} d(d - 2)\).
- 38.
de-Sitter-space \(d{S}_{d+1}\) is the hypersurface \({\eta }_{MN}{y}^{M}{y}^{N} = {\mathcal{l}}^{2}\) in \({\mathbb{R}}^{d,2}\).
- 39.
In other words, the boundary is the intersection of the hyperboloid with a plane which cuts through it in the asymptotic region. Topologically the intersection is \({S}^{1} \times {S}^{d-1}\).
- 40.
There are no additional Goldstone bosons associated with the broken rotations. The reason is that locally rotations cannot be distinguished from translations. It is a general fact that broken space-time symmetries are not accompanied 1–1 by Goldstone bosons. Another example is broken scale invariance. Even though dilations and special conformal transformations are broken, there is only one Goldstone boson, the dilaton (the field τ in Footnote 37).
- 41.
For N-extended superconformal theories with \(N\neq 4\) in four dimensions the superconformal group is SU(2, 2 | N) with bosonic subgroup \(SU(2, 2) \times U(N)\).
- 42.
The gravitational interaction is negligible at low energies. The dimensionless coupling constant is \({\kappa }_{10}{E}^{4}\), where E is the typical energy of the interaction. For \(E \ll 1/\sqrt{\alpha ^{\prime}}\) and \({\kappa }_{10} \propto {\alpha ^{\prime}}^{2}\) this is ≪ 1.
- 43.
This is a notion from the large-N expansion of YM theories for which we refer to the literature.
- 44.
There can be several saddle points which lead to the possibility of phase transitions in the CFT. We will not discuss this interesting issue.
- 45.
The choice of a lower limit of the ρ′ integral means that we do not consider diffeomorphisms of the boundary. They are of no interest here.
- 46.
If we include other bulk fields besides the metric, due to back-reaction the FG expansion (18.199) of the metric will, except for special masses of the fields, no longer be in integer powers of ρ only.
- 47.
Uniqueness is only true for the purely gravitational system. If we add other bulk fields, the situation changes; cf. also Footnote 46.
- 48.
The dimensions are fixed by the power of ρ which has dimension \({(length)}^{2}\) and \({g}_{\mu \nu }(x,\rho )\) is dimensionless.
- 49.
On dimensional grounds b n can at most be linear in g (n) as both carry length-dimension 2n. By assumption, f(R) is such that Anti-de-Sitter space is a solution of the equations of motion. Expand the action around this solution as \({g}_{\mu \nu }(x,\rho ) = {\eta }_{\mu \nu } + {\rho }^{n}{g}_{\mu \nu }^{(n)}(x)\). In this expansion the term linear in the fluctuations around the AdS-metric can only be a total derivative (or vanish altogether). Consider the terms \({\nabla }^{M}{\nabla }^{N}\delta {G}_{MN}\) and \({\nabla }^{M}{\nabla }_{M}\,\mathrm{tr\,}(\delta G)\). For fluctuations \(\delta {G}_{\mu \nu } = {\rho }^{n-1}{g}_{\mu \nu }^{(n)}\) the possibly dangerous terms, i.e. those which might contribute to b n , are of the type \({\rho }^{n}\mathrm{tr\,}{g}^{(n)}\). Explicit calculation shows that their coefficient is zero for d = 2n. Higher derivative terms in the variation of the action will involve coefficients g (m) for m < n.
- 50.
In superconformal field theories they also characterize the anomalous divergence of the \(\mathcal{R}\) current and the anomalous γ-trace of the supercurrent.
- 51.
For more general actions the equations of motion are of higher order and have more branches. The additional branches are spurious in the sense that, if the higher derivative terms are considered as small perturbations of the action (18.193), we should only consider perturbations of the two branches of solutions of the unperturbed equations of motion.
- 52.
This is the form in the Landau-frame where \({u}^{\mu }\) is the velocity of energy transport. It satisfies \({u}^{\mu }{T}_{\mu \nu }^{(1)} = 0\).
- 53.
For special values of the mass, there is only one power series solution and one solution containing \(\log (\rho )\). This always happens if \({\Delta }_{+} = {\Delta }_{-}\).
- 54.
In general one has to solve the coupled system of equations for the metric and the scalar field. Here we neglect the backreaction of the scalar on the metric.
- 55.
For ease of notation we use small Latin letters to label the coordinates of the boundary, rather than small Greek letters, as we did previously.
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Blumenhagen, R., Lüst, D., Theisen, S. (2012). String Dualities and M-Theory. In: Basic Concepts of String Theory. Theoretical and Mathematical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29497-6_18
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DOI: https://doi.org/10.1007/978-3-642-29497-6_18
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