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Low-Energy Effective Theories

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Introduction to String Theory

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Abstract

We construct the low-energy effective theories of the five 10d supersymmetric strings: their Lagrangians are essentially determined by SUSY. The effective theories yield a description which is non-perturbative in the sense that, while it is valid only asymptotically for small momenta, in this IR regime remains reliable for all values of the string coupling constant g. This remarkable property reflects the existence of powerful SUSY non-renormalization theorems for the relevant couplings and their dependence on g. We construct the half-BPS solutions of the effective theories: here the fundamental fact is that the SUSY central charges are not renormalized, so the results we get at the effective level are exact in the full string theory.

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Notes

  1. 1.

    \(\mu \), \(\alpha \) are \( Spin (d-1,1)\) vector and spinor indices, respectively. A is the SUSY extension index on which the R-symmetry group acts.

  2. 2.

    \(R\ne \textbf{UAut}\) only when the gravitational supermultiplet satisfies a reality constraint.

  3. 3.

    This follows from the representation theory of SUSY algebras [9]: it extends to the full connection in the covariant derivative \(D_\mu \) the well-known fact that in GR the spin-connection \({\omega _\mu }^{mn}\) is not a fundamental field but a composite operator of the vielbein \(e^m_\mu \) and its derivatives.

  4. 4.

    We refer to Sect.  11.1.1 for background on Riemannian holonomy groups and the facts alluded in the text. In this Chapter we use well-known results from differential geometry in an informal fashion.

  5. 5.

    For the definition and basic properties of Kähler, hyperKähler, and quaternionic Kähler geometries see Sect. 11.1.1 or [12,13,14].

  6. 6.

    Toroidal compactification changes the scalar manifold \(\mathcal {M}\) as Eq. (8.6) and Table 8.1 imply. However the scalar manifold of the higher dimensional theory is a totally geodesic subspace of the one in lower dimension, while all totally geodesic subspaces of a locally symmetric space is also locally symmetric. Hence the statement in the gray box holds in \(d\ge 3\) dimensions if it holds in \(d=3\). We shall illustrate this point explicitly in Chap. 13 when studying group disintegration and U-duality.

  7. 7.

    The border case of 8 supercharges will be considered in detail in Chap. 11.

  8. 8.

    A truncation of d.o.f. in a classical Lagrangian field theory is a consistent truncation iff all solutions of the truncated model are solutions to the original e.o.m. The truncation to configurations invariant under some bosonic symmetry is always consistent.

  9. 9.

    Warning. There is one way the argument in the text may fail: several short supermultiplets can recombine together to form a long one. This phenomenon does not happen when the BPS supermultiplets preserve many supercharges (the supermultiplets are “ultrashort”) which is the case of interest in the present chapter.

  10. 10.

    More generally one can consider geometries which are asymptotic to anti-de Sitter (AdS), see [26, 27] or Chap. 6 of [1].

  11. 11.

    The statement in the main text is slightly stronger than the Weinberg-Witten theorem.

  12. 12.

    Properly speaking, the argument in the text applies only for \(d\ge 4\), since for \(d\le 3\) there is no propagating graviton i.e. no state \(|2\rangle \). However the result remains true in 3d if we restrict to locally non-trivial theories, i.e. SUGRA models with propagating local d.o.f. [1].

  13. 13.

    By the standard relations between (universal) Clifford algebras in different signatures [32], the function defined here, \(\boldsymbol{N}(d)\), is equal to \(2\,\textbf{N}(d-2)\) where \(\textbf{N}(k)\) is the function defined in Eq. (2.57) of [1] i.e. the real dimension of the irreducible modules of the Clifford \(\mathbb {R}\)-algebra in dimension k.

  14. 14.

    Here ‘combine’ means identification at the linearized level (which is enough to identify and count d.o.f.). The full non-linear relation between 11d and 10d fields is, of course, more involved.

  15. 15.

    Here and below a symbol of the form \(X^{(k)}\) means that the field X is a k-form.

  16. 16.

    See, for instance, for 11 SUGRA in [33], for 10d IIA SUGRA in [34, 35], for 10d IIB SUGRA in [36], and for 10d \(\mathcal {N}=1\) SUGRA with matter in [37].

  17. 17.

    For Kaluza-Klein geometries in supergravity see the review [39].

  18. 18.

    Mutatis mutandis this is the same mechanism as discussed in BOX 6.2.

  19. 19.

    This formula holds up to a term which when multiplied by \(\sqrt{-G_{(11)}}\) becomes an irrelevant total derivative

  20. 20.

    But recall the crucial subtleties with the BRST invariance at zero momentum–see the final part of Sect. 3.7.3, BOX 3.3, and Sect. 5.6.

  21. 21.

    10d IIB SUGRA was first constructed in [43, 44].

  22. 22.

    For the theory of \(G(\mathbb {R})\)-invariant field-strenghts in extended SUGRA see Sect. 4.7.1 of [1]

  23. 23.

    For the theory of the Iwasawa gauge in SUGRA see Sect. 5.11 of [1]

  24. 24.

    We cannot gauge a continuous subgroup since there are no vector fields.

  25. 25.

    Here and below a subscript E means that the quantity is in the Einstein frame.

  26. 26.

    For exact solutions to Einstein’s equations see e.g. [54].

  27. 27.

    For heterotic/Type I BPS branes see Sect. 13.4.1.

  28. 28.

    In this section the IIB RR field \(C^{(0)}\) is seen as a 0-form gauge field rather than a scalar.

  29. 29.

    The charges are canonically normalized so that e is the Noether charge while m is the topological charge with Dirac’s normalization. The RR vertices (i.e. the Kähler-Dirac BRST condition) treat the field strengths \(F^{(k)}\) and their dual \(*F^{(10-k)}\) symmetrically, so in string theory it may be more natural to normalize the electric and magnetic charges with the same overall numerical coefficient.  Various normalization conventions are used in the literature.

  30. 30.

    The asymmetry is only apparent because it depends on which d.o.f. we use to describe the physics: electric fields or the dual magnetic ones. Going to the dual description electric and magnetic charges invert their roles. Recall from the Kähler-Dirac formulation of the R-R gauge fields in Chap. 3 that the superstring treats the gauge fields and their duals in a democratic way.

  31. 31.

    For the moment we consider branes carrying only one type of charge.

  32. 32.

    The original rotational-invariant ansatz, being the most general configuration invariant under a bosonic symmetry, is automatically a consistent truncation of SUGRA and we insert that ansatz directly into the action. That our slightly generalized ansatz is still a consistent truncation may be less obvious; however the property may be checked using the Ricci curvatures for a metric conformally equivalent to the warped product \(\mathbb {R}^m\times _{e^A} \mathbb {R}^{n-1,1}\) see BOX 8.6.

  33. 33.

    Here \(\epsilon _{\mu _1\cdots \mu _d}\) is the totally antisymmetric tensor with \(\epsilon _{0\cdots d-1}=1\) and the same for \(\epsilon _{m_1\cdots m_{\tilde{d}+1}n}\).

  34. 34.

    Note that the two exponents in (8.125) \(d\sigma +(D-2)B\) and \(-d\sigma +(D-2)B-a_k \phi +2C\) vanish when (8.126) holds. This fact greatly simplifies the algebraic manipulations.

  35. 35.

    We absorb an overall constant in the normalization of the coordinates.

  36. 36.

    The pairs (p, q) of coprime integers, taken modulo PCT, i.e. \((p,q)\sim (-p,-q)\), are in 1-to-1 correspondence with the points in the projective line over the rational \(\mathbb {P}^1(\mathbb {Q})\). Clearly \(\mathbb {P}^1(\mathbb {Q})\) is the modular group \(PSL(2,\mathbb {Z})\) modulo its unipotent subgroup  \(T^\mathbb {Z}\), \(\mathbb {P}(\mathbb {Q})\simeq PSL(2,\mathbb {Z})/T^{\mathbb {Z}}\).

  37. 37.

    This is a situation which happens only in Lorentzian signature: the holonomy group of a positive-signature metric is always reductive being a closed subgroup of the compact group O(d).

  38. 38.

    For the theory of Lorentzian holonomy groups see [74,75,76,77]; for a review Sect. 3.7 of [1].

  39. 39.

    \(i=1,\dots ,9\) for 11d and \(i=1,\dots ,8\) for 10d IIB.

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  1. Yanqi Lake Bei**g Institute of Mathematical Sciences and Applications (BIMSA), Bei**g, China

    Sergio Cecotti

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Cecotti, S. (2023). Low-Energy Effective Theories. In: Introduction to String Theory. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-031-36530-0_8

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