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Calabi–Yau Compactifications

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Introduction to String Theory

Part of the book series: Theoretical and Mathematical Physics ((TMP))

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Abstract

We study a large class of exact superstring vacua obtained by compactifying a 10d SUSY string down to 4d on Riemannian spaces of SU(3) holonomy a.k.a. Calabi–Yau (CY) 3-folds. These vacua are invariant under 4d Poincaré symmetry and one-quarter of the original supersymmetries. We discuss their deep geometry and physics, aiming to be didactical but also mathematically precise and self-contained. We review all geometry we need. One may study these vacua from three viewpoints:

(a):

in Algebro-Geometric terms, exploiting the description of the Calabi–Yau spaces as complex projective varieties. The main tool here is Griffiths’ theory of variations of Hodge structures (VHS) [1,2,3];

(b):

in terms of the Zamolodchikov geometry of the (2,2) SCFT living on the string world-sheet. Here the main tool is \(tt^*\) geometry [4,5,6];

(c):

in terms of their low-energy effective 4d \(\mathcal {N}=2\) SUGRA. The main tool here is “special Kähler geometry” [7,8,9,10,11,12].

These three approaches, despite the difference of languages, perspectives, and tools, turn out to be essentially equivalent [13].

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Notes

  1. 1.

    That is, a tensor in \(\mathbb {R}^n\) with the index structure \({T^{b_1b_2\cdots b_k}}_{a_1a_2\cdots a_\ell }\) with \(a_s,b_t=1,2,\dots , n\).

  2. 2.

    Properly speaking, this is the restricted holonomy groups, i.e. its connected component.

  3. 3.

    Recall that the Chern connection [22] is the unique connection on a holomorphic vector bundle with a Hermitian fiber metric which is both holomorphic and metric.

  4. 4.

    For more details see [24, 25]. For a readable summary for physicists see Chap. 3 of [7].

  5. 5.

    Here and below parallel stands for covariantly constant with respect to the Levi-Civita connection.

  6. 6.

    Here \(\Gamma ^i\) are Dirac matrices acting on a spin bundle S on the manifold M and \(\Gamma ^{ij}\equiv \tfrac{1}{2}[\Gamma ^i,\Gamma ^j]\). The generators of \(\mathfrak {spin}(\dim M)\) acting on a spinor \(\psi \in C^\infty (M,S)\) are the Dirac matrices \(\tfrac{1}{2}\Gamma ^{ij}\).

  7. 7.

    Indeed the spectrum of U(1) charges q of the states (11.15) should be symmetric under \(q\rightarrow -q\), and this fixes the charge of \(\psi _0\) to be \(-m/2\).

  8. 8.

    This is clear from the fact that a strict QK manifold is Einstein with a non-zero cosmological constant—hence has no parallel spinor by Theorem 11.6. A Ricci-flat QK manifold is hyperKähler.

  9. 9.

    A \(G_2\)-manifold has odd dimension (7) and there is no notion of chirality for parallel spinors.

  10. 10.

    The overbar stands for complex conjugation.

  11. 11.

    That is, a U(n)-structure, possibly with a torsion, whose underlying \(GL(n,\mathbb {C})\)-structure admits a torsionless connection.

  12. 12.

    Recall from BOX 1.6 that the Picard group \(\textsf{Pic}(X)\) is the group of isomorphism classes of line bundles \(L\rightarrow X\). For X smooth, it is also the group of divisors on X modulo linear equivalence.

  13. 13.

    By definition \(\mathcal {K}\) is a convex cone iff \(\omega _1,\omega _2\in \mathcal {K}\) implies \(\lambda _1 \omega _1+\lambda _2\omega _2\in \mathcal {K}\) for all \(\lambda _1,\lambda _2\in \mathbb {R}_{>0}\). It is strict if, in addition, \(0\ne \omega \in \mathcal {K}\) implies \(-\omega \not \in \mathcal {K}\).

  14. 14.

    A smooth function \(\Phi \) on a complex manifold is pluri-subharmonic iff the matrix \(\partial _j\partial _{\bar{k}}\Phi \) is non-negative [54]. In particular, a global Kähler potential (when it exists) is pluri-subharmonic.

  15. 15.

    With the condition that \((X,\mathcal {O}_X)\) is locally isomorphic to \((\mathbb {C}^n,\mathcal {O}_{\mathbb {C}^n})\) as a locally ringed space [55, 56], i.e. a complex manifold is locally modeled by \(\mathbb {C}^n\) with its local holomorphic functions.

  16. 16.

    We write \(\Lambda ^{p,q}(\mathcal {V})\) for the space of smooth forms of type (p, q) on \(X_0\) with coefficients in the holomorphic vector bundle \(\mathcal {V}\) [22] and simply \(T^{1,0}\) for the holomorphic tangent bundle \(T^{1,0}X_0\).

  17. 17.

    Warning. Here we are oversimplifying; for a precise treatment of m-CY, see [57,58,59].

  18. 18.

    For more details see e.g. [7].

  19. 19.

    The moduli space \(\mathcal {M}\), Eq. (11.69), has orbifold singularities. E.g. the 1-CY are elliptic curves, \(\mathcal {S}\) is the upper half-plane \(\mathcal {H}\), which is smooth, while \(\mathcal {M}_1=\mathcal {H}/SL(2,\mathbb {Z})\) has two orbifold points i, \(e^{2\pi i/3}\) fixed by (conjugacy classes of) finite subgroups of the modular group.

  20. 20.

    Indeed a locally symmetric space which is Ricci-flat is flat.

  21. 21.

    Through this chapter we see the bosonic configurations of the \(\sigma \)-model as maps \(\phi :\Sigma \rightarrow M\).

  22. 22.

    Complex manifolds of the form \(\{Z\in \mathbb {C}^n:\textrm{Im}\, Z\in V\}\) with \(V\subset \mathbb {R}^n\) a strict, convex cone are called tube domains or Siegel domains of the first kind [69]. 

  23. 23.

    By a Riemannian product we mean a product \(M_1\times M_2\) of manifolds equipped with the metric \(ds^2=ds_1^2+ds_2^2\) where \(ds_a^2\) is a Riemannian metric on \(M_a\) (\(a=1,2\)).

  24. 24.

    We are cheating a little bit: \(\widetilde{\mathcal {C}}\) is non-complete in the Zamolodchikov metric \(G_{ij}\). Perturbing \(G_{ij}\) by its Ricci tensor, \(G_{ij}\rightarrow G_{ij}+\epsilon R_{ij}\), we get a metric satisfying the conditions of the Theorem.

  25. 25.

    We stress that the distinction chiral vs. twisted-chiral is a matter of conventions.

  26. 26.

    Recall from Chap. 2 the notion of superconformal superfields: not all superfields are superconformal superfields. The chiral primaries are first components of superconformal superfields. More specifically, a general NS chiral state \(|\psi \rangle \) satisfies \(G^+_{-1/2}|\psi \rangle =0\), while a primary chiral state satisfies \(G^-_{+1/2}|\psi \rangle =G^+_{-1/2}|\psi \rangle =0\). A non-zero state of the form \(G^+_{-1/2}|\eta \rangle \) is a chiral descendent.

  27. 27.

    Notation. \(A_A\) (resp. \({}_AA\)) denotes the algebra A seen as a right (resp. left) module over itself. \(D(-)=\textrm{Hom}_\mathbb {C}(-,\mathbb {C})\) is the “classical” duality in the category of vector spaces over \(\mathbb {C}\). D maps right modules into left modules and left ones and viceversa. One has \(D^2=\textrm{Id}\).

  28. 28.

    Super-commutative means that the elements of the ring are distinguished in bosonic and fermionic; the bosonic ones commute and the fermionic ones anti-commute. Unital means that the algebra contains an identity element 1.

  29. 29.

    “Chiral” has two different meanings in this discussion. The reader should not confuse the two.

  30. 30.

    Note that the rhs of (11.141) is independent of \(z,\bar{z}\).

  31. 31.

    To simplify the notation, here and below TX (resp. \(T^*X\)) stands for the holomorphic tangent (resp. cotangent) bundle of vectors (resp. forms) of type (1, 0), \(T^*X=(TX)^\vee \).

  32. 32.

    Note that at weak coupling the operators (11.154), (11.155) saturate the unitarity bound \(h=|q|/2\) and so are chiral primaries; then they belong to short SUSY protected supermultiplets and their existence, quantum numbers, and dimensions are preserved even at strong coupling. The only aspect which may be renormalized by the world-sheet interactions is their product table.

  33. 33.

    The conventions were chosen to avoid clashes between the two meanings of the symbol q.

  34. 34.

    For an axiomatic approach to TFT see e.g. [74].

  35. 35.

    The gravitino vertices are the \(\gamma \)-traceless part of (11.174); the \(\gamma \)-traces are vertices for dilatini.

  36. 36.

    Not to be confused with the 10d one.

  37. 37.

    Although for \(H^1(X, T^*X)\) the identification is exact only at weak coupling, the counting of states is exact for all couplings since these states form short SUSY representations.

  38. 38.

    MUM=maximally unipotent monodromy [2]; physically it is the group of unitary automorphisms of the Lorentz algebra \(\mathfrak {spin}(3,1;\mathbb {R})\).

  39. 54.

    The subgroup \(U(h)\times U(1)\subset \textit{Sp}(2h+2,\mathbb {R})\) is compact, so contained in a maximal compact subgroup \(U(h+1)\subset \textit{Sp}(2h+2,\mathbb {R})\). The canonical projection \(\varpi :D_h\rightarrow \mathcal {H}_{h+1}\) maps an equivalence class modulo \(U(h)\times U(1)\) to the coarser equivalence class modulo the bigger group \(U(h+1)\).

  40. 55.

    A priori it is just a classical Poincaré invariant vacuum; but now the theory is supersymmetric, and no classical vacuum is lifted by quantum effects. So our vacua are actual quantum vacua.

  41. 56.

    For homogeneous bundles over Griffiths period domains see [7] or the original literature [114].

  42. 57.

    As a matter of notation, X represents the underlying smooth space stripped of the complex structure. \(X_s\) for \(s\in \mathcal {M}\) stands for the manifold endowed with the complex structure s.

  43. 58.

    The extend \(\mathcal {L}_\text {eff}\) is exact is specified by the SUSY non-renormalization theorems.

  44. 59.

    A vertical bar | stands for restriction of the differential form to the CY fiber.

  45. 60.

    For the notion of contact structure see [19] or [117, 118].

  46. 61.

    That is, satisfying the obvious transversality conditions.

  47. 62.

    The following statements hold for some sufficiently generic duality frame. The choice of a “good” frame may be obstructed if the SUGRA is gauged.

  48. 63.

    Since the considerations are strictly local, we do not distinguish between \(\mathcal {M}\) and its cover \(\widetilde{\mathcal {M}}\).

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Cecotti, S. (2023). Calabi–Yau Compactifications. In: Introduction to String Theory. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-031-36530-0_11

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