Moduli spaces of Calabi-Yau $d$-folds as gravitational-chiral instantons

Motivated by the swampland program, we show that the Weil-Petersson geometry of the moduli space of a Calabi-Yau manifold of complex dimension $d\leq4$ is a gravitational instanton (i.e. a finite-action solution of the Euclidean equations of motion of gravity with matter). More precisely, the moduli geometry of Calabi-Yau $d$-folds ($d\leq4$) describes instantons of (E)AdS Einstein gravity coupled to a standard chiral model. From the point of view of the low-energy physics of string/M-theory compactified on the Calabi-Yau $X$, the various fields propagating on its moduli space are the couplings appearing in the effective Lagrangian $\mathscr{L}_\text{eff}$.

The swampland program [1,2] (for reviews see [3,4]) looks for a characterization of the effective field theories which arise as low-energy limits of consistent theories of quantum gravity, separating them from the vast swampland of effective theories which "look" consistent from a low-energy perspective, but cannot be completed to a fully consistent theory of quantum gravity. The program has produced a dozen or so conjectural necessary conditions (the "swampland conjectures" [1][2][3][4]) that all effective theories of quantum gravity should satisfy.
Dually, there is an inverse-swampland procedure. If we know that a certain effective theory does arise from quantum gravity, we may apply the swampland ideas to predict properties of the model. Often such properties are too fancy for anybody to have enough fantasy to guess them, and they escaped us when we looked at these theories with pre-swampland eyes. With post-swampland insight we know better. This short note illustrates a simple application of the inverse-swampland strategy. The result echos the opening quotation, which was a major inspiration for Newton in formulating his own consistent theory of gravity.
We focus on the Weil-Petersson (WP) geometry of moduli spaces of compact Calabi-Yau (CY) manifolds X d of complex dimension d ≤ 4. These CY manifolds describe stable supersymmetric vacua in string/M-theory, and the quantum-consistent low-energy effective theories around these vacua are captured by the geometry of their moduli spaces. Hence inverse-swampland may yield new insights on the geometry of CY moduli.
We claim 1 that for d ≤ 4 the CY moduli geometry yields a finite-action solution to the classical equations of motion of a moduli-space field theory of the form where M is the CY moduli space, G αβ its metric, and R its scalar curvature. In d = 1 the only CY spaces are the elliptic curves; their moduli space has real dimension 2, so in this case the Einstein term in (1) is topological, while the cosmological constant Λ vanishes -eqn. (1) reduces for d = 1 to the classical Polyakov action of a string moving in the appropriate target space (which is also 2-dimensional). When d = 2 the moduli-space Newton constant κ 2 is an adjustable parameter (this freedom reflects the fact that the moduli metric is always Einstein for CY 2-folds). For d = 3, 4 the Newton constant depends only on the dimension d of the CY, while the cosmological constant Λ depends on d and the complex dimension m of the moduli space: When m = 1 the real dimension of the moduli space M is 2 and Λ = 0, and again (1) reduces to the classical Polyakov action. For m > 1 the moduli-space gravity is "dynamical", and the cosmological constant negative. For m = 2 we get instantons of a "realistic" fourdimensional gravity with matter. In the d = 2 case the matter decouples in the limit κ 2 → 0, and the moduli-space Einstein equations reduce to R αβ = −m G αβ . In this limit it is clear that the finiteness of the moduli volume (one of the swampland conjectures [2][3][4]) should be really understood as a finite action condition for the moduli-space field theory (1). This observation applies in general. The matter part of the action (1) is a standard σ-model with target a locally symmetric space Γ\G(R)/K, where the non-compact real Lie group G ≡ G(R) is the automorphism group of the corresponding Griffiths period domain, that is, explicitly where {h p,d−p prim } are the primitive Hodge numbers in middle dimension 2 K ⊂ G(R) is a maximal compact subgroup. For comparison, we recall that the Griffiths period domain D is the reductive coset G(R)/H where [5][6][7] One may replace G(R) with the Lie subgroup MT (R) ⊆ G(R) given by the real locus of the Mumford-Tate group MT [8][9][10] of the moduli of X d . Indeed the relevant matter field configuration φ : M → Γ\Γ(R)/K has image in the totally geodesic submanifold 3 The discrete group Γ ⊂ MT (Z) ⊆ G(Z) is the monodromy group of the CY period map.
2 H p,q (X d ) stands for the space of harmonic forms of type (p, q) on X d . 3 This statement follows from the structure theorem for the period map [8][9][10].
Very roughly speaking, the moduli space has the form M = G\ M , with M diffeomorphic to R 2m and G a discrete group with a neat subgroup of finite index. We call G the U-duality group. (In the present context it is isomorphic to the monodromy group Γ, but we denote them with distinct symbols for clarity).
A classical solution to (1) consists of two pieces of data: a metric G αβ on M admitting G as a group of isometries, and a harmonic mapφ : for some group homomorphism called the monodromy representation. When (7) holds, one also says that the mapφ is twisted by ρ. A ρ-twisted map descends to a map φ : M → Γ\G(R)/K, and we shall useφ and φ interchangeably. Our claim states that the CY moduli geometry is given by a pair (G αβ ,φ) which satisfies the equations of motion following from the action (1): where the derivative D α is covariant for the combined Levi-Civita connections of T * M and φ * T (G/K). Eqn.(9) just expresses the fact thatφ is a harmonic map M → G(R)/K for the source-space metric G αβ .
Remark 1. Eqn.(9) remains true when d ≥ 5. The WP metric G αβ still satisfies an "Einsteinlike" equation. However it seems that one cannot construct an off-shell action with positive kinetic terms whose canonical energy-momentum tensor yields the source term in the equation. This is to be expected since the moduli geometry of d ≥ 5 Calabi-Yau's is not required to have "magical" properties by swampland consistency conditions.
The moduli space fields (G αβ , φ) as effective couplings To make explicit contact with the swampland program, let us recall the low-energy 4d effective Lagrangian of Type IIB compactified on the (simply-connected) CY 3-fold X 3 where for brevity we wrote only the matter terms involving the bosonic fields of the vectormultiplets. The metric G αβ appearing in the vector-multiplet scalars' kinetic terms coincides with the Weil-Petersson metric on the moduli space M of X 3 [11,12]. For a fixed point ϕ ∈ M , the gauge coupling τ (ϕ) ab is a symmetric complex matrix with positive imaginary part, that is, a point in the Siegel upper half-space Hence the 4d gauge coupling may be identified with the map However this way of describing the gauge couplings is not intrinsic, since τ (ϕ) ab depends on a choice of duality frame. Even worse: the τ (ϕ) ab are multi-valued 4 functions on M because when we go around a non-trivial loop in M we come back with a rotated electromagnetic duality frame. The intrinsic description of the gauge couplings is instead given by the quotient map In other words, the lifted gauge coupling mapφ is twisted by the monodromy representation ρ as in eqns. (7), (8). Indeed, the U-duality group G acts both on M (by isometries) and on the vector field-strengths (by electro-magnetic dualities) while leaving the physical energymomentum tensor T µν invariant; this entails that the 'naive' gauge coupling mapφ is twisted by the monodromy representation ρ of G. This being understood, the on-shell configurations (11),(12) of the two fields (G αβ , φ) which propagate in the moduli space M are exactly the same as the couplings appearing in the Type IIB 4d effective Lagrangian L IIB : Remark 2. More generally, it is pretty obvious that all couplings appearing in the Lagrangian L eff of any 4d supergravity which is consistent with the swampland conjectures [1][2][3][4] and has ≥ 8 supercharges describe (as functions of the scalar fields) gravitational instantons. Again, the finite volume conjecture gets re-interpreted as the statement that the field configuration in moduli space which decribes the effective couplings in L eff has finite action.

Details and proofs
Of course, once we have strong reasons -such as the swampland story -to believe that something ought to be true, we look for actual proofs rather than relying on widely believed conjectures. Our treatment in this note will be totally rigorous (except that we do not discuss the singularities of the relevant solution -a crucial issue, but not one consistent with the purpose of writing a short note).
We present an informal discussion of the general picture in §. 0. Then in § §.1-9 we enter in the technical details, and write explicit expressions for all relevant quantities.

An informal sketch
There are two approaches (or languages) for the geometry of Calabi-Yau moduli spaces: (i) Griffiths theory of variations of Hodge structures (VHS) [5][6][7], and (ii) tt * geometry [16][17][18]. Equivalence of the two viewpoints (in the appropriate contexts) was proven in [19] [16] (and enshrined in the math literature as a theorem in [20]). We shall use both languages, with a preference for the second one.
On the moduli space of a Calabi-Yau d-fold there is an infinite family of a priori distinct canonical Kähler metrics. From the point of view of VHS this plethora arises because the Griffiths period domain D ≡ G(R)/H [5][6][7] carries several holomorphic homogeneous line bundles whose canonical connection has a curvature which is positive when restricted to the Griffiths horizontal tangent bundle. The pull-back to M, via the period map p : M → Γ\D, of any one of these curvatures yields a Kähler form on M. There is one horizontally-positive line-bundle which exists on the period domain D for all Hodge numbers {h p,q }, namely the Griffiths canonical line bundle [21]. The corresponding Kähler metric is called the Hodge metric K jk , and is the best behaved one in the family. In the case of Calabi-Yau d-folds, one has h d,0 = 1 and there is another important horizontally-positive line bundle whose sections are the holomorphic (d, 0)-forms. Its curvature defines the Weil-Petersson (WP) Kähler metric G jk . K jk , G jk do not exhaust the list of canonical VHS metrics. Taking linear combinations with positive coefficients of the several canonical metrics, we construct a convex cone C d of God-given Kähler metrics on the moduli M. The term "God-given" here has a precise technical meaning: the U-duality group G acts by isometries with respect to all Kähler metrics in the convex cone C d . This is quite remarkable, since G is a "huge" group: for CY 3-folds, say, unless the IIB 4d gauge couplings τ (ϕ) ab are numerical constants (i.e. the CY is rigid [13]), the Zariski closure of G is a semi-simple real Lie group of positive dimension [15]. We stress that C d is a cone of actual Kähler metrics, not just Kähler classes. We write K For d ≤ 4 the inequality is saturated provided the CY is not rigid.
The Ricci tensor R jk of the WP metric G jk is also the pull-back of the curvature of a homogenous line-bundle on D, hence R jk belongs to the linear span of the God-given metrics, i.e. R jk can be written as a linear combination of canonical VHS metrics. This is consistent since all Kähler metrics K (c) jk satisfy the same Bianchi identity as the WP Ricci tensor and D j is the Levi-Civita connection of the WP metric. On the moduli space of CY d-folds we have an identity of the form for certain numerical constants λ (c) (to be computed in §. 9 below). We conclude that the WP metric G jk is a solution to the Einstein equations provided the rhs of (20) may be written as a physically sound energy-momentum tensor plus a cosmological constant term.
For CY's of dimension d = 1, 2 one has dim C d = 1, so all canonical VHS metrics are multiples of the WP one which then must be Einstein, i.e. R jk = −Λ G jk for some Λ.
For d = 3, 4 there is a 2-parameter family of VHS Kähler metrics on M, and then the Ricci tensor of the WP metric must be a linear combination of the WP metric G jk and the Hodge one K jk . Now our Claim follows from the fact 5 that the combination entering in (20) is nothing else than the canonical energy-momentum tensor of the Γ\G(R)/K σ-model evaluated on the on-shell field configuration (11), (12).
Having sketched the general picture, let us now write the explicit formulae.  (3), is the fundamental one (2n and, respectively, s + t). The field E(x) is twisted by the monodromy representation (cfr. eqn. (7)) so it descends to a field (or map) E : M → Γ\G. Two field configurations, E(x) and E(x) ′ , which differ by the multiplication on the right by a position-dependent element of K, are declared to be gauge-equivalent (i.e. the same physical configuration) We have the K-principal bundle ̟ : G → G/K, and the physical gauge-invariant map is ̟ • E : M → G/K or, more precisely, its G-equivariant quotient We adopt the following notation: for a ∈ g (the Lie algebra of G), a e and a o denote, respectively, the projection on the even and odd parts under the Cartan involution θ. The action of the σ-model with target space Γ\G/K is (one checks that it is K gauge invariant). The energy-momentum tensor is The equations of motion say (by definition) that the field E is on-shell if and only if the corresponding physical map, ̟ • E : M → Γ\G/K, is harmonic.

Pluri-harmonic maps
Let M be a Kähler manifold and Y any Riemannian manifold.
Note that the covariant derivative D j contains only the Levi-Civita connection of f * T Y . If f is pluri-harmonic, G jk D j ∂kf ≡ 0, so f is a fortiori harmonic, hence a classical solution of the σ-model with target space Y and source space M. We stress that (27) does not contain the Kähler metric of M, so a pluri-harmonic map is harmonic for all choices of Kähler metric.
In our application Y is the locally symmetric space Γ\G/K which is non-compact of finite volume. 6 We assume M to be non-compact and the existence of some complete Kähler metric g jk on M of finite volume; "some" means that the reference metricg jk may have nothing to do with the physical metric G jk . 7 One shows that in these circumstances 8 any classical solution E 0 of the σ-model (defined with the space-time metricg jk ) which has finite action is automatically pluri-harmonic, hence a solution of the equations of motion for any other choice of Kähler metric G jk =g jk on the source space M.

Review of tt * geometry
A tt * geometry on the complex manifold M is just a pluri-harmonic map M → Γ\G/K of finite action, where G, K and Γ are as in §.1. As in that paragraph, the tt * map may be lifted to a map S : M → G. Again, we see S as a field on M taking value in the concrete matrix group G. In facts, S is just a special instance of the σ-model field E of §.1: S is not just an on-shell field configuration, it satisfies the stronger condition of being pluri-harmonic (this is essentially automatic in the present circumstances, see §. 2). S has a direct physical meaning: in the tt * literature [25] [18,26] S is called the BPS brane amplitude (for some value ζ = e iθ of the spectral parameter which depends on the chosen lift).
Since M is complex, we may decompose the differential forms into definite type where unbarred (barred) stands for type (1,0) (resp. (0,1)). We introduce the K-covariant Dolbeault differentials D = ∂ + A andD =∂ +Ā. We have the identity while the condition that the map is pluri-harmonic reads A short computation [27] [15] shows that the compatibility condition of (30) with (31), which expresses the fact that the tt * chiral ring R is commutative. Using (30),(31), and (32) one checks that the Maurier-Cartan identity (d + S −1 dS) 2 = 0 is equivalent to the statement that the 9 g C -valued connection is flat for all values of the spectral parameter ζ ∈ P 1 Eqn.(34) is the Lax form of the tt * PDEs [16][17][18]27].
The application of tt * geometry to 2d (2,2) QFT [16] works as follows. Let P be the complex space of F -term 10 parameters. Over P we have the vector bundle V whose fiber at p ∈ P is the space of susy vacua of the QFT with couplings p. The tt * connection D +D acts on V ; by eqn.(34) it endows V with a holomorphic structure. By construction the tt * connection is metric for the QFT Hilbert space inner product, and hence it is the unique Chern connection on V (and also the Berry one). In a holomorphic gauge we have where g ≡ (g ab ) is the Hilbert space Hermitian metric along the fibers (the tt * metric [16]). The spectral flow isomorphism [22] states that where R is the holomorphic bundle whose fiber R p is the chiral ring at p ∈ P . The last isomorphism in (36) is the reality structure 11 [16]. The holomorphic vacuum bundle V is then isomorphic to a sub-bundle of its tensor-square V ⊗2 . This yields an induction on the bundle metrics: start with the fiber metric g for V ; it induces a fiber metric for V ⊗2 , and its restriction to the sub-bundle R is then a second fiber metric h for V (one may iterate the process ad infinitum).

Superconformal tt * geometry
The discussion in §. 3 applies to all 2d (2,2) QFTs. When the (2,2) QFT is superconformal one is mainly interested in the tt * geometry restricted to the conformal submanifold M ⊂ P of (exactly) marginal deformations. When so restricted, the holomorphic bundles R → M and V → M get graded by the superconformal The decomposition of V is orthogonal for the tt * fiber metric g [16]. Conformal perturbation theory gives us the isomorphism 12 The Hodge metric is the metric on TM given by the induced metric on R 1 as a sub-bundle of V ⊗2 , while the WP metric is the normalized tt * metric restricted to V 1−ĉ/2 [16,24]: When the tt * geometry describes the complex moduli of a CY d-fold X d -that is, when the 2d (2,2) SCFT is the X d σ-model -one hasĉ = d and In this case the tt * Lie group G ≡ G(R), introduced in §. 3, is Sp(2n, R) or SO(s, t) forĉ odd, respectively, even; that is, the tt * group G(R) coincides with the VHS automorphism group (cfr. eqn. (3)). Moreover there is a U(1) grading element Q ∈ g ⊗ C such that 13 [24] [Q, C] = −C, 11 Equivalently, the topological metric η [23]. 12 In the VHS language this isomorphism is called the "local Torelli theorem". 13 The adjoint action of Q on g gets transported on the bundles V q → M because these bundles are the pull-back (via the period map) of homogeneous bundles on the Griffiths domain. See, e.g. chapter 11 of [28].
Refs. [16,19,20] show that the VHS geometry of the complex moduli of a CY d-fold is described by a tt * geometry which satisfies the additional conditions (37)-(41). The Lie sub-group H ⊂ G (cfr. eqn. (5)) is the centralizer of the U(1) charge operator Q in G. 14

Proof of eqn.(9)
The crucial fact is that a solution to tt * corresponds to a finite-action pluri-harmonic map hence, in particular, to a finite-action solution of the σ-model (25). Since ̟ • E is a solution for all Kähler metrics on M (cfr. §. 2), the tt * map ̟ • E is in particular harmonic for the WP metric G jk . This proves eqn. (9). The proof works for all dimensions d of the Calabi-Yau.

Review of [24]
The Hodge metric K jk was introduced in tt * geometry in ref. [24], and further studied in [29], for its relation with the τ -function of isomonodromic problems and, respectively, the Ray-Singer torsion. As already mentioned, in VHS theory the Hodge metric makes sense in the complex moduli space of any projective variety, Calabi-Yau or not. Correspondingly, from a tt * perspective the Hodge metric should be a good Kähler metric for all 2d (2,2) QFTs whether they are superconformal or not. When the 2d theory is superconformal, however, the metric K jk (restricted to the exactly marginal deformations) has nicer properties.
For a general (2,2) QFT the Hodge metric reads [24] K jk = tr C jCk , where C j andCk are the coefficients of the matrix-valued 1-forms C ≡ C j dt j andC ≡Ck dtk (cfr. (29)); {t j } are complex coordinates in the parameter space P of the (2,2) QFT [16].
In the superconformal case we restrict the 1-forms C,C to the conformal submanifold M ֒→ P , i.e. to marginal deformations. Conservation of the conformal U(1) charge yields and we can rewrite equation (43) in arbitrary real (that is, not necessarily holomorphic) local coordinates x α in the form where, in the second line, we used eqn. (29). From eqn. (26) we see that the energy-momentum tensor of the σ-model, evaluated on the particular tt * on-shell field configuration E = S, is In refs. [24,29] there is a second formula for the Hodge metric -this one valid only along the conformal manifold M of a superconformal tt * geometry. It is convenient to adopt the holomorphic gauge (35). We write g ab for the tt * Hermitian metric on the fibers of the vacuum bundle V written in a holomorphic trivialization, and (g q ) uv for its restriction to the sub-bundle V q ⊂ V of definite U(1) charge q, cfr.(37),(41). For the sub-bundle V 1−ĉ/2 ∼ = TM we use the holomorphic local frame {∂ z j } with z j complex coordinates on M; from now on indices from the middle of the latin alphabet j, k, l, . . . always refer to tensors defined in this holonomic holomorphic trivialization of V 1−ĉ/2 ⊂ V . Then, along the submanifold M ֒→ P , one has [24,29] K jk = ∂ i∂k q<0 2q log det g q .
The tt * equations yield a simple formula for the Riemann tensor of the WP metric on the conformal manifold of a (2,2) SCFT. Taking the trace, we get a universal formula for the Ricci tensor valid on M for allĉ where In the special caseĉ = 3 eqn.(50) is sometimes called the 'Strominger formula' [12].
Writing P q for the orthogonal projection V → V q , we havē ∂k∂ j log g q = −tr P q∂k (g∂ j g −1 ) = −tr P q [C j ,Ck] , where in the last equality we used eqn.(34) in the form 15 Setting q = −ĉ/2 in (52) we recover the formula (49). The next case, q = 1 −ĉ/2, yields The same result may be obtained more directly by the first equation in (39) using the general Kähler identity R jk = −∂ j∂k log det G, and eqn.(49).
From eqns. (47), (49) and (54) we read the linear relations between the three tensors G jk , K jk , and R jk on the moduli space of a Calabi-Yau d-fold. From the general discussion in §. 0 we know that there are two linear relations for d = 1, 2 and one for d = 3, 4: The first 3 lines are known to mathematicians [30] (eqn.(58) was first derived in [24,29] which is the classical Virasoro constraint of the Polyakov world-sheet string action.
In a holomorphic gauge where det η = 1 (they exist [23]) the reality constraint [16] implies where, as before, P q is the projector V → V q . K (c) jk is a positive Kähler metric when the coefficient function c(q) belongs to the appropriate convex cone C d ⊂ R [(d+1)/2] which manifestly includes the cone of increasing functions c(q + 1) > c(q).
We show that all metrics K (c) jk satisfy the "Bianchi identity" (19). This property is automatic since these tensors correspond to ∂∂-exact (1,1)-forms κ (c) . ∂κ (c) = 0 reads which is the explicit form of the linear relation (20). The tensor in the rhs is conserved by the "Bianchi identity". While the rhs looks as a valid energy-momentum tensor when evaluated on the on-shell configuration S, it is hard to find an off-shell action with positive kinetic terms which reproduces it. Our feeling is that it does not exist.