Ricci-flat metrics on vector bundles over flag manifolds

We construct explicit complete Ricci-flat metrics on the total spaces of certain vector bundles over flag manifolds of the group $SU(n)$, for all K\"ahler classes. These metrics are natural generalizations of the metrics of Candelas-de la Ossa on the conifold, Pando Zayas-Tseytlin on the canonical bundle over $\mathbb{CP}^1\times \mathbb{CP}^1$, as well as the metrics on canonical bundles over flag manifolds, recently constructed by van Coevering.


Introduction and main result
where L k are linear subspaces of C n , such that dim L k = m k . In place of m k we will often use the integers n i , defined by m k = k i=1 n i . In these terms, the manifold of flags (1.1) is a homogeneous space F n 1 ,...,ns = U (n) U (n 1 ) × ... × U (n s ) .
In what follows we will sometimes use the short notation F for this manifold. The complex geometry of flag manifolds was first studied in the classical work [13].
The authors of [14] use Calabi's ansatz to construct a Ricci-flat metric on the canonical bundle of the flag manifold, equipped with such a Kähler-Einstein metric. An important characteristic of Calabi's ansatz, that we review in Section 2, is that it produces a metric with a fixed Kähler class. On the other hand, by the Calabi-Yau theorem [15,16,17], in the case of a compact manifold with c 1 = 0 there should exist a Ricci-flat metric in every Kähler class. The relevant noncompact generalization of the theorem (to asymptotically-conical spaces), with the same statement, was constructed in [18,19]. and desingularizing Figure 1: The lower (shaded) part of the picture depicts the idea of the present paper. We obtain both the metrics on X K and on X V . The generalized vector a contains the Kähler moduli pertinent to the flag manifold, whereas b is the size of the projective space CP q−1 . 'Desingularizing' means removing a C q /Z q orbifold singularity that arises in the limit b → 0.
The Kähler cone of the total space of the canonical bundle over the flag manifold is the same as that of the underlying flag manifold. The Kähler moduli of the flag manifold, in turn, can be easily characterized geometrically: • As parameters defining an adjoint orbit, in which case the Kähler form is the Kirillov-Kostant form on this orbit (see [20] for a review).
• The most general Kähler metric can be directly constructed using the socalled quasi-potentials [23,24] that also featured in the physics literature in [25]. We will adopt this strategy throughout the paper.
As a result, for the s-step flag manifold (i.e. for the flags of type (1.1)) there are s − 1 real moduli. Therefore Calabi's ansatz does not capture the full moduli space of Ricci-flat metrics on the total space. This was taken into account in [26], where a full family of Ricci-flat metrics on the total space of K F was constructed, using a generalization of Calabi's ansatz. In fact, this generalization is analogous to the one that arose in the work [4] on the conifold and was also considered in [7]. Another interesting analogy may be observed, if one adds to the above the work [5], where essentially the Ricci-flat Kähler metric on the total space of K CP 1 ×CP 1 was constructed. As required by the Calabi-Yau theorem, this metric has two Kähler moduli (since H 2 (CP 1 × CP 1 , R) R 2 ), which geometrically correspond to the radii of the two spheres representing the 'zero section'. As one of the spheres shrinks (i.e., as we approach the boundary of the Kähler cone in a particular way), one gets a metric on a C 2 /Z 2 bundle over CP 1 . Such orbifold bundles were investigated in a more general context in [27]. Removing the orbifold singularity at the zero section, one obtains the total space of the vector bundle O(−1) ⊕ O(−1) over CP 1 [4] (the so-called 'conifold').
In the present paper we pursue a suitable generalization of this procedure to flag manifolds F n 1 ,...,ns . In this case, in place of CP 1 × CP 1 , we start with the manifold F n 1 ,...,ns × CP q−1 , and construct a Ricci-flat metric on In fact, alternatively one may view the manifold F n 1 ,...,ns × CP q−1 as a flag manifold of a semi-simple group SU (n) × SU (q), which allows to identify these metrics with a special case of the metrics constructed in [26]. We then take the limit, when the volume of CP q−1 vanishes, remove a C q /Z q -orbifold singularity and show that in the special case when the line bundle K 1/q F is well-defined (i.e., when q divides c 1 (F n 1 ,...,ns )), the resulting manifold is over F n 1 ,...,ns . The latter rank-q vector bundle will be denoted V . The logic just described is summarized in Fig. 1.
To formulate our result more precisely, we recall the expression for the first Chern class of the flag manifold: where U k are pullbacks of tautological bundles over Gr(m k , n) w.r.t. the forgetful projections π k : F n 1 ,...,ns → Gr(m k , n). We recall the derivation of this formula in section 3.1.
We prove the following statement: Proposition. There exists a complete Ricci-flat Kähler metric on X K in each Kähler class. If there exists a q ∈ N such that q|(n k + n k+1 ) (k = 1 . . . s − 1), then there exists a complete Ricci-flat Kähler metric on X V in each Kähler class. In both cases the line element of the metric is of the form where ds 2 F S is the Fubini-Study metric on CP q−1 , and ds 2 k is the Kähler-Einstein metric, satisfying R ij = ng ij , on the Grassmannian manifolds Gr(m k , n), where m k = k i=1 n i . Besides, c k = n k + n k+1 , A is the holomorphic connection of K Fn 1 ,...,ns ×CP q−1 and the constants (a k , α) determine the Kähler class. The action of the complex structure J is given by J 1 2 H µµ dµ = dφ + Im(A).
H µµ is of the form (1.8) If α + C > 0, a k + C > 0 for all k, and the angular variable φ takes values in [0, 2π] (1.6)-(1.8) describe a metric on X K for all Kähler classes.
The metrics so constructed are asymptotic to the Riemannian cone over the Sasaki-Einstein U (1) bundles over F n 1 ,...,ns × CP q−1 . For X K and X V these are the unit vector bundles of K Fn 1 ,...,ns ×CP q−1 and its q-th root respectively.
Comment. We note that the line element (1.6) has the form of Pedersen-Poon [28].
The structure of the paper is as follows. In section 2 we recall, how the Ricciflat Kähler metric on K CP 1 ×CP 1 is constructed, using a generalization of Calabi's ansatz. In section 3 we pass over to flag manifolds, starting in section 3.1 by explaining the expression for the first Chern class of a flag manifold and constructing the Kähler-Einstein metric in explicit form. Using the Kähler-Einstein metric, we construct in section 3.2 a generalization of Calabi's ansatz, which allows obtaining the Ricci-flat metric in every Kähler class. This generalized ansatz leads to an ODE, which is then solved in section 3.2. In the same section the topology of the manifold (the behavior near the zero section, as well as at infinity) is also analyzed. The appendix is dedicated to the calculation of the determinant of the Hermitian metric, which is used in writing out the Ricci-flatness equation.
2 Conifold and canonical bundle over The ansatz of Calabi may be succintly formulated as the requirement that the Kähler potential K on the total space assumes the form where K is the Kähler potential of the Kähler-Einstein metric on the underlying manifold M KE and u is the coordinate in the fiber. Throughout the paper we will assume that the Kähler-Einstein metric g KE on the base is normalized so that In this section as our principal example we will take M KE = CP 1 × CP 1 . Note that this manifold has two Kähler moduli (the sizes of the two spheres), so this is the simplest instance of the situation described in the introduction, namely the Calabi-Yau theorem requires the existence of two parameters in the metric on Y K := the total space of K CP 1 ×CP 1 . Calabi's ansatz (2.1) fails to capture the full moduli space, since the Kähler-Einstein condition on CP 1 × CP 1 requires that the radii of the two CP 1 's be equal.
Interestingly, Calabi's ansatz corresponds to a special point in the moduli space of metrics, namely the corresponding Kähler form lies in the compactly supported cohomology. This is characterized by a faster decay to the asymptotic form at infinity. For a more detailed discussion of this see [18,19,29].
Denoting the inhomogeneous coordinates on the two spheres by z and w, one introduces the following generalization of Calabi's ansatz (after a simple change of variables): Clearly K 1 and K 2 are the Kähler potentials of the two spheres, and K 1 + K 2 is the Kähler potential of the Einstein metric on CP 1 × CP 1 . Setting a 1 = a 2 = 0 would yield precisely the ansatz of Calabi. Now, we are dealing with a toric variety, the U (1) 3 holomorphic isometries being given by the rotations In such cases it is useful to pass to the moment map variables. The moment map is defined as the derivative ∇ J v K , where J is the complex structure, and v is the vector field corresponding to the holomorphic isometry. In practical terms, this is tantamount to replacing |z 1 | 2 +|z 2 | 2 → e s 1 +e −s 1 , |w 1 | 2 +|w 2 | 2 → e s 2 +e −s 2 , |u| 2 → e t in the Kähler potential, and differentiating it w.r.t. s i and t: One also computes the so-called symplectic potential H , which is the Legendre transform of the Kähler potential K w.r.t. the variables t, s, p: The nice feature of this potential is that the domain, on which it is defined, is precisely the moment polytope of the manifold. Clearly, since in our case the manifold is non-compact, the polytope is also unbounded. Up to inessential linear terms in µ, the dual potential reads: Note that the l log(l) structures appearing in the symplectic potential are typical for Kähler toric geometry [30]. Now, the function H(µ) is determined from the Ricci-flatness equation, which in this case reads Next we write down the explicit expression for the line element derived from the Kähler potential (2.3), using the dual variable µ: (2.10) Here A is the holomorphic connection on the canonical bundle K CP 1 ×CP 1 . Nonnegativity of the metric implies µ + a 1 ≥ 0, µ + a 2 ≥ 0, H µµ = P (µ) P (µ) ≥ 0. The latter requirement is equivalent to P (µ) ≥ 0. Note that P (µ 0 ) = 0 and P (µ) > 0 for µ > µ 0 , due to the first two inequalities. Completeness requires that the range of µ is µ ∈ [µ 0 , ∞). In fact, there are two distinct possibilities: • µ 0 > max(−a 1 , −a 2 ). This was considered by Pando Zayas and Tseytlin [5].
In the first case, one has H µµ = 1 µ−µ 0 + . . . as µ → µ 0 . The change of variables r = (µ − µ 0 ) 1/2 brings the metric to the asymptotic form (2.11) The absence of singularity at r = 0 requires that the angle φ has the range φ ∈ [0, 2π]. Then (2.11) shows that one has the canonical bundle, with connection Im(A), over CP 1 ×CP 1 , the two spheres having radii squared µ 0 +a 1 and µ 0 +a 2 . Varying µ 0 leads to changing the Kähler class of the metric, and all allowed classes (corresponding to the non-vanishing sizes of the CP 1 's) can be achieved in this way.
In the second case, let us assume that max(−a 1 , −a 2 ) = −a 1 . P (µ) now has a second-order zero at µ = µ 0 , so that H µµ = 2 µ−µ 0 + . . . Introducing the variable r = (2(µ − µ 0 )) 1/2 , in the vicinity of µ = µ 0 we can bring the metric to the form (2.12) Hereφ = φ 2 . If one sets z = const., the expression in round brackets is precisely the round metric on S 3 , provided one takes the periodicity of the angleφ to bẽ φ ∈ [0, 2π]. In this case the conical part dr 2 +r 2 (. . .) of the metric above describes the space R 4 C 2 . (Keeping instead the periodicity φ ∈ [0, 2π] would lead to a C 2 /Z 2 orbifold singularity at the zero section, r = 0.) Taking into account the stereographic z-variable of the remaining sphere CP 1 , one can show that the metric (2.12) describes the vicinity of the zero section in the O(−1) ⊕ O(−1) bundle over CP 1 (the so-called 'conifold') [4].
Since the varieties we have considered are toric, it is instructive to construct the respective toric polytopes. The polytopes corresponding to the Pando Zayas-Tseytlin solution and the Candelas-de la Ossa solution are schematically presented in Figs. 2 and 3, respectively. The sections of the moment polytope in the (µ, ν) plane are shown to the left of the full three-dimensional polytope.

Flag manifolds
In this section, apart from the unitary representation (1.2) for the flag manifold (1.1), it will be useful to recall the complex parametrization F n 1 ,...,ns = Gl(n; C)/P n 1 ,...,ns (with P n 1 ,...,ns a parabolic subgroup, stabilizing a given flag), which expresses the flag manifold as a homogeneous space of the complex linear group Gl(n; C).
We will start by recalling the explicit form of the Kähler-Einstein metric on the flag manifold, which is an essential ingredient in Calabi's ansatz. We then use a suitable generalization of this ansatz to construct the metric on X K -the total space of the canonical bundle over F n 1 ,...,ns × CP q−1 , and on X V -the total space of the vector bundle (the latter in the case when the q-th root of K F makes sense).

The canonical bundle
On F n 1 ,...,ns we can consider the vector bundles ξ j and U j (j = 1, ..., s) where the fiber of ξ j over the point is given by L j /L j−1 , and the fiber of U j is L j (U j are the tautological bundles).
Here, as before, L k ∼ = C m k . As is well-known [31] T F n 1 ,...,ns = 1≤i<j≤s We will be interested in the explicit expression for the first Chern class of the flag manifold: Rewriting in terms of U j gives Now, suppose there is a positive integer q that divides (n k +n k+1 ) for all k.  (F n 1 ,...,ns ). Let (u 1 , ..., u n ) ∈ U (n), where u i are column vectors. Then with J ij = mū jm du im . Therefore the line element of the Kähler-Einstein metric on F n 1 ,...,ns , satisfying (2.2), must take the form Note that Kähler-Einstein metric on flag manifolds were first discussed in [34]. It is useful for the following discussion to write out explicitly the Kähler potential corresponding to this metric. To this end consider the matrix W = (w 1 , .., w n ) ∈ Gl(n; C), where each w i is a column vector. We also define an n × m k -matrix Z k ∈ Hom(C m k , C n ) of rank m k Z k = (w 1 , ..., w m k ), where m k = k l=1 n l , (3.8) and introduce the function One can check that log(t k ) is the Kähler potential for the π-normalized canonical metric 1 on the Grassmannian Gr(m k , n). The potential of an arbitrary Kähler metric on the flag manifold [23,24] may then be written as In particular the Kähler-Einstein metric (3.6) arises when γ k = n k + n k+1 := c k .

Generalized Calabi Ansatz for all Kähler classes
From now on we assume there is a positive integer q that divides c k := n k + n k+1 for all k = 1, . . . , s − 1. Also we define the vector z = (z 1 , ..., z q−1 ) ∈ C q−1 . As a candidate for a Kähler potential on X K , we define . (3.11) From the discussion in the previous section it follows that K = q log(1 + | z| 2 ) + s−1 k=1 c k log(t k ) is the Kähler potential of the Kähler-Einstein metric on F n 1 ,...,ns × CP q−1 , and therefore the last term in (3.11) is the expression familiar from Calabi's ansatz (2.1). In (3.11) (α, a k ) are the parameters (Kähler moduli), akin to a 1 , a 2 from (2.3). Their range will be determined later. If we work on X K , the components z i of z are local coordinates on CP q−1 and u is a holomorphic coordinate on the fiber. If we work on X V , (u, z i ) are local coordinates on the fiber.
Notice that K 0 depends only on a single variable t. We will write K 0 and K 0 for its first and second derivatives w.r.t t. In fact, instead of dealing with the function K 0 , just like in section (2) it is convenient to perform a Legendre transform whence The meaning of µ is that it is the moment map for the U (1)-action u → e iα u. The line element then is given by where ds 2 F S has the form of a Fubini-Study metric on CP q−1 and Note that y i are the complex coordinates on the flag manifold, and are abbreviations for w lk (the components of the vectors w l ). The expression (3.14) may be brought to the form (1.6) used in the statement of the Proposition, if we introduce the angular variable φ = arg(u). Then, again using (3.24), we obtain Since t = H µ , we arrive at with Dφ = dφ + Im(A) as covariant derivative. Since the l.h.s. of the above equality is a one-form of type (1, 0), on which the complex structure J acts by multiplication by −i, we get The u-dependent part of the metric (3.14) is then We denote by g the Hermitian (Kähler) metric corresponding to (3.14). The Ricci-tensor is given by R ij = −∂ i∂j log det g , where ∂ i are derivatives w.r.t. the holomorphic coordinates. Therefore Ricci-flatness is satisfied if for a holomorphic function κ. It will be shown in App. A that where κ is a holomorphic function, and Using the definition from (3.11), as well as H µ = t, we may satisfy (3.21) by requiring

Solution of the Ricci-flatness equation
The ODE (3.25) is solved by . (3.27)

Behavior at ∞
Before analysing the effect of different choices for the integration constant C, we observe that H µµ → N µ as µ → ∞. (3.28) Here N = q + dim C (M ). Using (3.14) and (3.20) and making the substitution µ = 1 N r 2 , we find that the line element behaves at infinity as where ds 2 KE = qds 2 F S + s−1 k=1 c k π * k (ds 2 k ) is the Kähler-Einstein metric on F n 1 ,...,ns × CP q−1 with proportionality factor 1. Thus at µ → ∞ we obtain a cone over a Sasakian manifold, which is a U (1)-bundle over F n 1 ,...,ns × CP q−1 . As will be demonstrated in the remainder of this section the angle φ ranges from [0, 2π] on X K and from [0, 2πq] on X V . Therefore the U (1)-bundles are the unit -vector bundles of K Fn 1 ,...,ns ×CP q−1 and K 1/q Fn 1 ,...,ns ×CP q−1 respectively.

The metric on X K
We now analyze the dependence on the choice of C. If C is larger than the largest root of (α + µ) q−1 f (µ), then H µµ is strictly positive. The only possible issue would be a singularity at the zero section, but since that is to say the line element corresponds to a smooth metric. The complex variable re iφ parametrizes C -the fiber of a line bundle, and the connection A allows to identify this bundle with K F ×CP q−1 . Therefore the underlying manifold is X K . The condition a k + C > 0, α + C > 0, (3.32) makes the metric positive-definite. Upon restricting to the zero section µ = C, (3.14) reduces to with γ 0 = q(α + C), γ k = c k (a k + C). The definition (3.10) of the Kähler cone of the flag manifold, combined with γ 0 > 0 for CP q−1 , implies that all Kähler classes are obtained.

The metric on X V
We will now show that for C = −α the line element (3.14) describes a metric on X V . Notice that positive-definiteness of the metric (3.39) requires (since c k > 0 for all k) a k − α > 0. (3.34) It then follows easily from (3.23), (3.27) that the above condition also makes H µµ (and hence the metric (1.6)) positive for µ > −α.
One can visualize the limit µ → −α as sending the volume of CP q−1 to zero and embedding CP q−1 into the fiber C q . First we notice that Therefore to leading order in α + µ Inserting this in the full Kähler potential (3.11) and using (3.35), we get (up to an additive constant that we forget from now on) Here we have introduced the coordinates (x 0 , · · · , x q−1 ) on the C q fiber. They . This procedure changes the periodicity of arg(u) w.r.t the metric on X K in complete analogy to the Pando Zayas-Tseytlin/Candelas-de la Ossa metrics discussed in section 2 (keeping the original periodicity would result in a C q /Z q -singularity).
As we will now see, the above formula provides a Kähler potential in the vicinity of the zero section F n 1 ,...,nm ⊂ X V . The zero section is given by the equations x m = 0, m = 0, . . . q − 1.
Introducing the holomorphic connection A = k c k q ∂ log(t k ), we may write out explicitly the line element corresponding to the above potential, in the limit x m → 0: |dw ab | 2 .
The metric at W = 1 is therefore diagonal, with eigenvalues being equal to j−1 k=i c k (a k + µ), each of multiplicty n i n j . At an arbitrary point W ∈ F , the Hermitian metric g (µ) red is represented by a matrix of size dim C F × dim C F , whose entries are linear in µ. Since i<j n i n j = dim C F , we get det(g and ξ ({w mi ,w mi }) is independent of µ. To find ξ, we notice that in the limit µ → ∞ the metric (A.2) behaves as g (µ) red → µ g KE , where g KE is the Kähler-Einstein metric on F described in (3.6). Since its Kähler potential is c k log t k , from Ric = g KE we find