Abstract
Abstract. We study the Hessian geometry of toric Gibbons-Hawking metrics and their phase change phenomena via the images of their moment maps.
Introduction
This is a sequel to [20] in which techniques developed in string theory were applied to study the Kepler problem in classical gravity. We will apply the same techniques in this paper to study gravitational instantons in Euclidean gravity in dimension four of type A n−1 , i.e., the Gibbons-Hawking metrics.
Introduction
Since the advent of AdS/CFT correspondence [13, 19] in string theory, which are based supergravity solutions, there have appeared new constructions of SasakiEinstein metrics [9] which were soon realized to be toric [14] . Symplectic techniques developed for toric Kähler metrics on compact manifolds was generalized to the metric cones of the toric Sasaki-Einstein spaces, which are Kähler Ricci-flat. This leads to the applications of Hessian geometry [16] on convex cones to AdS/CFT [15] . These techniques were generalized and applied to the Kepler problem in [20] . First, in §6 of that work, we use the explicit construction of Kähler metrics with U (n)-symmetry [12, 5, 6 ] to obtain applications of symplectic coordinates and Hessian geometry. Next, these are applied to the A 1 case of Gibbons-Hawking metrics [8] (i.e., the Eguchi-Hanson metrics [7] ), as the Kepler metric on the Kepler manifold K 2 , in [20, §7.2] . An alternative treatment, based on Calabi ansatz [2] , is presented in [20, §8] . Generalizations to the Kepler metric and the Kähler Ricci-flat metrics on the resolved conifold are made in §9 and §10 of [20] , based on Calabi ansatz on O P 1 (−1) ⊕ O P 1 (−1) and O P 1 ×P 1 (−1, −1) respectively. The basic observations in [20] is that all these metrics obtained by such explicit constructions are toric and the images of their moment maps are polyhedral convex bodies, and one can recover the complex structures and Kähler structures by Hessian structures [16] on the convex bodies. In this paper we will show the same holds for toric GibbonsHawking metrics.
Introduction
As a direct consequence of the applications of moment maps and Hessian geometry, we will study some phase change phenomena of the Gibbons-Hawking metrics. We will describe a procedure in which "black hole" appears naturally such that outside the "blackhole" we still have Riemannian metric but inside it we get imaginary Riemannian metric. From a string theoretical point of view this is very natural. See the discussions in §6.
Introduction
The notion of a phase change for Kähler metrics were first introduced in [5, 6] . In [20, §10.6] , the author presented a new method to describe the flop of Kähler Ricciflat metrics on the resolved conifold [3] by embedding them in a two-parameter family of Kähler Ricci-flat metrics on the canonical line bundle of P 1 × P 1 . It then becomes natural to reexamine the phase change phenomena introduced in [5, 6] from the point of view of [20] . This has been carried out recently for local P 1 , local P 2 and local P 1 × P 1 cases in [18] . The underlying spaces for these case are toric Calabi-Yau 3-folds O P 1 (−1) ⊕ O P 1 (−1), O P 2 (−3) and O P 1 ×P 1 (−2, −2) respectively. In this paper, we will treat the case of toric Calabi-Yau 2-folds, i.e., the crepant resolutions of C 2 /Z n . They are also called the Gibbons-Hawking spaces, or the ALE spaces of type A. We conjecture that the phase change phenomena described in this paper can also occur on ALE spaces of type D and type E.
Introduction
The rest of the paper is arranged as follows. In §2, after recalling the GibbonsHawking construction we present some explicit choices of connection 1-forms involved in this construction. The applications of moment maps and Hessian geometry to toric Gibbons-Hawking metrics are presented in §3. Hessian geometry is used in §4 to explicitly construct local complex coordinates on Gibbons-Hawking spaces and to identify them with the crepant resolutions of C 2 /Z n . We describe the phase changes for Gibbons-Hawking metrics in §5. We end the paper by some concluding remarks in §6.
Gibbons-Hawking Construction
In this Section we first recall the Gibbons-Hawking construction [8] , then present some explicit choices involved in the construction, which will be crucial for computations in later Sections.
Gibbons-Hawking construction.
Given n distinct points p 1 , . . . , p n in R 3 , consider a function V defined by
Gibbons-Hawking construction.
Clearly, V is a solution of the Laplace equation
Gibbons-Hawking construction.
Denote by * the Hodge star-operator on R 3 . Since ∆V = − * d * dV , we have
Gibbons-Hawking construction.
. . , p n } be a domain on which we have
Gibbons-Hawking construction.
On the principal bundle U × S 1 → U with connection 1-form dϕ + α, where ϕ is the natural coordinate on S 1 , i.e., e iϕ ∈ S 1 , the horizontal lifts of vector fields 
Gibbons-Hawking construction.
is a local orthonormal frame for the metric
Gibbons-Hawking construction.
This metric is the Gibbons-Hawking metric in local coordinates {ϕ, x, y, z}.
2.2.
Complex structures on Gibbons-Hawking spaces. Consider the almost complex structure given by:
2.2.
The induced almost complex structure on the cotangent bundle in terms of the orthonormal frame is given by:
2.2.
Therefore, the space of type (1, 0)-forms are generated by:
2.2.
One can compute their exterior differentials and see that they do not contain (0, 2)-components as follows:
2.2.
therefore, by Newlander-Nirenberg theorem, the almost complex structure is integrable. We will address the problem of finding explicit local complex coordinates in §4. This complex structure is compatible with the Riemannian metric g, with the symplectic form given by:
2.2.
Since one has
2.2.
therefore, g, J gives a Kähler structure on the Gibbons-Hawking space. Indeed, by changing (x, y, z) to (y, z, x) and (z, x, y), one gets two more Kähler structures, making the Gibbons-Hawking metrics hyperkähler.
2.2.
2.3. Explicit expressions. Next we consider the explicit expressions. Let r = (x, y, z) and p j = (a j , b j , c j ), then
2.2.
Let us begin with the case of n = 1 and let c 1 = 0, i.e., p 1 = (0, 0, 0). Then (14)
2.2.
This is not an exact form on R 3 − {(0, 0, 0)}, but on U := R 3 − {(0, 0, z) | z ≥ 0} we will use the stereographic projection to get the following change of coordinates:
2.2.
Then we have
2.2.
we will use spherical coordinates
2.2.
Then we have
2.2.
xdy − ydx r(r + z) ..
2.2.
On U ∩Ũ we have the following coordinate change formula:
2.2.
One can then check that
2.2.
.
2.2.
It is easy to check that
2.2.
where θ is the argument in the (x, y)-plane:
2.2.
To get toric Gibbons-Hawking metrics, we will let the points p 1 , . . . , p n lie in a line, say, p j = (0, 0, c j ), j = 1, . . . , n, c 1 < c 2 < · · · < c n , we take U := R 3 − {(0, 0, z) | z ≥ c 1 } we take:
2.2.
Then we have
2.2.
.
2.2.
In particular,
2.2.
Then we have
2.2.
.
2.2.
On U ∩Ũ we have the following coordinate change formula:
2.2.
One can then check that
Hessian Geometry of Toric Gibbons-Hawking Spaces
In this Section we study the toric Gibbons-Hawking metrics from the point of view of moment maps and Hessian geometry. For this purpose, we need to introduce a 2-torus action and study its moment map. This leads to Hessian structures [16] on the convex bodies which arise as the image of the moment map.
Hessian Geometry of Toric Gibbons-Hawking Spaces
3.1. Torus action and moment map on a toric Gibbons-Hawking space. We now look at the 2-torus action given in local coordinates by:
Hessian Geometry of Toric Gibbons-Hawking Spaces
the associated vector field is X 1 = ∂ ∂ϕ and X 2 = −y
Hessian Geometry of Toric Gibbons-Hawking Spaces
. From the formula (13) for the symplectic form, it is easy to see that the moment map with respect to the torus action above is given by:
Hessian Geometry of Toric Gibbons-Hawking Spaces
where r j is defined by
Hessian Geometry of Toric Gibbons-Hawking Spaces
Indeed, we have
Hessian Geometry of Toric Gibbons-Hawking Spaces
3.2. Image of the moment map. Note:
Hessian Geometry of Toric Gibbons-Hawking Spaces
and since
Hessian Geometry of Toric Gibbons-Hawking Spaces
it is easy to see that the image of the moment map is the convex region given by the following inequalities:
Hessian Geometry of Toric Gibbons-Hawking Spaces
3.3. Symplectic coordinates and Hessian geometry for toric GibbonsHawking spaces. The following is our first main result: Proposition 3.1. If one takes the following local coordinates:
Hessian Geometry of Toric Gibbons-Hawking Spaces
then the symplectic form takes the following form:
Hessian Geometry of Toric Gibbons-Hawking Spaces
And in these symplectic coordinates, the Gibbons-Hawking metric takes the following form:
Hessian Geometry of Toric Gibbons-Hawking Spaces
where
Hessian Geometry of Toric Gibbons-Hawking Spaces
The complex potential and the Kähler potential are given by the following formulas respectively:
Hessian Geometry of Toric Gibbons-Hawking Spaces
And the Kähler potential is given by:
Hessian Geometry of Toric Gibbons-Hawking Spaces
for some constants C 1 and C 2 .
Hessian Geometry of Toric Gibbons-Hawking Spaces
Proof. By (32), (43) and (44), it is straightforward to get (48). By (44) we have
Hessian Geometry of Toric Gibbons-Hawking Spaces
and so we have
Hessian Geometry of Toric Gibbons-Hawking Spaces
The formulas for the complex potential and the Kähler potential will be proved in the next two subsections.
Hessian Geometry of Toric Gibbons-Hawking Spaces
Similar to [20] , write:
Hessian Geometry of Toric Gibbons-Hawking Spaces
where the coefficient matrices (G ij ) i,j=1,2 and (G ij ) i,j=1,2 are given by:
Hessian Geometry of Toric Gibbons-Hawking Spaces
It is easy to see that these matrices are inverse to each other.
3.4.
Complex potential functions. Next we will show that
3.4.
for some function ψ. To find ψ, we rewrite (44) as follows:
3.4.
From this we get:
3.4.
It follows that
3.4.
and
3.4.
Therefore, one has
3.4.
So we can take
3.4.
so that the equalities in (58) hold. We can write ψ in another form. Since
3.4.
is linear in µ 1 , µ 2 , we can take:
3.4.
Using the equalities
3.4.
we have
3.4.
3.4.1. The n = 1 case. One can take c 1 = 0. From the equation
3.4.
we can solve for ρ 2 :
3.4.
and (66)
3.4.
The complex potential is
3.4.
for some constants C 1 , C 2 , and the Kähler potential is
3.4.
Make the following change of coordinates:
3.4.
Then the image of the moment map is changed from the convex cone defined by:
3.4.
to the convex cone defined by:
3.4.
and the complex potential function becomes
3.4.
we can solve for ρ 2 :
3.4.
From this one finds that
3.4.
and by these the complex potential function becomes:
3.4.
for some constants C 1 , C 2 . We make the following change of coordinates:
3.4.
Then the image of the moment map is changed from the convex cone defined by:
3.4.
to the convex cone defined by:
3.4.
and up to a linear function in y 1 , y 2 ,
3.4.
where y = y 1 + y 2 , up to a term of the form C
3.4.
. Later we will fix these constants to be zero, so this is a special case of the following formula for the complex potential of Kähler Ricci-flat metrics on O P n−1 (−n) derived in [20, §7.1]:
3.4.
For n = 2, this gives us the Eguchi-Hanson metric [7] .
3.4.
3.5. Legendre transform and Kähler potential. As in [20] , the Kähler potentials of the toric Gibbons-Hawking metrics are:
Local Complex coordinates on Toric Gibbons-Hawking Spaces via Hessian Geometry
In this Section we study local complex coordinates and their relationships which arise naturally from the point of Hessian geometry. To arrive at the general case, we need to first study the n = 1 and n = 2 cases in detail.
4.1.
Hessian local complex coordinates. The function ψ is called the complex potential because one can find local complex coordinates z 1 and z 2 so that
4.1.
is of type (1, 0). By (56) and (57), we have
4.1.
Therefore, we take
4.1.
4.2. Toric Gibbons-Hawking metrics and Kähler form in Hessian local complex coordinates. We now express Riemannian metric, symplectic form and Kähler potential in terms of these local complex coordinates. First of all, by (84) and (85),
4.1.
From the last two equality we derive:
4.1.
Combined with (44), we get: 
4.1.
Proof. These can verified by straightforward computations as follows:
4.1.
4.3. The n = 1 case. From
4.1.
we get:
4.1.
And so
4.1.
When z 2 = 0, we have
4.1.
By (94) we then have:
4.1.
4.3.1. The (α, β)-coordinates. Make the following change of variables:
4.1.
The Gibbons-Hawking metric becomes:
4.1.
The Kähler form is
4.1.
and the Kähler potential can be taken to be
4.1.
One can check that
4.1.
And so we get
4.1.
This fixes the constants C 1 = C 2 = 0 in (67). Also note
4.1.
and so the β-axis in the (α, β)-plane is mapped by the moment map to
4.1.
similarly,
4.1.
i.e., the α-axis in the (α, β)-plane is mapped to
4.3.2.
Change of Hessian local complex coordinates in the n = 1 case. In the above we have focused on the local coordinate patch over U = R 3 − {(0, 0, z) | z ≥ 0}, now we switch the local coordinate patch overŨ = R 3 − {(0, 0, z) | z ≤ 0} where we have
4.3.2.
Recall in (26) we have seen thatα = α + θ 2 , there we can take
4.3.2.
so that dφ +α = dϕ + α. Now the symplectic form can be written as
4.3.2.
where we can takẽ
4.3.2.
From the complex potential one can find local complex coordinatesz 1 andz 2 so that
4.3.2.
By a direct computation, we have
4.3.2.
Therefore, we may takẽ
4.3.2.
By (100), the relationship between the (z 1 ,z 2 )-coordinates and the (α, β)-coordinates is given by:
4.3.2.
Note we have
4.3.2.
Therefore, whenz 2 = 0,
4.4.
The n = 2 case. Now we generalize the results in the case of n = 1 to n = 2. From the equations
4.4.
we solve for z and get two candidate solutions:
4.4.
One can fix one solution by requiring that as x → 0 and y → 0, z tends to a number < c 1 , since we are working over the domain U = R 3 − {(0, 0, z) | z ≥ c 1 }, so we have
4.4.
where R is defined by:
4.4.
From this one can check that:
4.4.
It follows that
4.4.
So we have:
4.4.
We also have
4.4.
Therefore,
4.4.
By (94) we then have:
4.4.
4.4.1. The (α, β)-coordinates. As in the n = 1 case, make the following change of variables:
4.4.
The Gibbons-Hawking metric becomes:
4.4.
And so the Kähler form becomes:
4.4.
This matches with [5] and [20] . The Kähler potential is
4.4.
In the next subsection we will use this to fix the constants in (51).
Moment map in (α, β)-coordinates.
We express many quantities in terms of these new variables:
Moment map in (α, β)-coordinates.
From these identities we easily get:
Moment map in (α, β)-coordinates.
where c is defined by c := c 2 − c 1 . It follows that the image of moment map is given by
Moment map in (α, β)-coordinates.
It is convex domain whose boundary consists of three linear pieces:
Moment map in (α, β)-coordinates.
Note when α = 0,
Moment map in (α, β)-coordinates.
and so the β-axix in the (α, β)-plane is mapped to L 1 . Similarly,
Moment map in (α, β)-coordinates.
, and so
Moment map in (α, β)-coordinates.
and so the α-axis in the (α, β)-plane is mapped to L 2 .
Moment map in (α, β)-coordinates.
Note now we have
Moment map in (α, β)-coordinates.
Therefore, by (51),
Moment map in (α, β)-coordinates.
and so by comparing with (122), C 1 = C 2 = 2 in (51).
4.4.3.
Change of Hessian local complex coordinates in the n = 2 case. In the above we have focused on the local coordinate patch over U = R 3 − {(0, 0, z) | z ≥ c 1 }, now we switch the local coordinate patch overŨ = R 3 − {(0, 0, z) | z ≤ c 2 } where we have
4.4.3.
By (39),α = α + 2θ, so we can take
4.4.3.
and so as in the n = 1 case,
4.4.3.
The complex potential becomes (132) ψ = (μ 2 + 2μ 1 ) log(μ 2 + 2μ 1 ) +μ 2 logμ 2 .
4.4.3.
From the complex potential one can find local complex coordinatesz 1 andz 2 so that
4.4.3.
By a direct computation, we have
4.4.3.
Therefore, we may takẽ
4.4.3.
By (112), introduce (α,β)-coordinates by:
4.4.3.
Together with (121), we derive the following relations:
4.4.3.
In the (α,β)-coordinates
4.4.3.
and therefore µ 2 + µ 1 = −c 1 , and theβ-axis in the (α,|beta)-plane is mapped by the moment map to the L 2 part of the boundary. Whenβ = 0,
4.4.3.
and so µ 2 + 2µ 1 = −(c 1 + c 2 ), therefore, theα-axis in the (α,β)-plane is mapped to the L 3 part of the boundary. There is another local coordinate patch, the one overÛ = R 3 −{(0, 0, z) | z ≤ c 1 } where we have
4.4.3.
whereα = α + θ, so we can takê
4.4.3.
and therefore take the following symplectic coordinates,
4.4.3.
Therefore, we may takê
4.4.3.
By (100), introduce (α,β)-coordinates by:
4.4.3.
Together with (121), we derive the following relations:
4.4.3.
It follows that we haveα =α,β =β. (142) 4.5. Hessian local complex coordinates on toric Gibbons-Hawking spaces. Generalizing the n = 1 and n = 2 case, one sees that for general n, the toric Gibbons-Hawking space is covered by n local coordinate patches, with local coordinates (α j , β j ), j = 1, . . . , n. The coordinates (α 1 , β 1 ) are given by:
4.4.3.
where z 1 , z 2 are defined by:
4.4.3.
And for i = 1, . . . , n − 1,
4.4.3.
In other words, the toric Gibbons-Hawking space can be obtained by gluing n copies of C 2 as follows. For i = 1, . . . , n, denote by U i the i-th copy of C 2 and let (α i , β i ) be the linear coordinates on it. Then U i and U i+1 are glued together by the above formula for change of coordinates. This space is nothing but the toric crepant resolution C 2 /Z n of the orbifold C 2 /Z n . Therefore, we can summarize our discussions so far as follows: Given n distinct real numbers c 1 < · · · < c n , one obtains via the Gibbons-Hawking construction a Kähler Ricci-flat metric on X n := C 2 /Z n , whose Kähler potential is given by:
4.4.3.
for some constants C 1 and C 2 . Here µ 1 , µ 2 are the two components of the moment map for Hamiltonian 2-torus action on X n ,
4.4.3.
By (40), the torus action is given by:
4.4.3.
in the (z 1 , z 2 )-coordinates, hence it is given by (150) (e it1 , e it2 ) · (α 1 , β 1 ) = (e i(t2−t1) α 1 , e it1 β 1 ).
4.4.3.
in the (α 1 , β 1 )-coordinates. The image of the moment map is a convex body whose boundary consists of consecutively a ray L 1 , n − 1 intervals L 2 , . . . , L n , and another ray L n+1 . Furthermore, the α i -axis in the (α i , β i )-plane is mapped to L i+1 , and the β i -axis is mapped to L i , for i = 1, . . . , n.
Phase Changes of Toric Gibbons-Hawking Metrics
In this Section we discuss the phase change [5, 6] of Gibbons-Hawking metrics as applications of the results in preceding Sections.
5.1.
A phase change of Eguchi-Hanson metrics. Let us now focus on the case of n = 2 and discuss the phase change phenomena of the Kähler Ricci-flat metric introduced in [5, 6] . We have seen that in this case the Kähler potential is given by
5.1.
and the Kähler form is
5.1.
With respect to the action (150), we can take
5.1.
The image of the moment map is the convex body defined by:
5.1.
Now we change c to b √ −1 for some b > 0, then the Kähler potential becomes:
5.1.
We have two cases to consider. (156) ψ
5.1.
This is still a purely real-valued function. Then one has
5.1.
and the second derivatives are:
5.1.
So in this case the Kähler form becomes:
5.1.
With respect to the action (150), we can take
5.1.
The image of the moment map is the convex cone defined by:
5.1.
In fact, in this case the second condition is implied by the first and the third conditions. 
5.1.
Up to an unimportant constant, ψ ∨ becomes purely imaginary, hence it does not define a pseudo-Kähler metric, but instead, a purely imaginary pseudo-Kähler metric. Its Kähler form can be found by taking its derivatives:
5.1.
And so the Kähler form is
5.1.
We can now make a comparison of all the cases in this subsection and make a summary. One can introduce a parameter T = c 2 which plays the role of the temperature, and consider the family ψ
5.1.
. A close examination of the computations in this subsection shows that one can use the same formulas for both the T ≥ 0 and the T < 0 cases. When T changes from a positive number to a negative number, a phase transition happens at T = 0. For T > 0, ω √ T is defined on the whole space of the line bundle O P 1 (−2); when T = 0, the metric blows up along the zero section of the line bundle; when T < 0, ω √ T defines a Kähler metric outside a circle bundle of O P 1 (−2), and it defines an imaginary pseudo-Kähler metric inside it.
5.1.
The appearance of imaginary pseudo-Kähler metrics suggests that one can consider ψ 2 ) defined for all c = a + bi ∈ C using the same formulas in the beginning of this subsection. This means in general we consider complex-valued Kähler form with nonvanishing real and imaginary parts.
5.2.
Phase changes of toric Gibbons-Hawking metrics. By the discussions in §4.5 we know that for c = (c 1 , . . . , c n ) with pairwise distinct components, one can define on X n = C 2 /Z n a Kähler metric ω c with a Kähler potential ψ . By allowing c to run in C n , one can then obtain many possibilities of phase changes of the toric GibbonsHawking metrics. In extending the formulas for c ∈ R n to c ∈ C n one may encounter the problem of taking different branches hence may have multi-valued solutions and may have to deal with interesting monodromy problems. We will leave these issues to future investigations.
Concluding Remarks
The method of moment maps and Hessian geometry have been developed in the mathematical literature in the context of toric canonical Kähler metrics on compact toric manifolds [10, 1, 4] , and they have been developed in the physics literature in the context of AdS/CFT correspondence involving toric Sasaki-Einstein metrics and noncompact Calabi-Yau metric cones [15] . More recent developments have generalized to Kähler metrics on noncompact toric spaces obtained by explicit constructions, and have been applied to the Kepler problem [20] . Earlier results on convexity of moment maps and Hessian geometry in the noncompact case have focused on the case of convex cones. In these new developments many examples having convex bodies as moment images have been found. Furthermore, new approach to the phase phenomena introduced in earlier work [5, 6 ] based on such convexity has been developed [18] .
Concluding Remarks
This work is the result of a natural continuation of the ideas in [20] . The theme is still to establish a link between gravity and string theory, guided by the basic principle of applying the intrinsic symmetry of the geometry. Therefore, we have focused on the toric Gibbons-Hawking metrics to have larger symmetry groups.
Concluding Remarks
The phase change we study in this paper leads us to imaginary Kähler metrics. This is actually not so surprising from the point of view of either general relativity or string theory. In general relativity, it is a common practice to change from a Lorentzian space-time to a Euclidean space-time by Wick rotation, i.e., changing to purely imaginary time. This complexification of the time variable led Penrose to consider the complexification of the whole space-time and metric. The physical spact-time is then often a real slice of this complexified space-time. Our work suggests the possibility of the following scenario when one takes a slice of the complexified space-time: At certain part of the space-time the metric is real-valued, but at certain part of the space-time the metric is purely imaginary and such a part is regarded as a blackhole. Of course, one can also encounter a slice along which the metric becomes complex-valued, i.e., the real part and the imaginary part of it can be both nonvanishing. But this is very natural from the point of view nonlinear sigma models or gauged linear sigma models in string theory [11] . When the real part is Kähler, one gets a complexified Kähler form ω − iB in the string theory literature.
Concluding Remarks
In the example of Eguchi-Hanson space, we introduce a parameter c = a + b √ −1 to induce the phase change. We regard this parameter as a "complexified temperature". Another motivation behind the work [20] and this paper is to search for a statistical reformulation of gravity by maximum entropy principle, partly supported by the appearance of expressions like p log p in many places in these two papers. The discussion of phase change in this paper seems to suggest that suitable "complexification of entropy", and more generally, "complexification of statistical mechanics", should be useful for such purpose.

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