## Abstract

Recent work of Harland shows that the *SO*(3)-symmetric, dimensionally reduced, charge-*N* self-dual Yang–Mills calorons on the hyperbolic space *N*-vortex solutions of an Abelian Higgs model as in the study of Witten on multiple instantons. In this paper, we establish the existence of such minimal action charge-*N* calorons by constructing arbitrarily prescribed *N*-vortex solutions of the Witten type equations.

## 1. Introduction

Instantons [1] are topological solitons of the zero-temperature Yang–Mills equations in the Euclidean space *T*>0, one needs to compactify the Euclidean time, *t*, which means that instantons become time-periodic [3] so that *t* is confined within the temporal cell
*k* is the Boltzmann constant, so that the normalized partition function assumes an apparently asymmetric form
*L*(*A*) is the action density of the gauge field *A* and *t*-periodic gauge field solutions can be stratified [3] by homotopy classes defined by maps from *S*^{2}×*S*^{1} into *S*^{3} and such finite-temperature instantons have been explicitly constructed by Harrington & Shepard [4], and called calorons, which approach the zero-temperature instantons in the limit *x* and *t*, in the partition function (1.2) disappears. Motivated by the work on hyperbolic monopoles [5–7] in the extreme curvature limit [8] in connection to the Euclidean monopoles, Harland [9] carried out a study of hyperbolic calorons and showed that, within Witten's *SO*(3)-symmetric dimensionally reduced ansatz [10], hyperbolic calorons may be obtained through constructing multiple vortex solutions of an Abelian Higgs Bogomol’nyi system over a cylindrical stripe. Specifically, a unit charge caloron is presented and large *β* period and large hyperbolic space curvature limits are discussed [9]. However, the existence of a general charge caloron solution remains unsolved.

Technically, the difficulty lies in the fact that the reduced governing equation is defined over an infinite stripe domain in

In §2, we follow Harland [9] to introduce the hyperbolic caloron problem. In particular, we recall the Bogomol’nyi equations of Harland [9] defined over a cylindrical stripe, similar to Witten's equations for the dimensionally reduced *SO*(3)-symmetric instanton problem [10]. In §3, we recall the governing elliptic equation in terms of the coordinate variables. In §4, we prove the existence of solutions by using a variational approach and a sub- and supersolution argument similar to the method used in constructing the Witten type instanton solutions in 4 m dimensions [13,14] systematically developed by Tchrakian [15–17]. Note that, unlike the problem in [13,14], the exponential decay of the curved metric and finite periodicity make it hard to gain precise information of a solution at infinity. Although we know that the solution remains bounded, we do not know whether it has a definite asymptotic value at infinity. In particular, we do not know its uniqueness. In §5, we deduce some suitable decay estimates near the boundaries of the stripe domain for the solution obtained which allow us to compute the associated topological charge realized as the total magnetic flux, or the second Chern class, explicitly. In §6, we obtain decay estimates for the gradient of the solution obtained. These estimates make the formal Bogomol’nyi reduction legitimate and lead us to the conclusion in §7 that the action of a charge *N* caloron represented by the gauge field *A* carries the anticipated minimum action, *S*(*A*)=2*π*^{2}*N*, in normalized units.

## 2. Hyperbolic calorons and vortices

Following Harland [9], the radial coordinate *R* of a point (*x*^{1},*x*^{2},*x*^{3}) in the hyperbolic ball *R*^{2}=(*x*^{1})^{2}+(*x*^{2})^{2}+(*x*^{3})^{2} which is confined in the interval 0≤*R*<*S* with *S*>0 the scalar curvature of *x*^{0}=*t* is of period *β*>0 and parametrizes *S*^{1}. Using d*Ω*^{2} to denote the metric on the standard 2-sphere and introducing the new variable
*Ξ*>0 is defined by the function

Let *A* be an *su*(2)-valued connection 1-form or the gauge field, *A*=*A*_{μ} d*x*^{μ}, with the associated curvature 2-form *F*_{A}=d*A*+[*A*,*A*], and * the Hodge star operator induced from the metric (2.2) over the manifold *S*(*A*), of *A*, is given in the standard form
*M* so that one can recognize the topological lower bound
*Q*(*A*)>0, saturates the lower bound (2.6), *S*(*A*)=2*π*^{2}*Q*(*A*) and satisfies the self-dual equation

As in [9], we are interested in the vanishing holonomy situation where *Q*(*A*) is an integer. To proceed further, we follow [9,18] to represent the *SO*(3)-symmetric gauge field *A* in terms of an Abelian gauge field *a*=*a*_{t} d*t*+*a*_{r} d*r* and a complex scalar Higgs field *ϕ*=*ϕ*_{1}−*iϕ* as
*q*=*x*^{j}*σ*^{j}/*R* with *σ*^{j} (*j*=1,2,3) denoting the Pauli spin matrices. Thus, in terms of the reduced Abelian curvature *F*_{a}=d*a* and connection d_{a}*ϕ*=d*ϕ*+*iaϕ*, the Yang–Mills action over *Q*(*A*) becomes the first Chern number
*ϕ*,*a*) satisfies the self-dual vortex equations over *SO*(3)-symmetric ansatz (2.8) as described in [9]. In view of such a connection, our main existence theorem for calorons may be stated as follows.

### Theorem 2.1

*For any integer N≥1, the self-dual Yang–Mills equation* (2.7) *over the hyperbolic space* *has a 2N-parameter family of smooth solutions, say {A}, realizing the prescribed topological invariant* *Q*(*A*)=*c*_{2}(*A*)=*N so that the action* (2.4) *saturates the topological lower bound stated in* (2.6), *S*(*A*)=2*π*^{2}*Q*(*A*)=2*π*^{2}*N. In fact, such solutions may be obtained by constructing multivortex solutions of* (2.13) *and* (2.14) *representing N vortices realized as zeros of the complex Higgs field ϕ over a cylindrical 2-surface* *defined in* (2.9) *and equipped with the metric* (2.10).

In the subsequent sections, we aim at solving the coupled equations (2.13) and (2.14), which belong to a category of gauge field equations over Riemann surfaces known as Hitchin's equations [21].

## 3. Elliptic governing equation

It will be convenient to rewrite (2.13) and (2.14) in terms of the (*r*,*t*)-coordinates as in [9]. Thus, these equations become
*D*_{r}*ϕ*=∂_{r}*ϕ*+*ia*_{r}*ϕ* and *D*_{t}*ϕ*=∂_{t}*ϕ*+*ia*_{t}*ϕ* are the gauge-covariant derivatives of *ϕ* and the field configurations are all *β*-periodic in the variable *t*.

Use complex variables to represent the equations with the convention
*ϕ*, we have *rt*-component of the curvature of *a* may be represented as

In view of (3.6), we can rewrite (3.2) away from the zeros of *ϕ* as

It is well known that the zeros of *ϕ* are discrete and have integer multiplicities. Let these zeros be
*m* appears in the list (3.8) *m* times. Set *u* stays bounded over

Conversely, the Higgs field *ϕ* has the amplitude |*ϕ*|^{2}=*e*^{u} in terms of a solution *u* of (3.9) and the Abelian gauge field *a*_{t},*a*_{r} may be constructed from using (3.1) to give us

Returning to the equation (3.9), staying away from the points *p*_{1},*p*_{2},…,*p*_{N}, and using the translation
*β* in its *t* variable. In the doubly periodic case, the solutions to the Liouville equation are considered by Olesen [12,22] in the context of non-relativistic Chern–Simons vortices and electroweak vortices over periodic lattices where one needs to use the elliptic functions [23–25] of Weierstrass as the holomorphic functions representing solutions of (3.14) are periodic. In our situation here, complication comes from both the periodicity of the solution *v* of (3.14) in the *t* variable and unboundedness of *v* in the *r* variable as *r*→0 and *Ξ* given in (2.3). Due to these difficulties, we choose to use analytic methods to study (3.9) directly, rather than treating it as an integrable equation, so that the desired boundary conditions and arbitrarily prescribed distribution of vortices, as well as their relations to the calculation of topological charges and minimal actions, can all be realized directly and readily.

## 4. Construction of solution to the vortex equation

For *x*=(*r*,*t*), set
*U*_{0}<0. Let *η*(*x*) be a smooth cut-off function such that 0≤*η*≤1, *η* is of compact support in the rectangle *u*_{0}=*ηU*_{0} is of compact support in *u*_{0}≤0 and

Rewrite *u* in (3.9) as *u*=*u*_{0}+*v*. Then we have
*r*>0 and the Euclidean time variable *t* of period *β*.

### Lemma 4.1

*The function* *v*^{+}=−*u*_{0} *is an upper solution of* (4.4).

### Proof.

The function *v*^{+} is clearly of period *β* in the variable *t*. Besides, in sense of distribution, we have from (4.3) the inequality

We next construct a lower solution of (4.4). For this purpose, we define

We consider the boundary value problem

The singular nature of the equation (4.7) does not allow us to approach it directly. Instead, we consider the approximate boundary value problem
*n*=1,2,…. Here, {*ε*_{n}} and {*K*_{n}} are monotone sequences of positive numbers with *ε*_{n}<*K*_{n}, supp(*G*)⊂(*ε*_{n},*K*_{n}), *n*=1,2,…, and

### Lemma 4.2

*The boundary value problem consisting of* (4.9) *and* (4.10) *has a unique solution which may be obtained by minimizing the functional*
*in the space*

### Proof.

Since *Ξ*(*r*)≥*Ξ*(*ε*_{n})>0 for *r*∈(*ε*_{n},*K*_{n}) and *e*^{w}−1−*w*≥0, we can use the Schwarz inequality and the Poincaré inequality to derive easily the coerciveness of the functional *I*_{n}, namely, *C*_{1},*C*_{2}>0. Hence, the existence of critical point of *I*_{n} in

In order to pass to the *f*_{r} and *f*′ interchangeably to denote the derivative of a function *f* with respect to the radial variable *r*.

### Lemma 4.3

*Let* *w*_{n} *be the unique solution of the boundary value problem* (*4.9*)–(*4.10*) *obtained in lemma* (*4.2*). *Then there hold the monotonicity relation*
*and the uniform coerciveness lower bound*
*where* *C*_{1},*C*_{2}>0 *are independent of* *n*. *Furthermore, the sequence* {*w*_{n}} *is monotone-ordered according to*

### Proof.

First we recall that for *n*≥1 the function *w*_{n} is the unique minimizer of the functional *I*_{n} in *w*_{n}=0 for *r*<*ε*_{n} and *r*>*K*_{n}. Then *I*_{n+1}(*w*_{n})=*I*_{n}(*w*_{n}). However, *w*_{n+1} is the global minimizer of *I*_{n+1} in *I*_{n+1}(*w*_{n+1})≤*I*_{n+1}(*w*_{n}) and (4.13) is established.

Let *f*(*r*) be a function so that *f*(*r*)=0 when *r*>0 is sufficiently small or large. Then, an integration by parts gives us
*e*^{w}−1−*w*≥0 again and (4.17), we have

Finally, applying the maximum principle and the condition *G*(*r*)≥0 in (4.9) and (4.10), we see that *w*_{n}<0 in (*ε*_{n},*K*_{n}). In particular, *w*_{n+1}<0 on [*ε*_{n},*K*_{n}]. Now in (*ε*_{n},*K*_{n}), the function *w*_{n+1}−*w*_{n} satisfies
*w*_{n+1}−*w*_{n})(*r*)<0 for *r*=*ε*_{n} and *r*=*K*_{n}. Applying the maximum principle to (4.19) gives us *w*_{n+1}<*w*_{n} in (*ε*_{n},*K*_{n}) or (4.15). ▪

### Lemma 4.4

*The sequence* {*w*_{n}} *constructed in lemma* (4.3) *is weakly convergent in* *The so-obtained weak limit, say* *w*, *is a classical solution of the equation* (4.7). *In fact, the convergence* *w*_{n}→*w* *may be achieved in any* *C*^{k}[*a*,*b*] *topology for arbitrary* *In particular, we have* *w*≤0 *everywhere*.

### Proof.

From (4.13) and (4.14), we see that there is an absolute constant *C*>0 such that
*w*_{n}} is bounded in *W*^{1,2}(*a*,*b*) for arbitrary *w*_{n}} is weakly convergent in *W*^{1,2}(*a*,*b*). Using extension, we can find a function *w*_{n}} converges to *w* in *W*^{1,2}(*a*,*b*) for any

Choose *n*_{0}≥1 such that (*a*,*b*)⊂(*ε*_{n},*K*_{n}) when *n*≥*n*_{0}. Thus, for any test function *w*_{n}} in *W*^{1,2}(*a*,*b*), we see that {*w*_{n}} is convergent in *C*[*a*,*b*] as well. We can take *w* is a weak solution of (4.7) over (*a*,*b*). Since (*a*,*b*) is arbitrary, we see that *w* is a weak solution of (4.7) over the full domain *r*>0. Standard elliptic theory then implies that *w* is a classical solution of (4.7).

The convergence in any *C*^{k}[*a*,*b*] topology for arbitrary *w*_{n}→*w* in *C*[*a*,*b*] as

### Lemma 4.5

*Let* *w* *be the solution of* (4.7) *obtained in lemma* (4.4). *Then it satisfies the boundary condition* (4.8) *for some unique number* *so that for arbitrarily small* *ε*>0, *we have* *w*(*r*)=*O*(*r*^{2−ε}) *as* *r*→0 *and* *as*

### Proof.

Let {*w*_{n}} be the sequence of solutions of (4.9) and (4.10) obtained in lemmas 4.2 and 4.3. Then for any *r*∈(*ε*_{n},*K*_{n}), we have by the Schwarz inequality and (4.20) the uniform bound
*C*>0 is independent of *n* and *r*. Hence *w*(*r*)=O(*r*^{1/2}) when *r*→0 which is a crude preliminary estimate. To improve it, we consider a comparison function
*U*=*w*+*W*. Choose *r*_{0}>0 small such that *G*(*r*)=0 for 0<*r*<*r*_{0}. In view of (4.7), (4.24),
*w*(0)=0, we have
*r*<*r*_{0} and *K*(*r*)→2 as *r*→0. Hence, when *r*_{0} is small, we have *K*(*r*)>(2−*ε*)(1−*ε*) for *r*∈(0,*r*_{0}). Inserting this condition into (4.26), we have
*U*(0)=*w*(0)+*W*(0)=0 and assuming *C*>0 in (4.24) is large enough so that *U*(*r*_{0})=*w*(*r*_{0})+*W*(*r*_{0})>0. Applying these in (4.27), we get *U*(*r*)>0,*r*∈(0,*r*_{0}). That is, we have obtained the estimate

In view of (4.20), we deduce that
*r*_{n}}, *G* in (4.7) is of compact support, there is some *K*>0 such that *G*(*r*)=0 for *r*>*K*. Thus, using *w*≤0 and the definition (2.3), we see that *w*_{rr} satisfies

To see the uniqueness of *W*_{1} and *W*_{2} such that
*W*=*W*_{1}−*W*_{2}. Then *W* satisfies
*ξ* lies between *W*_{1} and *W*_{2}. As *W*(0)=0 and *W*(*r*)>0 for all *r*>0. Otherwise, let *W*(*r*_{0})≤0 for some *r*_{0}>0. We may assume that *W* attains its global minimum at *r*_{0}. If *W*(*r*_{0})=0, then *W*_{r}(*r*_{0})=0, which implies *W*≡0 by the uniqueness of solution to the initial value problem of an ordinary differential equation, contradicting the condition *W*(*r*_{0})<0 but this contradicts the fact *W*_{rr}(*r*_{0})≥0. Hence *W*(*r*)>0 for all *r*>0.

Since *r*_{n}}, *W*_{r}(*r*_{n})→0 as *r*,*r*_{n}) and letting *W*(0)=0, *W*>0 and the above, we see that *W*(*r*) decreases. In particular,

The proof of the lemma is complete. ▪

Despite of the above uniqueness result, we are unable to show that

We are now ready to solve (4.4). We can state

### Theorem 4.6

*The equation* (4.4) *has a bounded solution v satisfying v*(*r,t*)= *O*(*r*^{2−ε}) *as r*→0, *where ε>0 is arbitrarily small.*

### Proof.

Let *w* be the solution of (4.7) stated in lemma 4.5. As *u*_{0}≤0, we have
*v*^{−}=*w* is a lower solution of the equation (4.4). Combining with lemma 4.1, we have *v*^{+}≥0≥*v*^{−}. Using elliptic method, we get a solution *v* of (4.4) satisfying *v*^{−}≤*v*≤*v*^{+}. As *v*^{+}=−*u*_{0} is of compact support in *v* is bounded and satisfies *v*(*r*,*t*)=O(*r*^{2−ε}) as *r*→0 for any small number *ε*>0. ▪

## 5. Calculation of topological charge

Let *v*=*v*(*r*,*t*) be the *β*-periodic solution of (4.4) obtained in theorem 4.6. Define the *β*-averaged function by
*v*(*r*,*t*)=O(*r*^{2−ε}) (when *r*>0 is small), *r*_{n}}, *r*,*r*_{n})×(0,*β*) and letting *v*+*u*_{0}≤0.

Integrating Δ*v* over

On the other hand, recall that *u*_{0}=*ηU*_{0} is of compact support and *U*_{0} may be decomposed as the sum of a regular and singular parts in the form
*S*_{ε} is the line element and ∂/∂*n* denotes the outnormal derivative on the circle |*x*−*p*_{j}|=*ε* (*j*=1,…,*N*).

Integrating (4.4) over *ϕ*|^{2}=*e*^{u}=*e*^{v+u0} and (3.2), we arrive at the anticipated flux quantization condition

In order to calculate the total action, we need to establish some suitable decay estimates for the gradient of a solution obtained, which will be considered in §6.

## 6. Decay estimates for the gradient of solution

In order to compute the action of a multiple instanton, we need to derive suitable decay estimates for the first derivatives of the solution *v* of (4.4) obtained earlier.

We shall first consider the decay estimates near *r*=0. As both *u*_{0}(*r*,*t*) and *g*(*r*,*t*) are compactly supported in *r*_{0}>0 sufficiently small so that the supports of *u*_{0} and *g* are contained in *v* satisfies

### Lemma 6.1

*Let* *v* *be the solution of* (4.4) *obtained earlier*. *For any arbitrarily small number* *ε*>0, *there is a constant* *C*(*ε*)>0 *independent of* *r*,*t* *such that the estimates*
*hold. In particular*, |∇*v*|→0 *uniformly as* *r*→0.

### Proof.

Since *Ξ*(*r*)=O(*r*), *v*(*r*,*t*)=O(*r*^{2−ε}) uniformly as *r*→0 for *ε*>0 arbitrarily small, and *v*≤0, we see that *p*>2 (say) where *L*^{p}-estimates indicate that *v*| is bounded over

For any *h*>0, consider the function
*v*^{h}} is uniformly bounded over *v*^{h}(*r*,*t*)→0 as *r*→0. Of course, in view of (6.1), *v*^{h} satisfies the equation
*u*^{h}(*r*,*t*) lies between *v*(*r*,*t*) and *v*(*r*,*t*+*h*). Using the comparison function *W* for fixed *ε*>0 defined in (4.24) and applying (4.25), we get
*r*_{0}>0 is sufficiently small and applied the uniform limit *v*(*r*,*t*+*h*)→0 as *r*→0. As {*v*^{h}} is bounded, we may also assume that *C*>0 in (4.24) is large enough so that
*v*^{h}+*W*=0 at *r*=0 and (6.6), the inequality (6.5), and the maximum principle together lead us to
*v*^{h}−*W*≤0, 0<*r*<*r*_{0}. Summarizing these results, we arrive at |*v*^{h}(*r*,*t*)|≤*W*(*r*), 0<*r*<*r*_{0}. Letting *h*→0, we obtain |*v*_{t}(*r*,*t*)|≤*W*, 0<*r*<*r*_{0}, as stated in (6.2).

In order to get the decay estimate for *v*_{r} as *r*→0, we note that
*v*≤0, the inequality
*C*_{1}>0 is an absolute constant and *K*=*r*^{2}(2/*Ξ*^{2}) *e*^{v} satisfies
*W* defined in (4.24), we have
*ε*<1 and *C* is as given in (4.24). We may choose *C* large enough so that *K*(*r*)≥*C*_{1}/*C*, 0<*r*<*r*_{0}. Then (6.14) gives us
*C*>0 in (4.24) is large enough, we have
*v*_{r} stated in (6.2). ▪

We now consider the decay estimate for |∇*v*| as *v*→0 as *v* may not lie in

Similar to (6.1), we know that *v* satisfies
*δ*>0 is sufficiently large. For convenience, we set

### Lemma 6.2

*We have*

### Proof.

We may extend *v* outside *w*, so that *w*=0 for *r*<*δ*/2 (say). Hence *w* satisfies
*h* is of compact support and smooth. Choose a smooth function *η*(*r*) in *r*≥0 such that
*η*_{ρ}=*η*(*r*/*ρ*) for *ρ*>0. Multiplying (6.18) by *C*>0 is a constant depending on *β* only. Letting

### Lemma 6.3

*There is a constant* *C*_{δ}>0 *such that*
*and* |∇*v*|(*r*,*t*)→0 as

### Proof.

Differentiating (6.17), we have
*v*_{t} is bounded and *v*_{t}→0 as

Similarly, differentiating (6.17) with respect to *r*, we have
*v*_{r} is bounded and *v*_{r}→0 as

The gradient decay estimates for the solution near the boundary of the domain

## 7. Calculation of action

Following [9], it will be convenient to express the dimensionally reduced action (2.11) of the gauge field *A* in terms of the (*r*,*t*)-coordinates explicitly as

By virtue of (3.12), lemmas 6.1 and 6.3, we see that the last integral on the right-hand side of (7.4) vanishes. Therefore, we obtain the quantized minimum action

In summary, we have seen that our main existence theorem for hyperbolic calorons of arbitrary scalar curvature, time period and topological charge stated in §2 is established in §4 through a construction of the solution of the multivortex equation derived by Harland [9], a computation of the associated topological charge in §6 and a calculation of the dimensionally reduced action in §7, which is based on the gradient boundary estimates of the solution obtained in §6.

## Data accessibility

This work does not have any experimental data.

## Author contributions

L.S. and R.S. offered ideas and insights in the mathematical formulation and conception of the problem. Y.Y. developed analytic methods to tackle the problem and wrote the paper. All authors gave final approval for publication.

## Funding statement

The research of Y.Y. was partially supported by National Natural Science Foundation of China under grant no. 11471100.

## Conflict of interests

We do not have competing interests.

- Received December 17, 2014.
- Accepted February 23, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.