Instantons in Lifshitz Field Theories

BPS instantons are discussed in Lifshitz-type anisotropic field theories. We consider generalizations of the sigma model/Yang-Mills instantons in renormalizable higher dimensional models with the classical Lifshitz scaling invariance. In each model, BPS instanton equation takes the form of the gradient flow equations for"the superpotential"defining"the detailed balance condition". The anisotropic Weyl rescaling and the coset space dimensional reduction are used to map rotationally symmetric instantons to vortices in two-dimensional anisotropic systems on the hyperbolic plane. As examples, we study anisotropic BPS baby Skyrmion 1+1 dimensions and BPS Skyrmion in 2+1 dimensions, for which we take Kahler 1-form and the Wess-Zumiono-Witten term as the superpotentials, respectively, and an anisotropic generalized Yang-Mills instanton in 4+1 dimensions, for which we take the Chern-Simons term as the superpotential.


Contents
Although the straightforward generalizations of those theories to higher dimensions are known to be perturbatively non-renormalizable, there are renormalizable higher dimensional generalizations of the sigma model and the gauge theory. Such theories, called the Lifshitz-type fields theories, are characterized by the following anisotropic scaling of the spacetime coordinates [1] t → λ z t, This scaling transformation is called the Lifshitz scaling with the dynamical critical exponent z.
Because of the anisotropy between space and time, the Lorentz symmetry is explicitly broken, whereas the spatial rotational symmetry is preserved. The spatial higher derivative terms in the Lagrangian improve the ultraviolet (UV) behaviors of the propagators so that the UV divergences of loop integrals become milder than those in the standard Lorentz symmetric theories. The absence of higher time derivative terms guarantees that the Lifshitz-type theories are free from the ghost problem associated with higher derivatives.
Using the idea of the Lifshitz scaling, Horava constructed a gravity theory [2,3], called "the Horava-Lifshitz gravity", which is expected to be renormalizable and unitary. In (d + 1) dimensional spacetime, non-linear sigma models with z = d and Yang-Mills theories with z = d − 2 are classically invariant under the Lifshitz scaling transformation, and expected to be renormalizable according to the weighted power counting [4][5][6] and a symmetry argument [7]. The various aspects of the Lifshitz-type theories have been discussed in the non-linear sigma models [8][9][10][11] and in the Yang-Mills theories [12][13][14] (see [15] for a review). In most of those cases, the theories are asymptotically free (or scale invariant) and it is likely that their strongly coupled low-energy dynamics have rich non-perturbative structures induced by instantons.
In this paper, we discuss instantons in classically scale invariant Lifshitz-type non-linear sigma models and gauge theories in the Wick-rotated spacetime. We focus on "supersymmetric theories" which satisfy "the detailed balance condition" characterized by a functional version of the superpotential. Such models admit BPS instantons described by the gradient flow equations for the superpotential (similar geometric flow equations have been discussed as instantons in the Horava-Lifshitz gravity [35,36] where g ab is the metric on M and V (φ, ∂ i φ, · · · ) is a function of the fields φ a and their spatial a sufficient number of spatial derivatives so that the propagator decreases sufficiently quickly in the UV regime. In the following, we discuss instantons in such models with higher spatial derivatives.

Derrick's scaling argument for Lifshitz-type sigma models
First, we discuss the stability of instantons in the Lifshitz-type theories by using Derrick's scaling argument [38]. Let us consider the following scaling transformation Here we assume that each term S (n 0 ,··· ,n d ) is finite and positive definite. Then the stability of the action under this scaling requires that there should be at least one pair of terms whose scaling weights have opposite signs. This is the necessary condition for the existence of non-trivial solutions of the equations of motion.
Let us first recall the Lorentz symmetric case in two dimensions. For a generic assignment of the scaling weights, the standard kinetic terms S (2,0) and S (0,2) have the weights with opposite signs w (2,0) = −w (0,2) = w 0 − w 1 . fixed sizes in the models with potential terms S (0,0) and higher derivative terms S (m,n) [39,40].

"Supersymmetric" models and BPS equations
In this section, we discuss the form of the Lifshitz-type action which admits instanton solutions.
We focus on the BPS case in which the action is bounded below by the topological charge * Note that Derrick's theorem does not guarantee the existence of non-trivial solutions of instantons. Such an action, which satisfies the so-called "detailed balance condition", is characterized by a functional defined on each time slice where W is a function of φ a and their spatial derivatives. In terms of the functional W , the action of the non-linear sigma model which admits BPS instantons is given by where the variation of W takes the form For simplicity, we assume that W does not contain the higher derivative of φ a in the following.
In (d + 1)-dimensional spacetime with d = 0, 1, 2, the action Eq. We call the functional W a "superpotential" since it is a generalization of the superpotential in the standard supersymmetric theories. The supersymmetry in the Lifshitz-type field theory in higher dimensions will be discussed elsewhere.
The action Eq. (2.8) can be rewritten into the Bogomol'nyi form where the topological charge T takes the form Since the topological charge T is a total derivative term, it has a fixed value for a given boundary condition. Therefore, the configurations with the least action in a fixed topological sector are the solutions of the following gradient flow equation This is the BPS equation which describes instantons in the Lifshitz-type sigma model.
Among various choices of the superpotential W , we focus on a specific class of W which has several special symmetries. Let ω be a closed (d + 1)-form on the target manifold M Then, we can find a functional W such that † The corresponding action takes the form where | · | denotes the norm with respect to the metric g ab . As we have discussed in the previous section, this type of action can have marginally stable instanton solutions. Their topological charge is given by where φ * ω denotes the pullback of ω with respect to the map φ from the (d + 1)-dimensional spacetime to the target space M.
One of the most significant properties of the action Eq. (2.16) is that it is invariant under the 18) and the spatial volume-preserving diffeomorphism depends only on the field on the boundary. If we assume that the fields φ a approaches to a single point on the target space M, the superpotential W can be viewed as a functional defined on the time slice at t whose variation gives Eq. (2.15).
As a consequence of these symmetry, instantons in this class of theories have continuous degeneracy associated with the symmetry broken by the configurations. The corresponding moduli parameters determines the shape of the instanton which can vary under the above transformations.
The action of the form Eq. (2.16) has some exotic properties: for example, it vanishes for an arbitrary static configuration which does not depend on one of the spatial coordinates. To obtain a physically reasonable model, we should modify the action by adding terms which would break the stability of the instantons. In the presence of such a modification, instantons are no longer true minima of the action. Nevertheless, the path integral in the modified model is dominated by the so-called constrained instantons [41] whose leading order forms are given by the BPS configurations in the original model. Therefore, it is still important to know the properties of the instantons in the model Eq. (2.16) and we will mainly focus on them in the following.

Anisotropic BPS baby Skyrmions
Let us first see the simplest example, the sigma model lump, in the Lorentz symmetric Kähler sigma model in two dimensions [42]. Then, we will see that it takes an anisotropic shape with a fixed size in the presence of additional potential and spatial higher derivative terms.
When the target space M is a Kähler manifold, the standard sigma model action can be obtained by setting where K is the Kähler potential which gives the target space metric g ab = ∂ 2 K ∂φ a ∂φ b . For this superpotential, the BPS equation takes the form This is known as the BPS equation which describes the sigma model lumps. The corresponding topological charge is given by the pullback of the Kähler form [42] Since the solution to the BPS equations is an arbitrary holomorphic map φ a = φ a (z), (z = t+ix) we can freely change the scale φ a (z) → φ a (λz) without changing the value of the action. This continuous parameter λ is the size modulus associated with the scale invariance of the sigma model action. As we have discussed in the previous section, the size of the BPS lump is fixed if we turn on a potential term and higher derivative terms. The baby Skyrmion is such a soliton with a fixed size in the Lorentz symmetric CP 1 sigma model. In our setup, we can introduce spatial higher derivative terms S (0,n) (n > 2) without breaking the BPS properties. An example of such model can be obtained by deforming W as where α is a constant parameter and w(φ,φ) is a function on M. Then, the BPS equation Note that the scaling transformation (t, x) → (λt, λx) is no longer the symmetry of the BPS equation. Fig. 1 shows a numerical solution of the anisotropic BPS configuration in the CP 1 model with the Kähler potential K = log(1 + |φ| 2 ) and the superpotential As shown in Fig. 1, the Lifshitz-type instanton has an anisotropic action density.

Anisotropic BPS Skyrmions
Next, we discuss the three dimensional Lifshitz-type non-linear sigma model whose target space This superpotential gives the following action with four spatial derivatives where ξ is the inverse coupling constant. Since ξ is dimensionless, this model is expected to be renormalizable according to the power-counting argument. The BPS equation takes the form This system is invariant under the z = 2 Lifshitz scaling and the spatial volume-preserving diffeomorphism. The topological charge is given by We call the instanton in this model "anisotropic Skyrmion" since it is characterized by the same topological charge (baryon number) as the Skyrmion in the Lorentz symmetric theories.

G = SU(2)
As an example, let us consider the case of G = SU(2). Let (r, θ) be polar coordinates of two- we can construct the following U(1) symmetric ansatz where k is an integer and φ is a complex-valued function which is independent of the spatial angle coordinate θ. ‡ Plugging this ansatz into the BPS equation Eq. (2.29), we obtain the following equation for φ: where we have defined the spatial radial coordinate by ρ = r 2 /4k = (x 2 1 + x 2 2 )/4k. Note that in this notation, the z = 2 Lifshitz scaling transformation becomes t → λt, ρ → λρ. (2.33) The scalar field φ can be viewed as a map from the (t, ρ)-plane (half plane ρ ≥ 0) to the unit disk |φ| ≤ 1. If we assume that φ takes a fixed value at infinity, the (t, ρ)-plane can also be regarded as a disk whose boundary corresponds to the line ρ = 0. Since the angle coordinate θ in the ansatz Eq.  Therefore, the set of solutions has two real parameters corresponding to the coordinates of the hyperbolic plane.

Deformation
We can also add the following term without breaking the z = 2 Lifshitz scaling invariance (2.37) The BPS equation becomes For α = 0, there is no longer the spatial volume-preserving diffeomorphism, and hence the We remark that the superpotential W now takes the form of the Wess-Zumino-Witten action.
As pointed out in [2], the ground state wavefuncitonal in the (2+1) dimensional theory reproduces the partition function of the 2d theory defined by the action W . A detailed study of this model would also be interesting.

Compactification
Let us next discuss instantons at finite temperature by compactifying the Euclidean time direction. In this case, we can consider an ansatz which is invariant under a combination of the time translation and the U(1) internal symmetry φ → e iα φ. The exact BPS solution which has such a symmetry is given by The topological charge of this solution is independent of the inverse temperature β under which the topological charge density in Eq. (2.40) transforms in such a way that x 2 1 + x 2 2 is replaced by where (ϑ, ϕ) are moduli parameters corresponding to SL(2, R)/SO(2). The exact solution discussed above is symmetric under the time translation. In general, such a symmetric configuration takes the form where v L and v R are fixed elements of the Lie algebras g L and g R , respectively. For this ansatz, This is the BPS equation in the two dimensional system described by the action This action is bounded below by the topological charge Note that although the action (energy) for a global vortex configuration is logarithmically divergent in the presence of the standard kinetic terms [46], it is finite in this type of model which has only potential and higher derivative terms.
where n is a unit vector n· n = 1. The Bogomol'nyi bound for this action is given by the following topological charge corresponding to π d+1 (S d+1 ) = Z where φ * Vol(S d+1 ) is the pullback of the volume form on S d+1 . The BPS equation is Let us assume that the single instanton solution is symmetric under the spatial rotation SO(d).
Combining with the SO(d) symmetry of the target space, we can find the following hedgehog ansatz n a = 1 − |φ| 2 x a r (a = 1, · · · , d),

Compactification
Assuming that the Euclidean time direction is compact, we can obtain the following implicit form of an exact solution which is symmetric under the time translation where ⌊x⌋ denotes the largest integer not greater than x. For any d, the scalar field φ has the asymptotic form The scalar field φ vanishes at the spatial infinity, so that n| ρ→∞ defines a map from S d−1 Therefore, this configuration is a global type soliton characterized by π d−1 (S d−1 ).

Other examples
Another class of higher dimensional models which admits BPS instantons is the Kähler sigma models. Since any Kähler manifold has the closed forms there can be instantons characterized by the topological charge of the form This topological charge gives the BPS bound for the action where V i is defined by The BPS equation have the scaling weights with the opposite signs w(S E i ) = −w(S B i ) = w 0 + w i − 3 j=1,j =i w j for generic w i (i = 0, 1, 2, 3), whereas both terms are scale invariant for w 0 = w i (i = 1, 2, 3). Therefore, the Yang-Mills instanton is stable under the generic scaling and marginally stable for the scaling with w 0 = w i .
As a generalization to higher dimensions, we can consider an analogue of Eq. (3.2). For d = 2n + 1, define E i and B i by Since S E i and S B i have the opposite weights there can be instanton solutions in the higher dimensional gauge theory described by Since this action is invariant under the z = d − 2 Lifshitz scaling (w 0 = d − 2, w i = 1 (i = 1, · · · , d)), the instantons are marginally stable and hence they have associated size moduli parameters. Note that the gauge coupling constant g is dimensionless and hence this model is expected to be renormalizable according to the power-counting argument.

Weyl rescaling and coset space dimensional reduction
An important property of the action Eq. (3.5) is that it is invariant under a generalized version of the Weyl transformation. Let us put the model on a curved space with a metric of the form The covariantized action § is invariant under the following change of the metric Using this anisotropic Weyl transformation with λ = r −1 , we can map the flat spacetime to the direct product of the hyperbolic plane and the (d − 1)-dimensional sphere H 2 × S d−1 where ρ = r d−2 /(d − 2). Therefore, the theory can be reduced to a two-dimensional system defined on the hyperbolic plane H 2 by taking an ansatz which is symmetric under a combination of the SO(d) rotation and the gauge transformation. It is known that the self-dual equation in the four dimensional pure Yang-Mills theory (d = 3) reduces to the vortex equation on the § The action which has covariance under the foliation-preserving diffeomorphism t → t ′ (t) and where E i and B i are given by hyperbolic plane [47]. This type of the dimensional reduction is called the coset space dimensional reduction [48].
We have already seen the examples of the coset space dimensional reduction for scalar fields (zero forms) in the previous section. In the following, we will see that the SO(d) symmetric instantons in the Lifshitz-type gauge theories can also be mapped to vortices on the hyperbolic plane by the dimensional reduction.

Generalized Yang-Mills Instantons
As in the case of scalar field theories, the superpotential formalism (detailed balance condition) can be used to construct actions which admit BPS configurations. The scale invariant action Eq. (3.5) for the SU(N) gauge field A can be obtained by using the Chern-Simons form as the superpotential (3.10) The corresponding topological charge is given by Tr(F ∧ · · · ∧ F ). (3.11) This gives the BPS bound for the action The BPS equation is E i = B i , that is, This is a generalization of the self-dual equation in the four dimensional Yang-Mills theory.

(5 + 1)-dimensional SU(4) gauge theory
Let us consider the simplest example in the (5+1)-dimensional SU(4) gauge theory, in which we can consider the SO(5) ⊂ SU(4) invariant ansatz. The action takes the form (3.14) This model has the classical z = 3 Lifshitz scale invariance The BPS equation and the topological charge are given by  (3.17) where our convention for the gamma matrices is {Γ i , Γ j } = 2δ ij and Γ † i = Γ i . Then, we can find the following basis of SO(5) invariant one-forms where r is the spatial radial coordinate andx i = x i /r. In terms of the SO(5) invariant one-forms, the most general SO(5) symmetric ansatz for the gauge field is given by where a t , a r and φ are functions of t and r. For this gauge field, the field strength takes the form (3.20) where f is the two-dimensional field strength and Dφ is the covariant derivative defined by Plugging into the BPS equation, we obtain the follwoing reduced equations for the two dimensional fields a and φ: where we have redefined the spatial radial coordinate by ρ = r 3 /9. These equations can be regarded as the vortex equations in the following system defined on the hyperbolic plane where V 4 is the volume of the unit 4-sphere and the spacetime indices are raised with the hyperbolic metric G This action can be obtained by substituting the rotationally symmetric ansatz Eq. (3.19) into the original action Eq. (3.14). Similarly, the 5d Chern-Simons term reduces to the superpotential for the two-dimensional system In terms of this superpotential, the two-dimensional action can be rewritten as where the topological charge is given by the first Chern number (up to trivial boundary terms) The two-dimensional system is anisotropic in the sense that the action Eq. It has been shown in [47] that the SO(3) invariant instantons ¶ in the 4d SU(2) Yang-Mills theory are described by the hyperbolic vortex equations in the two-dimensional system characterized by

Compactification
Next, let us consider the finite-temperature case. In the standard four dimensional SU(N) Yang-Mills theory, instantons at finite temperature are called calorons [49][50][51]. It is known that in the presence of a non-trivial holonomy around the Euclidean time circle, each caloron has N constituents which can be identified with magnetic monopoles [52][53][54][55][56]. Here, we look for an analogous object which constitutes the instanton in our setup.
Since the finite-temperature system has the additional U(1) symmetry corresponding to the time translation, we can consider an ansatz which is invariant under the U(1) × SO(5) transformations. By appropriately choosing the gauge, we can find the following ansatz Then, the BPS equations Eq. (3.22) reduces to The smoothness of the ansatz Eq. (3.19) at ρ = 0 requires that the profile functions ϕ(ρ) and a(ρ) should satisfy the boundary conditions On the other hand, a(∞) is a free parameter which determines the value of the Polyakov loop at the spatial infinity. For a(∞) = 0, the SU(4) gauge symmetry is broken to SU(2)×SU(2)×U(1).

Fig. 5: A numerical solution
The time independent solution discussed above can be identified with the BPS soliton in the following five dimensional system which can be obtained by the dimensional reduction along the periodic time direction: where Φ is the adjoint scalar field corresponding to A t . The topological charge of the BPS soliton in this system is given by The generalized monopole configurations characterized by this topological charge has been discussed in [57].
We can also perform the dimensional reduction along one of the spatial directions instead of the time direction. The resulting model is a five dimensional z = 3 Lifshitz-type gauge theory with an adjoint scalar. It is characterized by the superpotential The corresponding action takes the form This gauge theory is super-renormalizable since the coupling constant has the positive dimension The BPS soliton in this system has the same topological charge as Eq. (3.33) and obeys the BPS equations This equation can also be viewed as a generalization of the BPS monopole equation.

Summary and Discussion
In this paper, we have discussed BPS instantons in the Lifshitz-type non-linear sigma models and gauge theories. In the "supersymmetric models" in which the detailed balance condition is satisfied, the BPS equations for the instantons are described by the gradient flow equations for the "superpotential" W . For the Lifshitz-type non-linear sigma models, we have focused on Weyl rescaling and the coset space dimensional reduction have been used to map the instantons to Abelian vortices in the anisotropic system on the hyperbolic plane. As in the case of the instantons in four dimensions [58,59], it is also interesting to consider more general dimensional reductions which give non-Abelian theories in the two-dimensional spacetime and non-Abelian vortices in such theories.
Although the instantons in our setup are described by very simple BPS equations, the models we have considered have some exotic properties. For example, the action identically vanishes for any static configuration which is independent of one of the spatial coordinates. It is possible to modify the action to obtain a physically reasonable model without breaking the BPS properties and the Lifshitz scaling invariance. In particular, it is interesting to consider the anisotropic It should be very interesting to consider deformed supersymmetry algebras which allow us to compute the path integrals by means of the localization technique. For higher dimensions, the supersymmetry in the Lifshitz-type field theories has not yet been fully understood. We will discuss this topic in the near future.

Acknowledgments
The Note that Eq. (A.1) is a subalgebra in the standard 3d N = 2 SUSY algebra (4 real supercharges).
The supercharges can be represented by the following derivatives in the superspace The supercharges anti-commute with The non-linear sigma model discussed in Sec. 2.2 can be constructed in terms of the real multiplets Φ a = φ a + θψ a +ψθ a + θθF a ,Φ = Φ. (A.5) The simplest action for the real multiplets is given by where g ab is the Riemann metric on the target space and W is a function of Φ, ∂ i Φ and higher spatial derivatives. In terms of the component fields, the kinetic part takes the standard sigma model form −R abcd ψ a ψ bψcψd .
The superpotential part is given by where we have defined . (A.10) Eliminating the auxiliary fields F a , we obtain the following bosonic part This becomes the Lagrangian discussed in Sec. 2.2 after the Wick-rotation. The lower dimensional supersymmetric action can be obtained by the standard dimensional reduction along the spatial directions.