# Semi-Classical Stability of Supergravity Vacua

Abstract

We discuss the existence of instantonic decay modes which would indicate a semi-classical instability of the vacua of ten and eleven dimensional supergravity theories. Decay modes whose spin structures are incompatible with those of supersymmetric vacua have previously been constructed, and we present generalisations including those involving non trivial dilaton and antisymmetric tensor fields. We then show that the requirement that any instanton describing supersymmetric vacuum decay should admit both a zero momentum hypersurface from which we describe the subsequent Lorentzian evolution and a spin structure at infinity compatible with the putative vacuum excludes all such decay modes, except those with unphysical energy momentum tensors which violate the dominant energy condition.

## 1 Introduction

Supergravity theories exist in all spacetime dimensions with , and are currently regarded as effective field theories of superstring (and M) theories in some appropriate limit. Classical solutions of the theories can be found by setting to zero the fermionic fields together with their supersymmetric variations. We look for a vacuum in which the space-time is of the form where is a maximally symmetric four-dimensional space (de Sitter space, anti de Sitter space or Minkowski space) and K is a compact manifold; such a solution is consistent with the low-energy field equations, with the dilaton field constant and all other fields vanishing. The conditions for finding supersymmetric generators that leave the vacuum invariant restrict to be flat Minkowski space and to be a manifold that admits at least one covariantly constant spinor field. This in turn constrains the possible holonomy groups of ; for ten dimensional theories, must have a holonomy contained in [1], implying that must have a covering space that is , or a Calabi-Yau space . Similarly, for eleven dimensional supergravity, the holonomy of is contained in , and has a covering space that is , , or (where the latter is a manifold with the exceptional holonomy group ) [2].

It is important to have some criteria for determining whether is a reasonable candidate as the ground state of supergravity theories, but our incomplete understanding of string (and M) theory dynamics makes this question difficult to answer in full. The constraints above restrict the vacuum state to be Ricci-flat, with the requisite holonomy; we must, however, also show that the spectrum of the vacuum is stable and that there are no instantonic decay modes, ie. we must thus impose the conditions that should be stable at the classical and semi-classical level, which leads to non-trivial conditions on the vacuum manifold.

The first test of the stability of a space is to ask whether the space is stable classically against small oscillations. Small oscillations around will consist of a spectrum of massless states (the graviton, gauge fields, dilaton etc) and an infinite number of charged massive modes. The massless spectrum of the heterotic string theory, which is the theory that we will consider principally here, has been extensively discussed (see for example [1], [3], [4] and [2]); there are no exponentially growing modes with imaginary frequencies. The same applies to that of eleven dimensional supergravity.

Even if a state is stable against small oscillations, it may be unstable at the semiclassical level. This can occur if it is separated by only a finite barrier from a more stable state; it will then be unstable against decay by semiclassical barrier penetration. To look for a semiclassical instability of a putative vacuum state, one looks for a bounce solution of the classical Euclidean field equations; this is a solution which asymptotically at infinity approaches the putative vacuum state. If the solution is unstable, then the Gaussian integral around that solution gives an imaginary part to the energy of the vacuum state, indicating an instability.

The stability of Minkowski space at the semi-classical level as the unique vacuum state of general relativity was proved by the positive energy theorem of Schoen and Yau [5]. A completely different proof involving spinors satisfying a Dirac type equation on a three-dimensional initial value hypersurface was given by Witten [6], and shortly after re-expressed in terms of the Nester tensor [7]. Witten later demonstrated the instability of the vacuum of Kaluza-Klein theory [8]; the effective four-dimensional vacuum decays into an expanding bubble of “nothing”. However, this decay mode is excluded by the existence of (massless) elementary fermions.

Instabilities of non-supersymmetric vacua of string theories were discussed by Brill and Horowitz [9] who demonstrated that superstring theories admit instantonic decay modes that asymptotically resemble toroidal compactifications with constant gauge fields (but are incompatible with massless fermions). Mazur [10] showed that toroidal compactifications of multidimensional Minkowski space-time are semiclassically unstable due to topology change of the initial data hypersurface, and presented Euclidean Schwarzschild p-branes as possible instantons corresponding to tunnelling between different topologies. However, the instantons discussed by Mazur do not correspond to vacuum instabilities since they do not take account of the incompatible spinor structures of the instantons and (supersymmetric) vacua. Banks and Dixon [11] used conformal field theory arguments to show that spacetime supersymmetry cannot be continuously broken within a family of classical vacua and that two supersymmetric vacua are infinitely far away. It was suggested in [12] that if one takes into account target space duality all topology changing instabilities of toroidal vacua are impossible in the context of string theory.

More recently, another possible decay mode of the Kaluza-Klein vacua has been constructed by Dowker et al. [13], [14]. “Magnetic” vacua in Kaluza-Klein theory - vacua corresponding to static magnetic flux tubes in four dimensions - may decay by pair creation of Kaluza-Klein monopoles, though at a much smaller rate than for decay by bubble formation; the pair creation decay mode is however consistent with the existence of elementary fermions.

In this paper, we investigate further possible decay modes of the vacua of supergravity theories. Such instantons certainly do not preserve the supersymmetry of the vacuum; however, we cannot a priori exclude the possibility of the vacuum decaying into a state which asymptotically admits the supersymmetry generators of the vacuum. That is, there may exist solutions to the Euclidean field equations both whose geometry is asymptotic to that of the background vacuum state, and whose spin structure at infinity is compatible with that of the supersymmetric vacuum.

In §2 we describe the vacua of ten-dimensional supergravity theories, and discuss new examples of twisted compactifications which give rise to magnetic vacua in four dimensions. In the following section, we discuss Ricci-flat instantons which describe decay of toroidal vacua by bubble formation and pair creation of monopoles. In §4 we consider more general instantons, relaxing the assumption of Ricci flatness, and construct from five-dimensional charged black hole solutions decay modes involving non-vanishing dilaton and antisymmetric tensor fields.

In §5 we use extremal black hole solutions with non-degenerate horizons to describe decay modes whose topology is not inconsistent with asymptotically covariantly constant spinors, but whose energy momentum tensors are unphysical, violating the dominant energy condition.

In §6 we discuss more generally the existence of instantons describing the decay modes of the supersymmetric vacuum; we consider the formulation of Witten’s proof of the positive energy theorem, and show how this proof excludes the existence of physical decay modes of a supersymmetric vacuum. In §7 and §8 we extend the discussion to the Calabi-Yau vacuum of ten dimensional supergravity and to the vacua of eleven-dimensional supergravity. Finally, in §9 we present our conclusions.

Note that contrary to globally supersymmetric Yang-Mills theories, supergravity is not renormalisable. This puts the entire subject of instanton calculus in supergravity on a rather shaky basis; if however we regard supergravity theories as low energy limits of superstring theories, which are not expected to suffer from these deficiencies, to the order to which supergravity theories are formally renormalisable results from non-perturbative instanton calculations may be considered as the limiting values of the corresponding exact string theory results [15].

Since manifolds of many different dimensions will abound, we will adhere to the following conventions. The indices ; ; ; ; ; ; ; , , and . We use the mostly positive convention for Lorentzian metrics and will denote metrics in the string frame whilst denotes metrics in the Einstein frame. denotes induced metrics on boundaries at spatial infinity, whilst denotes induced metrics on spacelike hypersurfaces. Hatted indices refer to an orthonormal frame whilst unhatted indices refer to coordinate indices. denotes the Lorentzian action and denotes the Euclidean action. refers to the Newton constant in dimensions.

## 2 Vacua of supergravity theories

Our starting point is the action that arises as a low energy effective field theory from the heterotic string. We shall consider heterotic string theory for definiteness but most of the discussion depends only on the common sector of the low energy supergravity theory. For the massless bosonic fields of the theory (graviton , dilaton , antisymmetric tensor , 16 vector bosons ) the action (in the string frame) takes the form:

(2.1) |

As in general relativity, this action will give the correct field equations but in calculating the action we must also include the surface terms required to ensure unitarity. The corresponding action in the Einstein frame is:

(2.2) |

To find classical solutions of the supersymmetric theory, we set to zero the fermion fields (gravitino , dilatino , gaugino ) together with their supersymmetric variations. Assuming that all fields vanish except the graviton (and constant dilaton) the conditions for finding a supersymmetric generator that leaves the vacuum invariant reduce to:

(2.3) |

As is well-known, and was first shown in [1] (see also [3]), for a vacuum state of the form , where is a maximally symmetric four-dimensional space and is a compact six manifold, equation (2.3) implies that the maximally symmetric manifold must be flat Minkowski space, and must be a Ricci-flat compact six manifold, which admits at least one covariantly constant spinor of each chirality. The holonomy group of is thus constrained to be a subgroup of the generic holonomy group of a six-dimensional manifold , and hence the covering space of must be , or a Calabi-Yau space.

The simplest manifold which satisfies (2.3) and is Ricci-flat is the flat torus; we may also consider another class of toroidal vacua which asymptotically tend to static magnetic configurations in four dimensions. Although these solutions are non-trivial four dimensional configurations they are simply obtained from dimensional reduction of Euclidean space with twisted identifications. The construction of a static cylindrically symmetric flux tube in four dimensions by dimensional reduction of five dimensional Minkowski space in which points have been identified in a nonstandard way was discussed in [16], whilst more recently the construction was generalised to obtain sets of orthogonal fluxbranes in higher dimensional spacetimes [14].

We will now consider a four dimensional vacuum solution in which there are magnetic fields, arising from the ten-dimensional metric, associated with distinct isometry groups. We start with -dimensional Minkowskian space and identify points under combined spatial translations and rotations, ie.

(2.4) |

where we identify points and are integers. We will usually assume that the are identical. Since is already periodic, changing by an integer does not change the identifications; thus inequivalent spacetimes are obtained only for . Changing each by a multiple of leads to equivalent spacetimes in dimensions, though the four dimensional configurations are not equivalent. Geometrically. this spacetime is obtained by starting with (2.4) and identifying points along the closed orbits of the Killing vectors . Introducing a new coordinate , we may rewrite the metric as

(2.5) |

Dimensionally reducing along the Killing vectors , the four-dimensional metric in the Einstein frame is related to the -dimensional metric by:

(2.6) |

with Kaluza Klein gauge fields and the four-dimensional dilaton defined by . From (2.5), we find that:

(2.7) |

(2.8) |

(2.9) |

The off-diagonal terms in the internal metric imply that the torus is not a direct product of circles. The four-dimensional metric is given by

(2.10) |

where is defined by:

(2.11) |

The gauge fields in four dimensions are obtained by solving the simultaneous equations defined in (2.9); in the limit that only one field is non-zero, the solution reduces to the static magnetic flux tube. Asymptotically, each gauge field ; the gauge fields correspond to magnetic fields which are uniform at infinity.

We may also dimensionally reduce along the Killing vectors ; the corresponding four dimensional solution is unchanged, except that the magnetic field is modified to , and in this way all values of the four-dimensional magnetic fields associated with each can be obtained. For every , the proper length of the circles in the th direction grows linearly with for large ; thus, we can view the solution as an approximation to physical fields which is valid only for , in which range the three dimensional space is approximately flat, and the internal circles have approximately constant length. In order for the internal directions to remain unobservable, we must consider length scales which are large compared to their size: . The two restrictions imply a limited range of applicability of the spacetime, , which can include large magnetic fields only if the compactified dimensions are of the Planck scale. Since the different dimensional reductions change by multiples of , for given , at most one is physically reasonable.

These solutions can be obtained by the action of generating transformations on the original Kaluza-Klein solution of [16]; the required transformations are an subgroup of the T-duality group of the four-dimensional theory. The transformation acts as:

(2.12) |

where is an invariant matrix, and all other fields are left invariant. The particular transformation required here is (assuming that the radii of the compactified directions are identical) , a six-dimensional rotation that rotates an arbitrary six-dimensional vector in direction into a vector of the same magnitude with only one component non-zero.

Consider a solution for which fields are non-zero and equal to :

(2.13) | |||||

(2.14) |

This has the interpretation of a flux tube along the z-axis, associated with magnetic fields, and is the required background for nucleation of monopoles carrying charges with respect to p fields.

Even though these spacetimes are locally flat, the nontrivial identifications imply that if a vector is parallely transported around each , it will return rotated by an angle . It follows that for one spin structure, parallel propagation of a spinor around the th direction results in the spinor acquiring a phase , where is a generator of the Lie algebra of (spinor representation). For the other spin structure, parallel propagation gives a phase . For small , the natural generalization of the standard choice of spinor structure for a supersymmetric vacuum is the first choice. The magnetic vacua evidently admit no covariantly constant spinors, whereas for the standard metric on the torus, constant spinors are admitted.

Note that the invariance of the low energy effective action under the T-duality group, and the invariance of the equations of motion under S-duality transformations allows us to generate further solutions. We apply a particular S-duality transformation (corresponding to strong/weak coupling interchange), (with the dual of ) and . Then, rescaling to the string metric, , the four-dimensional solution becomes:

(2.15) | |||||

(2.16) |

Thus, the only non-vanishing gauge fields are those originating from the off-diagonal components of the two-form in ten dimensions. The solution describes an ‘electric’ flux tube, associated with gauge fields; each gauge field asymptotically approaches zero.

By applying a general generating transformation, we can obtain four-dimensional solutions describing tubes of magnetic flux associated with the gauge group of the heterotic theory; these are the required backgrounds for nucleation of other topological defects, such as H-monopoles.

## 3 Ricci-flat instantons

We firstly consider instanton solutions of the Euclidean field equations in ten dimensions, in which all fields except the graviton and the (constant) dilaton vanish, implying that . The asymptotic geometry of the instanton is ; we defer the discussion of Calabi-Yau vacua to §7. Evidently decay modes cannot preserve all the supersymmetry; 32 constant spinors requires trivial holonomy, implying that the solution admits a flat metric. However, if an instanton is to describe a possible vacuum decay mode, it must asymptotically admit the constant spinors of the background. Vacuum instability - which in many cases will correspond to physical formation processes - will hence result only from considering non-supersymmetric states which are not simple metric products, but rather contain topological defects such as monopoles or p-branes.

The instantons will usually globally admit a isometry group, as well as a hypersurface orthogonal Killing vector which we use to Wick rotate the solution to describe the subsequent Lorentzian evolution. We may also of course consider instantons which admit such isometries only asymptotically, although it is unclear how the effective four-dimensional solution can be interpreted in this case. Fixed points of these isometries will lead to apparent singularities such as bubbles and monopoles in the lower-dimensional spacetime; the structure of these fixed point sets determines whether the instanton has a spin structure consistent with that of the background.

For a ten-dimensional instanton, the fixed point set must have dimension , , , , , ; the classification of four-dimensional gravitational instantons in terms of the fixed points sets of a isometry was discussed in [17] and reviewed in [18]. This work has been generalised to higher dimensions in [14] and [19].

If the isometry admits no fixed point sets, there is a priori no obstruction to choosing the spin structure of the instanton to be consistent with that of the background. If however the fixed point set of the isometry is eight-dimensional, spinors must be antiperiodic about a closed orbit of the isometry at infinity, and the spin structure is incompatible with that of the (supersymmetric) vacuum. The twisted boundary conditions break supersymmetry, and, although this supersymmetry breaking can be made arbitrarily weak by taking the compactified directions to be arbitrarily large, the action of the instanton diverges as the radii approach infinity, implying that the rate of decay of the vacuum goes to zero.

The obvious example is the five-dimensional Euclidean Schwarzschild solution crossed with a flat torus (a decay mode first considered in [6]):

(3.17) |

where the periodicity of is and the range of is . Since the topology of the solution is , the spin structure is incompatible with that of the supersymmetric vacuum. The action for this instanton is obtained from the boundary term:

(3.18) |

with the trace of the second fundamental form of the boundary at infinity, and the corresponding term in the background. The action is hence:

(3.19) |

and thus the rate of vacuum decay does indeed vanish as the radii of the compactified directions increase, that is, as the supersymmetry breaking becomes arbitrarily weak. Furthermore, the decay rate is small if Planck length, and it is only in this case that the semiclassical calculation is reliable.

If the fixed point set of the isometry has dimension less than eight, the spinors need not necessarily be antiperiodic about a closed orbit of the isometry at infinity. As an example, we may consider dimensional reduction of (3.17) along the Killing vector [13] (where is the radius of the circle at infinity) which describes monopole pair creation in a background field. It is possible to choose the spin structure to be consistent at infinity with that of the solution describing the background field (2.5), but the magnitude of the magnetic field lies far outside the physical range of validity.

As also discussed in [13], we may consider instantons which are a product of and the five-dimensional Euclidean Kerr-Myers-Perry solution:

(3.20) | |||||

where the index runs over the coordinates of the remaining and . The most general such solution is labelled by one mass parameter and two angular momentum parameters (associated with the isometry group), but for simplicity we take only the mass parameter and one angular momentum parameter to be non-zero.

Reduction along , which has an eight-dimensional fixed point set, leads to decay of the four-dimensional vacuum by bubble formation, whilst reduction along 3.20) is: , we obtain fields of physical validity. The pair creation decay mode has a spin structure consistent with that of the magnetic vacuum and the action for the instanton ( , and for , which has a six-dimensional fixed point set, leads to decay by monopole pair production. In the latter case, the four-dimensional field

(3.21) |

so for physical the decay rate is very small.

In §2, we showed that by applying a generating transformation to a solution in which there was a single non-zero magnetic field in four dimensions we could obtain a solution in which there were several non-zero (metric) fields in four dimensions. Using the same generating techniques here, we expect to obtain more general solutions describing the pair creation of monopoles carrying charge with respect to gauge fields in a background of gauge fields (). Monopoles carrying several different charges have recently been constructed [20] within heterotic string theory by the supersymmetric uplifting of four-dimensional monopole solutions; we now discuss their nucleation.

The most general solution Ricci-flat solutions will be obtained from the three parameter five dimensional Euclidean Kerr-Myers-Perry solution (constructed in [21]) crossed with a flat five torus by first applying an transformation to and , and then identifying points along closed orbits of where the are chosen so that the action of the isometry is periodic.

We present an illustrative solution, describing pair creation of monopoles carrying two identical charges within two (equal) background fields. We identify points along closed orbits of , and in (3.20) and then introduce . The required generating transformation is:

(3.22) |

where acts on and as in §2.1, and generates a two-dimensional transformation by . The resulting solution is:

where . The four-dimensional solution obtained by dimensionally reduced along closed orbits of and is:

where is the determinant of the metric on the torus; the ten-dimensional solution is complete and non-singular.

If we take , then the metric is singular over all the horizon ; the solution describes a generalised bubble decay mode of the vacuum.

We may also take , in which case vanishes only at the poles of the horizon; this is the solution describing pair creation of monopoles. The horizon is a line, which is smooth provided that we take ; for , the singularities at the poles are joined by a string. Since

(3.25) |

to obtain four-dimensional magnetic fields of physical magnitude we need either negative, close to and or positive, close to and .

Consider the spin structure of the transformed solution. A spinor parallely transported about an orbit of can be shown to pick up a phase of ; the same phase is picked up by a spinor transported about an orbit of . The four-dimensional magnetic fields are , so this decay mode by bubble nucleation is incompatible with the vacuum spin +structure, defined by phases of .

If we take and dimensionally reduce along and , then the phase change about an orbit of is found to be which is consistent with the vacuum.

The action for this decay mode can be compared to (3.21); the transformation does not change the action, but after ensuring that the unit of charge is the same in each case, we find that

(3.26) |

where the notation specifies the charges carried by the monopoles. Thence the rate of decay by creation of monopoles carrying charges is approximately half as large as the rate of decay by creation of monopoles carrying charges (of the same magnitude); as we would expect, the higher the charge, the smaller the rate of decay.

By applying a more general transformation to these solutions, we might expect to obtain instantons describing the pair creation of other types of monopoles, such as H monopoles, within the backgrounds discussed in §2. Although a large class of solutions can be obtained by generating transformations, most of them will be singular and incomplete; the nature of the “dual” geometry depends on the fixed points of the isometry with respect to which we dualise and fixed points of the isometry in the original solution generically become singular points in the dual solution.

For example, the solution appropriate to H-monopole nucleation is given by the (Buscher) transformation (see [22] for a review of T-duality in string theory):

(3.27) | |||||

with all other fields invariant. Dualisation with respect to the isometry which has a fixed point set at , leads to a solution which is singular at these points (in both string and Einstein frames); thus we cannot interpret the solution as describing pair creation.

## 4 More general instantons

We have so far discussed only Ricci-flat instantons; evidently, more general decay modes of the vacuum involving non-zero gauge, antisymmetric tensor and dilaton fields should also be taken into account. Consistency with the background requires that all fields are asymptotically constant; these solutions were considered to some extent in [9] and we suggest generalisations here.

The decay modes presented in §3 involved five-dimensional Euclidean
black hole solutions, with a non-trivial topology and in looking for generalisations it is natural to consider
electrically charged black hole solutions in five dimensions.
^{1}^{1}1Five-dimensional black holes may carry a magnetic charge with respect
to the three form field strength, but the latter takes the form , and does not asymptotically vanish, so we need not
consider such solutions here.
In the following two
sections, we work with the effective five-dimensional action, and
implicitly take the product of the five-dimensional solution
with a flat torus.
Following the general prescription for dimensional reduction given in
[23], we obtain from (2.1) an
action in the string frame containing the terms:

(4.28) |

with deriving from the left current algebra, and where is the volume of the . Rescaling to the Einstein frame , we obtain an action:

(4.29) |

We may now invoke Poincaré string-particle duality in five dimensions to relate the three form field strength to its dual:

(4.30) |

which gives us the following action in terms of the axionic field strength :

(4.31) |

In §5 we shall consider more general solutions with both of these gauge fields non-trivial but we begin with a particularly simple five-dimensional (electrically) charged black hole solution:

(4.32) | |||||

where , and is the gauge potential associated with the field strength . Such a solution was first constructed in [24], and the Euclidean section was discussed in [9]. We define the charge as:

(4.33) |

and, with this convention, . We now look for a Euclidean section on which all the fields are real, by rotating . To obtain a real gauge potential on the Euclidean section, we must also rotate and hence , giving the solution:

(4.34) | |||||

where and we have added a pure gauge
term to the potential so that is non singular at . ^{2}^{2}2Evidently the potential approaches a non-zero
constant at
infinity, and thence a (purely gauge) Maxwell potential must exist
in the background also; this presents no problem since
the most general supersymmetric vacua may have constant gauge potentials.
The coordinate is now restricted to
and we must identify with period
. The limit
corresponds to an uncharged solution, whilst in the limit
the solution becomes singular. Since , the radius
at infinity becomes smaller as approaches its maximum value.

By rotating one of the coordinates on the sphere, we obtain:

(4.35) |

Since there are no terms of order in the fields, this solution has zero mass and charge, which is consistent with the fact that it results from the decay of a vacuum which certainly has zero mass and charge. Since the topology of the solution is , and the Killing vector has a fixed point set of dimension three at , spinors must be antiperiodic about the imaginary time direction, which prevents the solution from describing the decay of a supersymmetric vacuum.

It is straightforward to calculate the Euclidean action for this solution; this is most easily done in the string frame, since we can convert the volume term (4.28) to a surface term [9] using the dilaton field equation:

(4.36) |

where we now include the appropriate surface terms (and is the asymptotic value of the dilaton field). The action is thus:

(4.37) |

with ; this is consistent with the Schwarzschild action given in §3 when as required. Re-expressing this in terms of the radius at infinity:

(4.38) |

As before, the decay rate goes to zero as the supersymmetry breaking becomes arbitrarily small, and, for given radius , the vacuum decay described by this solution is slower than decay via the Ricci-flat solutions of §3. If we let , with the radius finite, the action diverges, and the radius of the “horizon” approaches infinity. Letting with constant gives a finite action, decreasing radius at infinity and an unstable horizon (since the horizon is singular for ). In the limit of small charges, , it is straightforward to show that:

(4.39) |

with the radius at infinity approximately .

We obtain the effective four-dimensional solution using the procedure given in [23] as:

(4.40) |

where is the metric in the Einstein frame and . Then the four-dimensional metric in the Einstein frame is:

(4.41) |

The solution describes the formation and subsequent expansion of a hole at , and differs from Witten’s original decay mode [6] only by the presence of an additional scalar field in four dimensions (originating from ).

As in §3, we can also consider dimensional reduction along the Killing vector where . The four-dimensional fields obtained are:

(4.42) | |||||

describing the pair creation of monopoles within a background magnetic field and background scalar fields and (which are asymptotically constant). However, the magnetic field once again lies outside the range of validity , and we need to consider rotating black holes to obtain magnetic fields of physical magnitude.

For simplicity, we consider a five-dimensional black hole solution with only one electric charge , and one rotational parameter non-zero. Such a solution may be obtained from boosting the (Lorentzian) Myers-Perry solution; the most general such solutions are discussed in [25]. Rotating , and , we obtain the Euclidean section (in the string frame):

(4.43) | |||||

where , is defined as previously and we have included a pure gauge term in so that is non-singular at the poles of the horizon . The charge using the same conventions as previously.

To avoid a conical singularity at the horizon, we choose the radius at infinity to be ; the Euclidean angular velocity is . The action is easily calculated from (4.36), with the background subtraction facilitated by the flatness of the solution for all values of ; then:

(4.44) |

As usual, we rotate a coordinate on the sphere to obtain the subsequent Lorentzian evolution. Then, dimensional reduction along leads to a bubble decay mode, and reduction along describes monopole pair production, with the decay rate of the latter suppressed since the action is greater. For the latter, magnetic fields of physical magnitude and avoidance of conical singularities in the four-dimensional solution require and .

We have considered only the most simple charged rotating solution; the prescription for obtaining the most general decay modes is as follows. Starting from the most general five-dimensional Lorentzian rotating, (electrically) charged black hole solution [25], we look for a Euclidean section on which all fields can be chosen to be real. If such a section exists, then by Witten rotating a coordinate on the sphere, we can obtain a vacuum decay mode. Dimensional reduction along a Killing vector with fixed point set of dimension three leads to decay by bubble formation, a decay process lying in a different superselection sector of the Hilbert space to the supersymmetric vacuum, whilst dimensional reduction along a Killing vector with a fixed point set of dimension one leads to decay by monopole pair production, a decay process consistent with the spin structure of the background. Finally, by rotating the torus coordinates (allowing for non-trivial angles between the generating circles), we obtain the generalisations of the solutions discussed in §3.

All such decay modes do not describe the decay of the supersymmetric vacuum, are incomplete at null infinity, and have actions greater than the action of the original decay mode of Witten described in §3.

## 5 Extremal black holes as instantons

The discussion in the previous sections has been based around five-dimensional black hole solutions of topology , whose asymptotic geometry is that of the background . However, extremal black holes are believed to have the topology , with the Killing vector in the circle direction having no fixed point sets [26]. In contrast to the choice for non-extremal solutions we must choose a spin structure such that spinors are periodic in this direction, and there is hence no obstruction to the analytically continued solutions asymptotically admitting the covariantly constant spinors of the background.

To illustrate this, we consider a particular five-dimensional extremal black hole solution to the equations of motion which follow from (4.31), carrying electric charges with respect to both gauge fields where:

(5.45) | |||||

For a spherically symmetric solution we have:

(5.46) | |||||

and there exist solutions with constant dilaton such that:

(5.47) |

The field equations then imply that the metric takes the Reissner-Nordstrom form:

(5.48) |

We consider these extremal solutions (with both charges non-zero) since extremal solutions with only one charge non-zero have degenerate horizons with zero area, and thus the Euclidean sections have naked singularities, and cannot be interpreted as an instantons. The above solution is the simplest extremal black hole with a non-degenerate horizon, and for this reason the corresponding dual solution in IIA theory compactified on was recently discussed in the context of the microscopic description of the entropy [27].

We now attempt to analytically continue the Lorentzian solution into the Euclidean regime, by rotating ; now, the gauge fields in the original solution are:

(5.49) |

When we rotate , if we impose the requirements that the dilaton field and are real, both and remain real and positive, so that the gauge fields become pure imaginary. If however we impose the requirements that the gauge fields are real on the Euclidean section, then becomes complex, and the metric is not real. Thence a Euclidean section on which all the fields are purely real does not exist.

If we take (electric) field strengths that are pure imaginary on the Euclidean section, our solution takes the form: