Dyons, Superstrings, and Wormholes: Exact Solutions of the Non-Abelian Dirac-Born-Infeld Action
We construct dyon solutions on coincident -branes, obtained by applying -duality transformations to type I superstring theory in 10 dimensions. These solutions, which are exact, are obtained from an action comprising the non-Abelian Dirac-Born-Infeld action and a Wess-Zumino-like action. When one spatial dimension of the -branes is taken to be vanishingly small, the dyons are analogous to the ’t Hooft/Polyakov monopole residing in a -dimensional spacetime, where the component of the Yang-Mills potential transforming as a Lorentz scalar is reinterpreted as a Higgs boson transforming in the adjoint representation of the gauge group. Applying a -duality transformation to the vanishingly small spatial dimension, we obtain a collection of -branes, not all of which are coincident. Two of the -branes, distinct from the others, acquire intrinsic, finite curvature and are connected by a wormhole. The dyons possess electric and magnetic charges whose values on each -brane are the negative of one another. The gravitational effects, which arise after the -duality transformation, occur despite the fact that the action of the system does not explicitly include the gravitational interaction. These solutions provide a simple example of the subtle relationship between the Yang-Mills and gravitational interactions, that is, gauge/gravity duality.
Theoretically appealing but experimentally elusive, the magnetic monopole has captured the interest of the physics community for more than eight decades. The magnetic monopole (an isolated north or south magnetic pole) is conspicuously absent from the Maxwell theory of electromagnetism. In 1931, Dirac showed that the magnetic monopole can be consistently incorporated into the Maxwell theory with virtually no modification to the theory . In addition, Dirac demonstrated that the existence of a single magnetic monopole necessitates not only that electric charge be quantized but also that the electric and magnetic couplings be inversely proportional to each other, the first suggestion of the so-called weak/strong duality. Subsequently, ’t Hooft  and Alexander Polyakov showed that, within the context of the spontaneously broken Yang-Mills gauge theory , topological magnetic monopole solutions of finite mass must necessarily exist. Furthermore, these solutions possess an internal structure and also exhibit the same weak/strong duality discovered by Dirac. Consequently, Montonen and Olive conjectured that there exists an exact weak/strong electromagnetic duality for the spontaneously broken gauge theory . More recently, this conjecture has become credible within the broader context of or Super-Yang-Mills theories. Despite the lack of experimental evidence for the existence of magnetic monopoles, physicists still remain optimistic of their existence. Indeed, Guth proposed the inflationary model of the universe, in part, to explain why magnetic monopoles have escaped discovery .
The focus of our investigation is electrically charged magnetic monopole (dyon) solutions within the context of superstring theory. In Section 2, we construct dyon solutions which are exact and closed to first order in the string theory length scale. We, first, begin with a type I string theory in ten dimensions, six of the spatial dimensions being compact but arbitrarily large. We, then, apply the group of -duality transformations to five of the compact spatial dimensions to obtain 16 -branes, some of which are coincident. The five -dualized dimensions of each -brane constitute the internal dimensions of a -dimensional spacetime. Making an appropriate ansatz, we obtain dyon solutions residing on the -branes. The solutions are based on an action which includes coupling of the -branes to NS-NS closed strings, the non-Abelian Dirac-Born-Infeld action, and coupling to closed strings, a Wess-Zumino-like action. We next apply a -duality transformation to the -branes, resulting in a collection of -branes, some of which are coincident and two of which are connected by a wormhole. Finally, we interpret the dyon solutions in the context of gauge/gravity duality.
Because of the differences in the literature among the systems of units, sign conventions, and so forth, we present in Appendix A the conventions chosen by us so that direct comparisons can be made between our results and those of other authors.
2. Dyons and Dimensional Reduction of Type I Theory
In this section, we construct dyon solutions based on superstring theory. We begin with type I superstring theory in ten dimensions , six of the spatial dimensions of which are compact. Next, we apply the group of -duality transformations to five of the compact dimensions letting the size, , of the dimensions become vanishingly small; that is, . These five dimensions are the internal dimensions of spacetime. Strictly speaking, spacetime consists of 16 -branes, bounded by orientifold hyperplanes. Each of the -branes comprises four spatial dimensions, three unbounded and one compact. In what follows, we assume that none of the -branes are close to the orientifold hypersurfaces. Thus, the theory describing the closed strings in the vicinity of any of the -branes is type II oriented, rather than type II unoriented. In this particular case, since we have applied the -duality transformation to an odd number of dimensions, the closed string theory is the type IIa oriented theory. Furthermore, each end of an open string must be attached to a -brane, which may be the same -brane or two different -branes. If we assume that the number of coincident -branes is (), then a gauge group is associated with the open strings attached to the coincident -branes. Given these prerequisite conditions, we now construct dyon solutions which reside on these coincident -branes. These solutions are derived from the -brane action comprising two parts, the Dirac-Born-Infeld action, , which couples NS-NS closed strings to the -brane and the Wess-Zumino-like action, , which couples closed strings to the -brane.
2.1. Dyon Solutions on -Branes
The dyon solutions are obtained from the equations of motion derived from the action, , which describes the coupling of closed string fields to a general -brane (which in our case is ). The action is  whereHere, is the physical tension of the -brane, and is its R-R charge (see Appendix B for a discussion of the relationships among the various string parameters).
The dyon solutions are based on the following ansatz. The dilaton background, , is constant: And background field vanishes: The metric is given by where, for our purposes, is restricted so that and .
For , we can reexpress the determinant in (2) as where See Appendix C, (C.21), for further details.
The term is the -dimensional identity matrix. The value of is the dimension of the group associated with the gauge fields residing on the -branes. All potentials vanish, except for the one-form potential , which is a constant background field, for some constant value . The gauge field, , is obtained from the gauge potential , whereNote that the gauge potentials are static; that is, they do not depend on time, , and also do not depend on the spatial coordinate . The gauge field , a Lie algebra-valued two-form, is given by (See Appendix A.) The components of the potentials and are constrained in accordance with the condition so that . To facilitate its interpretation, we express as a five-dimensional matrix which is explicitly partitioned into electric and magnetic fields which reside in four-dimensional spacetime and an additional component of the magnetic field which resides in the additional space dimension; that is,We are seeking dyon solutions. Therefore, with foresight, we make the following assumptions:The parenthetical index indicates that there is no summation of that index; however, if an expression contains two indices without parentheses, then summation of these two indices is implied. Furthermore, each matrix element in (13) includes a generator of ; for example, . Because we are seeking dyon solutions, we may assume without loss of generality that each is a generator in the fundamental representation of a local subgroup of (see (44a), (44b), (44c), and (44d)).
The action, (2), can be more straightforwardly interpreted from the perspective of four-dimensional spacetime. Since the action does not depend on the coordinate , we can trivially eliminate from the action by integrating the coordinate. As a result of the integration, the tension of the -brane, , and the Yang-Mills coupling constant, , are replaced by those of the -brane, and (see (B.3) and (B.5)). Let the size, , of the -dimension become vanishingly small; that is, . Then, the field becomes a Lorentz scalar transforming as the adjoint representation of the gauge group, and (14) gives the covariant derivative of . From the perspective of four spacetime dimensions, assumes the role of a Higgs boson transforming as the adjoint representation of the gauge group.
Substituting (4)–(6) and (13) into (1) and then integrating the coordinate, we obtain where The function is defined as We have used the fact that . In (16), the ordering of the generators of the algebra, , corresponds with the order of the fields as they appear in the equation; for example,Note that “STr” indicates that the trace is calculated symmetrically; that is, the trace is symmetrized with respect to all gauge indices [6, 7]. The implication is that the evaluation of the trace requires that after the expansion of (16) in powers of the field strengths, all orderings of the field strengths are included with equal weight; that is, products of are replaced by their symmetrized sum, before the trace is evaluated. This is discussed in detail in [6, 7].
In (16), the dot product and cross product of two 3-vectors, for example, and , are defined as and .
In obtaining (16), we have reexpressed the dilaton , on a -brane, in terms of the dilaton , on a -brane, both of which are related by a -duality transformation in the -dimension. Specifically, and are related by . The constant dilaton background has been incorporated into the physical tension (see Appendix B).
Substituting (5), (9), (6), and (13) into (3), we obtain Integrating the coordinate in (19), we obtain wherewhere . Here, In obtaining (21), we have explicitly evaluated using (B.4). Equation (21) is associated with the Witten effect. Witten has demonstrated that adding term (21) to the Lagrangian of Yang-Mills theory does not alter the classical equations of motion but does alter the electric charge quantization condition in the magnetic monopole sector of the theory [5, 8, 9]. In summary, the action, , for the -brane is given bywhereThe equations of motion which are obtained from (23) are where In addition, the fields satisfy the Bianchi identity
To facilitate the ensuing analysis, we transform the Lagrangian density, , to the Hamiltonian density, , using the Legendre transformationwhere where After performing detailed calculations, we obtain where
The electric field can be expressed as a function of : The term is given by
We seek dyon solutions which are BPS states, that is, whose energy () is a local minimum. First, we reexpress
The mixing angle, , between the electric and magnetic fields of the dyon is defined as The quantities and are the electric and magnetic charges, respectively, of the dyon. The energy, , is minimized by constraining the dyon solutions to satisfyIn (35), the second and third squared terms are zero as a consequence of the constraint. Since , the fourth squared term is also zero by virtue of Thus, simplifies so that the energy is Substituting (37a) and (37b) into (32) through (34) and using (39), we findIn (39), there are two terms which contribute to the mass of the system. The first term within the trace, that is, , corresponds to the volume of each coincident -brane (or -brane), which is infinite because the -branes are not compact. The second term, by virtue of the equations of motion, (26), and the Bianchi identity, (27), can be expressed as a divergence and is therefore a topological invariant. The second term corresponds to the mass of the dyon and is proportional to as discussed below.
The solutions to (25) and (27) can be straightforwardly obtained from the dyon solutions derived in . Adapting the notation of  to the notation used here, we express the vector potential , (10), in the form (in accordance with our conventions, the Yang-Mills coupling constant appears explicitly in the Lagrangian (A.1). In [8, 10], the coupling constant has been incorporated into the Yang-Mills fields. Thus, to compare results here with those in the references, the fields and related fields should be divided by )where is an arbitrary constant. For the Lie group
Here, constitute a representation of the subalgebra and commutes with each . The quantities are the spherical polar coordinates in three dimensions. The elements are related to :For , the -dimensional matrices , and are given by, , , and are suitable linear combinations of specific elements of the Cartan subalgebra of (see  for details). The value of the integer in (43a), (43b), and (43c) is the integer multiple of the fundamental unit of dyon’s magnetic charge.
These results differ from those of . For a direct comparison, first replace the azimuthal angle, , in  with and extend the domain from to ; that is, . Now, perform the change of variables to the dyon solutions of  to obtain those given in (41). In addition, apply the same change of variables to the metric in  to obtain the metric : Here, , , and . This generalizes the results of  which only applies to dyons with one unit of magnetic charge; that is, .
The solutions , , and are obtained as in where the dimensionless variable is related to the radial coordinate :
The field tensor of a dyon with electric charge and magnetic charge , can now be obtained from (41). Specifically,
We now show that gauge invariance of action, (23), implies invariance. Consider gauge transformations which are constant at infinity and are also rotations about the axis , specifically the gauge transformations  Action (23) is invariant under these gauge transformations. According to the Noether method, the generator of these gauge transformations, , is given by Substituting the Lagrangian density (24) into (52), we obtain whereare the magnetic and electric charge operators. Since rotations of about the axis must yield the identity for physical states, that is, applying the transformation on the left side of (55) to states in the adjoint representation of , we find that the eigenstates of are quantized with eigenvalue where is an arbitrary integer. Substituting (56) into (53), we obtain where we have defined by and used the fact that Taking in (57), we obtain the quantization condition for the electric charge The electromagnetic contribution to the mass (rest energy) of the dyon, , can be obtained by substituting (54a) and (54b) into (39) and integrating the second term within the trace to obtain
We can now make symmetry explicit. We first define If , then the weak/strong duality condition is equivalent to In (57), the transformation results in identical physical systems with only states being relabeled. The transformation is equivalent to Transformations (63) and (64) generate the group . See [8, 9] for further details.
Note that in (54a) and (54b) is, strictly speaking, not the electric charge operator because is not the electric field but rather is its conjugate; however, according to (33) and (34), if and become vanishingly small for asymptotically large values of the radial coordinate, then approaches . Thus, in the asymptotic limit is the electric charge operator. This distinguishing feature is a direct consequence of the fact that our analysis is based on the Born-Infeld action rather than the Yang-Mills-Higgs action. In our case, this point is inconsequential since , exactly.
2.2. Dyon Solutions on -Branes
As emphasized previously, the dyon solutions derived in Section 2.1, when interpreted from spacetime dimensions, that is, the compactified theory in which , are the ’t Hooft/Polyakov magnetic monopole or dyon, with the potential being a Higgs boson transforming in the adjoint representation of the gauge group . Here, our purpose is to reinterpret these dyon solutions in which from the equivalent -dual theory. In the -dual theory, the radius is replaced by () so that the radius of the -dimension . In addition, the potential is reinterpreted as the -coordinates of the -branes embedded in 4 + 1 dimensional spacetime. These coordinates can be directly obtained by diagonalizing , (41), using a local gauge transformation which rotates into . The -coordinates are the diagonal elements of the matrix; that is (we are assuming that after the -duality transformation the -branes are far from any orientifold hyperplanes. This can always be accomplished by adding the to component of the gauge potential a constant gauge transformation , being a suitable constant (see Appendix A)),whereOf the -branes of the -branes, denoted by , are coincident. The -coordinate of each is the constant value . For the remaining two -branes, denoted by and , the -coordinate of each is a function of the radial coordinate . Specifically, for and for , and as a consequence these two -branes have nonvanishing intrinsic curvature. This occurs despite the fact that before the application of the -duality transformation no gravitational interaction is explicitly present. We now introduce the length scale which is the separation between and , in the asymptotic limit as the radial coordinate . It is related to previously defined parameters by Another relevant length scale is the size of the dyon, that is, the region of space where all components of the Yang-Mills field, , are nonvanishing. According to (49), (46a), (46b), (46c), and (47), only the radial components of the electric and magnetic fields are long range, with the remaining components of the fields vanishing exponentially for . Thus, additional structure of the dyon becomes apparent whenever or equivalently whenever . We can therefore define the size of dyon , as measured from asymptotically flat space, that is, , to be In Figure 1, we show, for the gauge group , embedding plots of the 5 -branes as a function of the dimensionless radial coordinate, (). As , the -coordinate of approaches that of the coincident -branes, , in effect, joining them by a wormhole in an asymptotically flat region of space. At , is joined to by another wormhole. (See  for a recent discussion of thin shell wormholes exhibiting cylinder symmetry.) As we will show, in general, the intrinsic curvature of the two surfaces in the neighborhood of is relatively large but, nonetheless, finite. Although these features described in Figure 1 apply to the particular gauge group , they apply to all , . (For there are no coincident -branes.)
We now consider in detail the two -branes, and , whose -coordinates are radial dependent. The -duality transformation on the -dimension pulls back the metric onto and , inducing the metric, , Expanding the right-hand side of (69), we obtainwhere Here, and is given by (45). In obtaining (70), we have used (50) and the fact that the matrices , , and , (43a), (43b), and (43c), satisfy the relationship Each of the diagonal entries in the matrix corresponds to the metric on one of the -branes obtained from the -duality transformation. In the case of the first entries, corresponding to the -branes , the metric is flat. In the case of the last two entries corresponding to the -branes, and , their geometries are identical and intrinsically curved. The only feature which distinguishes these two -branes is that the function defining the -coordinate for -brane is replaced by for , as evidenced in (65) and (66a) and (66b) and (66c) and also in Figure 1. As a consequence, the electric and magnetic charges of the dyon on are minus the values on . The electric and magnetic field lines enter the wormhole from one -brane and exit from the other. Figure 2 is an embedding diagram showing the -branes and in the neighborhood of the radial coordinate . As there is no event horizon surrounding , the two -branes are joined by a wormhole at .
Of particular interest is the intrinsic scalar curvature of and , in the neighborhood of . The scalar curvature can be calculated from the metric , (70), and its value at is For a given value of , assumes its maximum value : when , where For either or , the scalar curvature ; that is, the geometry of and becomes flat, everywhere. The expression for the scalar curvature is a complicated function of and not amenable to straightforward interpretation and, therefore, will not be given. In Figure 3, we show a plot of the scalar curvature as a function of the radial coordinate. In this example, , and the dyon has only one unit of magnetic charge so that . Near , the scalar curvature is positive and finite. As increases, the scalar curvature becomes slightly negative and asymptotically approaches zero as . These features of the scalar curvature described for this specific example also apply in general.
Consider dyon solutions for which or less. The -strings connecting and would be in their ground state, a BPS state. In addition, assume that ; then as from an asymptotically flat region of space, within either or , the string length scale will be reached before the gravitational interaction becomes dominant at the length scale of . Thus, action, (2), which does not include the gravitational interaction, should apply, and consequently the dyon solutions derived should be accurate. On the other hand, let so ; then the -string, also a BPS state, becomes lighter than the -string. As a consequence of weak/strong duality, the dyon solutions should still be applicable with the -strings being replaced by -strings and the dyon electric and magnetic charges being interchanged.
After applying a -duality transformation to the dyon solutions obtained in Section 2.1, we have obtained dyon solutions residing on -branes where the effect of the gravitational interaction is apparent. This occurs despite the fact that action, (2), does not explicitly include the gravitational interaction. The presence of gravitational effects in this case is an example of how, in string theory, one-loop open string interactions, that is, Yang-Mills interactions, are related to tree level, closed string interactions, that is, gravitational interactions. In Figure 4, we depict, for illustrative purposes, two parallel -branes in close proximity. The two -branes can interact through open strings which connect the two -branes. In the figure, we show the one-loop vacuum graph for such an interaction which can be interpreted as an open string moving in a loop. Alternatively, the interaction can be interpreted as a closed string being exchanged between two -branes. In a certain sense, spin-2 gravitons, that is, the closed strings in their massless state, comprise a bound state of spin-one Yang-Mills bosons, that is, open strings in their massless state.
Prior to the application of the -duality transformation, the open strings can propagate anywhere in the -brane. After the -duality transformation, the open strings are constrained to propagate only within -branes, whereas the closed strings can still propagate in the bulk region between the -branes; that is, gravitons can propagate in spacetime dimensions not allowed for the Yang-Mills bosons. Based on certain general assumptions, Weinberg and Witten have shown the impossibility of constructing a spin-2 graviton as a bound state of spin-1 gauge fields . One of the assumptions on which the proof is based is that the spin-1 gauge bosons and spin-2 gravitons propagate in the same spacetime dimensions. In the example presented here, this assumption is violated so that the conclusion of their theorem is avoided. These dyon solutions, thus, provide a simple example of gauge/gravity duality, which is discussed in detail in the work of Polchinski .
We have investigated dyon solutions within the context of superstring theory. Beginning with type I superstring theory in ten dimensions, six of the spatial dimensions of which are compact, we have applied the group of -duality transformations to five of the compact dimensions. The result is 16 -branes, a number () of which are coincident. The five -dualized dimensions, whose size is taken to be vanishingly small, become the five internal spacetime dimensions while the remaining five dimensions correspond to the external -dimensional spacetime. Making a suitable ansatz for the gauge fields residing on the coincident -branes, we have obtained dyon solutions from an action consisting of two terms: the -dimensional, non-Abelian Dirac-Born-Infeld action and a Wess-Zumino-like action. The former action gives the low energy effective coupling of -branes to NS-NS closed strings and the latter of -branes to closed strings. The method of solution involves transforming the -dimensional action from the Lagrangian formalism to the Hamiltonian formalism and then seeking solutions which minimize the energy. The resulting dyon solutions, which are BPS states, reside on the -branes and are therefore associated with a supersymmetric gauge theory in spacetime dimensions. These dyon solutions can be alternatively understood in the limit when the size of the remaining compact spacetime dimension, , approaches zero. In this situation, the -dimensional spacetime is reduced to a -dimensional spacetime. As a consequence, the component of the vector potential becomes a Lorentz scalar with respect to -dimensional spacetime and can be interpreted as a Higgs boson transforming as the adjoint representation of the gauge group, analogous to the Higgs boson associated with the ’t Hooft/Polyakov magnetic monopole. Finally, we perform a -duality transformation in the -direction. As a result, of the -branes are transformed into coincident -branes, whose intrinsic geometry is flat. The remaining two -branes are transformed into two separate -branes whose intrinsic geometry is curved. As depicted in Figure 3, the two -branes are joined by wormhole at . The scalar curvature of each -brane reaches a maximum, finite value, at and approaches zero as . The dyon resides on these two -branes. Furthermore, the values of electric and magnetic charges of the dyon on one -brane are minus the values on the other -brane, and as a consequence the electric and magnetic field lines enter the wormhole from one -brane and exit from the other -brane.
The -duality transformation in the -direction causes two of the -branes to acquire intrinsic curvature. This occurs despite the fact that the Lagrangian density from which the dyon solutions have been obtained does not explicitly include the gravitational interaction. This can be understood heuristically from the open string, one-loop vacuum graph given in Figure 4. From one perspective, the graph describes an open string, whose ends are fixed on two different -branes, moving in a loop, or, alternatively, the exchange of a closed string between two -branes. Thus, the gravitational interaction, that is, the closed string interaction, and the Yang-Mills interaction, that is, the open string interaction, appear as alternative descriptions of the same interaction. This simple example is suggestive of the subtle, but profound, connection between the Yang-Mills and gravitational interactions, specifically gauge/gravity duality.
A. Units and Conventions
Concerning conventions, the Minkowski signature is , and the Levi-Cività symbols . Other relevant conventions are as follows: Greek letters denote four-dimensional spacetime indices, that is, 0, 1, 2, and 3, whereas capitalized Roman letters are used when the spacetime dimension is greater than four. The small Roman letters, , , , , and , are reserved for spatial dimensions in four-dimensional spacetime, that is, 1, 2, and 3. The small Roman letters, , are used to enumerate the generators of the Lie group. The Levi-Cività tensor in three space dimensions is , where are the spatial components of the metric tensor. We focus our attention on Yang-Mills theories based on the compact Lie groups, . A typical group element () is represented in terms of the group parameters as . The generators of this group in the fundamental representation are denoted by . The generator generates the portion of , and the remaining generate the portion. The Lie algebra of the group generators, , is , with being the structure constants of . The generators of are required to satisfy the trace condition . Thus, in particular, , being the -dimensional identity matrix. The Yang-Mills coupling constant is denoted by . We employ Lorentz-Heaviside units of electromagnetism so that ; consequently, the Dirac quantization condition is . The quantity is the magnetic charge of a unit charged Dirac monopole.
Consistent with our analysis, the Yang-Mills-Higgs Lagrangian is where is a potential function depending on the Higgs field, , and The covariant derivative, , is defined as Thus, The Higgs field is a scalar transforming as the adjoint representation of so that its covariant derivative is
B. Relationships among the Various String Theory Parameters
In principle, string theory has no adjustable parameters other than its characteristic length scale, ; however, various parameters of the theory do depend on values of the background fields. For reference, we provide explicit relationships between various string theory parameters and . Let be the vacuum expectation value of the dilaton background; that is, , being the dilaton background. The closed string coupling constant, , is The physical gravitational coupling, , is where is Newton’s gravitational constant in 10 dimensions and . The quantity is the gravitational constant appearing in the eleven-dimensional, low energy effective action of supergravity, and is the compactification radius for reducing the eleven-dimensional theory to ten dimensions. The physical -brane tension, , is where is -brane tension. The -brane charge, , isThe coupling constant of the Yang-Mills theory on a -brane is given by The ratio of the -string tension, , to the -string (1-brane) tension, , is
C. Evaluation of the Dirac-Born-Infeld Determinant in an Arbitrary Number of Dimensions
In this appendix, we provide a heuristic derivation of the formula for evaluating an -dimensional () determinant of the form , (2) and (5). (The notation used in this appendix does not adhere strictly to the font conventions defined in Appendix A.) Without loss of generality, we assume that the metric is diagonal. Consequently, we can express the determinant in the generic form where , and . Note: indices enclosed within parentheses are not summed.
Thus, the right-hand side of (C.1) comprises a sum of terms, each of which consists of products of the metric, or . We need only to consider terms in which the number of in each product is even, since terms containing products of an odd number of vanish because . Thus, , (C.1), can be reexpressed as a sum Each is a term in (C.1) which contains a product of which is even in number. The values of range from 0 to or depending on whether is even or odd. The value of , which contains no off-diagonal elements, , is evaluated as
In order to understand the structure of , for arbitrary , we first study the structure of :Equation (C.4) represents a term in (C.1) where the values of in the sum are restricted to the specific integer values from the set of integers . Multiplying (C.4) by the Levi-Cività symbol (which does not change value of the expression), we obtain We now rearrange the terms in (C.5) in a form which is more useful for the subsequent analysis. By virtue of the Levi-Cività symbols, none of the is equal to any of the others, and similarly for ; however, each of the is equal to one of the . Because of the antisymmetry of , some of the terms in the sum vanish, that is, whenever , or whenever , and so forth. By explicit construction or by a combinatorics argument, we can show that there are nine terms which are nonvanishing. We consider one typical term in the sum, for example, the term, , We now show how to reexpress (C.5) so half, that is, 2, of the four are associated with one of the Levi-Cività symbols and the other half are associated with the other Levi-Cività symbol. In order for the two to be associated with one Levi-Cività symbol, the four subscripts on the two must be different. We associate the first with the Levi-Cività symbol to its left. To determine the remaining associations, we proceed as follows. Since , we associate with the Levi-Cività symbol to the right. We now consider the second term in (C.5), that is, . Since , we associate with the Levi-Cività symbol to its right and with the Levi-Cività symbol to the right. We continue in this manner to the next remaining term, . Since does not equal any of the lowercase Roman subscripts associated with the Levi-Cività symbol to the left (), we assign to the Levi-Cività symbol to the left. This completes the process since each is associated with either of the two Levi-Cività symbols. Using (C.6) we reexpress the subscripts on the Levi-Cività symbols in (C.5):
Now, we permute the subscripts of the Levi-Cività symbols so that the corresponding lowercase and uppercase Roman letters are adjoining. Both uppercase and require the same number of movements as the uppercase and do. Since the number of permutations is even, no change in sign of the Levi-Cività symbols results from permuting the subscripts. Equation (C.7) becomes Each of the remaining can be expressed, similarly.
In order to understand, in generic terms, the structure of , consider the following expression : The expression differs from the individual term in that the repeated indices and are summed. By inspection, the term , as well as the remaining terms , is contained in . Overtly, the number of nonvanishing terms in (C.9) is , each factor of coming from each one of the Levi-Cività symbols. By a combinatorics argument, each factor of is eightfold redundant so that the number of independent terms associated with each Levi-Cività symbol is three. Consequently, the number of independent terms in is 9, that is, , with redundancy . Thus, Using similar reasoning, we can show that where In order to show (C.11), one needs to use the fact that the number of nonvanishing terms, , in is given by so that the number of independent terms, , associated with each of the two Levi-Cività symbols of is Thus, Both (C.13) and (C.14) are obtained from combinatorics arguments. Using the metric tensor to lower indices in and the relationship (C.15), we reexpress (C.11)