LBNL-41110

UCB-PTH-97/58

hep-th/9712103

May 24, 2021

Nonlinear Electromagnetic Self-Duality and Legendre Transformations
^{*}^{*}*Contribution to the Proceedings of the Newton Institute Euroconference
on Duality and Supersymmetric Theories, April, 1997
^{†}^{†}†This work was supported in part by
the Director, Office of Energy Research, Office of High Energy and Nuclear
Physics, Division of High Energy Physics of the U.S. Department of Energy under
Contract DE-AC03-76SF00098 and in part by the National Science Foundation under
grant PHY-95-14797.

Mary K. Gaillard and Bruno Zumino

Physics Department, University of California, and

Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720

We discuss continuous duality transformations and the properties of classical theories with invariant interactions between electromagnetic fields and matter. The case of scalar fields is treated in some detail. Special discrete elements of the continuous group are shown to be related to the Legendre transformation with respect to the field strengths.

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## 1 Duality rotations in four dimensions

The invariance of Maxwell’s equations under “duality rotations” has been known for a long time. In relativistic notation these are rotations of the electromagnetic field strength into its dual, which is defined by

(1.1) |

This invariance can be extended to electromagnetic fields in interaction with the gravitational field, which does not transform under duality. It is present in ungauged extended supergravity theories, in which case it generalizes to a nonabelian group [1]. In [2, 3] we studied the most general situation in which classical duality invariance of this type can occur. More recently [4] the duality invariance of the Born-Infeld theory, suitably coupled to the dilaton and axion [5], has been studied in considerable detail. In the present note we will show that most of the results of [4, 5] follow quite easily from our earlier general discussion. We shall also present some new results.

We begin by recalling and completing some basic results of [2, 3, 6]. Consider a Lagrangian which is a function of real field strengths and of some other fields and their derivatives :

(1.2) |

Since

(1.3) |

we have the Bianchi identities

(1.4) |

On the other hand, if we define

(1.5) |

we have the equations of motion

(1.6) |

We consider an infinitesimal transformation of the form

(1.7) | |||||

(1.8) |

where are real constant infinitesimal matrices and functions of the fields (but not of their derivatives), and ask under what circumstances the system of the equations of motion (1.4) and (1.6), as well as the equation of motion for the fields are invariant. The analysis of [2] shows that this is true if the matrices satisfy

(1.9) |

(where the superscript denotes the transposed matrix) and in addition the Lagrangian changes under (1.7) and (1.8) as

(1.10) |

The relations (1.9) show that (1.7) is an infinitesimal transformation of the real noncompact symplectic group which has as maximal compact subgroup. The finite form is

(1.11) |

where the real submatrices satisfy

(1.12) |

For the subgroup, one has in addition

(1.13) |

or, in finite form,

(1.14) |

Notice that the Lagrangian is not invariant. In [2] we showed, however, that the derivative of the Lagrangian with respect to an invariant parameter is invariant. The invariant parameter could be a coupling constant or an external background field, such as the gravitational field, which does not change under duality rotations. It follows that the energy-momentum tensor, which can be obtained as the variational derivative of the Lagrangian with respect to the gravitational field, is invariant under duality rotations. No explicit check of its invariance, as was done in [4, 5, 7, 8], is necessary. Using (1.7) and (1.9) it is easy to verify that

(1.15) |

so (1.10) is equivalent to the invariance of .

The symplectic transformation (1.11) can be written in a complex basis as

(1.16) |

where means complex conjugation and the submatrices satisfy

(1.17) |

The relation between the real and the complex basis is

(1.18) |

In [2, 3] we also described scalar fields valued in the quotient space . The quotient space can be parameterized by a complex symmetric matrix whose real part has positive eigenvalues, or equivalently by a complex symmetric matrix such that has eigenvalues smaller than 1. They are related by

(1.19) |

These formulae are the generalization of the well-known map between the Lobachevskiĭ unit disk and the Poincaré upper half-plane: corresponds to the single complex variable parameterizing the unit disk, to the one parameterizing the upper half plane.

Under

(1.20) |

or, infinitesimally,

(1.21) |

where

(1.22) |

The invariant nonlinear kinetic term for the scalar fields can be obtained from the Kähler metric [9]

(1.23) |

which follows from the Kähler potential

(1.24) |

which are equivalent up to a Kähler transformation. It is not hard to show that the metric (1.23) is positive definite. In this section the normalization of the fields has been chosen to be canonical when is set equal to the unit matrix, i.e., when the self-duality group reduces to the subgroup; the full self-duality can be realized when the matrix is a function of scalar fields. Throughout this paper we assume a flat background space-time metric; the generalization to a nonvanishing gravitational field is straightforward [2]–[5].

## 2 Born-Infeld theory

As a particularly simple example we consider the case when there is only one tensor and no additional fields. Our equations become

(2.1) |

(2.2) |

and

(2.3) |

We have restricted the duality transformation to the compact subgroup , as appropriate when no additional fields are present. So

Since is a function of alone, we can also write

(2.4) |

Comparing (2.3) and (2.4), which must agree, we find

(2.5) |

Together with (2.1), this is a partial differential equation for , which is the condition for the theory to be duality invariant. If we introduce the complex field

(2.6) |

(2.5) can also be written as

(2.7) |

Clearly, Maxwell’s theory in vacuum satisfies (2.5), or (2.7), as expected. A more interesting example is the Born-Infeld theory [7], given by the Lagrangian

(2.8) |

where

(2.9) |

For small values of the coupling constant (or for weak fields) approaches the Maxwell Lagrangian. We shall use the abbreviation

(2.10) |

Then

(2.11) |

(2.12) |

and

(2.13) |

Using (2.12) and (2.13), it is very easy to check that : the Born-Infeld theory is duality invariant. It is also not too difficult to check that is actually invariant under (2.2) and the same applies to (which in this case turns out to be equal to ). These invariances are expected from our general theory.

It is natural to ask oneself whether the Born-Infeld theory is the most general physically acceptable solution of (2.5). This was investigated in [4] where a negative result was reached: more general Lagrangians satisfy (2.5), the arbitrariness depending on a function of one variable. We discuss this in detail in Section 6.

## 3 Schrödinger’s formulation of Born’s theory

Schrödinger [8] noticed that, for the Born-Infeld theory (2.8), and satisfy not only (2.5) [or (2.7)], but also the more restrictive relation

(3.1) |

We have verified this by an explicit, although lengthy, calculation using (2.6), (2.12), (2.13) and (2.9). Schrödinger did not give the details of the calculation, presenting instead convincing arguments based on particular choices of reference systems. One can write (3.1) as

(3.2) |

where

(3.3) |

and Schrödinger proposed as the Lagrangian of the theory, instead of (2.8). Of course, is a Lagrangian in a different sense than , which is a field Lagrangian in the usual sense. Multiplying (3.1) by and saturating the unwritten indices , the left hand side vanishes, so that (2.7) follows. Using (3.1) it is easy to see that is pure imaginary: Schrödinger also pointed out that, if we introduce a map

(3.4) |

so that (3.1) or (3.2) can be written as

(3.5) |

the square of the map is the identity map

(3.6) |

This, together with the properties

(3.7) |

ensures the consistency of (3.1). Schrödinger used the Lagrangian (3.3) to construct a conserved, symmetric energy-momentum tensor. We have checked that, when suitably normalized, his energy-momentum tensor agrees with that of Born and Infeld up to an additive term proportional to .

Schrödinger’s formulation is very clever and elegant and it has the advantage of being manifestly covariant under the duality rotation which is the finite form of (2.2). It is also likely that, as he seems to imply, his formulation is fully equivalent to the Born-Infeld theory (2.8), which would mean that the more restrictive equation (3.1) eliminates the remaining ambiguity in the solutions of (2.7). This virtue could actually be a weakness if one is looking for more general duality invariant theories.

## 4 General solution of the self-duality equation

The self-duality equation (2.5) can be solved in general as follows. Assuming Lorentz invariance in four dimensional space-time, the Lagrangian must be a function of the two invariants

(4.1) |

Now

(4.2) |

where we have used the standard notation Substituting these expressions in (2.5) we obtain

(4.3) |

This partial differential equation for can be simplified by the change of variables

(4.4) |

which gives

(4.5) |

Alternatively one can use the variables

(4.6) |

to obtain the form

(4.7) |

The equation (4.5), or (4.7), has been studied extensively in mathematics and there are several methods to obtain its general solution [10]. (It is interesting that the same equation occurs in a study of 5-dimensional Born-Infeld theory [11].) In our case we must also impose the physical boundary condition that the Lagrangian should approximate the Maxwell Lagrangian

(4.8) |

when the field strength is small.

According to one of the methods given in Courant-Hilbert, the general solution of (4.7) is given by

(4.9) | |||||

(4.10) |

where the arbitrary function is determined by the initial values:

(4.11) | |||||

(4.12) |

One must solve for from (4.10) and substitute into (4.9). To verify [11] that these equations solve (4.7), differentiate (4.9) and (4.10):

(4.13) | |||||

(4.14) |

and eliminate between (4.13) and (4.14) to obtain

(4.15) |

i.e.,

(4.16) |

The condition that should approach the Maxwell Lagrangian for small field strengths implies that

(4.17) |

for small .

It is trivial to check the above procedure for the Maxwell Lagrangian, and we shall not do it here. The Born-Infeld Lagrangian (with for simplicity) is given by

(4.18) | |||||

(4.19) |

in terms of the variables and . Setting we see that this corresponds to

(4.20) | |||||

(4.21) |

Then (4.10) gives

(4.22) |

which is solved by

(4.23) |

Using (4.9), we reconstruct the Lagrangian

(4.24) |

Unfortunately, in spite of this elegant method for finding solutions of the self-duality equation, it seems very difficult to find new explicit solutions given in terms of simple functions. The reason is that, even for a simple function , solving the equation (4.10) for gives complicated functions .

## 5 Axion, dilaton and

It is well known that, if there are additional scalar fields which transform nonlinearly, the compact group duality invariance can be enhanced to a duality invariance under a larger noncompact group (see, e.g., [2] and references therein). In the case of the Born-Infeld theory, just as for Maxwell’s theory, one complex scalar field suffices to enhance the invariance to the noncompact duality invariance. This is pointed out in [5], but it also follows from the considerations of our paper [2]. In the example under consideration, is a single complex field, not an matrix. In order to agree with today’s more standard notation we shall use

(5.1) |

where is the dilaton and is the axion. For , the matrices are real numbers and and are independent. Then the infinitesimal transformation is

(5.2) |

and the finite transformation is

(5.3) |

For the subgroup,

(5.4) |

The scalar kinetic term, proportional to

(5.5) |

is invariant under the nonlinear transformation (5.2) which, in terms of , takes the form

(5.6) |

Since the scalar kinetic term is separately invariant, we assume from now on that does not depend on the derivatives of .

The full noncompact duality transformation on is now

(5.7) |

and we are seeking a Lagrangian which satisfies

(5.8) |

where

(5.9) |

and now

(5.10) |

Equating (5.8) and (5.9) we see that must satisfy

(5.11) |

This equation can be solved as follows. Assume that satisfies (2.1) and (2.5), i.e.

(5.12) |

where

(5.13) |

For instance, the Born-Infeld Lagrangian does this. Then

(5.14) |

satisfies (5.11). Indeed

(5.15) |

So

(5.16) |

(5.17) |

where we have defined

(5.18) |

and is given by (5.13). Now

(5.19) |

Using (5.12) in this equation we find

(5.20) |

We also have

(5.21) |

On the other hand, since

(5.22) |

we obtain

(5.23) |

In addition

(5.24) |

Using (5.20), (5.21), (5.23) and (5.24), together with (5.6), we see that (5.11) is satisfied. It is easy to check that the scale invariant combinations and , given by (5.18) and (5.13) have the very simple transformation law

(5.25) |

i.e., they transform according to the compact subgroup just as and in (2.2), but with the parameter replaced by . If is the Born-Infeld Lagrangian, the theory with scalar fields given by in (5.14) can also be reformulated à la Schrödinger. ¿From (5.17) and (5.18) solve for and in terms of and . Then must satisfy the same equation (3.1) that does when no scalar fields are present.

## 6 Duality as a Legendre transformation

We have observed that, even in the general case of , although the Lagrangian is not invariant, the combination [see (1.15)]

(6.1) |

is invariant. Here we restrict ourselves to the case of , one tensor and one complex scalar field . As in Section 5, we use the notation to denote the part of the Lagrangian that depends on the scalar fields, as well as on , but not on scalar derivatives. Then

(6.2) |

where

(6.3) | |||||

(6.4) |

There are several interesting special cases of this invariance statement. The first corresponds to arbitrary, which gives

(6.5) |

The second corresponds to arbitrary, which gives

(6.6) |

The third corresponds to arbitrary, which gives

(6.7) |

(6.8) |

Taking , we obtain

(6.9) |

which gives the dependence of on , in agreement with (5.14). This choice for the constant is allowed because this part of the Lagrangian, which does not include the kinetic term for the scalar fields, does not contain derivatives of the scalar fields. Using (6.6) in (6.2) we find

(6.10) |

i.e.,

(6.11) |

Setting in this equation, we see that is a function of , in agreement with the more precise statement (6.9). Setting instead , we find that is a function of , in agreement with (5.14).

We have shown that the Ansatz (5.14) of Section 5 is a natural consequence of the invariance of . Equation (6.14) with (6.4) can be interpreted as a Legendre transformation. Given a Lagrangian , define the dual Lagrangian , a function of the dual field , by

(6.15) |

(6.16) |

With these definitions, the dual of the dual of a function equals the original
function.^{1}^{1}1The unconventional factor 1/2 on the right hand side of
(6.15) is introduced to avoid overcounting when summing over the indices
of the antisymmetric tensors and .
In general, the dual Lagrangian is a very different function from the original
Lagrangian. For a self-dual theory, if we set

(6.17) |

we see from (6.14) that

(6.18) |

which must be independent of , since is.

The above argument can be inverted. Let the Legendre transformation (6.15) produce a dual Lagrangian given by (6.18) with , or

(6.19) |

It then follows that is invariant under (6.7) with , i.e.,

(6.20) |

If we now assume that it is also invariant under (6.5) with arbitrary , it follows that it is invariant under the entire group . Indeed, if we call the transformation (6.5) and the transformation (6.20), the product gives the most general transformation of .

If we normalize the scalar field differently, taking e.g., instead of ,

(6.21) |

(6.22) |

and write the Legendre transformation as

(6.23) |

we see that

(6.24) |

and

(6.25) | |||||

for a self-dual theory.

A standard normalization [12, 13] is , in which case the expectation value of the field is

(6.26) |

In the presence of magnetically charged particles and dyons (both electrically and magnetically charged) the invariance of the charge lattice restricts [14] the group to the subgroup generated by

(6.27) |

At the quantum level the Legendre transformation corresponds to the integration over the field in the functional integral, after adding to the Lagrangian a term .

## 7 Concluding remarks

Nonlinear electromagnetic Lagrangians, like the Born-Infeld Lagrangian, can be supersymmetrized [15, 16] by means of the four-dimensional superfield formalism, and this can be done even in the presence of supergravity. When the Lagrangian is self-dual, it is natural to ask whether its supersymmetric extension possesses a self-duality property that can be formulated in a supersymmetric way. We were not able to do this in the nonlinear case. When the Lagrangian is quadratic in the fields , the problem as been solved in [17], where the combined requirements of supersymmetry and self-duality were used to constrain the form or the weak coupling () limit of the effective Lagrangian from string theory, in which one neglects the nonabelian nature of the gauge fields.

The subgroup of that is generated by the elements and relates different string theories [18] to one another.

The generalization of [2] to two dimensional theories [19] has been used to derive the Kähler potential for moduli and matter fields in effective field theories from superstrings. In this case the scalars are valued on a coset space The kinetic energy is invariant under , and the full classical theory is invariant under a subgroup of . String loop corrections reduces the invariance to a discrete subgroup that contains the group generated by , where Re is the squared radius of compactification in string units.

Acknowledgements. We are grateful for the hospitality provided by the Isaac Newton Institute where this work was initiated. We thank Gary Gibbons, David Jackson, Bogdan Morariu, David Olive, Harold Steinacker, Kelly Stelle and Peter West for inspiring conversations. This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-95-14797.

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