On Electromagnetic Duality
Thomas B. Mieling Faculty of Physics, University of Vienna Boltzmanngasse 5, 1090 Vienna, Austria (Dated: November 14, 2018)
CONTENTS III. DUAL TENSORS
I. Introduction 1 This section introduces the notion of dual ten- sors, whose proper treatment requires some famil- II. Conventions 1 iarity with exterior algebra. Readers unfamiliar with this topic may skip section III A, since sec- III. Dual Tensors 1 tion III B suces for the calculations in the rest of A. The Hodge Dual 1 the paper, except parts of the appendix. B. The Complex Dual of Two-Forms 2
IV. The Free Maxwell-Field 2 A. The Hodge Dual V. General Duality Transformations 3 For an oriented vector space V of dimension VI. Coupled Maxwell-Fields 3 n with a metric tensor, the Hodge star opera- tor provides an isomorphism between the k-vectors VII. Applications 4 n − k-vectors. In this section, we discuss the gen- eral denition in Euclidean and Lorentzian vector VIII. Conclusion 5 spaces and give formulae for concrete calculations in (pseudo-)orthonormal frames. A. Notes on Exterior Algebra 6 The metric tensor g on V induces a metric tensor on ∧kV (denoted by the same symbol) by bilinear B. Collection of Proofs 7 extension of the map References 8 g(v1 ∧ · · · ∧ vk, w1 ∧ · · · ∧ wk) = det(g(vi, wj)). If the metric on V is non-degenerate, so is g on I. INTRODUCTION ∧kV . Let w be a k-vector. For every n − k-vector In abelian gauge theories whose action does not v the exterior product v ∧ w is proportional to depend on the gauge elds themselves, but only the unit n-vector Ω, since ∧nV is one-dimensional. on their eld strength tensors, duality transfor- We can thus write v ∧ w = ϕ(v, w)Ω, where ϕ mations are symmetry transformations mixing the is a bilinear function. As is shown in appendix eld tensors with certain dual tensors arising nat- A, there exists a unique element ∗w such that urally in symmetric formulations of the eld equa- g(v, ∗w) = s(g)ϕ(v, w), which is called the Hodge tions. dual of w. These notes are intended to clarify what is meant Denition. For a k-vector w, the dual ∗w is the by dual tensors and to shed some light upon the unique n − k-vector such that for all n − k-vectors group of such duality transformations. v it holds that v ∧ w = s(g) g(v, ∗w)Ω. (1) II. CONVENTIONS For a basis e it is convenient to introduce the This paper uses the Einstein summation conven- notation tion. The signature of the metric tensor in four eab... = ea ∧ eb ∧ ... dimensions is assumed to be (−, +, +, +). When g˜ = g(ea, ea)g(eb, eb) ... working with a general metric g, we denote by s(g) ab... the sign of its determinant1. For the permutation Since the Hodge dual may be equally constructed symbol , we us the sign convention . 01... = +1 on the dual space of an ordered vector space (where the volume form denes the preferred n-vector), we will make use of similar notation with indices raised. 1 Equivalently, one has s(g) = (−1)n, where n is the num- In [1] it is shown that the Hodge dual acts ber of negative signs in the signature of g. on an a right-handed orthonormal basis e in 2 the following way. For pairwise distinct indices Using the fact that ρσκλ κ λ λ κ , µνρσ = −2(δµδν − δµδν ) let be a completion such that i1, . . . , ik j1, . . . , jl it is readily veried that the operation F 7→ F˜ is (i1, . . . , ik, j1, . . . , jl) is a permutation of (1, . . . , n). its own inverse: Denote by sgn(π) its sign. Then i F˜ = − F˜ρσ (2) µν µνρσ ∗ei ...i = sgn(π)˜gi ...i ej ...j . 2 1 k 1 k 1 l 1 = − ρσκλF Example. In the three dimensional Euclidean 4 µνρσ κλ space we may choose a right-handed orthonor- E3 1 κ λ λ κ mal basis such that . We then = + (δµδν − δµδν )Fκλ = Fµν . e1, e2, e3 Ω = e123 2 have Every antisymmetric tensor F can be decom- posed into a self-dual and an self-anti-dual part ∗e12 = e3 1 1 and cyclic permutations thereof. The cross product F = F + + F − ≡ F + F˜ + F − F˜ . of two vectors a and b is given by 2 2 Note that self-dual two-forms exists only in a com- a × b = ∗(a ∧ b). plex setting. Furthermore, F + and F − are com- plex conjugate to each other, if F is real. Let e denote a right-handed orthonormal frame We give some useful algebraic properties of this in a four-dimensional Lorentzian vector space with complex duality transformation. For two antisym- volume form 0123. It is readily seen that the Ω = e metric tensors A and B we have the identity Hodge dual acts on the basis vectors as µρ ˜µρ ˜ 1 µ ρσ (7) 1 A Bρν + B Aρν = − δν A Bρσ , ∗eµ1...µk = µ1...µk eν1...νl . (3) 2 ν1...νl (n − k)! which shows that the contraction of two dual ten- Example. A two-form F in such a space has the sors can be expressed as contractions of the origi- decomposition nal tensors. A proof of this identity can be found in appendix B . Contracting the remaining pair of 1 indices, one nds F = F eµ ⊗ eν = F eµν . µν 2 µν ˜µν ˜ µν (8) A Bµν = A Bµν Using the fact that the Hodge star operator is lin- ear, we can compute the dual of F as From the previous equations it also follows that if both A and B are self-dual or anti-self dual 1 ρσ ρσ µν 1 2(∗F ) = Fρσ (∗e ) = Fρσ µν e , A (µBν)ρ = − ηµν A Bρσ. 2 ρ 4 ρσ We thus have Furthermore, if one of them is self-dual and the other one anti-self-dual it holds that µν 1 µνρσ (4) ∗F = Fρσ . µν 2 A Bµν = 0, [µ ν]ρ Aρ B = 0. B. The Complex Dual of Two-Forms
In general, the Hodge star operator is its own IV. THE FREE MAXWELL-FIELD inverse up to a sign, which depends (among other things) on the rank of the form acted upon. For Recall that the electromagnetic tensor F is de- two-forms, such as the electromagnetic eld tensor, rived from a covector potential A by means of it holds that F = dA, where d denotes the exterior derivative (in components ). Due to this Fµν = ∂µAν − ∂ν Aµ ∗ ∗ F = −F. fact, F satises a Bianchi identity
In order to have a self-inverse operation, the au- ∂ν Fρσ + ∂ρ Fσν + ∂σ Fνρ = 0. thors of [2] introduce a complex operator Contracting with µνρσ, one obtains the equivalent equation F˜ := −i(∗F ), (5) µν ∂ν F˜ = 0. in components (with respect to a right-handed or- thonormal frame) In the absence of sources, the eld equations take the form i F˜µν = − µνρσF . (6) µν 2 ρσ ∂ν F = 0, 3 for which we will write from now on div F = 0. Indeed, the variation of the action with respect to The pair of equations the potentials reads
div F = 0 Z (9) δS = d4x δL div F˜ = 0 i Z = d4x G˜ µν δF a are clearly invariant under the substitution 2 a µν Z ˜ 4 ˜ µν a F 7→ cos(α)F + i sin(α)F, = i d x (∂ν Ga )δA µ, which transforms the electric and magnetic eld as so the eld equations for the potentials are equiv- E cos α − sin α E alent to 7→ . B sin α cos α B ˜ (13) div Ga = 0. Recall that the eld energy density W and the Poynting vector S are given by We dene duality transformations as symmetry transformations of the eld equations, which are 1 W = E2 + B2 , linear transformations among the eld strengths 2 F and their associated tensors G (see [3]). S = E × B. Theorem 1. In four spacetime dimensions, a lin- Since both expressions are invariant, so is the ear transformation of the tensors F and G leaves energy-momentum tensor. the eld equations (including those of χ) invariant Remark. One might consider adding terms of the if, and only if, the transformation is symplectic. In form ˜ ˜µν or ˜µν to the action of electrody- other words, the group of duality transformations Fµν F Fµν F namics. But due to equation (8), the rst option is Sp(2n, R). does not add anything new, and due to the fact A proof of this theorem can be found in [3]. Fol- that div F˜ = 0 we have lowing [2], we do not give the general proof, but ˜µν ˜µν restrict our attention to a certain form of the La- Fµν F = (∂µAν − ∂ν Aµ)F grangian, which is common in supersymmetry and ˜µν = 2(∂µAν )F supergravity. µν = 2∂µ(Aν F˜ ), so second option simply adds a divergence, which VI. COUPLED MAXWELL-FIELDS does not contribute to the eld equations. a. The Lagrangian of Coupled Maxwell Fields V. GENERAL DUALITY We consider a collection of n covector potentials TRANSFORMATIONS A1, ..., An and a complex, symmetric n×n matrix- valued function with components (which may f fab The symmetry between the eld equations and depend on other elds), providing a coupling be- the Bianchi identities is a general feature of the- tween the various covector potentials by virtue of ories whose action depends on covector potentials the Lagrangian only via their eld strength tensors. 1 Consider covector potentials 1 n and a bµν n A , ..., A L = − (Re fab )F µν F a collection of additional elds χ1, .., χm together 4 (14) i with an action of the form + (Im f )F a F˜bµν , 4 ab µν ˜ (10) S [A, ∂A, χ, ∂χ] = S [F, χ, ∂χ], where a a a . We have seen that F µν = ∂µ A ν − ∂ν A µ where a a a . By construction, contractions of the form ˜ are divergences, but F µν = ∂µ A ν − ∂ν A µ F F the eld tensors F a satisfy the Bianchi identity since they are multiplied by a eld, the second term does not form a divergence as a whole (provided f div F˜a = 0. (11) is not constant). We will now show the following theorem. Furthermore, since S does not depend on the po- tentials explicitly, we have divergence free tensors Theorem 2. In the considered model, every linear ˜ with components Ga transformation of the elds F and G, which pre- serves their mutual relations and leaves the energy- δS˜ momentum tensor invariant, is a symplectic trans- G˜ µν = −2i . (12) a a formation, and vice versa. δF µν 4
b. The Field Equations The tensor G, as de- e. The Energy-Momentum Tensor In ap- ned in (12), evaluates to pendix B we show that the true energy- momentum tensor2 is given by b ˜b Ga = −(Im fab )F − i(Re fab )F . 1 The Bianchi idenities and the eld equations take T µ = (Re f ) F aµρF b − δµF aρσF b ν ab νρ 4 ν ρσ the symmetric form (21) There, it is also shown that the energy-momentum div F˜a = 0, (15) tensor transforms under (17) as ˜ div Ga = 0. T 7→ λT. (22) The relations between F and G are particularly simple, when expressed in terms of their self-dual Thus, the invariance of T under duality transfor- (or anti-dual) components mations is equivalent to λ = 1. f. Summary We conclude that duality trans- G − = +if F b− formations of this model are real transformations a ab (16) + ¯ b+ of the form Ga = −ifab F F AB F where the bar denotes complex conjugation. 7→ (23) c. Duality Transformations The system of G CD G equations (15) is clearly invariant under the sub- subject to the constraints stitution T T F a 7→ Aa F b + BabG A C − C A = 0 b b (17) b b BTD − DTB = 0 Ga 7→ Cab F + Da Gb ATD − CTB = , where the coecient matrices A, B, C and D are I real. Naturally, we assume this transformation to which can be written in the concise form be invertible. In matrix form, this substitution reads STΩS = Ω, (24) F AB F 7→ (18) where S is the transformation matrix and Ω is the G CD G symplectic metric which can be interpreted as the natural action of AB 0 +I GL(2n, ) on the vector (F,G)T. S = and Ω = . R CD −I 0 d. Transformation of the function f Since F and G are not independent, but related by equa- This completes our proof of theorem 2. tions (16), we require that for the transformed elds F 0,G0 there exist a transformed eld f 0 such that VII. APPLICATIONS
0 − 0 0b− G a = +if abF . a. Pauli-Coupling Given that gauge elds are This is clearly equivalent to the matrix equation commonly introduced by gauging global symme- tries, where minimal coupling arises naturally, one (C + iDf)F − = if 0(A + iBf)F −. might wonder what interaction with the electro- magnetic eld depends on the eld strength tensor Requiring this equation to hold for all possible val- F only. The simplest conceivable interaction term ues of −, we obtain F of a Dirac spinor ψ and F is given by the Pauli Coupling if 0 = (C + iDf)(A + iBf)−1. (19) ¯ µν Of course, we must assume A+iBf to be invertible. Fµν ψσ ψ, In appendix B, it is shown that such transfor- which leads to non-renormalizable theories [4, mations preserves the symmetry of arbitrary if, f p. 517f]. The Pauli coupling does not appear in and only if, the matrices and satisfy the A, B, C D the Standard Model of Particle Physics, but does system of equations have some applications in eective theories used to describe electromagnetic dipole moments. ATC − CTA = 0 BTD − DTB = 0 (20) ATD − BTD = λ I 2 i.e. the symmetric tensor acting as a source in the Einstein eld equations for some λ ∈ R. 5
b. Transformation of model parameters The VIII. CONCLUSION analysis of the coupled Maxwell elds still holds if the coupling coecients f are not determined by Considering abelian gauge elds A which couple elds, but constant parameters of the model. In at most via their eld strength tensors F , we saw this case, duality transformations allow to trans- that the Bianchi identities and the eld equations form these parameters according to (19). For ex- assumed the same form: ample, considering a single gauge eld and assum- ing to be real3, the duality transformation with ˜a ˜ f div F = 0, div Ga = 0, A = D = 0 and B = −C = 1 transforms f to 1/f. In [2, p. 82,85], this is interpreted as a transforma- which led us to introduce duality transformations, tion relating the regimes of weak and strong cou- mixing F and G. pling. Note, however, that the considered model The particular example of n such elds coupling is free of interactions and thus cannot be used as to scalar elds via the Lagrangian (14) showed, evidence for this statement. how physical restrictions to transformations mix- c. Dyons Electric and Magnetic Charges In ing F and G lead to the symplectic group. The- the presence of sources, described by an electric orem 1 guarantees that this is the maximal group current density j, Maxwell' equations read of such symmetry transformations. In [3], the analysis was performed at the level of div F˜ = 0, the Lagrangian, not a the level of the eld equa- tions, which has the advantage of admitting a sys- div G˜ = ij, tematic analysis for Noether currents. Indeed, the where G = iF˜. Applying the duality transforma- charge associated to duality transformations was tions as dened in the free case to this set found to be a gauge-invariant quantity. of equations, one arrives at a generalised form of The above considerations could be extended to Maxwell's equations with a magnetic current den- non-abelian gauge theories, describing for example sity k: gluons in the absence of quarks. The relevant La- grangian is exactly of the form (14) but the eld div F˜ = ik, strength tensor with components (25) div G˜ = ij. a a a a b c F µν = ∂µ A ν − ∂ν A µ + gfbc A µA ν Given a spacelike hypersurface Σ with volume form and future-pointing unit conormal , the total (f denotes the structure constants of SU(3)) is not dV n of the considered form, requiring thus further anal- electric charge q and total magnetic charge p can be dened as ysis. Z Z µ ˜νµ q = dV nµj = −i dV nµ∇ν G , Σ Σ Z Z µ ˜νµ p = dV nµk = −i dV nµ∇ν F , Σ Σ which implies that electric and magnetic charges are transformed by a duality transformation S ac- cording to
p0 p = S . q0 q In this sense, duality transformations allow the transition between dierent theories: in this case, they relate ordinary electrodynamics to a theory with magnetic charges. However, this relation re- quires the abuse of language: the duality transfor- mation is necessarily dened for the free theory, but applied to the interacting case4.
3 We can make this assumption without loss of generality, since the term ˜µν is a divergence, so the imaginary Fµν F part of f does not contribute to the eld equations. 4 This procedure is analogous to gauging of symmetries, where global symmetries are extended to local ones, leading to new theories. 6
Appendix A: Notes on Exterior Algebra Proof. For brevity, we use the notation a = a1 ⊗ · · · ⊗ ak, and likewise for b. We then have Denition (Action of permutations on tensors). a1 ∧ · · · ∧ ak = k! Alt a and thus
Let T be a k-tensor and π ∈ Sk a permutation. We dene the left-action of π on T as g(a1 ∧ · · · ∧ ak, b1 ∧ · · · ∧ bk) = (k!)2g(Alt a, Alt b) (σ T )(v , . . . , v ) = T (v , . . . , v ). B 1 k σ(1) σ(k) X = sgn(σ) sgn(π)g(σ a, π b) It is straight-forward to verify that this indeed B B σ,π∈S denes a left-action, meaning that k k 1 X Y = sgn(σ) sgn(π) g(a , b ) σ B (π B T ) = (σ ◦ π) B T. k! σ(j) π(j) σ,π∈Sk j=1 Denition. Let V be an n-dimensional vector 1 X = sgn(σ) det(a , b ) = det(g(a , b )), space, and T a contravariant tensor of rank k. We k! σ(i) j i j say that T is a k-vector, if σ∈Sk
σ T = (sgn σ)T where we have used the Leibniz expansion of the B determinant and the fact that permuting the rows for all σ ∈ Sk. The k-vectors form the vector space of a matrix changes its determinant by the sign of ∧kV , called the k-th exterior power of V . the permutation. Denition (Alternating Operator). For a k- For sets of increasing indices , tensor T , we dene its alternation as I = (i1, . . . , ik) we use the multi-index notation , etc. ei1,...,ik = eI 1 X Multi-indices of order with increasing sub-indices Alt T = sgn(π)(π T ). k k! B will be referred to as k-multi-indices. π∈Sk If T is a k-vector, it follows that Alt T = T . Proposition. If g is a non-degenerate metric ten- sor on V , then the extension of g to ∧kV is non- Denition (Exterior Product). For k- and l- degenerate as well. vectors, the exterior product (or wedge product) is dened as Proof. Let e be a (pseudo-)orthonormal basis of V , then the various (with a -multi-index) forms ∧ : ∧kV × ∧lV → ∧k+lV eI I k a basis of ∧kV . Then (k + l)! (A, B) 7→ A ∧ B = Alt(A ⊗ B) k!l! g(ei1 , ej1 ) . . . g(ei1 , ejk ) ...... Proposition. One can show that the exterior g(eI , eJ ) = . . . . product is associative. Furthermore, for a k-vector g(eik , ej1 ) . . . g(eik , ejk ) A and an l-vector B, it holds that If I and J dier, the matrix on the right hand side A ∧ B = (−1)klB ∧ A. contains a row full of zeros, so gIJ = 0. Otherwise, the matrix is diagonal and we have Denition (Extension of g to the tensor algebra). gII =g ˜I := g . . . g . We can thus write Let g be a metric tensor on V . We dene g on i1,i1 ik,ik tensors of arbitrary rank as follows: for tensors , A ( B of dierent rank, we set g(A, B) = 0, while for g˜ if I = J, g(e , e ) = I tensors of equal rank k we dene g by the bilinear I J 0 if I 6= J. extension of
k The matrix representing g is thus diagonal, and 1 Y g(a ⊗ · · · ⊗ a , b ⊗ · · · ⊗ b ) = g(a , b ). since g is non-degenerate on V the terms g˜I 1 k 1 k k! j j j=1 never vanish. Consequently, the matrix g(eI , eJ ) is invertible, which is equivalent to g being non- Remark. In a basis e, the inner product of two k- degenerate. tensors A, B evaluates to Proposition. Let be an -dimensional oriented 1 a ...a V n g(A, B) = A B 1 k , vector space. Denote by its preferred unit - k! a1...ak Ω n vector. For a k-vector w and an n − k-vector v, where indices are raised and lowered with g. the bilinear map Proposition. For simple k-vectors, the inner product takes the form v ∧ w = ϕ(v, w)Ω
g(a1 ∧ · · · ∧ ak, b1 ∧ · · · ∧ bk) = det(g(ai, bj)). is non-degenerate. 7
Proof. Let e be a (pseudo-)orthonormal basis. We show that the requirement of the trans- 0 Then e1...n = ±Ω. Every k-vector w can be de- formed matrix f , as dened in equation (19) being composed as symmetric for all possible symmetric f implies the three conditions (20). K w = w eK , Proof. By the denition of f 0 and the symmetry of where the sum extends over all k-multi-indices. For f, we have xed I = (i1, . . . ik), let J = (j1, . . . jn−k) be the multi-index such that I and J contain all indices if 0 = (C + iDf)(A + iBf)−1 from 1 to n. Then e ∧e = ±δI Ω. Consequently J K K if 0T = (AT + ifBT)−1(CT + ifDT) e ∧ Ω = e ∧ e wK = ±wI Ω. J J K Rewriting if 0 = if 0T as So if for all , then I for all ϕ(v, w) = 0 v w = 0 T T −1 −1 multi-indices I, implying that w = 0. (A + ifB ) M(A + iBf) = 0, Proposition. The Hodge Dual ∗ : ∧kV → ∧n−kV where is a well-dened bijection. M = (AT+ifBT)(C+iDf)−(CT+ifDT)(A+iBf), Proof. For a k-vector w, ϕ(·, w) is a linear form on we nd the equivalent condition . Expand- ∧n−kV . Since g denes a non-degenerate metric M = 0 ing the matrix product, one obtains on ∧n−kV , there exists a unique n − k-vector ∗w such that ATC − CTA g(·, ∗w) = ϕ(·, w). + (ATD − CTB)f + f(BTC − DTA) The mapping w 7→ ∗w is thus well-dened. + f(BTD − DTB)f = 0. The Hodge star operator is injective. Indeed, let Since is arbitrary, every line in this equation w and w0 be two k-vectors such that ∗w = ∗w0. It f follows that must vanish individually. The rst line yields the condition ϕ(v, w − w0) = 0 ATC − CTA = 0. for all n − k-vectors v. Since ϕ is non-degenerate, Requiring the second line to vanish for , one it follows that w = w0. Finally, since f = I nds that DTA−CTB is symmetric. We may thus n n equivalently write dim ∧kV = = = dim ∧n−kV, k n − k [ATD − CTB, f] = 0, it follows that ∗ is surjective as well. where [·, ·] is the matrix commutator. Since every symmetric matrix which commutes with all sym- Appendix B: Collection of Proofs metric matrices is a multiple of the identity, there exists a λ ∈ R, such that Proof of equation (7). Using the well-known iden- T T (B1) tity A D − C B = λI. Finally, requiring the last line to vanish for all sym- µ µ µ δσ δκ δλ metric clearly implies αµνρ ν ν ν f ασκλ = − δσ δκ δλ ρ ρ ρ T T δσ δκ δλ B D − D B = 0. and the notation µν... µ ν , we have δαβ... = δαδβ ... 1 We show that the energy-momentum tensor as- B˜µαA˜ = − µαρσ B Aκλ αν 4 ανκλ ρσ sociated to the Lagrangian density (14), dened via the Einstein equations, is given by (21). 1 αµρσ κλ = + ανκλBρσ A 4 Proof. Recall that the Einstein-Equations are de- 1 = − (δµρσ + δµρσ + δµρσ)B Aκλ rived from the Einstein-Hilbert action 4 νκλ κλν νκλ ρσ 1 1 Z + (δµρσ + δµρσ + δµρσ)B Aκλ S [g] = R ω, (B2) 4 νλκ λκν κνλ ρσ 2κ 1 µ αβ µα where is the Ricci scalar curvature, is the vol- = − δν A Bαβ − A Bαν . R ω 2 ume form induced by g and κ is Einstein's gravi- tational constant. To obtain a generally covariant 8
Lagrangian from (14), we write the associated ac- We show that under transformations of the form tion as (17), the energy momentum tensor (21) transforms as Z Z [A, ∂A] = ≡ L ω, S L T 7→ λT where Proof. Using equation (7), one obtains the equiva- lent expression 1 L = − (Re f )g(F a,F b)ω µ a+µρ b− a−µρ b+ 2 ab T ν = (Re fab ) F F νρ + F F νρ 1 (B5) + (Im f )g(F a, ∗F b)ω writing for the Lorentz tensor with indices 2 ab A · B µ µρ (the Lorentz indices are con- (A · B) ν = A Bρν 1 a b 1 a b = − (Re fab )g(F ,F )ω − (Im fab )F ∧ F . tracted as in ordinary matrix multiplication), we 2 2 can write In the last step, we have used the dening prop- T = −(Re f ) F a+ · F b− + F a− · F b+ erty of the Hodge dual. The action of the coupled ab (B6) a+ b− c.c. system is given by = −(Re fab )F · F +
Z Z Using the relations between G± and F ±, it is 1 1 a b S = R + Λ ω − (Im fab )F ∧ F , easy to see that under duality transformations we 2κ 2 have where a+ b− a+ b− (B7) (Re fab )F · F 7→ ξab F · F 1 Λ = − (Re f )g(F a,F b). where 2 ab ξ = (A − iBf¯)T(Re f 0)(A + iBf) We see that the last term in the action does not depend on g, so we may equally derive the eld Using the symmetry of f 0 and using the notation equations for for the gravitational eld from the A−T = (A−1)T = (AT)−1, we nd action 2 Re f 0 = f¯0T + f 0 Z 1 S˜ = R + Λ ω. ¯ −T ¯ T 2κ = +i(A − iBf) (C − iDf) − i(C + iDf)(A + iBf)−1 This is the standard form of the Einstein-Hilbert action with sources, leading to Consequently, we have
¯ T G = κ T, (B3) 2ξ = +i(C − iDf) (A + iBf) − i(A − iBf¯)T(C + iDf) where G is the Einstein tensor and T is the energy- T T ¯ T T momentum tensor, dened as = i(C A − A C) + if(D B − B D) + f¯(DTA − BTC) + (ATD − CTB)f −2 ∂ ¯ T = p−|g| Λ = λ(f + f) = 2λ Re f. µν p−|g| ∂gµν (B4) Thus, the energy momentum tensor transforms as ∂Λ = −2 + Λg , ∂gµν µν T 7→ λT (B8) where is an abbreviation for . Insert- |g| det(gµν ) ing Λ from above, one arrives at equation (21), as claimed.
[1] H.-J. Dirschmid, Tensoren und Felder. Springer Vi- [3] M. K. Gaillard and B. Zumino, Duality rotations enna, 1995. for interacting elds, Nuclear Physics B, vol. 193, [2] A. V. P. Daniel Z. Freedman, Supergravity. Cam- pp. 221244, dec 1981. bridge University Pr., 2012. [4] S. Weinberg, The Quantum Theory of Fields, vol. 1. Cambridge University Press, 1995.