A Non-Abelian Self-Dual Gauge Theory in 5+1 Dimensions
aa r X i v : . [ h e p - t h ] M a y UT-11-11
A Non-Abelian Self-Dual Gauge Theoryin 5+1 Dimensions
Pei-Ming Ho † , Kuo-Wei Huang † , Yutaka Matsuo ‡ † Department of Physics and Center for Theoretical Sciences,National Taiwan University, Taipei 10617, Taiwan, R.O.C. ‡ Department of Physics, Faculty of Science, University of Tokyo,Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
Abstract
We construct a non-Abelian gauge theory of chiral 2-forms (self-dual gauge fields)in 6 dimensions with a spatial direction compactified on a circle of radius R . It hasthe following two properties. (1) It reduces to the Yang-Mills theory in 5 dimensionsfor small R . (2) It is equivalent to the Lorentz-invariant theory of Abelian chiral2-forms when the gauge group is Abelian. Previous no-go theorems prohibiting non-Abelian deformations of the chiral 2-form gauge theory are circumvented by introducingnonlocality along the compactified dimension. e-mail address: [email protected] e-mail address: [email protected] e-mail address: [email protected] Introduction
The generalization of Abelian gauge theories with 1-form potentials to higher form potentialsis straightforward. For a p -form potential A ( p ) , we define its gauge transformation by δA ( p ) = d Λ ( p − , (1)where the gauge transformation parameter Λ ( p − is a ( p − -form, and the field strengthdefined by F ( p +1) = dA ( p ) (2)is an invariant ( p +1) -form. However, the generalization of higher form potentials A ( p ) ( p > to non-Abelian gauge theories have been a tough challenge to both theoretical physicists andmathematicians.In this paper we attack this problem for the case p = 2 , with the goal of describing thesystem of multiple M5-branes. The multiple M5-brane system has been the most challengingand mysterious brane system in string theory and M theory [1]. (For a review of M theorybranes, see [2].) The salient nature of the M5-brane theory is that it contains a self-dual gaugefield (also called a chiral 2-form potential). Some believed that a Lagrangian formulationfor the self-dual gauge theory was impossible, because the self-duality condition imposesfirst order differential equations on the gauge potentials, while an ordinary kinetic term ( ∂ µ B νλ )( ∂ µ B νλ ) always leads to a 2nd order differential equation. It turns out that the trickis to avoid using some of the components of the gauge potential, so that even though we get2nd order differential equations from varying the action, the self-duality condition appearsonly after integrating once the equations of motion. Those components which do not appearin the action appear as integration “contants”. Hence the Lagrangian for a single M5-branewas first constructed without manifest Lorentz symmetry, and a Lorentz-covariant versionis possible only by introducing an auxiliary field [4, 5].The gauge symmetry for a single M5-brane in the trivial background is Abelian. Thefirst non-Abelian gauge theory for self-dual 2-form potentials was found for an M5-branein a large C -field background [6]. A double dimension reduction of this M5-brane theory,called NP M5-brane theory, is in agreement to the lowest order with the noncommutativeD4-brane action in large NS-NS B field background [6]. If the NP M5-brane theory canbe deformed such that it agrees with the noncommutative D4-brane theory to all orders, itwould resemble the multiple M5-brane theory. However, it turns out that it is extremely This trick was later generalized in [3] so that for a given spacetime dimension D , one can write down aLagrangian for the self-dual gauge field for an arbitrary division of D into two positive integers D ′ and D ′′ ( D ′ + D ′′ = D ). We refer to it as the ( D ′ + D ′′ ) -formulation of the self-dual gauge theory. “NP” stands for “Nambu-Poisson”. A Nambu-Poisson structure is used to define the non-Abelian gaugesymmetry for the 2-form potential on the M5-brane. The physical origin of the Nambu-Structure is thecoupling of open membranes to the C -field background [7] The NP M5-brane theory was first derived fromthe BLG model [8]. Its gauge field content was further explored in [9, 10]. For a brief review, see [11]. B taking values in a Lie algebra, and the corresponding geometricalstructures are called “non-abelian gerbes". The immediate problem to construct such a modelis that we need to define covariant derivatives D µ , which need to be specified by a 1-formpotential A . For example, in [13], the gauge transformations of A and B are defined by A ′ = gAg − + gdg − + Λ , (3) B ′ = gBg − + [ A ′ , Λ] ∧ + d Λ + Λ ∧ Λ , (4)where g ∈ G is the gauge parameter and Λ ∈ g is a 1-form. Mathematically such gaugetransformations are well-defined, and suitable to describe some system such as the non-Abelian generalization of the BF model [14]. It is, however, not clear if it is relevant todescribe multiple M5. Physically, the introduction of A increases the physical degrees offreedom of the system. For the M5-brane system, there is no physical degree of freedomcorresponding to A . Furthermore, with the addition of A , the field B is not a genuine 2-formpotential in the sense that we can gauge away A by Λ , and then B is not independent of itslongitudinal components. The result is similar to spontaneous symmetry breaking.An independent attempt to construct non-Abelian 2-form gauge theory is to define it onthe loop space [15]. This approach introduces infinitely many more degrees of freedom to theusual Abelian chiral gauge theory even in the Abelian limit. Instead, our goal is to have anon-Abelian gauge symmetry which includes the Abelian theory as the special case when theLie algebra involved is Abelian. This criterium is not matched by any existing constructionin the literature.While a consistent algebra of non-Abelian gauge transformations for a higher form gaugetheory is already difficult to get, an action for a chiral gauge boson is even more difficult.Assuming the existence of an action and gauge transformation algebra, a no-go theorem [16]states that there is no nontrivial deformation of the Abelian 2-form gauge theory. One of theirassumptions was locality for the action and the gauge transformation laws. In particular,Lorentz symmetry was not assumed.The non-existence of the local action for multiple M5-branes was argued in another wayby Witten [17]. The M5-brane system is known to have conformal symmetry, which impliesthat upon double dimension reduction, the 4+1 dimensional action should be proportionalto Z d x R . (5)On the other hand, the reduction of a 5+1 dimensional local action on a circle should give Z d x = Z d x πR, (6)which has the opposite dependence on R . As long as we assume a Lorentz-covariant formula-tion for M5-branes without explicit reference to the compactifiation radius R except through2he measure of integration, this gives a strong argument against the Lagrangian formulationof multiple M5.Recently there are proposals [18, 19] claiming that the multiple M5-brane system com-pactified on a circle of finite radius R is described by the U ( N ) super Yang-Mills theory for N D4-branes even before taking the small R limit. This would be a duality between twotheories in 5 and 6 dimensions, respectively, but it can not be viewed as an example of theholographic principle of quantum gravity, because there is no gravitational force in thesetheories. Their proposal, if correct, would be revolutionary. However we will point out itsdifficulties in Sec. 5.2.These developments suggest that it is already a tremendous progress to have a theory formultiple M5-branes compactified on a circle of finite radius R , if the following two criteriaare satisfied:1. In the limit R → , the theory should be approximated by the gauge field sector of themultiple D4-brane theory, which is U ( N ) Yang-Mills theory in 5 dimensions.2. When the Lie algebra of the gauge symmetry is Abelian, the theory reduces to N copiesof the theory of 6 dimensional Abelian self-dual gauge field.In view of the no-go theorem [16], the absence of 6 dimensional Lorentz symmetry due tocompactification of the 5-th direction does not necessarily make the task much easier. Onthe other hand, the 2nd criterium ensures the 6 dimensional Lorentz symmetry in the brokenphase in the limit R → ∞ . In the following we will construct an interacting theory satisfyingboth criteria. The cost we have to pay to meet these criteria is a nonlocal treatment of thecompactified dimension, as we will see in the following sections. Such a description mayseem exotic, but it might be justified in view of the special role played by the compactifieddirection in defining M-theory as the strong coupling limit of type II A string theory.We organize this paper as follows. In section 2, we define the non-Abelian gauge trans-formation for an anti-symmetric two-form. A characteristic feature is to introduce separatetreatments for zero-mode and non-zero mode (KK mode) in the compactified direction. Insection 3, after a brief review of Lagrangian formulation of self-dual two-form, we explainhow to modify the equation of motion to include the non-Abelian gauge symmetry. In section4, we proposes an action which produces the equation of motion. We also describe how D4brane action can be derived in the small radius limit. Finally in section 5, we discuss somerelated issues such as the recovery of Lorentz symmetry in large R limit and comparisonwith the D4-brane approach. 3 Gauge Symmetry
As explained above, here we consider the case where the world volume of M5 is R × S where S is a circle of radius R with a coordinate x ∼ x + 2 πR . We will use the notation thatthe superscript “ (0) ” represents zero modes and “(KK)” represents Kaluza-Klein (non-zero)modes. For an arbitrary field Φ , we have the decomposition Φ = Φ (0) + Φ ( KK ) . (7)Obviously, ∂ Φ (0) = 0 , ∂ Φ = ∂ Φ ( KK ) . (8)We can define a non-local operator ∂ − on the space of KK modes, so that, for instance, ∂ − Φ ( KK ) is well defined.After a lot of trial and error, we find a consistent non-Abelian generalization of the gaugetransformation for a 2-form gauge potential B µν as δB i = [ D i , Λ ] − ∂ Λ i + g [ B ( KK ) i , Λ (0)5 ] , (9) δB ij = [ D i , Λ j ] − [ D j , Λ i ] + g [ B ij , Λ (0)5 ] − g [ F ij , ∂ − Λ ( KK )5 ] , (10)where i, j = 0 , , , , . The covariant derivative is defined in terms of the zero-mode: D i ≡ ∂ i + gB (0) i . (11)The parameter g is the coupling constant for the 2-form gauge interaction. The field strengthis F ij ≡ g − [ D i , D j ] = ∂ i B (0) j − ∂ j B (0) i + g [ B (0) i , B (0) j ] . (12)Eqs. (9) and (10) can be decomposed into their zero modes and KK modes as δB (0) i = [ D i , Λ (0)5 ] , (13) δB ( KK ) i = [ D i , Λ ( KK )5 ] − ∂ Λ ( KK ) i + g [ B ( KK ) i , Λ (0)5 ] , (14) δB (0) ij = [ D i , Λ (0) j ] − [ D j , Λ (0) i ] + g [ B (0) ij , Λ (0)5 ] , (15) δB ( KK ) ij = [ D i , Λ ( KK ) j ] − [ D j , Λ ( KK ) i ] + g [ B ( KK ) ij , Λ (0)5 ] − g [ F ij , ∂ − Λ ( KK )5 ] . (16)All quantities B i , B ij , Λ i , Λ take values in a Lie algebra h . The last term in (10) (or (16)) isthe only explicit nonlocality in these expressions. But in fact there is additional nonlocalityintroduced by how the gauge transformations are defined separately for the zero modes andKK modes.If h is abelian, the transformations (9) and (10) are equivalent to the conventional gaugetransformation of two-form gauge potential. For our non-abelian generalization, the gaugetransformation laws explicitly distinguish the zero modes from the KK modes and we treat4hem as if they are independent fields. All the commutators involve at most one KK mode.There is no term of the form [ B ( KK ) , Λ ( KK ) ] in the transformation laws. Some of the physicalmeanings of these peculiar features will be explained in Sec. 5.Because of these choices, the gauge transformations (13, 15) are closed by zero-modefields/gauge parameters. While they resemble the gauge transformation of non-abelian gerbe(3, 4) if one replaces B (0) i by A i , these are different since the Λ term in (3) and the nonlinearterm in (4) are absent in (13). In a sense, the transformation by the vector gauge parameter Λ (0) i is abelian (no Λ term) and the noncommutativity comes in through the transformationby Λ (0)5 . Our choice is more useful to realize the self-dual field after the transformations ofKK modes (14, 16) are included.There are 6 gauge transformation parameters Λ i , Λ , but only 5 of the KK modes are inde-pendent because the gauge transformation parameters are defined up to the transformation δ Λ ( KK ) i = [ D i , λ ( KK ) ] , δ Λ ( KK )5 = ∂ λ ( KK ) . (17)This “gauge symmetry of gauge symmetry” is crucial for the gauge symmetry to be justifiedas a deformation of the Abelain gauge symmetry. We can use this redundancy to “gaugeaway” Λ ( KK )5 . That is, the gauge transformation rules (9, 10) are equivalent to δB i = − ∂ Λ ′ i + g [ B i , Λ (0)5 ] , (18) δB ij = [ D i , Λ ′ j ] − [ D j , Λ ′ i ] + g [ B ij , Λ (0)5 ] , (19)where Λ ′ i ≡ Λ i − [ D i , ∂ − Λ ( KK )5 ] . (20)However, the zero mode Λ (0)5 can not be gauged away. Note that the only nonlocal termin the gauge transformation (the last term in (10)) is gauged away through this change ofvariables.The 3-form field strengths are defined as H (0) ij ≡ F ij ≡ g − [ D i , D j ] , (21) H ( KK ) ij ≡ [ D i , B ( KK ) j ] − [ D j , B ( KK ) i ] + ∂ B ij , (22) H (0) ijk ≡ [ D i , B (0) jk ] + [ D j , B (0) ki ] + [ D k , B (0) ij ] , (23) H ( KK ) ijk ≡ [ D i , B ( KK ) jk ] + [ D j , B ( KK ) ki ] + [ D k , B ( KK ) ij ]+ g [ F ij , ∂ − B ( KK ) k ] + g [ F jk , ∂ − B ( KK ) i ] + g [ F ki , ∂ − B ( KK ) j ] . (24)5hey satisfy the generalized Jacobi identities X (3) [ D i , H (0) jk ] = 0 , (25) X (3) [ D i , H ( KK ) jk ] = ∂ H ( KK ) ijk , (26) X (4) [ D i , H (0) jkl ] = 0 , (27) X (4) [ D i , H ( KK ) jkl ] = g X (6) [ H (0) ij , ∂ − H ( KK ) kl ] , (28)where P ( n ) represents a sum over n terms that totally antisymmetrizes all the indices. Thefield strength transforms as δH (0) ij = g [ H (0) ij , Λ (0)5 ] , (29) δH ( KK ) ij = g [ H ( KK ) ij , Λ (0)5 ] , (30) δH (0) ijk = g [ H (0) ijk , Λ (0)5 ] + g [ H (0) ij , Λ (0) k ] + g [ H (0) jk , Λ (0) i ] + g [ H (0) ki , Λ (0) j ] , (31) δH ( KK ) ijk = g [ H ( KK ) ijk , Λ (0)5 ] . (32)The components B ( KK ) i can be gauged away using the gauge transformations parametrizedby Λ ( KK ) i . In this gauge, B ( KK ) i = 0 , we have H ( KK ) ij = ∂ B ( KK ) ij . This motivates us to define ˆ B ( KK ) ij ≡ ∂ − H ( KK ) ij , (33)which transforms covariantly as δ ˆ B ( KK ) ij = g [ ˆ B ( KK ) ij , Λ (0)5 ] , (34)and then (22) and (24) are equivalent to H ( KK ) ij ≡ ∂ ˆ B ( KK ) ij , (35) H ( KK ) ijk ≡ [ D i , ˆ B ( KK ) jk ] + [ D j , ˆ B ( KK ) ki ] + [ D k , ˆ B ( KK ) ij ] . (36)The algebra of gauge transformations is closed and given by [ δ, δ ′ ] = δ ′′ , (37)with Λ ′′ = g [Λ (0)5 , Λ ′ ] , (38) Λ ′′ KK ) = g [Λ (0)5 , Λ ′ KK ) ] − g [Λ ′ , Λ ( KK )5 ] , (39) Λ ′′ i = g [Λ (0)5 , Λ ′ i ] − g [Λ ′ , Λ i ] . (40)6 .2 Coupling to Antisymmetric Tensors Apart from the application to multiple M5-branes, let us also consider applications to non-Abelian 2-form gauge theories which are not self dual. A potential problem is that thetransformation of H (0) ijk (31) is different from the usual covariant form like other componentsof H . If we couple other tensor fields to the gauge field, they will have to transform in asimilar way. A straightforward generalization of the transformation laws for H leads to thedefinition of gauge transformations of a totally antisymmetrized tensor field φ µ ··· µ n ( n ≤ ),which can be decomposed into a multiplet ( φ (0) i ··· i n − , φ ( KK ) i ··· i n − , φ (0) i ··· i n , φ ( KK ) i ··· i n ) , to be δφ (0) i ··· i n − = g [ φ (0) i ··· i n − , Λ (0)5 ] , (41) δφ ( KK ) i ··· i n − = g [ φ ( KK ) i ··· i n − , Λ (0)5 ] , (42) δφ (0) i ··· i n = g [ φ (0) i ··· i n , Λ (0)5 ] + g X ( n ) [ φ (0) i ··· i n − , Λ (0) i n ] , (43) δφ ( KK ) i ··· i n = g [ φ ( KK ) i ··· i n , Λ (0)5 ] , (44)where P ( n ) represents a sum of n terms that totally antisymmetrizes all indices.The transformation law (43) for the component φ (0) i ··· i n is different from all other compo-nents. It is defined to mimic the gauge transformation of H (0) ijk . We should check whether thiscomplication will prevent us from constructing a gauge field theory. First, products of thesefields φ (0) i ··· i n will also transform in the form of (43) when all indices are antisymmetrized onthe products. Secondly, the action of D i on φ (0) i ··· i n does not transform covariantly, but wecan define a covariant exterior derivative for φ (0) i ··· i n as ( D φ ) (0) i ··· i n +1 ≡ X ( n +1) [ D i , φ (0) i ··· i n +1 ] − ( − n X (( n +1) n/ [ B (0) i i , φ (0) i ··· i n +1 ] . (45)(This expression is nontrivial only if n ≤ .) This covariant exterior derivative is indeedcovariant, that is, δ ( D φ ) (0) i ··· i n +1 = g [( D φ ) (0) i ··· i n +1 , Λ (0)5 ] + g X ( n +1) [( D φ ) (0) i ··· i n , Λ (0) i n +1 ] , (46)where the exterior derivative of φ i ··· i n − is defined by ( D φ ) (0) i ··· i n = g X ( n ) [ D i , φ (0) i ··· i n ] . (47)It seems possible to down covariant equations of motion using exterior derivatives and totallyantisymmetrized tensors.The real problem with the transformation law (43) lies in the definition of an invariantaction. For example, to define a Yang-Mills like theory, the Lagrangian should look like Tr ( H (0) ijk H (0) ijk + 3 H (0) ij H (0) ij + H ( KK ) ijk H ( KK ) ijk + 3 H ( KK ) ij H ( KK ) ij ) . (48)7nly the first term is not gauge invariant. It is not clear how to modify the action to makeit invariant. Similarly it is hard to define the usual kinetic term for the components φ (0) i ··· i n of a matter field.In the following we will see that in a Lagrangian formulation of the non-Abelian self-dualgauge theory in 6 dimensions, we do not have to use the variables B (0) ij explicitly, so theanomalous covariant transformation law of H (0) ijk (31) will never be used. In fact we cansimply define H (0) ijk to be the Hodge dual of F ij , so that its gauge transformation is the sameas other components. As a result the covariant transformation laws for matter fields can beuniformly defined as δ Φ = g [Φ , Λ (0)5 ] (49)for all components of a matter field. The linearized Lorentz-covariant action for an Abelian chiral 2-form potential is [4, 9] S = 14! T M T − M Z d x (cid:20) ∂ ρ a∂ ρ a ) ∂ µ a ( H − ˜ H ) µλσ ( H − ˜ H ) νλσ ∂ ν a − H µνλ H µνλ (cid:21) , (50)where H µνλ = ∂ µ B νλ + ∂ ν B λµ + ∂ λ B µν , (51) ˜ H is the Hodge dual of H and a is an auxiliary field. In addition to the usual gauge symmetryfor a 2-form potential δB µν = ∂ µ Λ ν − ∂ ν Λ µ , (52)it is invariant under two gauge transformations δB µν = ( ∂ µ a )Φ ν ( x ) − ( ∂ ν a )Φ µ ( x ) , δa = 0 , (53)and δB µν = ϕ ( x )( ∂a ) ( H − ˜ H ) µνρ ∂ ρ a, δa = ϕ ( x ) . (54)Using the gauge symmetry (54), one can impose the gauge fixing condition a = x , (55)so that the action becomes S = 14 T M T − M Z d x (cid:18) ǫ ijklm H ijk (cid:20) H lm + 16 ǫ lmnpq H npq (cid:21)(cid:19) . (56)The 6 dimensional Lorentz symmetry is still preserved but with a modified transformationlaw for a boost parametrized by v k as δB ij = x v k ∂ k B ij − x k v k ∂ B ij − x k v k ( H − ˜ H ) ij . (57)8he gauge transformation (53) reduces in this gauge to δB i = Φ i . (58)Now we consider the compactification of the Abelian theory on a circle of radius R along x . All fields can be decomposed into their zero modes and KK modes, and the actionbecomes S = S (0) + S ( KK ) , (59)where S (0) = 2 πR T M T − M Z d x H (0) ijk H (0) ijk , (60) S ( KK ) = 14 T M T − M Z d x (cid:18) ǫ ijklm H ( KK ) ijk (cid:20) H ( KK ) lm + 16 ǫ lmnpq H ( KK ) npq (cid:21)(cid:19) . (61)The zero modes B (0) ij are 5 dimensional 2-form potential, and we can carry out the standardprocedure of electric-magnetic duality for S (0) to get an action for the dual 1-form potential S (0) dual = 2 πR T M T − M Z d x F ij F ij , (62)where F ij = H (0) ij is the field strength of the dual 1-form potential B (0) i .Let us check that the equations of motion derived from the new action S (0) dual + S ( KK ) lead to configurations satisfying self-duality conditions. For the zero modes, the equation ofmotion derived from the action S (0) dual is ∂ j F ij = 0 . (63)Defining a 3-form field H by H (0) ijk = 12 ǫ ijklm F lm , (64)we see that, due to the equation of motion (63), a 2-form potential B (0) exists locally suchthat H (0) = dB (0) . Since F also satisfies the Jacobi identity dF = 0 , we find ∂ k H (0) ijk = 0 . (65)Note that (64) is identical to the self-duality condition for the zero modes H (0) ijk = 12 ǫ ijklm H (0) lm (66)by identifying A i with B i . Hence we see that the zero modes of the self-dual gauge field canbe simply described by the Maxwell action S (0) dual .It is natural to non-Abelianize the equation of motion (63) for the zero modes by [ D j , F ij ] = 0 + · · · , (67)9p to additional covariant terms that vanish when the Lie algebra h is Abelian. In the nextsection we will derive the complete equation from an action principle.For the non-Abelian theory described in Sec. 2.1, we could also have defined H (0) ijk simplyas the Hodge dual of F ij , hence it is not necessary to introduce the components H (0) ijk whichhas the unusual transformation law (31). The transformation of F ij would then imply that H (0) , defined as the Hodge dual of F ij , transforms simply as δH (0) ijk = [ H (0) ijk , Λ (0)5 ] . (68)This would also lead us to redefine the transformation laws of matter fields as δφ = [ φ, Λ (0)5 ] (69)for all components of φ .For the KK modes, the equations of motion derived from varying S ( KK ) is ǫ ijklm ∂ k (cid:18) H ( KK ) lm + 16 ǫ lmnpq H ( KK ) npq (cid:19) = 0 . (70)This implies that ǫ ijklm (cid:18) H ( KK ) lm + 16 ǫ lmnpq H ( KK ) npq (cid:19) = ǫ ijklm Φ ( KK ) lm (71)for some tensor Φ ( KK ) lm satisfying ǫ ijklm ∂ k Φ ( KK ) lm = 0 . (72)We can redefine B ( KK ) lm by a shift B ( KK ) lm → B ′ ( KK ) lm ≡ B ( KK ) lm + ∂ − Φ ( KK ) lm (73)such that, due to (72), H ( KK ) lm → H ′ ( KK ) lm ≡ H ( KK ) lm + Φ ( KK ) lm , (74) ǫ lmnpq H ( KK ) npq → ǫ lmnpq H ′ ( KK ) npq ≡ ǫ lmnpq ( H ( KK ) npq + 3 ∂ n Φ ( KK ) pq ) = ǫ lmnpq H ( KK ) npq . (75)As a result, (71) is turned into the self-duality condition H ( KK ) lm = − ǫ lmnpq H ( KK ) npq . (76)Let us define the non-Abelian counterpart of (70) as ǫ ijklm (cid:20) D k , (cid:18) H ( KK ) lm + 16 ǫ lmnpq H ( KK ) npq (cid:19)(cid:21) = 0 . (77) As (72) implies that Φ ( KK ) lm = ∂ l Φ ( KK ) m − ∂ m Φ ( KK ) l for some vector field Φ ( KK ) l , in the Abelian theory theeffect of shifting B ( KK ) lm is equivalent to the shift of B ( KK ) l in the gauge symmetry (58), up to the usual gaugetransformation (52). ǫ ijklm (cid:18) H ( KK ) lm + 16 ǫ lmnpq H ( KK ) npq (cid:19) = ǫ ijklm Φ ( KK ) lm , (78)where Φ ( KK ) lm satisfies ǫ ijklm [ D k , Φ ( KK ) lm ] = 0 . (79)This again can be absorbed into a shift of B ( KK ) lm B ( KK ) lm → B ′ ( KK ) lm ≡ B ( KK ) lm + ∂ − Φ ( KK ) lm , (80)so that the self-duality condition (76) is arrived.The transformation (80) should also be viewed as a gauge transformation of the theory.The gauge transformation parameter Φ ( KK ) lm has to transform covariantly under the transfor-mation defined in Sec. 2.1 as δ Φ ( KK ) lm = [Φ ( KK ) lm , Λ (0)5 ] , (81)because the constraint (79) is covariant. It can then be checked that (80) commutes withthe gauge transformation (9, 10) defined in Sec. 2.1. Our task in the next section is to givean action that would lead to the non-Abelian equations of motion (67) and (77). Let us consider the following action for the non-Abelian chiral 2-form potential S = S (0) + S ( KK ) , (82)where S (0) = 2 πR T M T − M Z d x Tr ( F ij F ij ) , (83) S ( KK ) = 14 T M T − M Z d x Tr (cid:18) ǫ ijklm H ( KK ) ijk (cid:20) H ( KK ) lm + 16 ǫ lmnpq H ( KK ) npq (cid:21)(cid:19) . (84)This invariant action is a straightforward generalization of the action (61) and (62) for theAbelian theory.For small R , the M5-branes should be approximated by D4-branes in type II A theory, so S (0) should be identified the Yang-Mills theory for multiple D4-branes S (0) = 14 T D T − s Z d x Tr ( f ij f ij ) , (85)where the field strength f ij for multiple D4-branes is f ij ≡ [ ∂ i + A i , ∂ j + A j ] = ∂ i A j − ∂ j A i + [ A i , A j ] . (86) The meaning of the transformation of a gauge transformation parameter is this: Φ should be viewed as afunction depending on the gauge potential B (0) i as well as some free parameters corresponding to integrationconstants when we solve the constraint (79). The transformation of B (0) i induces a transformation of Φ .
11t is known that the gauge potential A in D4-brane theory is related to the gauge potential B in M5-brane theory via the relation A i = 2 πRB (0) i . (87)Plugging in the values of the parameters involved, T M = 12 π T M , T D = 1(2 π ) g s ℓ s , T s = 12 πℓ s , R = g s ℓ s , (88)we find that the coupling constant should be given by g = 2 πR. (89)This factor can also be obtained by demanding that the soliton solutions which resembleinstantons in the spatial 4 dimensions have momentum equal to n/R for some integer n inthe x direction.Notice that the overall factor of πR due to the integration over x in (83) is multipliedby a factor of /g for g is the Yang-Mills coupling for the zero mode field strength F ij ,giving an overall factor of /R in (85), in agreement with the requirement of conformalsymmetry in 6 dimensions. Witten’s argument [17] mentioned in the introduction around(5) and (6) that M5-brane action does not exist is resolved by allowing the coupling of a6 dimensional theory to depend on the compactification radius R . Normally the couplingconstant of an interacting field theory is independent of whether the space is compactified.Our strategy is to define a 6 dimensional field theory as the decompactification limit of acompactified theory, and the coupling depends on the compactification radius. In some sense,the coupling constant g is not really the coupling of the decompactified theory, which is aconformal field theory without free parameter. Witten’s argument should be understood asthe non-existence of another formulation of the 6 dimensional theory which does not referto the compactification radius. In other words, our model may be as good as it gets if wewant to describe multiple M5-branes with an action.Assuming that we will be able to show in future works that a well defined theory doesexist in uncompactified 6 dimensional spacetime as the decompactification limit of our model,one would still wonder how such a theory can be fully Lorentz invariant, while its definitioninvolves the choice of a special direction. We will discuss more on this point in the nextsection.The variation of S (0) leads to Yang-Mills equations, which can be interpreted as the self-dual equation for the zero modes. The full equation of motion for the zero modes B (0) i shouldalso include variations of S ( KK ) , which modifies the Yang-Mills equation by commutators thatvanish in the Abelian case.Contrary to the proposal of [18, 19], we have an explicit appearance of the KK modesthrough the action S ( KK ) (84). A useful feature of S ( KK ) is that it depends on B ( KK ) i and B ij only through ˆ B ( KK ) ij . Therefore, although S ( KK ) depends on both B ( KK ) i and B ( KK ) ij (unlike the12belian case where the action is independent of B i ), we only need to consider the variationof ˆ B ( KK ) ij . Variation of the action S ( KK ) with respect to ˆ B ( KK ) ij leads to the equation of motion(77), which is equivalent to the self-duality condition (76) via a shift in B ( KK ) lm , as we explainedin the previous section. Explicitly, the equations of motion are [ D j , F ij ] = R πR dx h ˆ B jk , (cid:16) H ( KK ) ijk − ǫ ijklm H ( KK ) lm (cid:17)i , (90) ǫ ijklm h D k , (cid:16) H ( KK ) lm + ǫ lmnpq H ( KK ) npq (cid:17)i = 0 . (91) R → ∞ The reader may find it strange that in the gauge transformation laws (14) and (16) wehave avoided commutators involving two KK modes, e.g. terms of the form [ B ( KK ) , Λ ( KK ) ] .Correspondingly, there is no term of the form [ B ( KK ) , B ( KK ) ] in the equations of motion. Allgauge interactions are mediated via zero modes. Here is our interpretation. In the limit R → ∞ , the Fourier expansion of a field approaches to the Fourier transform Φ( x ) = X n Φ n e inx /R −→ Φ( x ) = Z dk π ˜Φ( k ) e ik x . (92)The coefficients Φ n approach to ˜Φ( k ) as ˜Φ( k ) = 2 πR Φ n ( k = n/R ) . (93)According to this expression, the value of a specific Fourier mode Φ n must approach to zeroin the limit R → ∞ . In particular, the amplitude of the zero mode approaches to zero.While all interactions are mediated via the zero mode, this does not imply that there is nointeraction in the infinite R limit, because the coupling g = 2 πR → ∞ . The product of theamplitude of the zero mode with the coupling is actually kept finite in the limit.In the limit R → ∞ , the KK modes B ( KK ) µν should be identified with the 2-form potentialin uncompactified 6 dimensional spacetime. In uncompactified space, the constant part of B µν is not an observable, hence physically the KK modes B ( KK ) µν do not miss any physicalinformation a 2-form potential can carry. The zero modes B (0) µν approach to zero but a newfield A i replacing πRB (0) i survives the large R limit. The field A i can not be viewed aspart of the 2-form potential, in the sense that, due to the infinite scaling of B (0) i by R ,it can not be combined with B ( KK ) µν in a Lorentz covariant way to form a new tensor in 6dimensions. Rather it should be understood as the 1-form needed to define gerbes (or somesimilar geometrical structure) together with the 2-form potential. However this does notincrease the physical degrees of freedom of the 6 dimensional theory in the sense that thenumber of physical degrees of freedom in the 5 dimensional field A i is negligible comparedwith that of a 6 dimensional field. 13he fact that gauge transformation laws do not have terms of the form [ B ( KK ) , Λ ( KK ) ] ,and the fact that the equations of motion do not have terms of the form [ B ( KK ) , B ( KK ) ] , areboth telling us that our model is linearized with respect to the 2-form potential. No self-interaction of the 2-form potential is present, and all interactions are mediated by the 1-formpotential A i .As the decompactification limit R → ∞ is also the strong coupling limit g → ∞ , we donot expect the classical equations of motion (90)–(91) to give a good approximation of thequantum theory. We leave the problem of finding the decompactification limit of our theoryfor future study. The interpretation above allows us to understand some puzzles about the proposal of [18, 19]that the 5 dimensional D4-brane theory is already sufficient to describe the 6 dimensional M5-brane system even for finite R . In their proposal, the momentum p in the 5-th (compactified)direction is represented by the “instanton” number on the 4 spatial dimensions. The firstproblem with this interpretaion is that, in the phase when U ( N ) symmetry is broken to U (1) N , there is no instanton solution. But physically this corresponds to having M5-branewell separated from each other, and they should still be allowed to have nonzero p . Thisproblem does not exist in our model. In our model p is carried by the KK modes when theLie algebra of the gauge symmetry is Abelian. Furthermore, the Abelian case of our modelis already known to be equivalent to a 6 dimensional theory which has the full Lorentzsymmetry in the large R limit.The second problem of the proposal in [18, 19] is that the instanton number only givesthe total value of p of a state, but it is unclear how to specify the distribution of p overdifferent physical degrees of freedom. For example, the state with m units of p contributedfrom the scalar field X and n units of p from X cannot be distinguished from the statewith the numbers m and n switched. On the other hand, in our model, the instanton numberof the 1-form A i ≡ RB (0) i (94)should only be interpreted as the value of p of the field A i . (In other words, the so-called“zero-modes” B (0) i can still carry nonzero p . The 5-th momentum of the 2-form potential ismanifest as the KK mode index.) The scalar fields X I and the fermions Ψ , when they areintroduced into our model, would have their own KK modes to specify their p contribution.There is no ambiguity in the momentum carrier for a given instanton number.The reader may wonder whether it is redundant or over-counting for A i to be able tocarry nontrivial p . After all, A i is just B (0) i rescaled. Has not the KK modes B ( KK ) i alreadytaken care of the contribution of B i to p ? How can a field carry momentum in the x -direction if it has no fluctuation (e.g. propagating wave) in that direction? The answer issimple. It is well known in classical electrocmagnetism that the simultaneous presence of14onstant electric and magnetic fields carry momentum, because the momentum density p i isproportional to F j F ij . In the temporal gauge A = 0 , the conjugate momentum of A j is Π j ≡ ∂ A j , and the momentum density p i is proportional to F j F ij = Π j ( ∂ i A j ) − Π j ( ∂ j A i ) . (95)The first term is the standard contribution of a field to momentum p i . We also have ( ∂ φ )( ∂ i φ ) for a scalar field φ . But there is no analogue of the 2nd term for a scalar field. Itis possible for the 2nd term to be present because A i has a Lorentz index. The zero modeof A i in the x i direction can also contribute to p i through this term. Similarly, for a 3-formfield strength H , the momentum density of p is proportional to H ab H ab ( a, b = 1 , , , ),which includes the zero mode contribution H (0)0 ab F ab = 16 ǫ abcd F ab F cd (96)because H (0) ab = F ab . This is precisely the same expression as the instanton number density.Note that there are also contributions to p from the KK modes H ( KK )0 ab H ( KK ) ab in additionto the zero mode contribution, analogous to the first term in (95). In non-Abelianizing the gauge transformations of a 2-form potential, the zero mode Λ (0)5 plays a special role. We associate the special role played by Λ (0)5 to its topological nature:while Λ ( KK )5 can be “gauged away”, the zero mode Λ (0)5 corresponds to the Wilson line degreeof freedom for the gauge transformation parameter Λ along the circle in the x direction. Inthis section we generalize this association to construct non-Abelian gauge transformationsfor a 3-form potential.First we study the Abelian gauge theory for a 3-form potential on the spacetime of R d × T . Let the torus T extend in the directions of x and x . We can decompose a field Φ as Φ = Φ (0) + Φ ( KK ) , (97)where the zero mode Φ (0) has no dependence on T ∂ a Φ (0) = 0 , (98)and the KK mode Φ ( KK ) can be obtained from Φ as Φ ( KK ) = (cid:3) − (cid:3) Φ , (99)where (cid:3) ≡ ∂ a ∂ a ( a = 1 , . (100)15he Abelian gauge transformations of a 3-form potential B are given by δB i = ∂ i Λ − ∂ Λ i + ∂ Λ i , (101) δB ija = ∂ i Λ ja − ∂ j Λ ia + ∂ a Λ ij , (102) δB ijk = ∂ i Λ jk + ∂ j Λ ki + ∂ k Λ ij , (103)where a = 1 , and i, j, k = 0 , , , · · · , ( d + 1) . There is redundancy in the gauge transfor-mation parameters Λ ia , Λ ij so that the gauge transformation laws are invariant under thetransformation δ Λ = ∂ λ − ∂ λ , (104) δ Λ ia = ∂ i λ a − ∂ a λ i , (105) δ Λ ij = ∂ i λ j − ∂ j λ i . (106)Apparently there is also a redundancy in using λ to parametrize the redundancy in Λ . Thereare ( d + 2) components in λ , but only ( d + 1) of them are independent. Using the redundancyof Λ , we can “gauge away” ( d + 1) of the gauge transformation parameters. For instance, wecan set ρ i ≡ ∂ a Λ ia = 0 , Λ = 0 , (107)and use the following gauge transformation parameters ξ i ≡ ǫ ab ∂ a Λ ib , Λ ij , (108)so that Λ ( KK ) ia = − ǫ ab (cid:3) − ∂ b ξ i , (109)and the gauge transformation laws become δB i = − ξ i , (110) δB ija = − ǫ ab (cid:3) − ∂ b ( ∂ i ξ j − ∂ j ξ i ) + ∂ a Λ ij , (111) δB ijk = ∂ i Λ jk + ∂ j Λ ki + ∂ k Λ ij . (112)Viewing Λ ia as d copies of 1-form potentials on T , the ξ i ’s are the corresponding fieldstrengths, and so their integrals over T are quantized. It implies that ξ (0) i is quantized, andso we have to set ξ (0) i = 0 (113)when we use ξ i as infinitesimal gauge transformation parameters. In the following, we have ξ i = ξ ( KK ) i .To retrieve from (110)–(112) the original gauge transformation laws with redundancy,one can simply carry out the replacement ξ i → ξ i − ∂ i Λ , (114) Λ ij → Λ ij + (cid:3) − ( ∂ i ρ j − ∂ j ρ i ) . (115)16n the torus T , the gauge transformation parameter Λ (0)12 corresponds to a Wilson surfacedegree of freedom for the 2-form Λ . It should play the same role as Λ (0)5 in (9, 10). Toconstruct a consistent non-Abelian gauge transformation algebra for the 3-form potential,we only need to consider transformation laws for the parameters ξ ( KK ) i , Λ ij and Λ (0)12 . In theend we get the full gauge transformation laws through the replacement ξ ( KK ) i → ξ ( KK ) i − [ D i , Λ ( KK )12 ] , (116) Λ ( KK ) ij → Λ ( KK ) ij + (cid:3) − ([ D i , ρ ( KK ) j ] − [ D j , ρ ( KK ) i ]) , (117)where the covariant derivative D i should be defined as D i = ∂ i + B (0) i , (118)and ξ ( KK ) i ≡ ǫ ab ∂ a Λ ( KK ) ib , ρ ( KK ) i ≡ ∂ a Λ ( KK ) ia . (119)Here we have scaled B i to absorb the coupling constant g , which is expected to be givenby the area (2 π ) R R of the torus.We define the non-Abelian gauge transformations as δB i = [ D i , Λ (0)12 ] − ξ i + [ B ( KK ) i , Λ (0)12 ] , (120) δB ija = − ǫ ab (cid:3) − ∂ b ([ D i , ξ j ] − [ D j , ξ i ]) + ∂ a Λ ij +[ D i , Λ (0) ja ] − [ D j , Λ (0) ia ] + [ B ija . Λ (0)12 ] , (121) δB ijk = [ D i , Λ jk ] + [ D j , Λ ki ] + [ D k , Λ ij ] + [ B ijk , Λ (0)12 ] . (122)The algebra of gauge transformations is closed [ δ, δ ′ ] = δ ′′ , (123)with the parameters of δ ′′ given by Λ (0)12 ′′ = [Λ (0)12 , Λ (0)12 ′ ] , (124) ξ ′′ i = [ ξ i , Λ (0)12 ′ ] − [ ξ ′ i , Λ (0)12 ] , (125) Λ ′′ ij = [Λ ij , Λ (0)12 ′ ] − [Λ ′ ij , Λ (0)12 ] . (126)The field strengths should be defined as H (0) ij = [ D i , D j ] , (127) H ( KK ) ij = [ D i , B ( KK ) j ] − [ D j , B ( KK ) i ] + ∂ B ( KK ) ij − ∂ B ( KK ) ij , (128) H ijka = [ D i , B jka ] + [ D j , B kia ] + [ D k , B ija ] − ∂ a B ijk − ǫ ab (cid:3) − ∂ b (cid:16) [ F ij , B ( KK ) k ] + [ F jk , B ( KK ) i ] + [ F ki , B ( KK ) j ] (cid:17) , (129) H ijkl = X (4) [ D i , B jkl ] − X (6) [ F ij , β kl ] , (130)17here β ij ≡ (cid:3) − ∂ a B ija , (131)so that all the field strength components transform as δH ij = [ H ij , Λ (0)12 ] , (132) δH ijka = [ H ijka , Λ (0)12 ] + X (3) [ F ij , Λ (0) ka ] , (133) δH ijkl = [ H ijkl , Λ (0)12 ] + X (6) [ F ij , Λ (0) kl ] . (134)It may be possible to define a non-Abelian self-dual gauge theory for a 3-form potential in 8dimensional Euclidean space. We leave the problem for future study.Apparently, the same idea can be used to define a non-Abelian gauge symmetry for p -formpotentials on R d × T p − . In this paper, we found a consistent, closed algebra of non-Abelian gauge transformations forthe 2-form potential with 1-form gauge transformation parameters in 6 dimensions. There is agauge symmetry in the gauge transformations parametrized by a 0-form. The transformationlaw is nonlocal in the direction which is compactified to a circle.We also found an action which passes the two major tests for it to be relevant to theM5-brane theory: It is equivalent to a Lorentz-invariant chiral 2-form theory in 6 dimensionsin the Abelian phase, and it is equivalent to a Yang-Mills theory in 5 dimensions in the limit R → .There are several additional tests this model has to pass in order to prove that it is thecorrect theory for multiple M5-branes:1. In some sense there is a well defined limit R → ∞ in which the theory describesmultiple M5-branes in uncompactified spacetime.2. 6 dimensional Lorentz symmetry in the limit R → ∞ .3. The existence of a supersymmetric extension of the theory.In the above we have given persuasive arguments on these issues, but we leave the fullanswers to these questions for future investigation. Acknowledgment
The authors thank Chong-Sun Chu, Kazuyuki Furuuchi, Hirotaka Irie, Sheng-Lan Ko, To-mohisa Takimi, and Chi-Hsien Yeh for helpful discussions. Y. M. would like to thank thehospitality of people in Taipei during his stay in last September. K. W. H. is grateful to18ung-Hong Huang for encouragements. The work of P.-M. H. and K. W. H. is supportedin part by the National Science Council, the National Center for Theoretical Sciences, andthe LeCosPA Center at National Taiwan University. Y. M. is partially supported by Grant-in-Aid (
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