Aspects of Effective Theory for Multiple M5-Branes Compactified On Circle
aa r X i v : . [ h e p - t h ] N ov UT-14-39
Aspects of Effective Theoryfor Multiple M5-BranesCompactified On Circle
Pei-Ming Ho † and Yutaka Matsuo ‡ † Department of Physics and Center for Theoretical Sciences,Center for Advanced Study in Theoretical Sciences,National 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
A supersymmetric non-Abelian self-dual gauge theory with the explicit introductionof Kaluza-Klein modes is proposed to give a classical description of multiple M5-braneson R × S . The gauge symmetry is parametrized by Lie-algebra valued 1-forms with theredundancy of a 0-form, and the supersymmetry transformations without gauge-fixingare given. We study BPS configurations involving KK modes, including M-waves andM2-branes with non-trivial distributions around the circle. Finally, this supersymmetricgauge theory of two-forms can be equipped with more general non-Abelian gerbes infive dimensions. e-mail address: [email protected] e-mail address: [email protected] Introduction
In the past few years, there has been growing interest in finding a low energy effectivetheory for multiple M5-branes in M theory [1]–[43]. One of the many approaches to theproblem is to consider M5-branes compactified on a circle of finite radius R . (In the end,you can take the decompactification limit R → ∞ for the uncompactified theory.) Via doubledimensional reduction, when R → , M5-branes become D4-branes, which are described bythe 5-dimensional super Yang-Mills theory. This duality serves as an important constrainton the model for multiple M5-branes.At the same time, a model of multiple M5-branes should admit the configurations inwhich all M5-branes are well separated from each other so that they are all decoupled. Inthis limit, the model should be described by multiple copies of the single M5-brane effectivetheory, which has a well-known Lagrangian [44, 45] with an Abelian gauge symmetry (withor without compactification). This is another important constraint on the multiple M5-branetheory.In addition, the world-volume theory of M5-branes is expected to have the (2 , -superconformalsymmetry in 6 dimensions. Although it is possible that only part of the supersymmetry ismanifest in a Lagrangian formulation [14, 15, 24, 30], the same field content (more preciselythe dynamical degrees of freedom) should agree with that of the (2 , -theory.In our opinion, the most important feature of M5-branes is the gauge symmetry of a 2-form gauge potential. While the Abelian theory for such a gauge symmetry is well understoodboth in physics and mathematics [46], the non-Abelian counterpart is rather mysterious. Inmathematics, there is still no consensus about the precise definition of non-Abelian gerbes[47, 48]. In physics, the construction of a satisfactory theory for non-Abelian 2-form gaugepotential is usually obstructed by various no-go theorems [49, 50, 1, 2, 3, 51, 19]. A crucialdifference between the ordinary gauge symmetry for 1-form potentials and that for 2-formpotentials is that the latter has a redundancy in the gauge transformation laws. How to non-Abelianize the gauge symmetry without losing this “gauge symmetry of gauge symmetry”is the key issue of the problem in order to have the correct number of degrees of freedom.This is perhaps directly connected to the core of the mysteries about M5-branes, which offeran opportunity to guide us to significantly expand our understanding of the notion of gaugesymmetry.In fact, there is already a non-Abelian gauge theory for a 2-form potential. It is theeffective theory for a single M5-brane in the background of large C -field [54]. The non-Abelian algebra is characterized by the Nambu-Poisson bracket as a result of the C -fieldbackground.In previous works [12, 26, 27], a model was proposed for the gauge field degrees of freedomin a system of multiple M5-branes. Its 6-dimensional base space is compactified on a circle See also [52, 53] as a different class of applications of 2-form gauge theory in physics so that the no-gotheorems are not relevant.
1f finite radius R , and it satisfies the following criteria:1. When KK modes are removed on dimensional reduction, it reduces to the Yang-Millstheory, the gauge field sector of multiple D4-branes.2. When the gauge group U ( N ) is replaced by U (1) N , it reduces to decoupled multiplecopies of the 6D self-dual gauge theory.3. It has a consistent non-Abelian gauge symmetry algebra for a self-dual 2-form po-tential in 6 dimensions, without any excessive physical degrees of freedom (such as anextra 1-form potential).This proposal [12, 26, 27] stands out as the only existing model that has been shown tosatisfy all three criteria above. However, it misses the ingredients of matter fields andsupersymmetry. One of the purposes of this paper is to show that an existing proposal[30] of the supersymmetric theory for multiple M5-branes can be viewed as the gauge-fixed version of the supersymmetrization of this non-Abelian self-dual gauge theory. It hasthe right field content, although only part of the (2 , -supersymmetry is manifest. WithSUSY, one may proceed to study various aspects of the system in more detail, such assupersymmetric classical configurations, which are the other focus of the paper.The plan of the paper is as follows. We review and elaborate on the non-Abelian 2-form gauge theory [12, 26, 27] in Sec. 2. We show that there are infinitely many conservedcharges associated with the translation symmetry in the compactified direction. In Sec. 3we extend the supersymmetry algebra proposed in Ref. [30] to a larger algebra that closes onthe gauge transformation, so that the former can be viewed as the gauge-fixed version of thelatter. In Sec. 4, we construct BPS configurations which involve KK modes, including thosedescribing M2-branes lying along a non-compactified direction with non-trivial distributionin the compactified direction. We take the large R limit of these BPS solutions and evaluatetheir behavior. We also find BPS states corresponding to M-waves, that is, propagating wavesin the compactified direction. In Sec. 5, we point out that supersymmetric gauge theoriescan be defined for more general set-up of non-Abelian gerbes in 4+1 dimensions [27]. Finally,in Sec. 6, we comment on other approaches to multiple M5-branes and conclude. In this section, we review the gauge symmetry and action for the non-Abelian self-dual gaugefield proposed in Refs. [12, 26, 27]. We will also analyze the theory in more detail, givingexpressions for the Hamiltonian, Poisson brackets and conserved charges. An interesting The gauge transformation should be parameterized by a 1-form in the adjoint representation of thegauge group, with a redundancy parametrized by a 0-form. The theory of [30] is derived from the framework of supersymmetric theories developed in a series ofworks [55].
The base space of the theory is R × S . The coordinates x µ ( µ = 0 , , , , are used for R and x for S . Naturally, the 2-form potential B MN ( M, N = 0 , , , , , ) is decomposedinto two sets of components B µ and B µν . It is also natural to decompose all fields into zeromodes and Kaluza-Klein (KK) modes as Φ = Φ (0) + Φ ( KK ) . (1)The gauge potential B MN and gauge transformation parameter Λ M take values in a non-Abelian Lie algebra. It should be u ( N ) for N M5-branes in flat spacetime. We identify theWilson loop (zero mode) of the one-form gauge parameter Λ M as a 0-form gauge parameter λ λ ≡ πR Λ (0)5 = I dx Λ , (2)which is independent of x . With respect to this 5D gauge parameter λ , we shall treat thezero mode of B µ as the corresponding 5D 1-form potential A µ .The non-Abelian gauge transformation law for the 2-form potential is defined by [12] δB µ = [ D µ , Λ ] − ∂ Λ µ + [ B ( KK ) µ , λ ] , (3) δB µν = [ D µ , Λ ν ] − [ D ν , Λ µ ] + [ B µν , λ ] − [ F µν , ∂ − Λ ( KK )5 ] , (4)where the 5-dimensional covariant derivative and field strength are D µ = ∂ µ + A µ , (5) F µν = [ D µ , D ν ] , (6)with the gauge potential A µ identified with the zero mode of B µ through the relation A µ ≡ πRB (0) µ = I dx B µ . (7)The coefficient πR shows up from the relation between the field theories on M5 and D4 andmay be interpreted as the coupling constant g on D4 [12]. The appearance of ∂ − in (4) isneeded for a closed gauge symmetry algebra. In fact, a quantity only has to be periodic up to a gauge transformation, so the decomposition of afield into zero modes and KK modes as in (1) is not always possible. We will comment on twisted boundaryconditions in Sec. 2.1.1. F µν here is different from the F µν in Ref. [12] by an overall factor of πR . δA µ = [ D µ , λ ] , (8) δB ( KK ) µ = [ D µ , Λ ( KK )5 ] − ∂ Λ ( KK ) µ + [ B ( KK ) µ , λ ] , (9) δB (0) µν = [ D µ , Λ (0) ν ] − [ D ν , Λ (0) µ ] + [ B (0) µν , λ ] , (10) δB ( KK ) µν = [ D µ , Λ ( KK ) ν ] − [ D ν , Λ ( KK ) µ ] + [ B ( KK ) µν , λ ] − [ F µν , ∂ − Λ ( KK )5 ] . (11)This gauge symmetry algebra is closed. It is [12] [ δ, δ ′ ] = δ ′′ (12)with the corresponding gauge parameters related via the following relations: λ ′′ = [ λ, λ ′ ] , (13) Λ ′′ KK ) = [ λ, Λ ′ KK ) ] − [ λ ′ , Λ ( KK )5 ] , (14) Λ ′′ µ = [ λ, Λ ′ µ ] − [ λ ′ , Λ µ ] . (15)As the case of Abelian gauge symmetry for 2-form potentials, there is a redundancy inusing Λ µ and Λ to parametrize the non-Abelian gauge transformations defined above. Thegauge transformation is unchanged when Λ µ and Λ are changed by δ Λ ( KK ) µ = [ D µ , ξ ( KK ) ] , δ Λ ( KK )5 = ∂ ξ ( KK ) (16)for an arbitrary function ξ ( KK ) that has no zero mode. Note that Λ (0)5 (equivalently λ ) is nottransformed because it is the Wilson-loop degree of freedom of the gauge parameters. Thistopological nature of λ is the qualification of its special role in the gauge transformationlaws.The field strength H MNP is defined by [12] H µν = 12 πR F µν + ∂ B µν + [ D µ , B ( KK ) ν ] − [ D ν , B ( KK ) µ ] , (17) H ( KK ) µνκ = [ D µ , B ( KK ) νκ ] + [ D ν , B ( KK ) κµ ] + [ D κ , B ( KK ) µν ]+[ F µν , ∂ − B ( KK ) κ ] + [ F νκ , ∂ − B ( KK ) µ ] + [ F κµ , ∂ − B ( KK ) ν ] . (18)In terms of the zero modes and KK modes, eq. (17) is equivalent to H (0) µν = 12 πR F µν , (19) H ( KK ) µν = ∂ B µν + [ D µ , B ( KK ) ν ] − [ D ν , B ( KK ) µ ] . (20)All the components of the field strength H defined above transform covariantly in theform δ Φ = [Φ , λ ] . (21)4lthough the definition of the component H (0) µνκ is missing, luckily, in the self-dual gaugetheory, we can completely ignore H (0) µνκ by focusing on its Hodge dual H (0) µν [12], which isessentially the ordinary Yang-Mills field strength F µν in 5 dimensions. The self-dualitycondition for the zero modes is replaced by the Yang-Mills equation [12].From the definitions of the field strengths, it is straightforward to derive the Bianchiidentities [12]: X (3) [ D µ , H (0) νκ ] = 0 , (22) X (3) [ D µ , H ( KK ) νκ ] = ∂ H ( KK ) µνκ , (23) X (4) [ D µ , H ( KK ) νκρ ] = X (4) [ F µν , ∂ − H ( KK ) κρ ] , (24)in which the 2-form potential B MN appears only through the field strength H MNP except thezero-mode B (0) µ (or equivalently the 1-form potential A µ ). Here P (3) and P (4) refer to sumsover permutations to totally anti-symmetrize all of the (3 or 4) indices in each expression.The gauge symmetry defined above has the following properties:1. The gauge symmetry reduces to (multiple copies of) that for the Abelian 2-form gaugepotential when the Lie algebra is Abelian.2. The “gauge symmetry of gauge symmetry” is consistently promoted to the non-Abeliancase. That is, the gauge transformation law (3) and (4) parametrized by Λ µ , Λ hasthe redundancy (16).It will be useful in computations below to define the covariant quantity ˆ B µν ≡ ∂ − H ( KK ) µν . (25)While ˆ B µν = B ( KK ) µν in the gauge B ( KK ) µ = 0 , the quantity ˆ B µν transforms covariantly beforegauge fixing. In terms of ˆ B µν , the field strengths can be expressed as H ( KK ) µν = ∂ ˆ B µν , (26) H ( KK ) µνκ = [ D µ , ˆ B νκ ] + [ D ν , ˆ B κµ ] + [ D κ , ˆ B µν ] . (27) Let us recall that, in gauge theories with one-form potentials, Wilson loop arises as a newdegree of freedom when a spatial direction is compactified on a circle along x . It can berepresented by the zero mode of the gauge potential A (0)5 , which behaves as a gauge-covariantscalar. In our two-form gauge theory, the analogue of A (0)5 is B (0) µ , which behaves as a one-form potential in 5D.Furthermore, when there is a compactification of an additional circle along x in theone-form gauge theory, it is possible to turn on a quantized flux on the torus of ( x , x ) .5t can be described by linear terms in the potential A , A . Although linear terms are notperiodic functions, they are allowed because the potential only needs to be periodic up togauge transformations. Similarly, if we add a linear term to a periodic two-form potential B MN as B µν → B ′ µν = B µν + Σ µν ( x ) x , (28) B µ → B ′ µ = B µ , (29)where Σ µν is independent of x . If B µν is periodic, B ′ µν is no longer of the form (1) andsatisfies the twisted boundary condition B ′ µν ( x + 2 πR ) = B ′ µν ( x ) + 2 πR Σ µν . (30)It is easy to check that if the tensor Σ µν satisfies the relation X (3) [ D µ , Σ νλ ] = 0 , (31)all components of the new 3-form field strength, H ′ µνλ = H µνλ , H ′ µν = H µν + Σ µν , (32)are still periodic as the old 3-form field strength. In particular, all gauge-invariant quantitiesare periodic. Incidentally, the condition (31) is equivalent to the Bianchi identity if Σ µν isproportional to F µν .By analogy, the gauge transformation parameters Λ M , λ only need to be periodic up tothe transformation (16).Since the operator ∂ − is well defined only when (1) holds, twisted boundary conditionsrequires an extension of our formulation on the 2-form potential. We leave the completetheory including twisted boundary conditions for future works.As a comment related to the issue of boundary conditions, the non-Abelian self-dualgauge theory can be equivalently reformulated by adding a linear piece to B µν so that B µν ≡ B (0) µν + 12 πR F µν x + B ( KK ) µν (33)and simplifying (17) by dropping the first term H µν ≡ ∂ B µν + [ D µ , B ( KK ) ν ] − [ D ν , B ( KK ) µ ] , (34)while keeping all other definitions intact. The linear term in B µν does not affect the period-icity of the field strength H . 6 .2 Lagrangian The action for Abelian self-dual gauge fields (also called “chiral bosons”) [57, 44, 59, 60, 61]can be found in various forms in the literature. Having a non-Abelianized gauge symmetryfor 2-form potentials, one would like to construct a gauge-invariant action.To write down a Lagrangian for self-dual gauge fields in a manifestly Lorentz-covariantway, one needs to introduce auxiliary fields. For simplicity, one often considers non-Lorentz-covariant expressions for Lagrangians without auxiliary fields. They can be thought of as thegauge-fixed versions of certain Lorentz-covariant formulations. For our purpose of describingM5-branes compactified on a circle, the compactification partially breaks Lorentz covariance,and it is natural to pick the compactified direction x as a special direction in the Lagrangianformulation, with all Lorentz symmetry in the remaining directions ( x , x , · · · , x ) intact.The action considered in Ref. [12] for multiple M5-branes compactified on a circle is anextension of the Abelian version [58] S = − Z d x ˜ H µν ( H µν − ˜ H µν ) (35)(up to an overall normalization), where ˜ H µν ≡ − ǫ µνλσρ H λσρ . (36)Decomposing the fields into zero modes and KK modes, we note that this action for anAbelian 2-form potential is equivalent to S = − πR Z d x F µν F µν − Z d x ˜ H µν KK ) ( H ( KK ) µν − ˜ H ( KK ) µν ) (37)by suitably integrating out H (0) µνκ and redefining the gauge field A µ via (7). The zero mode H (0) µνκ disappears from this action, but its existence is guaranteed by the Maxwell equation ∂ µ F µν = 0 . (40)In this case, the Maxwell equation is equivalent to the self-duality condition for the zeromodes.The action for the non-Abelian theory is then taken to be of the same form but with anoverall trace [12] S = − πR Z d x
14 Tr[ F µν F µν ] − Z d x Tr[ ˜ H µν KK ) ( H ( KK ) µν − ˜ H ( KK ) µν )] , (41) The Maxwell equation implies that there exits a tensor B (0) µν such that F µν = 12 ǫ µνκσρ ∂ κ B (0) σρ . (38)One can then define H (0) µνκ by H (0) µνκ = ∂ µ B (0) νκ + ∂ ν B (0) κµ + ∂ κ B (0) µν , (39)and (38) is of the same form as the self-duality condition for the zero modes. δB ( KK ) µν = Φ ( KK ) µν , δB µ = 0 , (42)where Φ ( KK ) µν satisfies the constraint ǫ µνκσρ [ D κ , Φ ( KK ) σρ ] = 0 . (43)This gauge symmetry is responsible for establishing the 1-1 correspondence between theequivalence classes of solutions to the equations of motion for the KK modes ǫ µνκσρ [ D κ , ( H ( KK ) σρ − ˜ H ( KK ) σρ )] = 0 (44)and the self-dual configurations defined by H ( KK ) µν = ˜ H ( KK ) µν . (45)Analogous additional gauge symmetries also appeared in other M5-brane actions in theliterature [44, 45]. It is a universal feature of the Lagrangian formulation of chiral bosontheories.There are other equivalent formulations of Abelian self-dual gauge fields that one canstart with and extend it to the non-Abelian theory. In particular, another choice of theaction is S = − π Z d x H µν ( H µν − ˜ H µν ) , (46)where x ∈ [0 , πR ) , as a small modification of the previous action (35). It is different from(35) only in the first factor H µν of the Lagrangian. We study this formulation in more detailnow.Like the previous action, this action also enjoys an additional gauge symmetry δB µν = Φ µν , (47)where Φ µν is an arbitrary function independent of x . This gauge symmetry implies thatthe zero mode of B µν is a pure gauge artifact. The equation of motion for the KK modesderived from this new action is ∂ (cid:16) H ( KK ) µν − ˜ H ( KK ) µν (cid:17) = 0 , (48)and it is equivalent to the self-duality condition (45). The advantage of this choice is that theequivalence between equations of motion and self-duality condition is particularly simple. Itis also very easy to check that the action (46) reduces directly to S = − πR Z d x F µν F µν − Z d x H µν KK ) ( H ( KK ) µν − ˜ H ( KK ) µν ) , (49)8ithout having to use the gauge symmetry (47) or integrating out any field.The non-Abelian counterpart of (49) is S = − πR Z d x
14 Tr[ F µν F µν ] − Z d x Tr[ H µν KK ) ( H ( KK ) µν − ˜ H ( KK ) µν )] . (50)Since the solutions to the equations of motion can be matched with self-dual configurations,this Lagrangian is equivalent to the previous Lagrangian (41) at the classical level. It is notclear how they may be related to each other at the quantum level. In general, there are manyclassically equivalent Lagrangians for a self-dual gauge field [59, 61]. It will be interesting toinvestigate the quantum theories for these actions. In this subsection, we provide basics of the Lagrangian and Hamiltonian formulations of thetheory.Let us repeat the Lagrangian (50) here for convenience of the reader: S = − πR Z d x
14 Tr[ F µν F µν ] − Z d x Tr[ H µν KK ) ( H ( KK ) µν − ˜ H ( KK ) µν )] . (51)When the gauge-fixing condition B ( KK ) µ = 0 (52)is imposed, this action is identical to the gauge field part of the supersymmetric actionproposed in Ref. [30]. Note that B ( KK ) µ appears in the action only through ˆ B µν (25). In terms of A µ and ˆ B µν , theaction (51) is S = − πR Z d x Tr [ F µν F µν ] − Z d x Tr [ ∂ ˆ B µν ( ∂ ˆ B µν + 12 ǫ µνκσρ [ D κ , ˆ B σρ ])] . (53)The equation of motion for the KK modes ˆ B µν is ∂ (cid:18) ∂ ˆ B µν + 12 ǫ µνκσρ [ D κ , ˆ B σρ ] (cid:19) = 0 . (54)It is equivalent to ∂ ˆ B µν + 12 ǫ µνκσρ [ D κ , ˆ B σρ ] = 0 , (55)as ∂ − is well defined on KK modes.The equation of motion for the zero modes A µ is πR [ D ν , F µν ] + 12 I dx ǫ µνκσρ [ ∂ ˆ B νκ , ˆ B σρ ] = 0 . (56)This is of the form of the Yang-Mills equation with a source term. It reduces to the pureYang-Mills equation when KK modes vanish.9 .3.2 Hamiltonian Formulation In the Lagrangian as well as the equations of motion, the KK modes of B µν and B µ areencoded in ˆ B µν , and the zero modes are present in terms of A µ . All physical gauge degreesof freedom in the theory reside completely in ˆ B µν and A µ .As there is no time-derivative terms of the temporal components A and ˆ B i ( i, j =1 , , , ) in the Lagrangian (51), they are Lagrange multipliers. The corresponding con-straints are H ( KK )0 i = ˜ H ( KK )0 i (57)for ˆ B ( KK )0 i ( i, j = 1 , , , ), and a modified Gauss’ law for A . As the canonical formulationof Yang-Mills theory is well known, we will focus our attention on the KK modes.The BRST anti-field formulation of the theory was already given in [26]. Here we providea simpler, more elementary Hamiltonian formulation. To describe the Hamiltonian formu-lation for the KK modes, we first solve the constraints (57), which determine uniquely thevalues of the Lagrange multipliers ˆ B i = − ǫ ijkl ∂ − H jkl ( KK ) (58)in term of the dynamical fields ˆ B ij . We can thus replace ˆ B i everywhere in the Lagrangianby this expression, so that the only dynamical fields of the KK modes are ˆ B ij .As there is no more unsolved constraints, we can define the conjugate momentum of ˆ B ij simply as ˆΠ ij ≡ δ S δ∂ ˆ B ij = − ǫ ijkl H kl KK ) . (59)Denoting the Fourier modes of a field ΦΦ = X n ∈ Z Φ ( n ) e − inx /R (60)by Φ ( n ) ( n ∈ Z ), the Poisson bracket is given by { ˆ B ( m ) ij , ˆ B ( n ) kl } = i Rn δ m + n ǫ ijkl . (61)Here the superscripts ( m ) , ( n ) are labels for the KK modes.The Hamiltonian for the KK modes is H ( KK ) = Z d x ˆΠ ij ∂ ˆ B ij − S. (62)It can be simplified using self-duality conditions as H ( KK ) = − Z d x (cid:16) H ( KK )0 AB H AB ( KK ) (cid:17) = 2 Z d x (cid:16) H ( KK )0 ij H ( KK )0 ij + H ( KK ) ijk H ( KK ) ijk (cid:17) , (63)where A, B = 1 , , , , and i, j = 1 , , , . It is positive-definite.10 .4 Conserved Currents Apart from the Hamiltonian, the momentum P is also conserved due to translation symme-try in the x -direction. The contribution of the KK-modes of the gauge field is P ( KK )5 = Z d x (cid:16) H ( KK )0 ij H ij KK ) (cid:17) . (64)In fact, due to the property that KK modes only interact through zero modes, there areinfinitely many conserved charges. For any positive integer n , the KK modes labelled by n and − n can be simultaneously created or annihilated by a zero mode. The number ofexcitations of the KK mode with label n minus the number of excitations of the KK modewith label − n is constant. There is thus a conserved current for each integer n > .Formally, both actions (41) and (50) take the form B ( − n ) KB ( n ) ( K is an operator inde-pendent of fields), so they are invariant under the transformation δB ( n ) = ǫ n B ( n ) , δB ( − n ) = − ǫ n B ( − n ) ( n > . (65)This is proportional to the transformation induced by a translation in x if all parameters ǫ n are given by ǫ n = nǫ . But the transformation parameters ǫ n ( n > for different Fouriermodes are allowed to be independent. The translation symmetry in x induces infinitelymany symmetries because of the peculiar interaction feature of the theory.These infinitely many symmetries lead to an infinite number of conserved currents, j µ ( n ) = πRǫ µνλρσ Tr (cid:16) H ( n ) νλ B ( − n ) ρσ − H ( − n ) νλ B ( n ) ρσ (cid:17) = nπi Tr (cid:16) ǫ µνλρσ B ( n ) νλ B ( − n ) ρσ (cid:17) (66)for n = 1 , , , · · · . The self-duality condition implies that they indeed satisfy the conser-vation law ∂ µ j µ ( n ) = 0 in 5D. P is written in terms of them as (by taking ǫ n = − inǫ/R ) P = − X n> Z d x inR j n ) . (67) A supersymmetric gauge theory in 5 dimensions for the gauge-fixed fields A µ and B ( KK ) µν in thegauge (52) were proposed in Ref. [30] to describe multiple M5-branes. Like our formulationof the gauge theory for the 2-form potential, the zero modes and KK modes are treatedseparately in the supersymmetric theory. We will show that the super-algbera in Ref. [30]can be viewed as the gauge-fixed version of a super-algebra with the full gauge symmetry.The extension of the supersymmetry to be fully consistent with the gauge symmetry isnecessary for the completeness of the M5-brane theory proposed in Refs. [12, 26, 27].From the viewpoint of 5D SUSY, upon the compactification on a circle of radius R , the11assless fields on M5-branes is composed of the following SUSY multiplets [30]: ( A (0) µ , φ (0) , χ (0) a , Y (0) ab ) = a massless vector multiplet , (68) ( F ( n ) µν , φ ( n ) , χ ( n ) a , Y ( n ) ab ) = tensor multiplets with mass m n , (69) ( h (0) a ˙ b , ψ (0)˙ b ) = a massless hypermultiplet , (70) ( h ( n ) a ˙ b , ψ ( n )˙ b ) = hypermultiplets with mass m n . (71)The indices a, b, ˙ a, ˙ b (taking values , ) are the labels of the fundamental representationsfor two SU (2) groups as part of the rotation symmetry of the transverse dimensions of theM5-branes. The 5-dimensional uncompactified spacetime indices are µ, ν = 0 , , , , . Thefermions χ (0) a , ψ (0)˙ b , χ ( n ) a , ψ (0)˙ b are 5D spinors representing 6D Weyl spinors.The mass of a field with KK-mode index n is m n = nR , (72)and the auxiliary bosonic field Y ab has symmetrized indices: Y ab = Y ba .In this theory of multiple M5-branes, all the fields are in the adjoint representation ofthe gauge group. All the scalars φ, h a ˙ b and fermions χ a , ψ ˙ b are covariant (21) under gaugetransformations. The field F ( n ) µν in Ref. [30] should be identified with our gauge field strengththrough the relation F ( n ) µν ≡ RH ( n ) µν = in ˆ B ( n ) µν . (73)In comparison with the notation of Ref. [30], other fields are also rescaled in a similar way. We will use the totally antisymmetrized tensors ǫ ab , ǫ ˙ a ˙ b to raise or lower SU (2) indices,and we will use the NW-SE convention for contraction. For example, Φ a = ǫ ab Φ b , Φ a = Φ b ǫ ba . (75)There is an additional massless vector multiplet ( A (0) µ , φ (0) , χ (0) a , Y (0) ab ) defined in this model[30]. But it is fixed to be a constant (see eq.(3.12) in Ref. [30]), and hence will be ignored.Although only the 5D N = 2 SUSY is manifest, but we hope that the rest of the desiredsymmetry is hidden. In fact, a method is proposed in Ref. [30] to upgrade this model toanother one with the full 6D N = (2 , superconformal symmetry. We will focus on thesimpler model in this work for clarity and simplicity. In view of the 6D theory, it is natural to rescale the fields in Ref. [30], which are labelled with thesuperscript [ BGH ] : χ [ BGH ]( n ) a = Rχ ( n ) a , φ [ BGH ]( n ) = Rφ ( n ) , Y [ BGH ]( n ) ab = RY ( n ) ab . (74)The variables on the right hand side are those used in this paper. δA µ = −
12 ¯ ǫ a γ µ χ (0) a , (76) δφ ( n ) = i ǫ a χ ( n ) a , (77) δH ( n ) µν = ¯ ǫ a γ [ µ D ν ] χ ( n ) a − i φ ( n ) , ¯ ǫ a γ µν χ (0) a ] + i ǫ a γ µν ( D φ χ ( n ) a ) , (78) δχ ( n ) a = 14 γ µν H ( n ) µν ǫ a − i D/ φ ( n ) ǫ a − Y ( n ) ab ǫ b −
12 ( D φ φ ( n ) ) ǫ a , (79) δY ( n ) ab = −
12 ¯ ǫ ( a D/ χ ( n ) b ) + i [ φ ( n ) , ¯ ǫ ( a χ b ) ] − i ǫ ( a ( D φ χ ( n ) b ) ) , (80) δh ( n ) a ˙ b = − i ¯ ǫ a ψ ( n )˙ b , (81) δψ ( n )˙ b = i D/ h ( n ) a ˙ b ǫ a + 12 ( D φ h ( n ) a ˙ b ) ǫ a , (82)where we have used the notation D φ defined by ( D φ Φ ( n ) ) ≡ − im n Φ ( n ) + [ φ (0) , Φ ( n ) ] , (83)and Φ [ µν ] ≡ (Φ µν − Φ νµ ) , Φ ( ab ) ≡ (Φ ab +Φ ba ) for symmetrized and anti-symmetrized indices.The covariant derivative is D µ = ∂ µ + A µ . For any field Φ , its zero mode is denoted as Φ (0) . All the equations above are valid for n = 0 (the zero modes) as well.Notice that the zero mode of the scalar φ (0) appears only through the operator D φ inthe gauge transformation laws. (The same is true for the Lagrangian.) It is tempting tointerpret D φ as the covariant derivative D in the Fourier basis, and φ (0) as the (missing) 5thcomponent A of the 1-form gauge potential. It is peculiar that a transverse coordinate φ (0) of the M5-brane also resembles a component of the 1-form potential upon compactification.Our task here is to find the SUSY transformation law for the component B ( KK ) MN , which isabsent in the (gauge-fixed) SUSY transformation laws (76)–(82). The SUSY transformationof A (0) µ (76) more or less suggests that, before gauge fixing, δB ( n ) µ = −
12 ¯ ǫ a γ µ χ ( n ) a . (84)In addition, the SUSY transformation law (78) for the gauge-covariant field H ( n ) µν suggeststhat we define the gauge transformation of the rest of the gauge potential components B ( n ) µν by δB ( n ) µν = − i ǫ a γ µν χ ( n ) a − R n [ φ ( n ) , ¯ ǫ a γ µν χ (0) a ] + R n [ φ (0) , ¯ ǫ a γ µν χ ( n ) a ] − iRn [ B ( n )[ µ , ¯ ǫ a γ ν ] χ (0) a ] . (85)To summarize, the SUSY transformation laws for the zero modes are the same as thatfor the 5D super Yang-Mills theory, and the SUSY transformation laws for the KK modes The convention in Ref. [30] is that D µ = ∂ µ − A µ . As a result, A µ , F µν here differ from those in Ref. [30]by a sign. δB ( KK ) µ = −
12 ¯ ǫ a γ µ χ ( KK ) a ,δB ( KK ) µν = − i ǫ a γ µν χ ( KK ) a − i ∂ − φ ( KK ) , ¯ ǫ a γ µν χ (0) a ] + i φ (0) , ¯ ǫ a γ µν ∂ − χ ( KK ) a ] + [ ∂ − B ( KK )[ µ , ¯ ǫ a γ ν ] χ (0) a ] , (86)together with (77) and (79) –(82).Let us check whether the super-algebra for the SUSY transformations defined above isclosed up to gauge transformations. It is straightforward to check that SUSY transformationson B ( n ) µ satisfy the closure relation [ δ , δ ] B ( n ) µ = α ν ∂ ν B ( n ) µ + β inR B ( n ) µ + [ D µ , Λ ( n )5 ] + [ B ( n ) µ , λ ] − inR Λ ( n ) µ , (87)or equivalently, [ δ , δ ] B ( KK ) µ = α ν ∂ ν B ( KK ) µ + β∂ B ( KK ) µ + [ D µ , Λ ( KK )5 ] − ∂ Λ ( KK ) µ + [ B ( KK ) µ , λ ] , (88)where the coefficients are given by α µ = 12 ¯ ǫ a γ µ ǫ a , (89) β = i ǫ a ǫ a , (90) Λ ( KK )5 = − α µ B ( KK ) µ + βR φ ( KK ) , (91) Λ ( KK ) µ = α ν B ( KK ) µν + βB ( KK ) µ + α µ R φ ( KK ) + h ∂ − (cid:16) βB ( KK ) µ + α µ R φ ( KK ) (cid:17) , φ (0) i , (92) λ = − Rα µ B (0) µ + βφ (0) . (93)On the right hand side of (88), the first term is a translation in the x µ direction, the secondterm is a translation in the x direction, the third and fourth terms are gauge transformationsby Λ ( KK ) i , Λ ( KK )5 and the last term is a gauge transformation by λ (the 5D gauge transfor-mation parameter). The gauge transformation pieces in the super-algebra agree nicely withthe gauge transformation of B ( KK ) µ (9).It can be checked that the same super-algebra observed for B ( KK ) µ in (88), that is, [ δ , δ ] = α i p i + βp + δ Λ + δ λ (94)applies to all other fields, with the parameters α i , β , Λ and λ defined by (89) – (93). Here ( δ , δ ) are the SUSY transformations with parameters ( ǫ , ǫ ) , δ Λ is the gauge transformationfor the KK modes of the 2-form potential, δ λ is the 5D SYM gauge transformation, and p i , p are generators of translations, which are for our case just derivatives ∂ i , ∂ .14 Solitonic Solutions
All BPS states invariant under translation along x survives dimensional reduction and canbe represented by configurations in the 5D SYM theory. They are all automatically includedin the theory studied here, including those discussed in Refs. [7] and [17]. In the following,we will look for BPS solutions involving KK modes.According to the SUSY transformation laws (76) – (82), a BPS configuration for the KKmodes should allow nontrivial solutions of the SUSY parameter ǫ to the following equations γ µν H ( KK ) µν ǫ a − i D/ φ ( KK ) ǫ a − Y ab ( KK ) ǫ b − D φ φ ( KK ) ǫ a , (95) i D/ h a ˙ b ( KK ) ǫ a + 12 D φ h a ˙ b ( KK ) ǫ a , (96)assuming that all fermionic fields vanish. Here the derivative D φ (83) and its complexconjugate are defined by ( D φ Φ ( n ) ) ≡ − im n Φ ( n ) + [ φ (0) , Φ ( n ) ] , (97) ( ¯ D φ Φ ( n ) ) ≡ + im n Φ ( n ) + [ φ (0) , Φ ( n ) ] . (98)In various circumstances, the BPS conditions are not sufficient to guarantee the satis-faction of all equations of motion. Hence we list here for reference the equations of motionfor the KK modes derived from the supersymmetric action of Ref. [30]: nR H µν n ) − i ǫ µνλσρ D λ H ( n ) σρ − i ([ φ (0) , H µν n ) ] − [ φ ( n ) , F µν ]) = 0 , (99) D µ D µ φ ( n ) − n R φ ( n ) + iR n [ F µν , H ( n ) µν ] − iRn [ D µ φ (0) , D µ φ ( n ) ] − iRn [ φ (0) , D µ D µ φ ( n ) ] − iRn [ Y ab (0) , Y ( n ) ab ] − inR [ φ (0) , φ ( n ) ] − [ φ (0) , [ φ (0) , φ ( n ) ]] − iRn [ φ (0) , [ φ (0) , [ φ (0) , φ ( n ) ]]] = 0 , (100) Y ( n ) ab − iRn ([ φ (0) , Y ( n ) ab ] − [ φ ( n ) , Y (0) ab ]) = 0 , (101) D µ D µ h ( n ) a ˙ b − n R h ( n ) a ˙ b − [ h b ( n )˙ b , Y (0) ab ] − inR [ φ (0) , h ( n ) a ˙ b ] + [ φ (0) , [ φ (0) , h ( n ) a ˙ b ]] = 0 . (102)In the above we have set all fermions to zero.In terms of D φ (83) and ¯ D φ (98), they are simplified as ¯ D φ H µν n ) − i ǫ µνλσρ D λ H ( n ) σρ + i [ φ ( n ) , F µν ] = 0 , (103) ¯ D φ ( D µ D µ φ ( n ) + D φ D φ φ ( n ) ) + iR n [ F µν , H ( n ) µν ] − iRn [ D µ φ (0) , D µ φ ( n ) ] − iRn [ Y ab (0) , Y ( n ) ab ] = 0 , (104) ¯ D φ Y ( n ) ab − [ φ ( n ) , Y (0) ab ] = 0 , (105) ( D µ D µ + D φ D φ ) h ( n ) a ˙ b − [ h b ( n )˙ b , Y (0) ab ] = 0 . (106) For example, see [62]. .1 M2 Along x An M2-brane stretched between two M5-branes separated by a finite distance in a transversedirection intersects with either M5-brane on a one-dimensional space, and it is described asa self-dual string from the viewpoint of the M5-brane worldvolume theory. The descriptionfor these states is known for the zero modes (in SYM theory) [7], however this descriptionmay not be complete. If the self-dual string lies along the x -direction, it can certainly bedescribed by zero modes. But if it lies along other directions, say x , the zero modes canonly describe the state when the self-dual string is smeared over the circle along x . We willconsider the extension of these zero-mode BPS solutions by turning on KK modes, in orderto describe a self-dual string that is localized in the x -direction. Our strategy is to first findzero-mode BPS solutions, and then consider small fluctuations of the KK modes with thezero-mode solution as a background, ignoring back-reactions. If an M2-brane is not wrapped around x , but lies along x , it is described by a staticstring-like configuration [7] F i ′ j ′ = ǫ i ′ j ′ k ′ D k ′ Φ , A = 0 , A = sin θ Φ , X = cos θ Φ , (107)where i ′ , j ′ , k ′ = 1 , , and Φ satisfies D Φ = 0 . To regulate the total energy and momentum,we compactify x on a circle of radius R . Let r ≡ qP i ′ =1 x i ′ . At large r , the solution of Φ is approximately Φ = φ σ − qσ πr + · · · , (108)where φ is an arbitrary constant and q ∈ Z . For cos θ = v/ p v / π n /q , sin θ = 2 π n/q p v / π n /q , φ = p v / π n /q , (109)the momentum and magnetic charge are P = − πR π ng Y M , Q M = vqg Y M , (110)where the YM coupling is related to the radius of the circle of x by π g Y M = 1
R . (111)The energy is E = 2 πR q Q M + ( P / πR ) . (112)This solitonic solution preserves half of the SUSY in the 5D SYM theory.16n Ref. [7], it is claimed that the zero-mode solution above represents all the BPS config-urations for a self-dual string winding around the circle of x . This has to be the case if the5D SYM theory is indeed the complete description of multiple M5-branes. In our approach,while there are independent KK-mode degrees of freedom, one may wonder if it is possibleto find BPS states in which KK modes are excited on top of this zero-mode configuration sothat the self-dual string is not uniformly smeared over x . Despite the lack of a complete theory with Lagrangian and SUSY transformation rules, fieldequations for the 2-form gauge potential and BPS conditions were proposed in Ref. [29] forM2-branes ending on multiple M5-branes. A solution exists [29] to represent an open M2-brane stretched between two M5-branes, lying along the x -direction, with an x -dependentdistribution. (It can be extended to more general solutions for more than two M5-branes[35].) The similarity and differences between the theory of Ref. [29] with our theory ofmultiple M5-branes will be discussed later in Sec. 6.2, but coincidentally the solution found inRef. [29] can be adopted for our calculation. (We will see later that the equations consideredin Ref. [29] are only a subset of all the equations one needs to check in the model studiedhere.)For simplicity, we consider the special case of θ = 0 in (107) for the zero modes F i ′ j ′ = − iǫ i ′ j ′ k ′ D k ′ φ (0) , A = A = 0 , (113)where i ′ , j ′ , k ′ = 1 , , . This implies that F i ′ = F = F i ′ = 0 . For the purpose of includingKK modes in a way that will be convenient for our discussions below, let us re-calculate thezero-mode solution by starting with the ansatz for the ’t Hooft-Polyakov monopole: A i ′ = ǫ i ′ j ′ k ′ f ( r ) x j ′ σ k ′ , (114) φ (0) = h ( r ) x · σ, (115)where x · σ ≡ x i ′ σ i ′ , r ≡ √ x i ′ x i ′ (116)and σ i ′ represents generators of the su (2) Lie algebra with the commutation relations [ σ i ′ , σ j ′ ] = ǫ i ′ j ′ k ′ σ k ′ . (117)(Repeated indices are summed over even when they are both subscripts or both superscripts.)Eq. (113) then implies that r dfdr + f = r dhdr + f h, (118) r dfdr + r f = f h − r h. (119)17These two equations can be combined to give a single (non-linear) second order differentialequation for h ( r ) .) An explicit solution to these equations was given in Ref. [29]: f ( r ) = 1 r − cr sinh( cr ) , (120) h ( r ) = 1 r − cr coth( cr ) (121)with a constant parameter c . The solution above is singular at r = 0 , the location of theM2-brane. The fact that it has to be singular somewhere is expected from the equation D i ′ D i ′ φ (0) = 0 , (122)which can be derived by taking covariant derivative on the first equation of (113), since thesecond order differential operator D is negative definite.To compare this solution with the expression (108) in the previous subsection, one cancarry out a gauge transformation φ → U φU − , F i ′ j ′ → U F i ′ j ′ U − (123)by the SU (2) matrix U ≡ exp " − ( x σ − x σ ) p ( x ) + ( x ) tan − p ( x ) + ( x ) x ! , (124)which is also singular at the origin to bring it to the form in which φ → ( c − r ) σ at large r .In fact, we will not need the explicit solution for the discussion below. All we need isthat the zero-mode solution can be put in the form (114) and (115).For the KK modes, as above, we focus on solutions with h ( n ) a ˙ b = 0 . First, we assume thatall interaction terms vanish in the equations of motion (to be verified later) by demanding [ φ (0) , H µν n ) ] − [ φ ( n ) , F µν ] = 0 , (125) [ F µν , H ( n ) µν ] − D µ φ (0) , D µ φ ( n ) ] = 0 , (126) [ φ (0) , φ ( n ) ] = 0 , (127) Y ( n ) ab = 0 , (128)so that the equations of motion are linearized nR H µν n ) − i ǫ µνλσρ D λ H ( n ) σρ = 0 , (129) D µ D µ φ ( n ) − n R φ ( n ) = 0 . (130)We also extend the BPS conditions (113) for the zero-mode solution to the KK modesby H ( n ) i ′ j ′ = − iǫ i ′ j ′ k ′ D k ′ φ ( n ) , ˆ B ( n )0 i ′ = ˆ B ( n ) i ′ = 0 . (131)18ccording to the BPS conditions for the KK modes (95), (96), this ansatz (131) preserves1/4 of the SUSY for SUSY parameters ǫ a satisfying the conditions γ ǫ a = − ǫ a , γ ǫ a = ǫ a . (132)(Recall that the solutions for M2-branes wrapped around x are also 1/4-BPS states.)Eq. (131) implies that H ( n )0 i ′ = H ( n ) i ′ = 0 . ( H ( n )045 will not be zero.) The self-dualitycondition then implies that H ( n )0 i ′ j ′ = H ( n ) i ′ j ′ = 0 .The equations of motion (129) and (130) would be valid if ˆ B ( n )04 = φ ( n ) , (133) D i ′ D i ′ φ ( n ) = n R φ ( n ) . (134)What we need to do now is to find explicit solutions for (134). Then we can determine thevalues of ˆ B ( n ) i ′ j ′ and ˆ B ( n )04 using (131) and (133).Following (114) and (115), we take the ansatz φ ( n ) = h ( n ) ( r ) x · σ (135)to find solutions to the equation (134), which implies that h ( n ) satisfies the equation d h ( n ) dr + 4 r dh ( n ) dr + 4 f h ( n ) − r f h ( n ) = n R h ( n ) . (136)An explicit solution to this equation was found in Ref. [29]: h ( n ) ( r ) = c n e −| n | r/R r (cid:18) cR | n | coth( cr ) (cid:19) (137)for arbitrary parameters c n . Since all KK modes are decoupled from all other KK modes,we have infinitely many parameters c n to parametrize the amplitude of each KK modeindependently.It can now be checked that the assumptions (125)–(128) are valid. As only the ansatz(114), (115) and (135) are needed for this check, all solutions of f ( r ) , h ( r ) , h ( n ) ( r ) to thedifferential equations (118), (119) and (136) give legitimate BPS states in the multiple M5-brane theory.Note that the zero-mode solution in SYM theory discussed in the previous subsectionalso represents an M2-brane along x , but it is smeared over the circle of x . Here we havefound the solutions with an arbitrary distribution along x , including those localized arounda point in the x -direction. This allows us to consider the localization of the M2-brane in alltransverse directions. R Limit
We take the BPS solution above for M2-branes in the x -direction as an example to demon-strate how the theory of multiple M5-branes for finite radius R can also be used to obtaininformation about infinite R , the uncompactified space.19n the limit of small R , the zero mode φ (0) dominates over the KK modes. For a localizedsource in three large transverse dimensions ( x , x , x ) , the massless field φ (0) should scale as /r with r = p ( x ) + ( x ) + ( x ) at small r , when the kinetic term dominates over thepotential term in its field equation. This is indeed the case in the solution of φ (0) above in(115) and (121). (Note that the factor ( x · σ ) /r can be gauge-transformed to σ via U (124).)Similarly, the KK modes φ ( n ) behave as massive fields in three large transverse dimensionsand scales like e −| n | r/R /r in the UV limit. This can be verified by examining the solution of φ ( n ) in (135) and (137).On the other hand, for a large radius R of the compactified circle, φ should behave asa massless field in four large transverse dimensions ( x , x , x , x ) . Hence one expects that,in the UV limit when the field equation is dominated by the kinetic term, φ scales like /ρ ( ρ = p r + ( x ) ) as a result of rotation symmetry in ( x , x , x , x ) . Note that, since the5D Lorentz symmetry in ( x , x , x , x , x ) is manifest in the theory, this rotation symmetryimplies the full 6D Lorentz symmetry.In the limit of large R , it is more convenient to replace the index n for KK modes by thewave number k ≡ nR . (138)In this limit, the sum over KK modes P n is approximated by an integral over k : X n ∈ Z F ( n ) ≃ R Z ∞−∞ dkF ( Rk ) (139)for any function F ( n ) . For a delta-function source at x = 0 , we superpose all KK modeswith equal amplitude since R dk e − ikx = 2 πδ ( x ) . That is, we choose c n = α to be indepen-dent of n in the solution for each KK mode (137), and sum over n to find φ = X n ∈ Z φ ( n ) e − inx /R ≃ αR Z dk r e −| k | r − ikx (cid:18) c | k | coth( cr ) (cid:19) x · σr = 2 αRρ x · σr − αRcr coth( cr ) log( ρ/ Λ) x · σr , (140)where Λ is an IR cut-off parameter, and the factor x · σr can be transformed to σ by a gaugetransformation through the matrix (124). We should take α → as R → ∞ such that thesolution φ is finite in the limit of large R .In the UV limit, φ is dominated by the first term which indeed demonstrates the /ρ behavior implied by the 6D Lorentz symmetry. The second term in the expression of φ depends on the parameter c which characterizes the profile of the soliton solution in the ( x , x , x ) -directions. Since we have taken a Dirac δ -function profile for the solution inthe x -direction, we do not expect this term to be invariant under the 4D rotations in ( x , x , x , x ) . For a nontrivial evidence of the 6D Lorentz symmetry, one should find asolution (with a nontrivial x -profile) invariant under the 4D rotation at finite r in the large R limit. 20 .2 BPS States for Pure KK Modes Since all KK modes interact only with zero modes, they are all decoupled if we set all zeromodes to zero. The system becomes equivalent to an infinite set of free fields.Setting the zero modes to zeros, the equations of motion (99)–(102) are simplified to i ∗ dF ( n ) − m n F ( n ) = 0 , (141) ( ∂ µ ∂ µ + m n ) φ ( n ) = 0 , (142) ( ∂ µ ∂ µ + m n ) h ( n ) a ˙ b = 0 , (143) Y ( n ) ab = 0 , (144)where F ( n ) ≡ RH ( n ) µν dx µ ∧ dx ν is a two-form in 5D, and ∗ denotes the Hodge dual in 5D.Even though all the KK modes are decoupled in the equations of motion, they are relatedby the BPS conditions for a BPS state. The BPS conditions (95) and (96) are simplified as − H ( n ) µν γ µν ǫ a − i γ µ ∂ µ φ ( n ) ǫ a − Y ab ( n ) ǫ b + im n φ ( n ) ǫ a = 0 , (145) γ µ ∂ µ h a ˙ b ( KK ) ǫ a − nR h a ˙ b ( KK ) ǫ a = 0 . (146)In general it relates the KK modes H ( n ) µν , φ ( n ) and Y ( n ) ab to one another. There are KK modes representing uniform sinusoidal waves propagating along the x direc-tion are BPS states. These M-waves solutions that we will present below were first obtained[37] for a different proposal of the M5-brane theory [22]. But here these solutions are tobe checked against the field equations and BPS conditions of a complete theory with asupersymmetric Lagrangian and gauge symmetry.Consider the ansatz of self-dual configurations H ij n ) = 12 ǫ ijkl H ( n ) kl . (147)All equations of motion are satisfied by H ( n ) ij = c ( n ) ij e inx /R , φ ( n ) = const × e inx /R , h ( n ) a ˙ b = const × e inx /R , (148)where c ( n ) ij is a self-dual constant matrix c ( n ) ij = 12 ǫ ijkl c ( n ) kl , (149)and the equation of motion (141) implies that H ( n )0 i = 0 . There are no relations among theamplitudes as all KK modes are decoupled.These independent waves of gauge fields and scalars are 1/4-BPS states symmetric forSUSY parameters ǫ a satisfying γ ǫ a = ǫ a , γ ǫ a = − iǫ a . (150)Obviously these solutions also survive in the large R limit by replacing the KK mode index n by k (138). 21 Supersymmetric Gerbe
The discussion in the previous sections can be straightforwardly generalized to the set-upin Ref. [27] where a formulation of non-Abelian gerbes was proposed. Let G be an arbitraryLie group and ρ be an arbitrary (not necessarily irreducible) representation. We write g torepresent the Lie algebra of G and ρ to be the representation of g . Let V be the representationspace where ρ acts. V can be regarded as an Abelian group by the action of addition. Forthe example of N M5-branes, G = U ( N ) and V is the space of KK modes in the adjointrepresentation.In this set-up, we define the one-form ( A, ˜ A ) to take values in the semi direct product g ⋉ V ( A ∈ g , ˜ A ∈ V ) and the two-form B ∈ V . The pair g ⋉ V and V is an example ofcrossed module, which is the standard ingredient to define a non-Ablian gerbe. In Ref. [27],we argued that a system with the structure of non-Abelian gerbe is often limited to free ortopological theory. Indeed, such topological theory was used to classify the phases of non-Abelian gauge theory in four dimensions [52, 53]. Our example seems to be the only exceptionwhere some modification of the gauge transformation enables us to define an interacting fieldtheory.We also need to include a mass matrix M which is a linear map acting on V and commutewith the action of ρ . Suppose V is decomposed into the invariant subspaces V = ⊕ i V i , so that M = ⊕ i m i I i , where I i is an identity matrix acting on V i . Our discussion so far correspondsto a specific choice V = ⊕ ∞ n =1 ( V n ⊕ V − n ) , m n = n/R and V ± n is the adjoint representationof g . We introduce the zero-form gauge parameters Λ ∈ g and ˜Λ ∈ V and the one-form gaugeparameter ˜ a ∈ V . The gauge transformation proposed in Ref. [27] is, δA i = ∂ i Λ + [ A i , Λ] , (151) δ ˜ A i = ∂ i ˜Λ + ρ ( A i )( ˜Λ) − ρ (Λ)( ˜ A i ) + M ˜ a i , (152) δ ˜ B ij = ∂ i ˜ a j − ∂ j ˜ a i + ρ ( A i )(˜ a j ) − ρ ( A j )(˜ a i ) − ρ (Λ)( ˜ B ij ) + M − ρ ( F ij )( ˜Λ) . (153)The last term in (153) is a modification necessary to have homogeneous gauge transformationof field strength, F ij = ∂ i A j − ∂ j A i + [ A i , A j ] , (154) ˜ F ij = ∂ i ˜ A j − ∂ j ˜ A i + ρ ( A i )( ˜ A j ) − ρ ( A j )( ˜ A i ) − M ˜ B ij , (155) ˜ Z ijk = X (3) (cid:16) ∂ i ˜ B jk + ρ ( A i ) ˜ B jk − M − ρ ( F ij )( ˜ A k ) (cid:17) , (156)such that F ij = ( F ij , ˜ F ij ) ∈ g ⋉ V . 22ransformation of curvature becomes δF ij = [ F ij , Λ] , (157) δ ˜ F ij = − ρ (Λ)( ˜ F ij ) , (158) δ ˜ Z ijk = − ρ (Λ)( ˜ Z ijk ) . (159)In order to see the correspondence with the previous sections, one may consider takingone of V n in V = ⊕ ∞ n =1 ( V n ⊕ V − n ) and translate the notation in the previous sections by thefollowing rules: B (0) µ → A µ , B ( KK ) µ → ˜ A µ , B ( KK ) µν → i ˜ B µν (160) Λ (0)5 → Λ , Λ ( KK )5 → ˜Λ , Λ ( KK ) µ → i ˜ a µ , (161) H ( KK ) µν → ˜ F µν , H ( KK ) µνκ → i ˜ Z µνκ , (162)with ∂ → iM ( M = m n ). We use different indices i, j instead of µ, ν since parts of thissection can be applicable to other dimensions. The homogeneity of the gauge transformations enables us to write the gauge invariant action, L = − Tr ( F ij ) − h ˜ F ij , ˜ F ij i − h ˜ Z ijk , ˜ Z ijk i . (163)Here h· , ·i is an inner product in V which is invariant under the action of G .For our interest in self-dual gauge theories, a covariant action which leads to the self-dualequation is S = Z d x Tr ˜ F µν ( ˜ F µν − i ( ∗ ˜ Z ) µν ) . (164)The gauge field part of our action for the multiple M5-branes is a special case of this expres-sion. The equation of motion derived from (164) is ˜ F − i ∗ ˜ Z = 0 . (165) SUSY relates A i with χ a , φ and ˜ B, ˜ A with ˜ χ, ˜ φ , so χ a , φ ∈ g and ˜ χ, ˜ φ ∈ V . In Ref. [30], SUSYtransformation closes without the extra gauge parameters ˜Λ , ˜ a , so it is natural to define thefermion transformation to be homogeneous as the field strength, δχ a = [ χ a , Λ] , δ ˜ χ a = − ρ (Λ) ˜ χ a . (166) Note that A µ and F µν in this section are different from A µ and F µν in other sections of this paper bya factor of πR . Y, φ, ˜ φ, ˜ h, ˜ ψ should be similar to χ, ˜ χ .The SUSY transformation laws for the general case of non-Abelian gerbes are a straight-forward extension of the SUSY transformation laws (76)–(82) given the special case of mul-tiple M5-branes first given in Ref. [30]. They are δA µ = −
12 ¯ ǫ a γ µ χ a , (167) δ ˜ A µ = −
12 ¯ ǫ a γ µ ˜ χ a , (168) δ ˜ B µν = −
12 ¯ ǫ a γ µν ˜ χ a − i M ¯ ǫ a γ µν (cid:16) ρ ( χ a )( ˜ φ ) + ρ ( φ ) ˜ χ a (cid:17) + 1 M ¯ ǫ a γ [ ν ρ ( ˜ χ a ) ˜ A µ ] , (169) δχ a = 14 γ µν F µν ǫ a − i D/ φǫ a − Y ab ǫ b − D φ φǫ a , (170) δ ˜ χ a = 14 γ µν ˜ F µν ǫ a − i D/ ˜ φǫ a − ˜ Y ab ǫ b − D φ ˜ φǫ a , (171) δ ˜ Y ab = −
12 ¯ ǫ ( a D/ ˜ χ b ) + iρ ( ˜ φ )(¯ ǫ ( a χ b ) ) − i D φ ¯ ǫ ( a ˜ χ b ) ) , (172)where D φ ˜Φ = − iM ˜Φ + ρ ( φ ) ˜Φ . Thus we see that the 5D supersymmetric gauge theory formultiple M5-branes allows us to choose any non-Abelian gerbe defined in [27]. Some [6, 7] proposed that the same 5-dimensional SYM theory for D4-branes can be inter-preted as a theory for M5-branes even at finite radius. It was claimed that all momentummodes on M5-branes are described by zero-mode configurations with non-zero 4D instantoncharges. This proposal attracted a lot of attention and was investigated by many (see forexample [13, 18, 20, 28, 31, 33]). On the other hand, we believe that, although the 4Dinstantons carry P -charge, there are other independent KK-mode degrees of freedom. TheKK modes should be kept explicitly in the M5-brane theory. Our arguments are as follows.Roughly speaking, for a matter field Φ , the momentum density p is of the form Π ∂ Φ , (173)where Π is the conjugate momentum of Φ . In the free field theory of a single M5-brane, themomentum density p due to B ij is proportional to H ij H ij = 12 ǫ ijkl H kl H ij . (174)While the zero mode contribution ǫ ijkl H kl H ij = 12 ǫ ijkl F ij F kl (175)24f the 2-form potential B ij is indeed the instanton density, the question is whether the KKmode contribution in (173) and (174) should all be discarded in the multiple M5 theory. Ifwe accept the single M5-brane theory as a correct low energy effective theory (which canbe verified by studying solitonic solutions corresponding to M5-branes in the 11 dimensionalsupergravity), both instantons on D4-branes and KK-modes of matter fields contribute tothe momentum p through (173) and (174), at least in the limit when all M5-branes are farapart and decoupled.Some may argue that KK modes in M theory are identified with D0-branes in string the-ory, and D0-branes are identified with instantons on D4-branes, so KK modes are equivalentto instantons. However, the identification of D0-branes with instantons is justified only inthe low energy, small R limit, because the SYM theory is only a low energy effective theoryin the limit of small R . More precisely, D4-brane is the KK reduction of M5-brane compact-ified on a small circle. As KK reduction removes KK modes, zero-modes carrying P charge(such as instantons) survive KK reduction and persist in the D4-brane theory. The fact thatD0-branes on D4-branes can be identified with instantons does not imply that all D0-branesare described as instantons before taking the low energy, small R limit.To claim that the instanton configurations of gauge fields accounts for all possible sourcesof P requires a new type of gauge symmetry in which the KK-mode degrees of freedom isgauge-equivalent to the instanton configurations. There has never been such an example infield theory.Finally, without the KK modes, it would be hard to imagine how one can describe the BPSstates we considered in Sec. 4.1.2, that is, parallel M2-branes lying in the x -direction (orother large spatial directions) when they are not uniformly smeared over in the x -direction. A central idea in our formulation is to identify the vector field A µ needed in a non-Abeliangauge theory with certain components of the tensor field B MN by choosing a special direc-tion (the compactified direction x ), to avoid excessive physical degrees of freedom. Afterthe proposal of Ref. [12], a similar strategy was taken in Ref. [22], followed by a series ofpublications [29, 35, 37, 39]. We explain here the differences between their model and ours,and hopefully through this discussion the reader will also understand our model better, inparticular about the zero mode sector.The main difference of Ref. [22] from our proposal lies in the treatment of the zero modesof B MN , which leads to a difference in the equation of motion for A µ . In our approach, theequation of motion for A µ reduces to the standard 5D YM equations when we set all KKmodes to zeros. This is not the case for the theory proposed in Ref. [22].In the discussion below, we label a quantity defined in Ref. [22] by the symbol “[CK]”. Thework of Ref. [22] defined the 1-form potential A [ CK ] µ via the equation (Eq.(3.19) in Ref. [22]): F [ CK ] µν = Z dx ˜ H [ CK ] µν , (176)25here ˜ H [ CK ] µν is defined as (denoted as ˜ H µν in eq.(3.2) of Ref. [22]) ˜ H [ CK ] µν = 12 ǫ µνκσρ [ D κ , B σρ [ CK ] ] . (177)In Ref. [22], eq.(176) restricts the zero-mode B (0) [ CK ] µν . In contrast, the zero-modes in ourmodel are defined only in terms of B (0) µ , without explicitly referring to B (0) µν .The problem with eq.(176), or the reason why we have avoided explicit reference to B (0) µν in our model, is its deviation from the Yang-Mills equation when KK modes are removed ondimensional reduction. Taking the covariant derivatives on both sides of (176), we get [ D ν , F [ CK ] µν ] = 14 Z dx ǫ µνκσρ [ F νκ [ CK ] , B σρ [ CK ] ] . (178)After removing all KK modes, the Yang-Mills equation is still modified by a term of theform πR [ F [ CK ] , B (0)[ CK ] ] on the right hand side. (Note that B (0)[ CK ] is constrained by (176) soone cannot set it to zero at will.) They need to prove that somehow the correction term isnegligible in the low energy limit in order for their model to be consistent with D4-branephysics.Another difference is that, in our model, we have a free 1-form parameter Λ for thenon-Abelian gauge transformations, while it is strongly constrained to a much smaller gaugesymmetry in Ref.[22]. In fact, if one does not demand the explicit presence of such a gaugesymmetry and an invariant action at the same time, the no-go theorem [1, 2, 3] would notbe applicable, and the introduction of nonlocality may not be fully justified.Incidentally, despite their claim, the 6D Lorentz symmetry in the model of Ref. [22]is not a genuine Lorentz symmetry in the usual sense, as the definition of the angularmomentum involves an integral over the whole space-time. Furthermore, their proposedLorentz transformation can be defined even after adding more symmetry-breaking terms inthe Lagrangian.An interesting question is whether it is possible to write down an uncompactified theoryfor multiple M5-branes. There are strong constraints on the S -matrix [19] for self-interactionsof the self-dual tensor multiplet from Lorentz symmetry and supersymmetry in 6D. On theother hand, various physical aspects of the uncompactified theory can be extracted in thelarge R limit of the compactified theory, as we did in Sec. 4.1.3. An uncompactified theoryis not in crucial need unless it has some advantages such as manifest covariance in Lorentzsymmetry, supersymmetry and gauge symmetry. In addition to the works mentioned above, there are many other attempts to formulate aneffective theory for multiple M5-branes, or just to explore potentially interesting higher-formgauge theories in 6D. Some approached the problem through the mathematical notion of3-algebra [5, 9, 11, 16, 32], higher gauge theory or twistor space [4, 24, 36, 38, 41]. Some26sed holographical principle as a tool [21, 34, 42, 43]. The interest in multiple M5-branetheory has also inspired new theoretical frameworks for higher gauge theories [14, 66, 67, 68],which are interesting by themselves.The model studied in this paper based on [12, 26, 27, 30] satisfies the following criteria foran effective theory of multiple M5-branes: (i) It agrees with 5D SYM in the absence of KKmodes. (ii) It agrees with 6D single M5-brane when the gauge group is Abelian. (iii) It hasthe full gauge symmetry for a 2-form potential. (iv) It has the correct field content. It is theonly model satisfying all of those requirements. However, only part of the supersymmetry,and part of the rotation symmetry in the transverse directions of the M5-brane are manifest.The full 6D Lorentz symmetry in the UV limit is also not yet proven.More tests on the model should be carried out, especially on its hidden Lorentz symmetryand supersymmetry. It will also be interesting to study the large R limit in more detail,including scattering processes, and to compare the results with the no-go conclusion basedon supersymmetry of Ref. [19].We believe that a good comprehension of the multiple M5-brane system will be considereda significant breakthrough not only in string theory, but also in the context of general fieldtheories, as it will open a door to a new class of symmetries and related new physics thatwe know very little of. Acknowledgement
The authors would like to thank Heng-Yu Chen, Chong-Sun Chu, Kazuo Hosomichi, Yu-Tin Huang, Takeo Inami for their interest and discussions. PMH is supported in part bythe National Science Council, Taiwan, R.O.C. and by the National Taiwan University NSC-CDP-102R3203. YM is partially supported by Grant-in-Aid (KAKENHI
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