aa r X i v : . [ h e p - t h ] J a n BPS states in (2 ,
0) theory on R × T M˚ans Henningson
Department of Fundamental PhysicsChalmers University of TechnologyS-412 96 G¨oteborg, Sweden [email protected]
Abstract:
We consider (2 ,
0) theory on a space-time of the form R × T , where the first factordenotes time, and the second factor is a flat spatial five-torus. In addition to theirenergy, quantum states are characterized by their spatial momentum, ’t Hooft flux,and Sp(4) R -symmetry representation. The momentum obeys a shifted quantizationlaw determined by the ’t Hooft flux. By supersymmetry, the energy is bounded frombelow by the magnitude of the momentum. This bound is saturated by BPS states,that are annihilated by half of the supercharges. The spectrum of such states isinvariant under smooth deformations of the theory, and can thus be studied by ex-ploiting the interpretation of (2 ,
0) theory as an ultra-violet completion of maximallysupersymmetric Yang-Mills theory on R × T . Our main example is the A -series of(2 ,
0) theories, where such methods allow us to study the spectrum of BPS states formany values of the momentum and the ’t Hooft flux. In particular, we can describethe R -symmetry transformation properties of these states by determining the imageof their Sp(4) representation in a certain quotient of the Sp(4) representation ring. Introduction
Understanding the conceptual foundations of the six-dimensional quantum theorieswith (2 ,
0) supersymmetry [1] remains, at least for the present author, an outstand-ing challenge. In this paper, we will consider such a theory defined by an elementΦ of the
ADE -classification on a space-time of the form R × T = R × R / Λ . (1.1)Here the factor R denotes time, we identify the spatial R factor with its dual ( R ) ∗ by means of the standard flat metric, and Λ ⊂ R is a rank five lattice.A basic question is what the possible values of the spatial momentum p are. Onemight think that these should be given by the latticeΛ ∗ ≃ H ( T , Z ) ⊂ H ( T , R ) (1.2)dual to Λ, but this is not quite true. To explain this point, we consider a furtherdiscrete quantum number f ∈ H ( T , C ) (1.3)known as the ’t Hooft flux. Here the finite abelian group C is isomorphic to thecenter subgroup of the simply connected Lie group G corresponding to the elementΦ of the ADE -classification. Thus C ≃ Γ weight / Γ root , (1.4)where Γ weight and its dual Γ root are the weight- and root-lattices of G respectively.The inner product on the weight space of G induces a perfect pairing on C withvalues in R / Z ≃ U (1). As we will explain in more detail later, it is then possible todefine a product f · f ∈ H ( T , R / Z ) , (1.5)and the correct quantization law for the momentum p ∈ H ( T , R ) can be shown tobe p − f · f ∈ H ( T , Z ) ≃ Λ ∗ . (1.6)The main theme of this paper is to analyze the implications of supersymmetry.The generators Q i , i = 1 , . . . , of the Sp(4) R -symmetry group. In six-dimensionalMinkowski space, they also transform as a Weyl spinor under the SO(1 ,
5) Lorentzgroup, and obey a symplectic Majorana condition. (On R × T , the Lorentz group isof course broken to a discrete (and generically trivial) subgroup of the SO(5) spatialrotation group, so actually only the Sp(4) representation content ⊕ ⊕ ⊕ of the supercharges is relevant. But to avoid cluttering the notation, it is stillconvenient to present some formulas in an SO(1 ,
5) covariant way.) The equal-timeanti-commutation relations of the supercharges can be written in the form { Q i , Q † j } = δ ij (cid:0) E − γ γ · p (cid:1) , (1.7)2here E and p denote the energy and the momentum respectively. ( γ and γ arethe temporal and spatial Dirac matrices). Unitarity requires the matrix on the righthand side to be positive semi-definite, from which follows that E ≥ | p | . (1.8)We may thus distinguish between three broad classes of states: • Vacuum states have E = p = 0 , (1.9)so that the right hand side of (1.7) is identically zero. In view of (1.6), anecessary requirement for such a state is that f · f = 0. It is then consistent toimpose that these states are annihilated by all supercharges. The spectrum ofsuch states was investigated in [2] (in greatest detail for the A - and D -series). • BPS states generically have E = | p | > , (1.10)so that the right hand side of (1.7) has half maximal rank. Indeed, E − γ γ · p = E (1l − γ p ) , (1.11)where the transverse chirality matrix γ p defined by γ p = | p | − γ γ · p (1.12)has eigenvalues +1 and − representation of Sp(4) and the cor-responding annihilation operators (also in the representation). For givenvalues of the momentum p and the ’t Hooft flux f , we start with a set ofstates which are annihilated by the annihilation operators and transform insome (in general reducible) representation R ( f,p ) of Sp(4). Acting with thecreation operators then builds up a multiplet of states transforming as( B ⊕ F ) ⊗ R ( f,p ) , (1.13)where the Sp(4) representations B and F (for ‘Bosonic’ and ‘Fermionic’), aregiven as direct sums of the even and odd alternating powers of respectively: B = ⊕ ( ) a ⊕ ( ) a = ⊕ ⊕ ⊕ F = ⊕ ( ) a = ⊕ . (1.14)As usual, the spectrum of such states, as described by the Sp(4) representa-tion R ( f,p ) , can be expected to be invariant under a large class of continuous3eformations of the theory, notably including deformations of the flat metricon T . There is thus some hope of determining it explicitly, at least in somecases. This is the goal of the present paper. An important point is that therepresentation R ( f,p ) can only depend on the orbit [ f, p ] of the pair ( f, p ) underthe SL ( Z ) mapping class group of T . • Non-BPS states have
E > | p | , (1.15)so that the right hand side of (1.7) has maximal rank. The fermionic creationoperators, as well as the annihilation operators, then transform in the ⊕ representation of Sp(4), and build up multiplets of the form( B ⊕ F ) ⊗ ( B ⊕ F ) ⊗ R ′ ( f,p ) (1.16)for some Sp(4) representation R ′ ( f,p ) . Understanding the structure of this repre-sentation and the corresponding energies E appears out of reach at the present,though.We will now briefly describe the strategy to compute the BPS spectrum andoutline the rest of the paper. As we discuss in section two, the key point is toconsider T as a product T = T × S . (1.17)Type Φ (2 ,
0) theory on R × T can then be regarded as the ultraviolet completionof maximally supersymmetric Yang-Mills theory on R × T with gauge group G adj = G/C (1.18)of adjoint type, and a coupling constant determined by the radius of the S factor[3]. The component of the momentum p of (2 ,
0) theory along the S direction isgiven by the instanton number k over T of the Yang-Mills theory. The quantizationlaw (1.6) then amounts to certain topological facts concerning principal G adj bundlesover T .In section three, we take the weak coupling (small radius) limit. For non-zeroinstanton number k , the theory then formally reduces to a version of supersym-metric quantum mechanics on the corresponding moduli space of (anti) instantonconfigurations. But in practice, this model is difficult to analyze.In section four, we instead turn to the case of zero instanton number k = 0. Inthe weak coupling limit, the theory then localizes on the moduli space of flat con-nections. Isolated flat connections are particularly useful, since fluctuations aroundthem can be reliably analyzed by semi-classical methods. In this way, we can deter-mine the contribution of such a connection to the BPS representation (1.13) modulorepresentations of the form (1.16). In other words, we may determine the image[ R ( f,p ) ]of R ( f,p ) in the corresponding quotient of the Sp(4) representation ring. Fur-thermore, assuming that all representations of the form (1.16) in fact belong tonon-BPS states allows us to make a concrete proposal for the actual representation R ( f,p ) of the BPS states. 4 would think that it should eventually be possible to understand also the contri-butions from flat connections that are not isolated, but we will not pursue this here.Instead, in section five, we restrict our attention to the A -series, so that G = SU( n )for some n . The reason is that, for certain principal G adj bundles, all flat connec-tions are then isolated and can be treated as described above. (The case when n isprime is particularly convenient.) In this way, we arrive at a proposal for the BPSrepresentations R ( f,p ) for many values of the pair ( f, p ) subject to (1.6). (2 , theory to supersymmetric Yang-Mills theory As discussed in the introduction, the flat five-torus T can be constructed as T = R / Λ , (2.1)where R is endowed with the standard flat metric, and Λ ⊂ R is a rank five lattice.It follows that Λ ≃ H ( T , Z ) . (2.2)Let now λ ∈ Λ be a primitive lattice vector, which we complete to a basis λ , . . . , λ of Λ. The dual basis of the dual latticeΛ ∗ ≃ H ( T , Z ) (2.3)is denoted λ , . . . , λ . We decompose the lattice Λ asΛ = ˜Λ ⊕ ( λ ⊗ Z ) , (2.4)where ˜Λ is the rank four lattice generated by λ , . . . , λ . (In general, we denotefour-dimensional quantities with a tilde.) After an SO(5) spatial rotation, we mayassume that R = ˜Λ ⊗ R is the standard four-dimensional subspace R = { ( x , . . . , x ) ∈ R | x = 0 } (2.5)of R . Finally we define a flat four-torus T as T = R / ˜Λ . (2.6)So topologically, T = T × S , (2.7)where S = ( λ ⊗ R ) / ( λ ⊗ Z ) (2.8)is a circle in the direction of λ . The (2 ,
0) theory can now be regarded as anultra-violet completion of maximally supersymmetric Yang-Mills theory on R × T , (2.9)5here the first factor denotes time.The gauge group of the Yang-Mills theory is G adj = G/C, (2.10)where the G is the simply connected (and simply laced) Lie group correspondingto the element Φ of the ADE -classification. It is important to note that the gaugegroup is not simply connected (unless Φ = E ). Indeed, π ( G adj ) ≃ C. (2.11)The next step in defining the theory is to choose a gauge bundle P , i.e. a principal G adj bundle over T . The isomorphism class of P is determined by two characteristicclasses: The magnetic ’t Hooft flux m = m λ ∪ λ + . . . + m λ ∪ λ ∈ H ( T , C ) , (2.12)which is the obstruction against lifting P to a principal G -bundle over T , and the(fractional) instanton number k ∈ H ( T , R ) ≃ R . (2.13)The classes m and k may not be chosen independently, but are correlated as k − m · m ∈ Z . (2.14)Here we have defined the product m · m ∈ R / Z by m · m = m m + m m + m m , (2.15)with the multiplications given by the pairing on C . (The components of m areantisymmetric in the sense that m = − m , et cetera in additive notation.) Clearlythe definition of m · m is invariant under the SL ( Z ) mapping class group of T . (Wecaution the reader that the quantity m · m is denoted as m · m in many papersincluding [2].)The group ˜Ω = Aut( P ) of gauge transformations may be identified with thespace of sections of the associated bundleAd( P ) = P × Ad G adj , (2.16)where Ad denotes the adjoint action of G adj on itself. We let Ω denote the connectedcomponent of ˜Ω, and define the quotient group Ω of ‘large’ gauge transformationsas Ω = ˜Ω / Ω ≃ Hom( π ( T ) , C ) ≃ H ( T , C ) . (2.17)A physical state must be invariant under Ω , but may transform with non-trivialphases under Ω. These transformation properties are described by the electric’t Hooft flux e = e λ ∪ λ ∪ λ + . . . + e λ ∪ λ ∪ λ ∈ H ( T , C ∗ ) (2.18)6f the state. Here C ∗ = Hom( C, U (1)) (2.19)is the Pontryagin dual of C . Indeed, we have H ( T , C ∗ ) ≃ Hom( H ( T , C ) , U (1)) ≃ Hom(Ω , U (1)) . (2.20)Furthermore, the pairing on C induces an isomorphism C ≃ C ∗ , so e can also beregarded as an element of H ( T , C ).We can now describe the relationship between (2 ,
0) theory and Yang-Mills theoryin somewhat more detail. By the K¨unneth isomorphism H ( T , C ) ≃ H ( T , C ) ⊕ H ( T , C ) , (2.21)the ’t Hooft flux f = f λ ∪ λ ∪ λ + . . . + f λ ∪ λ ∪ λ ∈ H ( T , C ) (2.22)of the (2 ,
0) theory decomposes into the magnetic and electric ’t Hooft fluxes m and e of the Yang-Mills theory: f = m ∪ λ + e. (2.23)Similarly, under the K¨unneth isomorphism H ( T , R ) ≃ H ( T , R ) ⊕ H ( T , R ) , (2.24)the five-dimensional momentum p decomposes into the instanton number k over T and the four-dimensional momentum ˜ p : p = k ∪ λ + ˜ p. (2.25)We are now in a position to understand the shifted quantization law (1.6). In analogywith (2.15), we define a product m · e ∈ H ( T , R / Z ) as m · e = ( m · e ) λ + . . . + ( m · e ) λ , (2.26)where e.g. ( m · e ) = m e + m e + m e ∈ R / Z (2.27)and similarly for the components ( m · e ) , . . . , ( m · e ) . Consider now a continuousspatial translation along a closed curve representing some homology class˜ λ ∈ H ( T , Z ) ≃ ˜Λ . (2.28)For a bundle with magnetic ’t Hooft flux m , this is equivalent to a large gaugetransformation parametrized by ω = m [˜ λ ] ∈ H ( T , C ) ≃ Ω , (2.29)i.e. the 2-cohomology class m is partially evaluated on the one-cycle ˜ λ resulting ina 1-cohomology class ω . A state with electric ’t Hooft flux e then transforms witha phase ω ∪ e ∈ H ( T , U (1)) ≃ U (1) , (2.30)7hich in fact equals ( m · e )[˜ λ ] ∈ R / Z ≃ U (1) . (2.31)So the four-dimensional momentum ˜ p ∈ H ( T , R ) obeys the quantization law˜ p − m · e ∈ H ( T , Z ) ≃ ˜Λ ∗ . (2.32)Together with the relation (2.14), this is equivalent to (1.6), where the componentsof f · f = ( f · f ) λ + . . . + ( f · f ) λ ∈ H ( T , R / Z ) (2.33)are given by ( f · f ) = f f + f f + f f ∈ R / Z (2.34)and similarly for ( f · f ) , . . . , ( f · f ) . Again, although we have expressed variousquantities relative to the chosen basis λ , . . . , λ of Λ, the formalism is actuallycovariant under the SL ( Z ) mapping class group of T . For a given gauge bundle P , we introduce an associated vector bundle over T ad( P ) = P × ad g adj , (2.35)where ad denotes the adjoint action of G adj on its Lie algebra g adj (which of courseequals the Lie algebra of G ). Maximally supersymmetric Yang-Mills theory on R × T with gauge group G adj contains the following fundamental fields: • A connection D on P , locally represented by a connection one-form A withvalues in ad( P ). (We work in temporal gauge, so the time-component of theconnection one-form is identically zero.) The magnetic field strength F = dA + A ∧ A is a global section of Ω ( T ) ⊗ ad( P ). • Five sections φ of ad( P ) transforming in the representation of Sp(4) . • Four fermionic sections ψ of S ⊗ ad( P ), where S = S + ⊕ S − is the sum ofthe positive and negative chirality spinor bundles over T (for the trivial spinstructure). They transform in the representation of Sp(4).The complete Lagrangian is most easily obtained by dimensional reduction from1 + 9 to 1 + 4 dimensions [4]. But to begin with, we will focus our attention on theterms involving only the connection one-form A , i.e.12 g Z T Tr (cid:16) ˙ A ∧ ∗ ˙ A − F ∧ ∗ F (cid:17) , (2.36)where Tr denotes a suitably normalized bilinear form on g adj , and ∗ : Ω k ( T ) → Ω − k ( T ) (2.37)8s the Hodge duality operator constructed from the flat metric on T . (Time deriva-tives are denoted with a dot.) The coupling constant g is related to the circumferenceof the S factor of T = T × S , i.e. to the magnitude of the lattice vector λ : g = | λ | . (2.38)Here we have for simplicity assumed that λ is orthogonal to the subspace R spanned by the rank four lattice ˜Λ. Otherwise, the Lagrangian would contain afurther CP-violating term, analogous to the familiar theta-angle term in (1 + 3)-dimensional Yang-Mills theory.The canonical conjugate to the connection one-form A is a section E (the electricfield strength) of Ω ⊗ ad( P ) given by E = 1 g ˙ A. (2.39)The Hamiltonian becomes H = g Z T Tr ( E ∧ ∗ E ) + 12 g Z T Tr ( F ∧ ∗ F ) , (2.40)and the momentum corresponding to a translation by a constant spatial vector v on T is ι v ˜ p = Z T Tr ( ι v F ∧ ∗ E ) . (2.41)One way to view this theory is to regard it as describing a fictitious particle ofmass µ = g moving on the flat infinite-dimensional Euclidean space of connections A , under the influence of a potential V = 12 g Z T Tr( F ∧ ∗ F ) . (2.42) For a fixed non-zero value of the instanton number k = 12 Z T Tr( F ∧ F ) , (3.1)the potential (2.42) is bounded from below by V ≥ | k | g . (3.2)For positive (negative) k , the bound is saturated for connections A with anti self-dual(self-dual) curvature F , i.e. F = − ∗ F ( F = ∗ F ). Furthermore, by the Cauchy-Schwarz inequality, the magnitude square | ˜ p | of the four-dimensional momentum isthen bounded from above by | ˜ p | = X v ( ι v ˜ p ) ≤ Z T Tr( F ∧ ∗ F ) Z T Tr ( E ∧ ∗ E ) = | k | Z T Tr ( E ∧ ∗ E ) , (3.3)9here v is summed over an orthonormal basis of tangentvectors to T . So for fixedpositive k and fixed ˜ p , the energy is bounded from below by H ≥ kg + g k | ˜ p | + O ( g ) . (3.4)This agrees with the magnitude | p | of the five-dimensional momentum p decomposedas in (2.25) into components ˜ p and k along T and S respectively, and thus confirmsthe correctness of this decomposition.When k = 0, we instead have the inequality H ≥ Z T Tr (cid:18) g ι v F ∧ ∗ ι v F + g E ∧ ∗ E (cid:19) = ι v ˜ p + 12 Z T Tr (cid:18) g ι v F − gE (cid:19) ∧ ∗ (cid:18) g ι v F − gE (cid:19) ≥ ι v ˜ p, (3.5)where v is an arbitrary tangent vector on T . So the energy bound in this case is H ≥ | ˜ p | , (3.6)again in agreement with (2.25). For given values of the characteristic classes k and m with k > k < M k,m denote the corresponding moduli space of (anti) instanton solutions. Inthe weak coupling limit g →
0, the theory formally reduces to a supersymmetricquantum mechanical model describing a particle of mass µ = g moving on M k,m .(The zero-point fluctuations of the modes of the connection one-form A transverseto M k,m together with the contributions of the scalar fields φ cancel against thecontributions of the fermionic fields ψ , since the corresponding eigenvalues agree[5].) But this model is not easy to analyze, one reason being that, according to(3.4), the energy gap between states of different momenta ˜ p vanishes in the weakcoupling limit.We will thus not attempt a complete treatment of this quantum mechanics, butcontent ourselves with a few remarks on its degrees of freedom. In the weak couplinglimit g →
0, the terms in the Yang-Mills Lagrangian involving the scalar fields φ and the spinor fields ψ become Z T Tr (cid:16) ˙ φ ∧ ∗ ˙ φ − Dφ ∧ ∗ Dφ (cid:17) . (3.7)and Z T Vol T Tr (cid:16) ¯ ψ ( γ ˙ ψ + γ · Dψ ) (cid:17) (3.8)respectively. Here γ and γ denote the time-like and spatial Dirac matrices. At ageneric point A in M k,m , the scalar Laplacian∆ = ∗ D ∗ D : Γ(ad( P )) → Γ(ad( P )) (3.9)10s strictly positive, so we need not take the scalar fields into account. For positive k , there are no negative chirality zero modes for the spatial Dirac operator γ · D : Γ( S ⊗ ad( P )) → Γ( S ⊗ ad( P )) . (3.10)The number of positive chirality zero modes is thus given by the Atiyah-Singer indextheorem as c (ad( P )). (The signature and Euler characteristic of T both vanish.)Let now ψ be a positive chirality zero-mode, i.e. a solution to the spatial Diracequation γ · Dψ = 0 . (3.11)We can then construct two tangent vectors δA to M k,m at the point A . In back-ground Coulomb gauge D · δA = 0, they take the form δA = ¯ ηγψ, (3.12)where ¯ η is an arbitrary constant spinor of positive chirality. Indeed, the inducedvariation δF of the field strength δF = ¯ η ( γ ∧ D ) ψ (3.13)is easily seen to be anti-self dual by using the chirality condition on ¯ η and the Diracequation (3.11). In fact, M k,m is a curved hyperk¨ahler manifold of real dimensiondim M = p (ad( P )) = 2 c (ad( P )) , (3.14)so, since there are two linearly independent choices of ¯ η , these δA span the wholetangent space of M k,m at A .We may construct two distinguished fermionic zero modes ψ as the Cliffordproduct of F and an arbitrary constant spinor κ of positive chirality: ψ = F κ. (3.15)Indeed, it follows from the Bianchi identity DF = 0 and the anti self-duality of F that such a ψ fulfills (3.11). The corresponding tangent vectors δA to M k,m at A are δA = ι v F, (3.16)where the constant vector v on T is given by v = ¯ ηγκ. (3.17)Using the Bianchi identity DF = 0, the induced variation of the field strength comesout to be given by the Lie derivative of F along v : δF = L v F, (3.18)so this tangent vector δA represents a translation of the instanton configuration on T . Such translations generically act non-trivially on M k,m , which thus can be seenas a fibration with T fiber over a hyper K¨ahler manifold M ′ k,m of real dimension4( nk − M k,m might be empty [6].) The wavefunction of a BPS state of four-dimensional momentum ˜ p is (locally) constant on M ′ k,m and depends on the fiber coordinates ˜ x ∈ T = R / ˜Λ as exp(˜ p · ˜ x ). Because ofthe quantization of the fermionic zero-modes, this will in fact be a multi-componentwave function. 11 Flat connections
A bundle P of vanishing instanton number, i.e. k = 0 , (4.1)necessarily has a magnetic ’t Hooft flux m ∈ H ( T , C ) obeying m · m = 0 . (4.2)For such a bundle, the potential (2.42) attains its minimum value V = 0 for con-nections A with vanishing field strength F = 0 . (4.3)In the weak coupling limit g →
0, the theory thus localizes on a neighborhood ofthe moduli space M ,m of such flat connections.The moduli spaces M ,m are described in [2] for the A - and D -series. (For a moregeneral theoretical discussion, which however focuses on bundles over T rather than T , see [7].) In general, M ,m is a disconnected sum of several components: M ,m = [ α M α . (4.4)(The range of the label α depends on the magnetic ’t Hooft flux m .) Each component M α is of the form M α = ( T r α × T r α × T r α × T r α ) /W α (4.5)for some number r α known as the rank of the component, and some discrete group W α , which acts on the torus T r α .The simplest example of such a component is obtained for an arbitrary group G adj by considering a topologically trivial bundle P , i.e. not only k = 0 but also m = 0. There is then a component M of M ,m for which r equals the rankof G adj , the torus T r is a maximal torus of G adj , and the discrete group W is thecorresponding Weyl group. But even for such a trivial bundle P , there are in generalalso other components M α of M ,m .Returning now to the general case, the moduli space M ,m of flat connectionsmay be parametrized by the holonomies U ˜ x ∈ Hom( π ( T ) , G adj ) (4.6)of the connection D based at some point ˜ x ∈ T modulo simultaneous conjugationby elements of G adj . (Such conjugations represent the connected component Ω ofthe group ˜Ω of gauge transformations.) A concise way of describing the holonomy U ˜ x is to evaluate it on the basis elements λ i , i = 1 , . . . , ≃ H ( T , Z ) ≃ π ( T ) . (4.7)12he resulting group elements U ˜ x [ λ i ] commute in G adj . But an arbitrary lifting ˆ U ˜ x [ λ i ]of them to the simply connected group G is in general only almost commuting inthe sense that ˆ U ˜ x [ λ i ] ˆ U ˜ x [ λ j ]( ˆ U ˜ x [ λ i ]) − ( ˆ U ˜ x [ λ j ]) − = m ij , (4.8)where m ij ∈ C denotes the corresponding component in the expansion of the mag-netic ’t Hooft flux m .Letting ˜ x vary over T and evaluating the holonomy U ˜ x on a fixed cycle ˜ λ ∈ ˜Λgives a covariantly constant section U ˜ x [˜ λ ] of Ad( P ), i.e. DU ˜ x [˜ λ ] ≡ dU ˜ x [˜ λ ] + U ˜ x [˜ λ ] A − AU ˜ x [˜ λ ] = 0 . (4.9)In other words, a large gauge transformation parametrized by U ˜ x [˜ λ ] leaves the con-nection one-form A invariant: A → U ˜ x [˜ λ ] A ( U ˜ x [˜ λ ]) − + dU ˜ x [˜ λ ]( U ˜ x [˜ λ ]) − = A. (4.10)Finally, we note that the topological class of this gauge transformation is given by(2.29). The quantization of the theory on a component M α of positive rank r α >
0, is rathersubtle. At a general point on this component, the holonomies U spontaneously breakthe gauge group G adj to a subgroup of rank r α . Generically, the Lie algebra h ofthis unbroken subgroup is abelian, but in general it may be of the form h ≃ s ⊕ u (1) r , (4.11)for some number r , 0 ≤ r ≤ r α , and some semi-simple algebra s of rank r α − r . Givensuch an algebra h , we let M h denote the closure of the corresponding subspace of M ,m . In a neighborhood of M h , the degrees of freedom corresponding to s aremodeled by supersymmetric quantum mechanics with 16 supercharges based on theLie algebra s [8]. The latter theory has no mass-gap, but is believed to have a finitedimensional linear space V s of normalizable zero-energy states [9][10]. An explicitconstruction of these quantum mechanical states is notoriously difficult, essentiallybecause the system has a potential with flat valleys extending out to infinity in fieldspace. But this property of the supersymmetric quantum mechanics implies thatthe Yang-Mills theory has dim V s normalizable zero-energy states supported near M h . The validity of this picture was confirmed in rather much detail in [2].Somehow, there should also be a spectrum of BPS states with non-zero momen-tum ˜ p supported near M h , but it is not clear to the present author precisely howthis could be determined. A better understanding of this issue would certainly bevery useful, and we hope that further progress can be made, but we will not pursueit here. 13 .2 Isolated flat connections The situation is better for an isolated flat connection D , i.e. a component M α of M of rank r α = 0. As we will now explain, fluctuations around D can be reliablyanalyzed by semi-classical methods in the weak coupling limit. We thus expand theconnection one-form A around the connection one-form A of D as A = A + ga, (4.12)where the fluctuation a is a global section of Ω ( T ) ⊗ ad( P ).We let Γ(ad( P )) denote the space of L -sections of the vector bundle ad( P ) withrespect to the sesqui-linear inner product( α, β ) = Z T Tr( ¯ α ∧ ∗ β ) . (4.13)Our first task is to define a convenient basis of this space. By flatness of D , thecovariant derivatives i D v : Γ(ad( P )) → Γ(ad( P )) (4.14)commute with each other other for different constant vector fields v ∈ H ( T , R ) on T . We can then introduce an orthonormal basis b ˜ p of Γ(ad( P )) of their simultaneouseigensections. The label ˜ p can be thought of as taking its values in a subset ˜ P of H ( T , R ), and is defined so that the corresponding eigenvalues are 2 π ˜ p [ v ]: i D v b ˜ p = 2 π ˜ p [ v ] b ˜ p (4.15)(We are suppressing any further labels that are possibly needed to distinguish dif-ferent sections with the same eigenvalues. But for the A -series, which we will treatin more detail in the next section, there is in fact no such degeneracy.) The complexconjugate of b ˜ p is ( b ˜ p ) ∗ = b − ˜ p . The b ˜ p are also eigensections of the adjoint actionof the holonomy U ˜ x evaluated on a cycle ˜ λ ∈ H ( T , Z ) ≃ ˜Λ based at some point˜ x ∈ T : U ˜ x [ λ ] b ˜ p ( U ˜ x [ λ ]) − = ˜ z [ λ ] b ˜ p , (4.16)where the eigenvalue ˜ z [ λ ] is a phase determined by˜ z = exp(2 πi ˜ p ) ∈ H ( T , U (1)) . (4.17)The set ˜ P of possible values of ˜ p is actually best described by giving the subset˜ Z ⊂ H ( T , U (1)) of possible values of ˜ z . The cardinality of this set equals thedimension of the group G , and, since the flat connection D is isolated, it does notcomprise the trivial element of H ( T , U (1)). We then have˜ P = { ˜ p ∈ H ( T , R ) | exp 2 πi ˜ p ∈ ˜ Z } . (4.18)The fluctuation a in (4.12) can now be expanded as a = X ˜ p ∈ ˜ P a ˜ p b ˜ p , (4.19)14ith some coefficients a ˜ p that are vectors on T . In background Coulomb gauge D · a = 0, a ˜ p is constrained by the condition ˜ p · a ˜ p = 0, and thus takes its values in a3-dimensional linear space transforming as ⊕ ⊕ under the Sp(4) R -symmetry.Similarly, we expand the scalar fields φ and the spinor fields ψ as φ = X ˜ p ∈ ˜ P φ ˜ p b ˜ p ψ = X ˜ p ∈ ˜ P ψ ˜ p b ˜ p . (4.20)Here the coefficients φ ˜ p are a set of spatial scalars transforming in the represen-tation, and the coefficients ψ ˜ p are a set of spatial spinors transforming in the ⊕ representation of Sp(4). In the g → g Z T Tr( ˙ A ∧ ∗ ˙ A − F ∧ ∗ F ) = X ˜ p ∈ ˜ P ( ˙ a ˜ p · ˙ a − ˜ p + ˜ p · ˜ pa ˜ p a − ˜ p ) Z T Tr( ˙ φ ∧ ∗ ˙ φ − Dφ ∧ ∗ Dφ ) = X ˜ p ∈ ˜ P ( ˙ φ ˜ p ˙ φ − ˜ p + ˜ p · ˜ pφ ˜ p φ − ˜ p ) Z T Vol T Tr( ¯ ψγ ˙ ψ + ¯ ψγ · Dψ ) = X ˜ p ∈ ˜ P ( ψ ˜ p γ ˙ ψ − ˜ p + ψ ˜ p ˜ p · γψ − ˜ p ) . (4.21)So for each ˜ p ∈ ˜ P , there is a set of bosonic harmonic oscillators a ˜ p and φ ˜ p withtemporal frequency | ˜ p | transforming in the representation B = ⊕ ⊕ ⊕ , (4.22)and a set of fermionic harmonic oscillators ψ ˜ p with the same frequency transformingin the representation F = ⊕ . (4.23) We will now quantize the fluctuations around an isolated flat connection D . Let |Ai denote the corresponding vacuum state of vanishing energy and momentum.Acting on this state with a string of bosonic and fermionic creation operators of theharmonic oscillators associated to a single value ˜ p ∈ ˜ P builds up a (pre-) Hilbertspace H ˜ p . The complete Hilbert space H is (the Hilbert space completion of) thetensor product H = O ˜ p ∈ ˜ P H ˜ p . (4.24)There is a further subtlety that needs to be considered: If ˜ λ ∈ H ( T , Z ) is suchthat m [˜ λ ] = 0, a gauge transformation parametrized by the holonomy U ˜ x [˜ λ ] belongsto the connected component Ω of the group of gauge transformations ˜Ω. We mustthen project the Hilbert space H onto the subspace H inv of states that are invariant15nder such transformations. According to (4.16), this is a non-trivial projection ifthere are ˜ z ∈ ˜ Z such that the phase ˜ z [˜ λ ] is non-trivial. However, as we will see inthe next section, this does not happen for the A -series.In the g → p ∈ ˜ P adds an amount | ˜ p | to the energy and an amount ˜ p to the momentum. The totalenergy E and momentum ˜ p of a state constructed by acting on the vacuum |Ai witha string of creation operators labeled by ˜ p , . . . ˜ p k ∈ ˜ P are thus E = | ˜ p | + . . . + | ˜ p k | ˜ p = ˜ p + . . . + ˜ p k . (4.25)If ˜ p , . . . , ˜ p k are all parallel vectors, the state is light-like in the sense that E = | ˜ p | .As discussed in the introduction, this is a necessary condition for it to be BPS. Butit is not sufficient: For a non-zero value of the coupling g , the above expression forthe momentum ˜ p of the state will still be correct, but the energy E might be higherthan the value stated above so that E > | ˜ p | . Indeed, already the known termsin the Lagrangian can be expected to generate such corrections at higher ordersin perturbation theory, and there are presumably further unknown terms in theLagrangian of arbitrarily high powers in the fields multiplied by appropriate powersof g , which may give further contributions. So most of these states, while light-likeat tree level, can actually be expected to be non-BPS.To gain more information about which states actually are BPS, we need toconsider the transformation properties under the Sp(4) R -symmetry. For a fixed˜ p ∈ ˜ P , the states of H ˜ p with total momentum k ˜ p for a positive integer k transformin the representation Z k given by Z k = F a ⊗ B ks ⊕ F a ⊗ B k − s ⊕ . . . ⊕ F a ⊗ B k − s . (4.26)Here the subscripts a and s denote the alternating and symmetric products of thebosonic and fermionic representations respectively. (If k <
8, the terms with nega-tive powers of B are absent.) The dimension of this representation isdim Z k = (cid:18) (cid:19)(cid:18) k + 77 (cid:19) + (cid:18) (cid:19)(cid:18) k + 67 (cid:19) + . . . + (cid:18) (cid:19)(cid:18) k − (cid:19) = 16315 (132 k + 154 k + 28 k + k ) . (4.27)We will give a more precise description of Z k in the next subsection, but at themoment we just note that by supersymmetry, Z k = ( B ⊕ F ) ⊗ W k (4.28)for some representation W k . We decompose W k = ( B ⊕ F ) ⊗ R ′ k ⊕ R k , (4.29)where the representations R ′ k and R k are chosen so that R ′ k is as large as possible.Thus Z k = ( B ⊕ F ) ⊗ R ′ k ⊕ ( B ⊕ F ) ⊗ R k , (4.30)16he states transforming according to the second term must be BPS. Presumablymost of the states transforming according to the first term are non-BPS, unlesssome additional symmetry or other principle that we have not taken into accountforces them to be BPS. Somewhat more cautiously, our conclusion might be phrasedas follows: While the BPS states are not necessarily given by the representation R k appearing in (4.30), at least the true BPS representation agrees with R k wheninterpreted as an element in the quotient ring R C (Sp(4)) /I. (4.31)Here R C (Sp(4)) is the representation ring of Sp(4), and I is the ideal generated bythe representation B ⊕ F .Sofar, we have considered the states of a single factor H ˜ p for ˜ p ∈ ˜ P . A stateof total momentum ˜ p ∈ H ( T , R ) (which is not necessarily an element of ˜ P ) inthe complete Hilbert space H is clearly non-BPS if oscillators of different values˜ p , . . . , ˜ p k ∈ ˜ P are excited, since such a state is not even light-like at tree level. Butif only k oscillators of momentum ˜ p/k ∈ ˜ P are excited, we can apply our previousreasoning and argue that (at least) a set of states transforming as ( B ⊕ F ) ⊗ R k areBPS. Taking all possible values of k into account, we thus conjecture that the BPSstates of some fixed momentum ˜ p ∈ H ( T , R ) transform as ( B ⊕ F ) ⊗ R where R = X k | ˜ p R k . (4.32)Here the sum is over all positive integers k that divide ˜ p in the sense that ˜ p/k ∈ ˜ P .Again, we have a rigorous statement in the quotient ring (4.31). Sp(4) representation theory
We will now work out the precise form of the representation R k for a given value of k . An irreducible representation V k ,k of the Sp(4) R -symmetry group is labelledby two non-negative integers k and k (the Dynkin labels). By the Weyl dimensionformula, the dimension of this representation isdim V k ,k = 16 (1 + k )(1 + k )(2 + k + k )(3 + 2 k + k ) . (4.33)We have e.g. V , = V , = V , = . (4.34)In particular, the symmetric powers of the representation are given by ks = (cid:26) V , ⊕ V , ⊕ . . . ⊕ V k, for even kV , ⊕ V , ⊕ . . . ⊕ V k, for odd k. (4.35)17t follows that B ks = k M k =0 N k − k × V k , , (4.36)where the multiplicities N l are given by N l = (cid:26) (24 + 34 l + 15 l + 2 l ) for even l (21 + 34 l + 15 l + 2 l ) for odd l. (4.37)After some more work, it transpires that the representations R k and R ′ k definedabove are (almost) given by R k = 3 × V k − , ⊕ V k − , ⊕ k − M k =0 (4 k − k − × V k , ⊕ k − M k =0 (2 k − k − × V k , (4.38)and R ′ k = M k =0 ,k − N k − − k × V k , . (4.39)Their dimensions are dim R k = 115 (8 k + 5 k + 2 k ) (4.40)and dim R ′ k = 17! ( k − k − k − k ( k + 1)( k + 2)( k + 3) . (4.41)These representations obey equation (4.30), but not quite the requirement that R ′ k be as large as possible. In fact, for k ≥ k − V , to R ′ k and remove the same number of representations B ⊕ F = 3 × V , ⊕ V , ⊕ × V , (4.42)from R k . I do not know of any argument to prove that these terms indeed correspondto BPS states, and thus should be included in R k rather than R ′ k , except that theformulas (4.38) and (4.39) look simpler that way. A -series To be able to use the analysis of the previous section, we must consider a gaugegroup G adj and a magnetic ’t Hooft flux m ∈ H ( T , C ) such that the correspondingmoduli space M ,m of flat connections only consists of components M α of rank r α = 0, i.e. of isolated flat connections. This will restrict us to the A -series andprime values of the magnetic ’t Hooft flux. But again, one could hope to eventually18e able to analyze also components of positive rank, so that arbitrary groups and’t Hooft fluxes could be treated.Consider thus the case Φ = A n − for some positive integer n . The correspondingsimply connected group G = SU( n ) consists of unimodular n × n matrices. Its centersubgroup C consists of matrices of the form exp(2 πic/n )1l n , where c ∈ Z n ≃ C . Theinner product on C is given by c · c ′ = 1 n cc ′ ∈ R / Z , (5.1)for c, c ′ ∈ Z n .For m ∈ H ( T , Z n ), we define the SL ( Z ) invariant u as the greatest commondivisor of the components of m and n : u = gcd( m , . . . , m , n ) . (5.2)We can then write m = um ′ (5.3)for some m ′ ∈ H ( T , Z n/u ), and define a further SL ( Z ) invariant m ′ · m ′ ∈ Z n/u as m ′ · m ′ = m ′ m ′ + m ′ m ′ + m ′ m ′ . (5.4)One can show (see e.g. [2]) that all the connected components M α of the modulispace M ,m of flat connections have the same rank r α given by r α = u (cid:30) n/u gcd( m ′ · m ′ , n/u ) − , (5.5)provided that this quantity is an integer. (Otherwise, M ,m is empty.) So we get r α = 0 if e.g. u = 1 (5.6)and m · m = 0 . (5.7)In fact, there is then a unique isolated flat connection A . (There are, up to si-multaneous conjugation, n different quadruples ˆ U ˜ x [ λ i ] ∈ G , i = 1 , . . . , U ˜ x [ λ i ] ∈ G adj .) We can describe this connection via the adjoint action of itsholonomies on the space of sections Γ(ad( P )) as in (4.16). The set ˜ Z in which ˜ z takes its values can then be regarded as a subset of the cohomology group H ( T , Z n ) ≃ H ( T , n Z / Z ) ⊂ H ( T , R / Z ) ≃ H ( T , U (1)) . (5.8)In fact, ˜ Z = n ˜ z ∈ H ( T , Z n ) (cid:12)(cid:12)(cid:12) ˜ z = 0 , ˜ z ∧ m = 0 o . (5.9)Note that the cardinality of ˜ Z equals the dimension n − G = SU( n ). It is notdifficult to check that for a given value of ˜ z ∈ ˜ Z , there exists a corresponding value19f the electric ’t Hooft flux e ∈ H ( T , Z n ), unique modulo terms of the form m ∧ t for t ∈ H ( T , Z n ), such that˜ z = m · e ∈ H ( T , n Z / Z ) . (5.10)According to (2.31), it is then consistent to declare that the corresponding harmonicoscillators are invariant under topologically trivial gauge transformations, and trans-form under ‘large’ gauge transformations according to e . There is thus no need toproject onto states invariant under gauge transformations in the connected compo-nent Ω . The four-dimensional momentum ˜ p of the harmonic oscillators takes itsvalues in the set ˜ P defined in (4.18):˜ P = (cid:26) ˜ p ∈ H ( T , n Z ) (cid:12)(cid:12)(cid:12) n ˜ p = 0 , ( n ˜ p ) ∧ m = 0 (cid:27) , (5.11)where n ˜ p ∈ H ( T , Z ). n prime The conditions on u and m ′ · m ′ are not as restrictive as they may seem: For eachpossible value of m , there are n different values of e ∈ H ( T , C ), and the resultingvalues of f = m + e actually give representatives of many SL ( Z ) orbits of f . E.g.for n prime (so that u = 1), there are n + 1 SL ( Z ) orbits of m :orbit cardinality m = 0 1 m = 0 , m · m = 0 n + n − n − m · m = 1 /n n − n . . . . . .m · m = ( n − /n n − n n (5.12)But there are only 3 SL ( Z ) orbits of f :orbit cardinality f = 0 1 f = 0 , f · f = 0 n + n − n − f · f = 0 n − n − n + n n (5.13)The relationship between these orbits is m = 0 m · m = 0 m · m = 1 /n . . . m · m = ( n − /nf = 0 1 . . .f · f = 0 n − n . . .f · f = 0 n − n n . . . (5.14)20here the entries denote the number of e -values for a given m in the correspondingSL ( Z ) orbit that gives an f in the corresponding SL ( Z ) orbit. So although onlythe m = 0, m · m = 0 orbit has an isolated flat connection, this makes calculationspossible for all f , except the single value f = 0.Finally, we consider the action of SL ( Z ) on the set of pairs ( f, p ) obeying (1.6):For the f = 0 orbit, we have p ∈ H ( T , Z ). There is then one orbit for each positiveinteger value of gcd( p ) = gcd( p , . . . , p ) . (5.15)But as discussed above, we have no results for the corresponding BPS spectrum.Also for the f = 0, f · f = 0 orbit, we have p ∈ H ( T , Z ). But here there are twoorbits for each positive integer value of gcd( p ), distinguished by f ∧ ( p/ gcd( p )) ∈ H ( T , C ) (5.16)being zero or non-zero. In both cases, the BPS spectrum could be determined asdescribed above by choosing a decomposition (2.4) of Λ such that the component k in the decomposition (2.25) of p vanishes. In fact the BPS spectrum is emptywhen f ∧ ( p/ gcd( p )) = 0. For the f · f = 0 orbit, we consider np ∈ H ( T , Z )instead of p ∈ H ( T , n Z ). There is then one orbit for each finite value of gcd( np ) =gcd( np , . . . , np ). (Necessarily f ∧ ( np/ gcd( np )) = 0 in this case.) 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