Supersymmetric Noether Currents and Seiberg–Witten Theory

Annals of Physics 290, 156–183 (2001) doi:10.1006/aphy.2001.6151, available online at http://www.idealibrary.com on

Supersymmetric Noether Currents and Seiberg–Witten Theory1 A. Iorio Department of Physics, University of Salerno, Via S. Allende, 84081 Baronissi (SA), Italy; and School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland E-mail: [email protected]; [email protected]

L. O’Raifeartaigh School of Theoretical Physics, Dublin Institute for Advanced Studies,10 Burlington Road, Dublin 4, Ireland E-mail: [email protected]

and S. Wolf Institut de Physique Th´eorique, BSP, 1015 Lausanne, Switzerland E-mail: [email protected] Received October 4, 2000

The purpose of this paper is twofold. The first purpose is to review a systematic construction of Noether currents for supersymmetric theories, especially effective supersymmetric theories. The second purpose is to use these currents to derive the mass-formula for the quantized Seiberg–Witten model from the supersymmetric algebra. We check that the mass-formula of the low-energy theory agrees with that of the full theory (in the broken phase). °C 2001 Academic Press

1. INTRODUCTION Most aspects of supersymmetry (Susy) have by now been extensively developed. However, supersymmetric Noether currents do not appear to have been treated in a systematic manner in the literature. This is a pity because many aspects of Susy could be clarified by considering the currents, an example being the computation of the mass formula in the Seiberg–Witten (SW) model [1] (see also [2] for a review). The Susy-Noether currents correspond to space-time symmetries and hence are by no means trivial, as discussed in more detail below. Furthermore, in models, such as the SW model, the Susy theories are effective ones, and in effective theories the Noether currents have particular difficulties, as is also discussed below. So far, there does not seem to be any systematic approach to constructing the currents. In two previous letters [3, 4], an attempt was made to remedy this situation, using the SW model as a non-trivial test case. The purpose of the present paper is to give a more detailed review of this more systematic construction of Susy-Noether currents and to apply it to the SW model. 1 Sadly, while this paper was under the process of publication, Lochlainn O’Raifeartaigh passed away. A.I. dedicates this work to the memory of his maestro and will always remember his genious, his patience, and his dedication to physics.

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From the point of view of the Noether currents the essential feature of the SW model is the existence of a central charge Z , which is obtained as an anti-commutator of the Susy charges and provides the mass formula. Thus the Noether currents are directly connected to the mass-spectrum. The mass formula is given by √ (1) M = |Z | = 2|n e a + n m a D | and it is of paramount importance for the exact solution of the quantum theory. In Eq. (1), n e and n m are the electric and magnetic quantum numbers, respectively, and a and a D are the vev’s of the scalar field and its dual, surviving the Higgs phenomenon in the spontaneously broken phase SU(2) → U(1). Classically a D = τ a, where τ = τ R + iτ I = θ/2π + i4π /g 2 is the complex coupling, g is the renormalizable SU(2) coupling, and θ is the CP violating vacuum angle. In the quantum theory a D is identified with ∂F/∂a, where F is the holomorphic prepotential. In a nutshell the important features of Z in the SW model are: (1) It allows for spontaneous symmetry breaking (SSB) of the gauge symmetry within the Susy theory; (2) It gives the complete and exact mass spectrum for the elementary particles as well as the topological excitations; (3) It exhibits an explicit SL(2, Z) duality symmetry, which is not a symmetry of the theory in the Noether sense; (4) In the quantum theory it is the most important global piece of information at our disposal on the space of the gauge inequivalent vacua (the moduli space Mq ). The first two results in the list date back to the early days of the mass formulae. In fact, in their classic paper Witten and Olive [5] proved (a simplified version of) Eq. (1) for a semi-classical N = 2 Susy Georgi–Glashow model with gauge group SO(3). In contrast, the results (3) and (4) are the crucial ingredients to solve the quantum corrected theory. It is then not surprising that, following the paper of Seiberg and Witten, there have been several attempts to compute the mass formula for the quantum case [6, 7]. In fact, in [6, 7] there is no direct proof of this formula but only a brief check that the bosonic terms of the SU(2) √ high-energy effective Hamiltonian for a magnetic monopole admit a BPS lower bound given by 2|F (1) (a)| (here and in the following by F (n) we mean the nth derivative of F ). A similar type of BPS computation, only slightly more general, has been performed in [6]. There the authors considered again the SU(2) high-energy effective Hamiltonian H but this time for a dyon, namely also the electric field contribution was considered. Of course the topological term Htop is the lower of the fields satisfy bound for H , the inequality H ≥ Htop being saturated when the configurations √ the BPS equations [8]. Thus the authors in [6] found that Htop = 2|n e a + n m F (1) (a)|; therefore they identified the r.h.s. with the modulus of the central charge |Z |. A common feature of all those computations is that they only consider the bosonic contributions to |Z | and this is rather unsatisfactory since, due to Susy, one might expect fermionic terms to play a role. Furthermore the computations briefly described above are rather indirect. The complete and direct computation has to involve the Noether supercharges constructed from the correspondent Lagrangian. Two independent complete computations appear now in [3, 4]. The aim of this paper is to show what we have learned about Susy-Noether currents by performing those computations. Susy-Noether currents present quite serious difficulties due to the following reasons. First Susy is a space-time symmetry; therefore the standard procedure to find Noether currents does not give a unique answer. A term, often called improvement, has to be added to the term one would obtain for an internal symmetry. The additional term is not unique; it can be fixed only by requiring the charge to produce the Susy transformations one starts with, and for non-trivial theories it is by no

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means easy to compute. Second the linear realization of Susy involves Lagrange multipliers called dummy-fields, which of course have no canonical conjugate momenta. On the other hand, if dummyfields are eliminated to produce a standard Lagrangian, then the variations of the fields are no longer linear and the Noether currents are no longer bilinear. A further problem is that the variations of the fields involve space-time derivatives and this happens in a fermion-boson asymmetric way (the variations of the fermions involve derivatives of the bosons but not conversely). This implies some double-counting solved only by a correct choice of the current. We have solved these problems by implementing a canonical formalism in the different cases under consideration. First we construct the Noether currents for the classical limit of the U(1) sector of the theory. In this case Susy is linearly realized regardless of the dummy-fields, no complications arising in the effective case are present, and the fields are non-interacting. When the procedure is clearly stated in this case we move to the next level, the effective U(1) sector, and we see what is left from the classical case and what is new. Now the currents are very different and, for instance, we cannot use the classical formula to overcome the above mentioned fermi-bose asymmetry in the transformations of the fields. Nevertheless the constraints imposed by Susy are strong enough to force the effective centre to have an identical form as the classical one,2 proving the SW conjecture that a D = F (1) (a). The last step is to consider the SU(2) sector. There we find that the canonical procedure implemented in the U(1) sector does not need any further change and our analysis confirms that U(1) is the only sector that contributes to the centre. Naturally future work would be the generalization of our results to any Susy theory, possibly to obtain a general Susy–Noether Theorem. The task is by no means easy due to the above mentioned problems and other difficulties. For instance, as is well known, for ordinary space-time symmetries the energy momentum tensor Tµν can be obtained by embedding the theory in a curved space-time with metric gµν , defining Tµν = δS/δg µν and then taking the flat-space limit. In Susy the situation is much more complicated because the embedding has to be in a curved superspace, thus pointing to supergravity which is much more involved than simple gravity. One may also want to investigate the (non-holomorphic) next-to-leading order term in the superfield expansion of the SW effective Action [9]. The presence of derivatives higher than second spoils the canonical approach and the Noether procedure cannot be trivially applied. The interest here is to understand how the lack of canonicity and holomorphy (a crucial ingredient for the solution of the SW model) affects the currents and charges, and therefore the whole theory itself. Of course this analysis is somehow more general and it could help to understand how to handle the symmetries of full effective Actions.

2. NOETHER CURRENTS FOR SPACE-TIME SYMMETRIES It is a well-known fact that the Noether theorem does not give a unique conserved current when applied to space-time symmetries. This can be seen immediately by defining as symmetry of a field theory R 4 the transformation of the fields and/or of the coordinates leaving invariant the Action A = d xL(8i , ∂8i ), where 8i are fields of arbitrary spin and the invariance is obtained algebraically, i.e., without the use of the Euler–Lagrange equations for the fields (off-shell). Thus, this definition clearly distinguishes the two cases of (rigid) internal symmetry, δL = 0, and space-time symmetry, δL = ∂ µ Vµ . In the first case no ambiguity arises and the on-shell conserved current is Nµ = 5iµ δ8i , where 5iµ = ∂L/∂(∂ µ 8i ). In the second case the current is Jµ = Nµ − Vµ ,

(2)

Of course this does not mean that quantum corrections are not present, as is expected to be the case for N = 4, but only that having a dictionary we could replace classical quantities by their quantum correspondents with no other changes. 2

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and is clearly not unique since any other Vµ0 = Vµ + ∂ ν W[µν] still satisfies the requirement above introduced. Often the terms ∂ ν W[µν] have to be added ad hoc, by requiring the charges to reproduce the symmetry transformations or, for instance, to fit the current into a Susy multiplet. It is interesting to remember here that there are two Noether theorems [10], the first for rigid symmetry, and the second for local symmetry. The theorem we used in the previous discussion is the first theorem, where the parameter of the infinitesimal transformation ² is taken to be constant. On the other hand one can also produce a Noether current by letting ² become local and varying the Action with respect to this parameter one obtains Z (3) δlocal A = − d 4 x J µ ∂µ ², R where a surface term d 4 x∂µ (J µ ²) is discarded. This alternative method relies anyway on the first Noether theorem since one is entitled to identify J µ with a conserved current only if the Action is invariant under the rigid transformation. Or, alternatively, it is sufficient to remember that in general even if the Action is locally symmetric the (improper, according to Noether’s terminology) conservation law descends from the rigid symmetry alone. Another point is that in general the currents in Eq. (2) and in Eq. (3) can differ by a ∂ν W [µν] type of term. Thus only when internal symmetries are concerned (Vµ = 0) do the two alternative procedures give the same answer. On the contrary for space-time symmetries this is not the case. Another problem is the canonical structure of the current.3 For space-time currents the problem is to express Vµ canonically and this is exactly where the issue of the improvement terms comes into play (see also [11]). In this paper we want to address the question of Noether currents for a highly non-trivial space-time symmetry: Susy. The difficulties of Susy–Noether currents are • Susy is a superspace-time symmetry. Thus it shares with standard space-time symmetries all the problems above outlined and, furthermore, the non-commutative structure of the superspace makes it highly non-trivial to obtain the currents by embedding the theory in a curved superspace. • Linear realization of Susy involves dummy-fields. This poses some delicate questions as when to eliminate those Lagrange multipliers and how to treat them in a canonical setting. • Susy variations involve space-time derivatives in a way not symmetrical with respect to fields of different spin. This means that Susy maps bosonic fields into fermionic fields and fermionic fields into conjugate momenta of the bosonic fields. This problem causes some double counting essentially solved by partially integrating in the fermionic sector. For the SW model, the situation is even more complicated due to the following problem that closes the list of difficulties encountered in the computations illustrated later: • Effective Lagrangians, even non-Susy. As we shall see, in SW theory we have to deal with effective Lagrangians and renormalization does not constraint the fermionic terms to be bilinear and the coefficients of the kinetic terms to be constant. As a matter of fact, the SW effective Lagrangian is quartic in the fermionic fields and has coefficients of the kinetic terms that are non-polynomial functions of the scalar field. Because of this, the Noether procedure requires a great deal of care. For example, we shall encounter equal time commutations (Poisson brackets, see Appendix A.2.) between fermions and bosons such as {ψ, πφ } = f (φ)ψ

from

{πψ, ¯ πφ } = 0,

(4)

R By canonical here we mean that we can write the current as 5µ δ8; therefore the charge is Q = d 3 x5δ8, where 0 5 = 5 . Thus we can immediately generate the transformation of the field 8 by acting on it with its canonically conjugate momentum 5. 3

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where f (φ) is a non-polynomial function of the scalar field related to the coefficient of the kinetic terms. This reflects the difficulty of treating Noether currents in a quantum context [12, 13]. Let us note here that in SW theory most of the information on the quantum corrections is contained in the expressions of the dummy-fields on-shell. Thus by keeping them explicitly in our Lagrangian we can implement the Noether procedure in a similar fashion in both cases, classical and quantum. The difference will show up when we have to explicitly write down the currents that generate the transformations, namely when we need to express our charges in terms of canonical variables. These problems are addressed in our analysis and we can give here a “working recipe” we have found: • The Susy–Noether charge that correctly reproduces the Susy transformations is the one obtained from J µ = N µ − V µ where δL = ∂µ V µ and V µ has to be extracted as it is; i.e., no terms like ∂ν W [νµ] have to be added. Furthermore V µ has to be expressed in terms of momenta and Susy variations of the fields. • The variation δL has to be performed off-shell by the definition of symmetry. Nevertheless the dummy-fields, and only them, automatically are projected on-shell. • The full current J µ contains terms of the form πψ δψ, that generate the fermionic transformations. The same term can be written as πφ δφ + · · ·; therefore it also generates the bosonic transformations. The situation is more complicated for effective theories. • When the effective theory allows for a canonical description (for instance in SW this happens only at the first order in the momentum expansion of the effective Action), the canonical commutation relations are preserved even if some of the usual assumptions, such as that at equal time all fermions and bosons commute, are incorrect. Noether currents at the effective level do not exhibit the same simple expressions as at the classical level. Of course a recipe is not a final solution and a lot of work has to be done to fully understand the issue of Susy–Noether currents or more generally space-time Noether currents. Nevertheless our work surely is a guideline in this direction and successfully solves the problem of the SW Susy currents that we intended to study.

3. SOME SIMPLE EXAMPLES We now want to make our recipe more explicit by applying it to two simple cases: the Wess– Zumino (WZ) massive model and the classical N = 2 Susy Yang–Mills (SYM) model. In the first case we shall show the recipe at work for the simplest Susy model with dummy-fields coupled to physical fields. In the second case we shall re-obtain the classical mass formula of Witten and Olive. This formula will be useful for exhibiting the formal resemblance of the classical and quantum expressions of the central charge Z . In both cases we shall start with a doubled fermionic phase-space and impose canonicity by partial integration and by means of Poisson brackets [3]. Of course similar results would be obtained by making use of second class constraints and the correspondent Dirac brackets [4, 14, 15]. 3.1. WZ Massive Model The Lagrangian density and Susy transformations of the fields for this model [16] are given by i m m 2 i L = − ψ ∂/ ψ¯ − ψ¯ ∂¯/ ψ − ∂µ A∂ µ A† + F F † + m AF + m A† F † − ψ 2 − ψ¯ , 2 2 2 2

(5)

SUSY NOETHER CURRENTS AND SEIBERG–WITTEN THEORY

161

and δA =

√ 2¯² ψ¯ √ µ α˙ √ α ˙ δ ψ¯ = i 2(σ¯ ²) ∂µ A† + 2¯² α˙ F † √ ¯ δ F † = i 2² ∂/ ψ,

√

δ A† =

2²ψ, √ µ √ δψα = i 2(σ ²¯ )α ∂µ A + 2²α F, √ δ F = i 2¯² ∂/¯ ψ,

(6) (7) (8)

where A is a complex scalar field, ψ is its Susy fermionic partner in Weyl notation (see Appendix A.1), and F is the complex bosonic dummy field. † ¯ πψ¯ ), is doubled fermionic because ψ and ψ¯ play the The phase-space (A, π A ; A† , π A |ψ, πψ ; ψ, ¯ πψ¯ ). role of fields and momenta at the same time. A proper phase-space would use only (ψ, πψ ) or (ψ, It is really a matter of taste when to integrate by parts to obtain a proper phase-space and implement the canonical Poisson brackets. In fact, even if N µ and V µ both change under partial integration, the total current J µ is formally invariant, when expressed in terms of the fields and their derivatives, but not in terms of the fields and their momenta. Let us keep (5) as it stands, define the following non-canonical momenta µ

πψα =

i µ¯ (σ ψ)α , 2

µα˙

πψ¯ =

µ

i µ α˙ (σ¯ ψ) 2

µ

π A = −∂ µ A† ,

π A† = −∂ µ A,

(9) (10)

and use Eq. (2) to obtain the supersymmetric current J µ . We compute V µ by varying (5) off-shell, under the given transformations, and obtain4 † µ µ µ ¯ µ¯ + δ F ψπψµ + δ F † ψπ ¯ µ¯ V µ = δ Aπ A + δ A† π A† − δ A ψπψ − δ A ψπ ψ ψ †

µ

µ

¯ ¯, − 2δ Fon ψπψ − 2δ Fon ψπ ψ

(11)

where δ X Y stands for the part of the variation of Y which contains X (for instance δ F ψ stands for √ † 2² F) and Fon , Fon are the dummy-fields given by their expressions on-shell (Fon = −m A† ). Note here that we succeeded in finding an expression for V µ in terms of the π µ ’s and variations of the † fields. Note also that the terms involving Fon and Fon were obtained automatically. Then we write the rigid part of the current µ

µ

µ

µ

¯ ¯, N µ = δ Aπ A + δ A† π A† + δψπψ + δ ψπ ψ

(12)

¡ ¢ µ ¯ µ¯ ; J µ = N µ − V µ = 2 δ on ψπψ + δ on ψπ ψ

(13)

and the current is given by

† therefore J µ = 2(N µ )on fermi , with obvious notation. Note that only the variations with F and F on-shell occur. Therefore we conclude that: (1) the dummy-fields are on-shell automatically and, if we keep the fermionic non-canonical momenta given in (9), (2) the full current is given by twice the fermionic rigid current (N µ )on fermi . The first (on-shell) result illustrates the second ingredient of the recipe given above. We shall see in the highly non-trivial case of the SW effective Action that the on-shell result still holds, and it seems to be a general feature of Susy–Noether currents.

Here and in the following we choose to keep the Susy parameters ²’s, since this simplifies some of the computations involving spinors. 4

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The second result, instead, is only valid for simple Lagrangians and it breaks down for less trivial cases.5 Nevertheless, when applicable, Eq. (13) remains a labour saving formula. All we have to do is to rewrite J µ in terms of fields and their derivatives √ ¡ ¢ (14) J µ = 2 ψ¯ σ¯ µ σ ν ²¯ ∂ν A + i²σ µ ψ¯ Fon + h.c. , then choose one partial integration √ √ µI † J µ = δon ψπψ + 2ψσ µ σ¯ ν ²∂ν A† + i 2¯² σ¯ µ ψ Fon √ √ ¯ µII , or = 2ψ¯ σ¯ µ σ ν ²¯ ∂ν A + i 2²σ µ ψ¯ Fon + δon ψπ ψ¯ µI

µI

µ II

(15) (16)

µ II

where πψ = iσ µ ψ¯ (πψ¯ = 0) and πψ¯ = i σ¯ µ ψ(πψ = 0) are the canonical momenta obtained by (5) conveniently integrated by parts, and perform our computations using canonical Poisson brackets. To integrate by parts in the effective SW theory a greater deal of care is needed due to the fact that the coefficients of the kinetic terms are functions of the scalar field. † Choosing, for instance, the phase-space (A, π A ; A† , π A |ψ, πψI ), what is left is to check that the charge Z Z √ √ ¡ ¢ 3 0 † (17) Q ≡ d x J (x) = d 3 x δon ψπψI + 2ψσ 0 σ¯ ν ²∂ν A† + i 2¯² σ¯ 0 ψ Fon correctly generates the transformations. This is a trivial task in this case since the current and the expression of the dummy-fields on-shell are very simple and the transformations can be read off immediately from the charge (17). We shall see that for more complicated models this is not the case. It is worthwhile to notice at this point that to generate the transformations of the scalar field A† one has to use √ (18) δ A ψπψI = δ A† π A† + 2ψ¯ σ¯ 0 σ i ²¯ ∂i A. Notice also that the transformation of ψ¯ is obtained by acting with the charge on the conjugate momentum of ψ : {Q, πψI }−. 3.2. N = 2 SYM Model There exist two massless N = 2 Susy multiplets with maximal helicity 1 or less: the vector multiplet and the scalar multiplet [17, 18]. We are interested in the vector multiplet 9, also referred to as the N = 2 SYM multiplet, for the moment in its Abelian formulation. Its spin content is (1, 12 , 12 , 0, 0) and, in terms of physical fields, it can accommodate 1 vector field vµ , 2 Weyl fermions ψ and λ, and one complex scalar A. The N = 2 vector multiplet can be arranged into two N = 1 multiplets, the vector (or YM) multiplet W = (λα , vµ , D) and the scalar multiplet 8 = (A, ψα , E), where E and D are the (bosonic) dummy fields,6 related by R-symmetry: ψ ↔ −λ, E † ↔ E and v µ → −v µ (charge conjugation). The N = 2 Susy transformations of these fields are well known. In our notation the first set of transformations is given by [19] √ δ1 A = 2²1 ψ √ (19) δ1 ψ α = 2²1α E δ1 E = 0 5 There are two reasons for that curious result: the fictitious double counting of the fermionic degrees of freedom and the the fermi-bose asymmetry in the canonical structure of the Susy transformations. 6 We use the same symbol E for the electric field and for the dummy-field. Its meaning will be clear from the context.

SUSY NOETHER CURRENTS AND SEIBERG–WITTEN THEORY

√ δ1 E † = i 2²1 ∂/ ψ¯ √ δ1 ψ¯ α˙ = −i 2²1α ∂/αα˙ A†

163

(20)

δ1 A † = 0

¢ β¡ δ1 λα = −²1 σβµν α vµν − iδβα D

¯ δ1 v µ = i²1 σ µ λ,

δ1 D = −²1 ∂/ λ¯

(21)

δ1 λ¯ α˙ = 0, where vµν = ∂µ vν − ∂ν vµ is the Abelian vector field strength. By R-symmetry one can obtain the second set of transformations by simply replacing 1 → 2, ψ ↔ −λ, v µ → −v µ , and E † ↔ E in the first set. The N = 2 SYM low-energy effective Lagrangian, up to second derivatives of the fields and four fermions, is given by [20] µ · µ ¶¸ 1 1 2 Im † (2) † µ µν ¯ ¯ −F (A) ∂µ A ∂ A + vµν vˆ + iψ ∂/ ψ + iλ∂/ λ − EE + D L= 4π 4 2 ¸ · · ¸¶ 1 1 i 1 + F (3) (A) √ λσ µν ψvµν − (E † ψ 2 + Eλ2 ) + √ Dψλ + F (4) (A) ψ 2 λ2 , (22) 2 4 2 2 ∗ = ²µνρσ v ρσ is the where F(A) is a holomorphic and analytic7 function of the scalar field; vµν i ∗ µν † ∗ is its antidual of v ; vˆ µν = vµν + 2 vµν is its self-dual projection, and vˆ µν = vµν − 2i vµν 8 self-dual projection. Susy constrains all the fields to be in the same representation of the gauge group as the vector field, namely the adjoint representation. In the U(1) case this representation is trivial, and the derivatives are standard rather than covariant. We notice here that v0 plays the role of a Lagrange multiplier, and the associate constraint is the Gauss law. Thus, by taking the derivative of L with respect to v0 we obtain the quantum modified Gauss law for this theory, namely

0=

∂L = ∂i 5i , ∂v0

(23)

where 5i = ∂L/∂(∂0 vi ) is the conjugate momentum of vi , given by ¢ 1 ¡ † 5i = −(I E i − RB i ) + √ F (3) λσ 0i ψ − F (3) λ¯ σ¯ 0i ψ¯ i 2

(24)

and F (2) = R + iI. The theory described by the Lagrangian in (22) is the Abelian sector of the SW model. Their achievement essentially consists in the exact determination of the function F in the three sectors of the space of gauge inequivalent vacua Mq .

7

By analytic, we mean that it can have branch cuts, poles, etc., but no essential singularities. If we define the electric and magnetic fields as usual, E i = v 0i and B i = 12 ² 0i jk v jk , respectively, we have vˆ 0i = E i + i B i and vˆ †0i = E i −i B i . 8

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U(1) Classical Model. We shall study, for the moment, the classical limit of this Lagrangian.9 In this limit the second line of (22) vanishes, and writing explicitly, the first line becomes µ µ ¶¶ 1 2 θ 1 1 † µν † µ vµν v ∗µν vµν v + ∂µ A ∂ A − EE + D − L= − 2 g 4 2 64π 2 µ ¶ 1 τ τ∗ ¯ ¯ τ ¯ τ∗ ¯ ¯ ¯ − ψ∂/ ψ − ψ ∂/ ψ + λ∂/ λ − λ∂/ λ . (25) 4π 2 2 2 2 ¯ πψ¯ ; λ, πλ ; λ, ¯ πλ¯ ), as in If we keep the improper phase-space (A, π A ; A† , π A† ; vi , 5i | ψ, πψ ; ψ, the WZ model, the non-canonical momenta are given by π Aµ = −

1 µ † ¡ µ ¢† ∂ A = π A† , g2

5µν = −

1 (τ vˆ µν − τ ∗ vˆ †µν ) 8πi

(26)

and ¡ µ¢ τ µ 4π πψ¯ α˙ = ψ α σαα˙ , 2

¡ µ ¢α τ∗ ˙ 4π πψ = − ψ¯ α˙ σ¯ µαα 2

(27)

similarly for λ. As explained earlier in order to compute the Susy-Noether currents we have to compute their Vµ part. In the classical case this is an easy matter, but for the effective theory it is not trivial. We obtain µ

µ

V1 = π A δ1 A + 5µν δ1 vν +

τ∗ ¯ µ τ µ µ µ δ1 ψπψ¯ + δ1 ψπψ + δ1D λπλ + ∗ δ1v λπλ , τ τ

(28)

µ

where δ1 L = ∂µ V1 , and again δ X Y stands for the term in the variation of Y that contains X (for instance δ1D λα ≡ i²1α D). Note that we have only to compute the first Susy current, since by Rsymmetry, charge conjugation, and complex conjugation we can obtain the other currents. The total µ current J1 is given by10 µ

µ

µ

J1 = N1 − V1

µ µ µ ¯ µ¯ − V1µ = π A δ1 A + 5µν δ1 vν + δ1 ψπψ + δ1 λπλ + δ1 ψπ ψ

=

2iτ I ¯ µ 2iτ I on µ δ1 ψπψ¯ − ∗ δ1 λπλ , τ τ

(29)

µ where N1 is the rigid current, δ1 λ¯ = 0, and δ1on λ stand for the variation of λ with dummy-fields ¯ In this case this means E = D = 0 on-shell (there are no dummy-fields in the variation of ψ). and one could also wonder if they are simply cancelled in the total current. But, in agreement with our recipe, we shall see later that indeed the dummy-fields, and only them, have been automatically projected on-shell. 9 At this end it is sufficient to recall that in the classical case there is no running of the coupling; therefore there is only one global description at any scale of the energy. Thus we can use the expression µ 2¶ ∞ X i h34k A 1 + A2 ck 4k (157) ln F(A) = τ A2 + A2 2 2 2π 3 A k=1

to write the classical limit as F(A) →

1 2 τA , 2

(158)

where τ is the complex coupling constant already introduced. 10 J µ stands for ² 1 J µ or ² J 1µ . In the following we shall not keep track of the position of these indices; they will be 1 1 1 simply treated as labels.

SUSY NOETHER CURRENTS AND SEIBERG–WITTEN THEORY

165

If we set θ = 0 in this non-canonical setting, we recover the same type of expression, J µ = µ 2(Nfermi )on , for the total current obtained in the massive WZ model, namely ¡ ¢ µ µ ¯ µ¯ . J1 |θ =0 = 2 δ1on λπλ + δ1 ψπ (30) ψ We see again that the double counting of the fermionic degrees of freedom provides a very compact formula for the currents. All the information is contained in the fermionic sector, since the variations of the fermions contain the bosonic momenta. Unfortunately this does not seem to be the case for the effective theory, as we shall see in the next section. We have now to integrate by parts in the fermionic sector of the Lagrangian (25) to obtain a proper phase space.11 Everything proceeds along the same lines as for the WZ model. Thus the canonical fermionic momenta are ¡

I µ¢ α˙

πψ¯

=

i α µ ψ σαα˙ , g2

¡

I µ¢ α˙

πλ¯

=

i α µ λ σαα˙ g2

(31)

Iµ Iµ and πψ = πλ = 0, where with I we indicate one of the two possible choices (ψ¯ and λ¯ are the fields). The partial integration changes V µ , but, of course, also N µ changes accordingly and they Iµ µ ¯ µ¯ and still combine to give the same total current J µ . Namely N1 = π A δ1 A + 5µν δ1 vν + δ1 ψπ ψ Iµ µ 1 ∗ ¯ ∗µν that give a total current τ ²1 σν λv V1 = π A δ1 A + 8πi

√ Iµ ¯ I¯ µ − i ²1 σν λˆ ¯ v µν = − 2²1 σ ν σ¯ µ ψπν A + 5µν δ1 vν − 1 τ ∗ ²1 σν λv ¯ ∗µν , J1 = δ1 ψπ ψ 2 g 8πi

(32)

where we used the identities √ ¯ I¯ µ = − 2²1 σ ν σ¯ µ ψπν A δ1 ψπ ψ −

i ¯ v µν = 5µν δ1 vν − 1 τ ∗ ²1 σν λv ¯ ∗µν . ²1 σν λˆ 2 g 8πi

(33) (34)

The point we make here is that the current is once and for all given by µ

J1 =

¢ 1 ¡√ ¯ v µν 2²1 σ ν σ¯ µ ψ∂ν A† − i²1 σν λˆ 2 g

(35)

but its content in terms of canonical variables changes according to partial integration. Furthermore one has to conveniently re-express the current obtained via the Noether procedure to obtain the relevant expression in terms of bosonic or fermionic momenta and transformations. Note also that θ does not appear in the explicit formula, as could be expected. Choosing the temporal gauge for the vector field, v 0 = 0, and defining the canonical momenta as usual (remember that our metric is ηµν = diag(−1, 1, 1, 1)) we can write down the first Susy charge Q 1α and the other charges are simply obtained by R-symmetry, charge conjugation, and complex conjugation. These charges correctly reproduce the Susy transformations. We can now compute the Poisson brackets that contribute to the centre Z = 4i ² αβ {Q 1α , Q 2β }+ obtaining ¸ µ · µ · ¶ ¸¶ Z √ ¡ √ ¢ 1 i † 1 ¡ ∗0i ¢ † 1 ∗ ¯ i0 ¯ Z = d 3 x ∂i i 2 5i A† + B AD − τ ψ σ¯ λ + i 2 ∂i 5i A† + ∂i v AD , 4π 4π 4π (36) 11 In this case there is no effect of the partial integration on the bosonic momenta since ∂ τ = 0; we shall see that this is µ no longer the case for the effective theory.

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IORIO, O’RAIFEARTAIGH, AND WOLF

where A D = τ A. By using the Bianchi identities, ∂i v ∗0i = 0, and the classical limit of the Gauss law (23), we are left with a total divergence. The final expression for Z is then given by µ ¶ √ Z 2 1 E † † E E B AD , (37) Z = i 2 d 6 · 5A + 4π 2 E is the measure on the sphere at infinity S∞ , and we have made the usual assumption that where d 2 6 −3/2 ¯ ¯ ψ and λ fall off at least like z . This is the U(1) version of the well-known result of Witten and Olive. Note that we ended up with the anti-holomorphic centre.

SU(2) Classical Model. Before moving on to the quantum theory we want to make full contact with Witten and Olive’s computation that was performed for the classical non-abelian model. At this end we see from the current in (32) that a straightforward generalization to the gauge group SU(2) leads to the charge µ ¶ Z i 3 a a a a0i ¯ ¯ = d x δ π − ² σ v ˆ ψ λ , (38) ²1 Q SU(2) 1 1 i 1 ψ¯ g2 where a is the SU(2) index, we dropped the index I , the identities (33) and (34) have to be used when necessary, and the SU(2) momenta and fields are defined as usual (see Appendix A.3). By using the same techniques as in the U(1) case, we see that this charge correctly generates the transformations that do not involve dummy-fields, namely δ1 via , δ1 Aa , δ1 A†a , δ1 ψ¯ a , and δ1 λ¯ a , but does not generate the transformations involving the dummy-fields (see, for instance [19] and Appendix A.3). Thus some terms must be missing. We can obtain the missing terms by considering the classical (microscopic) a SU(2) Lagrangian LSU(2) class and solving the Euler–Lagrange equations for D . The result is given by abc b c† A A , (D a )on class = i²

(39)

a on where we used the standard expression for LSU(2) class (see, for instance, [2, 19]). Note that (E )class = 0. From our recipe, we know that the charge has to produce the transformations with the dummy fields on shell. D a appears in the transformation of λa ; therefore we want to produce δ1D λa from , πλ¯a }, where πλ¯a = iτ I λa σ 0 is the classical SU(2) conjugate momentum of λ¯ a . Thus we {²1 Q SU(2) 1 conclude that a missing term in the classical charge is given by

iτ I ²1 σ 0 λ¯ a ² acd Ac Ad† .

(40)

Furthermore this term is the only missing term, because once it is added then we obtain all the correct Susy transformations. Thus the final expression for the classical SU(2) first Susy charge is µ ¶ Z i SU(2) 3 a a a a0i 0 ¯ a acd c d† ¯ ¯ = d x δ1 ψ πψ¯ − 2 ²1 σi λ vˆ + iτ I ²1 σ λ ² A A . (41) ²1 Q 1 g If we compute Z by taking the Poisson brackets of this charge with its R-symmetric counterpart we obtain an expression in any respect similar to (37) but this time with all the SU(2) contributions. This means, for instance, that the Gauss law has to be modified accordingly (see more on this in the next section). By breaking the gauge symmetry along one direction, say h0|Aa |0i = δ a3 a, we define the electric and magnetic charges a` la Witten and Olive [5], Z 1 E ·5 E 3 A3 d 26 ne ≡ (42) a Z Z 1 1 1 E3 3 1 E3 3 2E E · nm ≡ (43) d 6· B A = B AD, d 26 a 4π aD 4π where a D = τ a and only the U(1) fields remaining massless after SSB appear.

SUSY NOETHER CURRENTS AND SEIBERG–WITTEN THEORY

167

√ ¡ ¢ Z = i 2 n e a ∗ + n m a ∗D .

(44)

We finally have

4. QUANTUM CASE: SEIBERG–WITTEN MASS FORMULA After these preliminaries, we now wish to consider the quantum SW model. We shall first concentrate on the U(1) sector and then generalize our results to the full high-energy SU(2) sector. 4.1. The U(1) Sector We have now to consider the quantum corrected U(1) Lagrangian given in (22). This time the dummy-fields couple non-trivially to the fermions. Their expressions on-shell are given by ¢ 1 ¡ D = − √ f ψλ + f † ψ¯ λ¯ 2 2

and

E=

¢ i ¡ † ¯2 f λ − f ψ2 , 4

(45)

where f (A, A† ) ≡ F (3) /I. As we shall see in a moment, they represent the most important difference between the classical and the quantum case. µ As before, we can concentrate on the computation of the first Susy current J1 . The task, of course, µ is to find V1 and it turns out that its computation in the quantum case is by no means easy. We have µ µ found by direct computation V1 and V¯ 1 . Of course the first one has been the most difficult to find, since if one understands how to proceed in the first case, the other cases become only tedious checks. We do not want to explicitly show here all the details of this lengthy computation. In Appendix B, we µ shall give in some details only the simplest part of the computation of V1 , namely the contribution coming from the F-terms. ¯ πψ¯ ; λ, πλ ; λ, ¯ πλ¯ ), the If we keep the improper phase-space ( A, π A ; A† , π A† ; vi , 5i | ψ, πψ ; ψ, non-canonical momenta are given by ¡ µ ¢† µ π A = −I∂ µ A† = π A† 5µν = −

(46)

¢ ¢ 1 ¡ (2) µν 1 ¡ F vˆ − F (2)† vˆ †µν + √ F (3) λσ µν ψ − F (3)† λ¯ σ¯ µν ψ¯ 2i i 2

(47)

and ¡

πψµ¯

¢ α˙

=

1 (2) α µ F ψ σαα˙ , 2

¡

µ ¢α

πψ

1 ˙ = − F (2)† ψ¯ α˙ σ¯ µαα 2

(48)

similarly for λ. Note that for the moment we scale F by a factor of 4π . The result of the full µ computation of V1 (F-terms and F † -terms) is µ

µ

V1 = δ1 Aπ A +

F (2)† ¯ µ 1 1 µ µ ¯ ∗µν + √ F (3)† ²1 σ µ ψ¯ λ¯ 2 . δ1 ψπψ¯ + δ1 ψπψ + δ1 λπλ + F (2)† ²1 σν λv F (2) 2i 2 2 (49) µ

µ

Of course the rigid current is formally identical to the classical one, N1 = π A δ1 A + 5µν δ1 vν + µ µ ¯ µ¯ , but the F (3)† term in V1µ has no classical analogue. Thus, even if we δ1 ψπψ + δ1 λπλ + δ1 ψπ ψ µ µ µ conveniently rearrange the terms in the total current J1 = N1 − V1 to write µ

J1 =

2iI ¯ µ 2iI 1 1 µ δ1 ψπψ¯ − (2)† δ1on λπλ + √ F (3)† ²1 σ µ ψ¯ λ¯ 2 + √ F (3) ²1 ψλσ µ λ¯ (2) F F 2 2 2

(50)

168

IORIO, O’RAIFEARTAIGH, AND WOLF µ

we see that the classical formula, Jµ = 2(Nfermi )on , no longer holds. Therefore we can just move on to a proper phase space, by partially integrating in the fermionic sector. Some care is needed due to the fact that now the coefficient of the kinetic terms is a function and not a constant. By choosing, for instance, Iµ ψ¯ and λ¯ as fields, the canonical momentum of A† becomes π A† = −I∂ µ A− 12 F (3)† (ψ¯ σ¯ µ ψ + λ¯ σ¯ µ λ), Iµ µ Iµ and of course the fermionic momenta become (πψ¯ )α˙ = iIψ α σαα˙ , (πψ )α = 0, similarly for λ. µ µν Note that 5 and π A do not change. The first Susy charge for the U(1) effective theory is then given by · ¸ Z 1 (3)† 0 ¯ ¯2 ¯ I¯ + 5i δ1 vi − 1 F (2)† ²1 σi λv ¯ ∗0i − √ F ² σ . (51) ψ λ ²1 Q 1I = d 3 x δ1 ψπ 1 ψ 2i 2 2 Again we have to conveniently re-express it in terms of fermionic or bosonic variables when necessary. In this case it is not immediate to verify if this charge correctly reproduces the Susy transformation. In fact one could wonder if the presence of cubic fermionic terms and third derivatives of F would spoil the simple structure of the transformations generated by this charge [13]. This point is really delicate, since the most crucial requirement of the SW model is that Susy is preserved at quantum level. We leave to Appendix C the explicit proof that this charge generates exactly the transformations given12 in (19)–(21). The only difference between classical and quantum Susy transformations is the expression of the dummy-fields on-shell: classically they are all zero (in the U(1) sector); at the quantum level they are given by Eq. (45). Note also that in the effective case some of the standard assumptions about Poisson brackets do no longer hold, as can be seen by the following non-trivial bracket i {π A , πψ¯ }− = 0 ⇒ {π A , ψ}− = − f ψ. 2 Nevertheless the canonical structure survives (see Appendix A.2). We are now in the position to compute the central charge for this theory, µ · ¸ Z √ 1 i (1)† 1 (2)† ¯ i0 ¯ Z = d 3 x ∂i i 25i A† + BF − F ψ σ¯ λ 4π 4π · ¸¶ √ ¡ ¢ 1 ¡ ∗0i ¢ (1)† ∂i v F , + i 2 ∂i 5i A† + 4π

(52)

(53)

where we reintroduced the factor 4π and used the formula Z = 4i ² αβ {Q 1α , Q 2β }+ . Imposing the Bianchi identities and the Gauss law (23), and dropping the fermionic term as in the classical case, we can eventually write µ ¶ √ Z 1 E (1)† E · 5A E †+ BF , Z = i 2 d 26 4π

(54)

where B i = 12 ² 0i jk v jk as in the classical case. Note that from Eq. (54) we can define the SW dual of the scalar field A† as given by †

A D ≡ F (1)† (A† ).

(55)

12 In Appendix C we shall verify that the charge ² Q I , obtained by using ψ ¯ and λ¯ as fields, works. We do not show there 1 1 , with ψ and λ as fields work, but this is indeed the case. This proves that also at the effective that also the other charge ²1 Q II 1 level the final result is insensitive to partial integration.

SUSY NOETHER CURRENTS AND SEIBERG–WITTEN THEORY

169

Surprisingly enough the expression (54) is formally identical to the classical one given in (37). We see that the topological nature of Z is sufficient to protect its form at the quantum level. All one has to do is to use a little dictionary and replace classical quantities with their quantum counterparts. Thus we can apply exactly the same logic as in the classical case and define the electric and magnetic charges a` la Witten and Olive. The final expression for the U(1) effective central charge is √ ¡ ¢ Z = i 2 n e a ∗ + n m a ∗D ,

(56)

where h0|A† |0i = a ∗ , h0|F (1)† |0i = a ∗D , and n e , n m are the electric and magnetic quantum numbers, respectively. Eventually we proved the SW mass formula. At this end we can simply use the BPS type of argument given in [1, 6], noticing that our direct computation includes fermions but they occur as a 2 . Thus total divergence which falls off fast enough to give contribution on S∞ M = |Z | =

√ 2|n e a + n m F (1) (a)|.

(57)

A last remark is now in order. The U(1) low energy theory is invariant under the linear shift F(A) → F(A) + c A. This produces an ambiguity in the definition of Z . For this and other purposes we want also to analyse the SU(2) high-energy theory. 4.2. The SU(2) Sector We wish to generalize the U(1) charge (51) to the SU(2) case. We recall that for the classical version the transition U(1) → SU(2) required also the addition of the term (40). Clearly in the effective case this term has to be iI ab ²1 σ 0 λ¯ b ² acd Ac Ad†

(58)

and it reduces to the classical term (40) in the limit F(A) → 12 τ A2 . The addition of this term suffices also in the quantum theory. In other words the full SU(2) quantum first Susy charge is µ Z i 3 ²1 Q 1 = d x 5ai δ1 via + δ1 ψ¯ a πψa¯ + F †ab ²1 σi λ¯ a v ∗ib 2 ¶ 1 − √ F †abc ²1 σ 0 ψ¯ a λ¯ b λ¯ c + iI ab ²1 σ 0 λ¯ b ² acd Ac Ad† , (59) 2 2 where we have dropped the label “SU(2)” and, as usual, the other charges are obtained by means of R-symmetry, charge conjugation, and complex conjugation. This term is the only new term required to produce the Susy transformations, the Higgs and Yukawa potential in the Hamiltonian, and, as we shall see in a moment, it is responsible for most of the new terms in the centre. We want now to compute the Hamiltonian H with our charges. The main point here is to obtain the non-trivial Gauss law for the SU(2) theory from the Legendre-transformed of H . This will be crucial to obtain the correct centre for this sector of the theory. ˙ ¯ 1α˙ }+ where we defined H = P 0 = −P0 . This lengthy {Q 1α , Q The formula for H is H = − 4i σ¯ 0αα computation is illustrated in some details in Appendix D. Its final result is given by µ Z 1 1 3 H = d x − (I ab )−1 5ai 5bi − (I ab )−1 Rbc 5ai B ic − (I ab )−1 F †ab F e f B ib B i f 2 2 ¡ ¢ † − (I ab )−1 π Aa π Ab + I ab (Di Aa )(Di A†b )

170

IORIO, O’RAIFEARTAIGH, AND WOLF

¢ 1¡ + iI ab ψ a σ 1 Di ψ¯ b + iI ab λa σ i Di λ¯ b + ∂i F †ab λ¯ a σ i λb 2 1 1 − √ I ad ² abc (Ac ψ¯ d λ¯ b + A†c ψ d λb ) + I ab ² acd ² b f g Ac A†d A f A†g 2 2 i + √ (I a f )−1 F f eg ψ e σi0 λg (5ia + F †ab B ib ) 2 i − √ (I ec )−1 F †abc ψ¯ a σ¯ i0 λ¯ b (5ie + F ed B ib ) 2 1 1 †e f g cad gc −1 ¯ e ¯ f a c F (I ) ψ ψ ψ ψ + F †abc F e f g (I ae )−1 λ¯ b λ¯ c λ f λg + F 16 16 3 †bec †e f g ab −1 ¯ a ¯ c ¯ f ¯ g 3 + F F (I ) ψ ψ λ λ + F bec F e f g (I ab )−1 ψ a ψ c λ f λg 16 16 ¶¶ µ 1 1 abcd a b c d 1 †abcd ¯ a ¯ b ¯ c ¯ d − ψ ψ λλ − F ψ ψ λλ F 2i 4 4 µ ¶ Z 1 †ab ¯ a i b i ab ¯ a i b 3 F λ σ¯ λ − I ψ σ¯ ψ , + d x∂i 2 2

(60)

where E ai = v a0i and B ai = 12 ² 0i jk v ajk are the SU(2) generalization of the electric and magnetic fields, respectively. We notice here that, in the last line, we kept a total divergence to explicitly show ¯ λ; ¯ πψ¯ , πλ¯ ) that we partially integrated the fermionic kinetic terms, in order to fix the phase space (ψ, we started with.13 By Legendre transforming this Hamiltonian and reintroducing v 0 , one obtains the Lagrangian given in Appendix D, which gives the SU(2) modified Gauss law Di 5ig = −² gac I ab (Ac D0 A†b + A†c D0 Ab + iψ b σ 0 ψ¯ c + iλb σ 0 λ¯ c ).

(61)

We now come to one of the main purposes of our paper, which is to compute the central charge for the SU(2) theory, Z = 4i ² αβ {Q 1α , Q 2β }+ . After some fairly straightforward computations we obtain Z Z =

¡ £√ ¤ d 3 x ∂i i 2(5ai A†a + B ai F †a ) − F †ab ψ¯ a σ¯ i0 λ¯ b

√ £¡ ¢ ¤¢ + i 2 Di 5ai A†a + iI be ² bcd A†d (ψ e σ 0 ψ¯ c + λe σ 0 λ¯ c ) − ² abc Ab A†c π Aa ,

(62)

where the properties of F a...b listed in Appendix D were extensively used. We see that the terms which are not a total divergence, given in the second line above, simply cancel due to the Gauss law (61). Eventually we are left with the surface terms that vanish when the SU(2) gauge symmetry is not broken down to U(1). If we break the symmetry along a flat direction of the Higgs potential, say a = 3, we recover the same result we found in the U(1) sector. In other words we see that on the It turns out that this total divergence is not symmetric with respect to ψ and λ and this is reflected in the last term of the third line where only λ-terms appear. This means that the Lagrangian we shall obtain by Legendre transforming H will be slightly different from the one expected. Nevertheless the difference will not affect the conjugate momenta; therefore the Susy charges above constructed are not affected by this asymmetry. Furthermore this problem is entirely due to the non-trivial partial integration in the effective case. As explained earlier this does not affect the explicit expression of the currents and charges. 13

SUSY NOETHER CURRENTS AND SEIBERG–WITTEN THEORY

171

sphere at infinity µ ¸ · ¶ √ 1 E a †a 1 †ab ¯ a ¯ b ¢ E a A†a + E · i 2 5 d 26 B F F ψ σE λ − 4π 4π µ ¶ Z √ 1 E 3 †3 E · 5 E 3 A†3 + B AD , → i 2 d 26 4π Z

Z =

(63)

†3

where σE¯ ≡ (σ¯ 01 , σ¯ 02 , σ¯ 03 ), ∂F † /∂ A†3 = A D , and we reintroduced the factor 4π. We also made the usual assumption that the bosonic massive fields in the SU(2)/U(1) sector (a = 1, 2) and all the fermionic fields fall off faster than r 3 , whereas the scalar massless field (a = 3) and its dual tend to their Higgs vev’s a ∗ and a ∗D , respectively. Note that, in the broken phase, the formula (63) is the same as the U(1) formula (54). Thus we conclude that the fields in the massive sector have no effect on the mass formula. APPENDIXES A. Notation and Conventions In this appendix we explain the notation and conventions used for the spinors, the Poisson brackets, and the SU(2) gauge group. A.1. Spinor Conventions and Useful Algebra We follow Wess and Bagger [16] with no changes. The spinors are Weyl two components in Van der Waerden notation. In addition to the identities given in [16] we also find ψ α λβ − ψ β λα = −² αβ ψλ, α˙ ¯ β˙

β˙ ¯ α˙

α˙ β˙

ψ¯ λ − ψ¯ λ = ² ψ¯ λ¯ ,

ψα λβ − ψβ λα = ²αβ ψλ

(64)

ψ¯ α˙ λ¯ β˙ − ψ¯ β˙ λ¯ α˙ = −²α˙ β˙ ψ¯ λ¯

(65)

and the following identities for the σ matrices with free spinor indices ¢ 1¡ µ µ σδγ˙ ²γ α ² βδ − σαγ˙ δγβ 2 ¢ 1¡ ˙ ˙ = σ¯ µ αδ ²αδ ² βγ + σ¯ µ αβ δαγ 2 1 ¡ µδγ ˙ γ˙ ¢ ˙ = σ¯ ² α˙ γ˙ ²δ˙ β˙ − σ¯ µ αγ δβ˙ 2 ¢ 1 ¡ µ α˙ δ˙ µ = σ αδ˙ ² ²β˙ γ˙ + σαβ˙ δγα˙˙ 2

σαµν β σν γ γ˙ = ˙ σαµν β σ¯ ναγ

σ¯ µν αβ˙˙ σ¯ νγ˙ γ σ¯ µν αβ˙˙ σν αγ˙

(66) (67) (68) (69)

α˙ that imply σ µν σν = σν σ¯ νµ = − 32 σ µ , σ¯ µν σ¯ ν = σ¯ ν σ νµ = − 32 σ¯ µ , and σαµνβ σ¯ µν =−δβα˙˙ δαβ , σα0µβ σγδ 0µ = β˙ 0µα˙ γ˙ 1 1 α˙ γ˙ α˙ γ˙ βδ δ β − 4 (²αγ ² + δα δγ ), σ¯ β˙ σ¯ 0µδ˙ = − 4 (² ²β˙ δ˙ + δδ˙ δβ˙ ). Also useful are the identities

(σ ρσ σ µν )αβ vρσ vµν = − 12 δβα vµν vˆ µν , σ ρσ σ µ vρσ = vˆ µν σν , σ¯ µ σ ρσ vρσ = −ˆv µν σ¯ ν ,

(σ¯ ρσ σ¯ µν )αβ˙˙ vρσ vµν = − 12 δβα˙˙ vµv vˆ †µν

(70)

σ µ σ¯ ρσ vρσ = −ˆv †µν σν , σ¯ σρ σ¯ µ vσρ = vˆ †µν σ¯ ν , (71)

where vˆ µν = v µν + 12 v ∗µν , vˆ †µν = v µν − 12 v ∗µν , and v µν = −v µν .

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IORIO, O’RAIFEARTAIGH, AND WOLF

A.2. Graded Poisson Brackets We deal with c-number valued fields, i.e., non-operator, in the classical as well as effective case. Therefore the Susy algebra has to be implemented via graded Poisson brackets,14 namely with Poisson brackets {, }− and anti-brackets {, }+ . We define the following equal time Poisson (anti) brackets µ ¶ Z δ B1 (x) δ B2 (y) δ B2 (y) δ B1 (x) 3 − (72) {B1 (x), B2 (y)}− ≡ d z δ8(z) δ5(z) δ8(z) δ5(z) µ ¶ Z δ B(x) δ F(y) δ F(y) δ B(x) − (73) {B(x), F(y)}− ≡ d 3 z δ8(z) δ5(z) δ8(z) δ5(z) ¶ µ Z δ F1 (x) δ F2 (y) δ F2 (y) δ F1 (x) − , (74) {F1 (x), F2 (y)}+ ≡ d 3 z δ8(z) δ5(z) δ8(z) δ5(z) where the B’s are bosonic and the F’s fermionic variables and 8 and 5 span the whole phase space. From this definition it follows that the properties of the graded Poisson brackets are the same as for standard commutators and anti-commutators. The canonical equal-time Poisson brackets for a Lagrangian with bosonic and fermionic fields are given by the usual expressions. The same structure survives at the effective level even if a great deal of care is required. For instance the non-zero equal-time Poisson brackets for the U(1) sector of the effective SW theory are {A(x), π A (y)}− = {A† (x), π A† (y)}− = δ (3) (Ex − Ey ),

j

{vi (x), 5 j (y)}− = δi δ (3) (Ex − Ey )

(75)

and ¡ ¢β˙ ª © ¡ ¢β˙ ª β˙ ψ¯ α˙ (x), πψI¯ (y) + = λ¯ α˙ (x), πλ¯I (y) + = δα˙ δ (3) (Ex − Ey )

(76)

¡ ¢β ª © ¡ ¢β ª ψ¯ α (x), πψII (y) + = λα (x), πλII (y) + = δαβ δ (3) (Ex − Ey )

(77)

© or ©

depending on the phase-space one chooses. Otherwise, in both cases, one can write i {ψα (x), ψ¯ α˙ (y)}+ = − σα0α˙ δ (3) (Ex − Ey ) I

(78)

¯ λ, ¯ even if the effective 5i contains all (same for λ). Note that {5i , χ}− = 0, where χ ≡ ψ, λ, ψ, † the fermions of the theory. Finally, we have that {π A , I}− = − 2i1 F (3) and {π A† , I}− = 2i1 F (3) , thus the usual assumption that fermions and bosons should commute no longer holds i {π A , πψ¯ }− = 0 ⇒ {π A , ψ}− = − f ψ 2 i {π A† , πψ¯ }− = 0 ⇒ {π A† , ψ}− = + f † ψ 2

(79) (80)

with f = F (3) /I (same for λ). 14

Following a nice argument given by Dirac [14], this definition leads to the quantum anti-commutator for two fermions. The original argument relates classical Poisson brackets to the commutator [B1 (x), B2 (y)]− → i h- {B1 (x), B2 (y)}− , where the B’s stand for bosonic variables. The generalization to fermions is naturally given by [F1 (x), F2 (y)]+ → i h- {F1 (x), F2 (y)}+ (we shall use the natural units h- = c = 1), where the F’s are fermionic variables. See also [21].

SUSY NOETHER CURRENTS AND SEIBERG–WITTEN THEORY

173

β

Note that {ψα , πψ }+ = δαβ and {ψ α , πψβ }+ = δβα are compatible iff πα = −²αβ π β . Note also that we impose {ψα , πψβ }+ = ²αβ = {πψα , ψβ }+ . A.3. SU (2) Conventions E = 1 σ a X a with a = 1, 2, 3 and we follow the summaA generic SU(2) vector is defined as X 2 a tion convention. The σ ’s are the standard Pauli matrices satisfying [σ a , σ b ] = 2i² abc σ c , where the ² abc are the structure constants of SU(2), and Trσ a σ b = 2δ ab . The covariant derivative and E = ∂µ X E − i[Evµ , X E ], and vEµν = ∂µ vEν − ∂ν vEµ − i[Evµ , vEν ], the vector field strength are given by Dµ X respectively. Some authors keep the renormalizable SU(2) gauge coupling g even in the effective theory (for instance their covariant derivatives are defined as Dµ X a = ∂µ X a + g² abc vµb X c ). This is somehow misleading since in SW theory the effective coupling is once and for all given by τ (a) = F (2) (a). Of course the microscopic theory is scale invariant before SSB, and a redefinition of the fields g X → X does no harm. The matter is less clear in the effective theory, where even the definition of what is a field poses some problems and scale invariance is lost after SSB. Therefore we prefer to follow the conventions of [1], where already at a microscopic level the g is absorbed in the definition of the fields and only appears in the overall factor 1/g 2 (see also our expression for the U(1) classical Lagrangian in (25)). Nevertheless we can keep track of g since by charge conjugation g → −g (see for instance [22]), which in our notation becomes ² abc → −² abc . The SU(2) transformations for the first Susy are given by √ δ1 AE = 2²1 ψE √ δ1 ψE α = 2²1α EE (81) δ1 EE = 0 √ E¯ + 2i[ AE† , ²1 λE ] / ψ δ1 EE† = i 2²1 D √ / αα˙ AE† δ1 ψ¯E α˙ = −i 2²1α D

(82)

δ1 AE† = 0

¢ β ¡ µν E δ1 λE α = − ²1 σβ α vEµν − iδβα D E = −²1 D / λE¯ δ1 D

E¯ δ1 vE µ = i²1 σ µ λ,

(83)

δ1 λE¯ α˙ = 0. The SU(2) effective canonical momenta are given by ¡ ¢ 5ai = − 2i1 F ab vˆ 0ib − F ab† vˆ †0ib +

1 √ F abc λb σ 0i ψ c i 2

−

¯ b σ¯ 0i ψ¯ c √1 F abc† λ 2

a = − I ab ∂ 0 Ab − 12 F †abc (ψ¯ b σ¯ 0 ψ c + λ¯ b σ¯ 0 λc ) π Aa = −I ab ∂ 0 A†b , π A†

πψa¯ = iI ab ψ b σ 0 ,

πλ¯a = iI ab λb σ 0 ,

(84) (85) (86)

where we choose the setting I (see the correspondent U(1) expressions) and the temporal gauge for the vector field (thus D0 = ∂ 0 ). Their classical limit is obtained by simply using F ab → τ δ ab and F ab···c → 0. Finally, the SU(2) prepotential F is a holomorphic function of the SU(2) gauge Casimir Aa Aa , a = 1, 2, 3. Our F corresponds to the function H in Seiberg and Witten conventions [1].

174

IORIO, O’RAIFEARTAIGH, AND WOLF

Some care is necessary in handling its derivatives, the first four being given by F a = 2Aa F (1)

(87)

F ab = 2δ ab F (1) + 4Aa Ab F (2) F

abc

(88)

= 4(δ A + δ A + δ A )F ab

c

ac

b

bc

a

(2)

+ 8A A A F a

b

c

(3)

(89)

F abcd = 4F (2) (δ ab δ cd + δ ac δ bd + δ bc δ ad ) + 8F (3) (Aa Ab δ cd + Aa Ac δ bd + Aa Ad δ bc + Ab Ac δ ad + Ab Ad δ ac + Ac Ad δ ab ) + 16F (4) Aa Ab Ac Ad

(90)

similarly for F † . From the expressions (87)–(90) it is easy to obtain the following very useful identities, ² abc F bd Ac = ² adc F c ²

bcd

F A = −² be

d

bed

(91)

F A bc

d

F abc ² cde Ae = F be ² ade + F ae ² bde ,

(92) (93)

and similarly for F † . Properties (91)–(93) are extensively used throughout the SU(2) computations. B. Computation of V µ for the U(1) Theory We first notice that, by varying off-shell the Lagrangian (22) under the Susy transformations given in (19)–(21), there is no mixing of the F terms with the F † terms. The structure of this Lagrangian is 2iL = −F (2) [2B + 2F] + F (3) [1B2F] + F (4) [4F] + F (2)† [2B + 2F]† − F (3)† [1B2F]† − F (4)† [4F]† ,

(94) (95)

where B and F stand for bosonic and fermionic variables, respectively. For instance, if we vary the F terms under δ1 we have (δ1 F (4) )[4F] = 0, whereas the other terms combine as ¡ ¢ with δ1 F (3) [1B2F] ∼ F (4) (1B3F) F (4) δ1 [4F] ∼ F (4) (1B3F) ¡ ¢ with δ1 F (2) [2B + 2F] ∼ F (3) (2B1F + 3F). F (3) δ1 [1B2F] ∼ F (3) (2B1F + 3F) µ

Finally there are the terms F (2) δ1 [2B + 2F], which are the naive generalization of the classical V1 . The aim is to write these quantities as one single total divergence and express it in terms of momenta and variations of the fields. This computation is by no means easy. It is matter of • identifying similar terms and comparing them. • using partial integration cleverly: never throw away surface terms! • using extensively Fierz identities and spinor algebra. µ

What we shall show explicitly in this appendix is only the simplest part of the computation of V1 , namely the contribution coming from the F-terms. Let us apply the scheme discussed above. First we consider the F (4) type of terms. If we find contributions from these terms we know that they cannot be cancelled by terms coming from the rigid current Nµ and there is no hope to rearrange them in the form of on-shell dummy-fields

175

SUSY NOETHER CURRENTS AND SEIBERG–WITTEN THEORY

(they only contain F (3) type of terms). This would then be a signal that by commuting the charges we could have contributions that would spoil the SW mass formula. What we find is that the terms ¤ 1£ (δ1 ψ)ψλ2 + ψ 2 (δ1 λ)λ 2 ¤ 1 £√ 2²1 ψλ2 E − ²1 σ µν λvµν ψ 2 + i²1 λψ 2 D = F (4) 2

F (4) δ1 [4F] = F (4)

(96)

added to the terms ¡

δ1 F

(3)

·

¢

[1B2F] = F

²1 ψλσ

(4)

µν

¸

1 ψvµν − √ E²1 ψλ2 + i D²1 ψψλ 2

(97)

fortunately give zero. Let us then move to the next level, the F (3) terms. In principle these terms can be present, since they appear in the expression of the on-shell dummy-fields. We find that the terms ¸ · ¢ 1¡ 1 i (3) (3) µν † F [(δ1 1B)2F] = F √ λσ ψδ1 vµν − δ1 E ψψ + √ (δ1 D)ψλ 2 2 2 ¸ · i 1 i (3) µν ¯ ¯ ¯ =F √ λσ ψ2i²1 σν ∂µ λ − √ ²1 ∂/ ψψψ − √ ²1 ∂/ λψλ 2 2 2 i ¯ 1 ψ − ²1 ∂/ ψψψ) ¯ = √ F (3) (2λ∂/ λ² (98) 2 summed to the terms √ √ ¡ ¢ ¯ 1 ψ − i 2F (3) ψ∂/ ψ² ¯ 1 ψ) − δ1 F (2) [2F] = −i 2F (3) λ∂/ λ² again give zero. What is left are the other F (3) terms and the F (2) terms. There we find · 1 i −F (2) δ1 [2B + 2F] = −F (2) (∂µ A† )∂ µ (δ1 A) + (δ1 vµν )v µν + (δ1 vµν )v ∗µν 2 4 ¯ + i(δ1 λ)∂/ λ¯ − Eδ1 E † − Dδ1 D + i(δ1 ψ)∂/ ψ¯ + iψ∂/ (δ1 ψ)

(99)

¸

· = −F

(2)

√ ¯ µν (∂µ A† )∂ µ ( 2²1 ψ) + (i²1 σν (∂µ λ))v

√ √ ¡ ¢ 1 ˙ ¯ ∗µν + i 2²1 ∂/ ψ¯ E − 2ψ α ∂/αα˙ ∂/ αβ − (i²1 σν (∂µ λ))v A† ²1β 2 ¸ √ ¢ β ¡ µν α α α˙ ¯ ¯ ¯ − i²1 σβ vµν − iδβ D ∂/ αα˙ λ − i 2²1 ∂/ ψ E − D²1 ∂/ λ √ £√ ¤ 2²1 (∂ µ ψ)∂µ A† + 2²1 ψ ¤ A† £√ ¤ = − F (2) ∂ µ 2²1 ψ∂µ A† . = − F (2)

(100)

Thus we find the first non-zero contribution. Let us note that this term would already be a total divergence if we impose the classical limit F (2) → τ . Thus we can guess that the F (3) terms have to

176

IORIO, O’RAIFEARTAIGH, AND WOLF

combine to give the quantum piece missing in order to built up a total divergence when summed to the terms (100). We find that · ¸ √ 1 1 −δ1 F (2) [2B] = F (3) 2²1 ψ −∂µ A† ∂ µ A − vµν vˆ µν + E E † + D 2 4 2 summed to · 1 1 F (3) [1B(δ1 2F)] = F (3) √ (δ1 λ)σ µν ψvµν + √ λσ µν (δ1 ψ)vµν 2 2 ¸ ¢ 1¡ i i − E † ψδ1 ψ + Eλδ1 λ + √ D(δ1 ψ)λ + √ Dψδ1 λ 2 2 2 · i 1 = F (3) − √ ²1 σ µν σ ρσ ψvµν vρσ + √ ²1 σ µν ψvµν D 2 2 √ ¢ µν † + λσ ²1 vµν E − E E 2²1 ψ + E²1 σ µν λvµν

¸ i 1 2 µν − i²1 λD E + i D E²1 λ − √ D²1 σ vµν ψ − √ D ²1 ψ 2 2 ¸ · √ 1 1 = − F (3) √ ²1 σ µν σ ρσ ψvµν vρσ + E † E 2²1 ψ + √ D 2 ²1 ψ 2 2

give ¡ µ (2) ¢£ √ ¤ ∂ F − 2²1 ψ∂µ A† .

(101)

Collecting the two contributions (100) and (101) we end up with the wanted total divergence √ £ √ ¤ ¡ ¢£ √ ¤ ¡ ¢ F (2) ∂ µ − 2²1 ψ∂µ A† + ∂ µ F (2) − 2²1 ψ∂µ A† = ∂ µ −F (2) 2²1 ψ∂µ A† µ (2) ¶ F µ δ1 Aπ A , = ∂µ I

(102)

where the definitions of momenta and the Susy transformations were used. More labour is needed for the F † terms. We only give the result of that computation here. We have √ £ †√ † (2)† ¯ v µν† + F (2)† ²1 σ µ λD+F ¯ ∂µ F (2) 2²1 ψ∂ µ A† + iF (2) ²1 σν λˆ 2²1 σ ν σ µ ψ∂ν A† √ ¤ i † ¯ 2 . + i 2F (2) ψ¯ σ¯ µ ²1 E + √ F (3)† ²1 σ µ ψλ 2 Using the definitions of the non-canonical momenta given in (46)–(48) and the Susy transformations of the fields given in (19)–(21), these terms can be recast into the form · † † F (2) F (2) µ ¯ µ¯ + 2iδ1 λπλµ + F (2)† ²1 σν λv ¯ ∗µν ∂µ − δ1 Aπ A + 2i (2) δ1 ψπ ψ I F ¸ i † µ ¯ 2 . + 2iδ1 ψπψ + √ F (3) ²1 σ µ ψλ 2

(103)

SUSY NOETHER CURRENTS AND SEIBERG–WITTEN THEORY

177

Summing up the terms (102) and (103) and dividing by 2i we obtain the final expression µ

µ

†

V1 = δ1 Aπ A +

F (2) 1 † ¯ µ¯ + δ1 ψπψµ + δ1 λπλµ + 1 F (2)† ²1 σν λv ¯ ∗µν + √ F (3) ²1 σ µ ψ¯ λ¯ 2 . δ1 ψπ ψ (2) F 2i 2 2 (104)

As explained earlier this form is not canonical and has to be modified according to the rules given there. C. U(1) “Effective” Susy Transformations In this appendix we want to explicitly prove that the U(1) effective charge given in (51) indeed generates the Susy transformations (19)–(21). The component fields in the N = 2 SYM multiplet are ¯ λ(λ)). ¯ We have (A(A† ), v µ , ψ(ψ), Z

© ª I ¯ d 3 y A(x), δ1 ψ(y)π ψ¯ (y) −

11 A(x) ≡ {A(x), ²1 Q 1I }− = Z

√ © ª d 3 y A(x), 2²1 σ ν σ¯ 0 ψ(y)I(y)∂ν A† (y) −

= Z

√ © ª d 3 y A(x), 2²1 ψ(y)π AI (y) + irr. −

= =

√ 2²1 ψ(x) = δ1 A(x),

(105)

where “irr.” stands for terms irrelevant for the Poisson bracket, © ª 11 A† (x) ≡ A† (x), ²1 Q 1I − = 0 = δ1 A† (x) Z © © ª ª I 11 vi (x) ≡ vi (x), ²1 Q 1 − = d 3 y vi (x), 5 j (y) − δ1 v j (y) = δ1 vi (x) © ª 11 ψ¯ α˙ (x) ≡ ²1 Q 1I , ψ¯ α˙ (x) − =

Z

© I β˙ ª d 3 y δ1 ψ¯ β˙ (y) πψ¯ (y), ψ¯ α˙ (x) + = δ1 ψ¯ α˙ (x)

© ª 11 λ¯ α˙ (x) ≡ ²1 Q 1I , λ¯ α˙ (x) − = 0 = δ1 λ¯ α˙ (x).

(106) (107) (108) (109)

For 11 πψI¯ α˙ some attention is due to the fact that πψI¯ α˙ is a product of a bosonic function I and of a fermion ψ. On the one hand µ ¶ ª © 1 ¯ π I¯ (x) λ¯ 2 d 3 y − √ F (3)† ²1 σ 0 ψ, ψ α˙ − 2 2 1 = − √ F (3)† ²1α σα0α˙ λ¯ 2 , 2 2

© ª 11 πψI¯ α˙ (x) ≡ ²1 Q 1I , πψI¯ α˙ (x) − =

Z

(110)

on the other hand, writing explicitly πψI¯ α˙ we have 1 11 πψI¯ α˙ (x) = √ F (3) ²1 ψψ α σα0α˙ + iIσα0α˙ 11 ψ α , 2

(111)

178

IORIO, O’RAIFEARTAIGH, AND WOLF 1 (F (3) 11 A − F (3)† 11 A† ). 2i

where we have used 11 I = 11 πψI¯ α˙ we obtain 11 ψ α =

Thus, by comparing the two expressions for

µ ¶ √ α √ i i 2²1 − f ψ 2 + f † λ¯ 2 = 2²1α E on = δ1on ψ α , 4 4

(112)

where we have used the expression (45) for E on-shell and the Fierz identity ψα ψ β = − 12 δαβ ψ 2 . More labour is needed to compute 11 λ from 11 πλ¯Iα˙ . On the one hand © ª 11 πλ¯Iα˙ (x) ≡ ²1 Q 1 , πλ¯Iα˙ (x) − µ Z © ª 3 = d y 5i (y) δ1 vi (y), πλ¯Iα˙ (x) − © ª 1 (2)† ¯ F (y)v ∗0i (y) ²1 σi λ(y), πλ¯Iα˙ (x) − 2i ¶ ª © 2 1 ¯ − √ F (3)† (y)²1 σ 0 ψ(y) λ¯ (y), πλ¯Iα˙ (x) − . 2 2 −

(113)

On the other hand 1 11 πλ¯Iα˙ = √ F (3) ²1 ψλα σα0α˙ + iIσα0α˙ 11 λα . 2

(114)

Thus, writing explicitly 5i and collecting the terms according to the order of the derivative of F we have ¢ 1 ¡ (2) 0i i F vˆ − F (2)† vˆ †0i ²1α σiαα˙ + F (2)† v ∗0i ²1α σiαα˙ 2 2 1 1 + √ F (3) λσ 0i ψ²1α σiαα˙ − √ F (3) ²1 ψλα σα0α˙ 2 2 1 ¯ 1α σiαα˙ − √1 F (3)† ²1 σ 0 ψ¯ λ¯ α˙ . − √ F (3)† λ¯ σ¯ 0i ψ² 2 2

iIσα0α˙ 11 λα = −

(115)

The terms are arranged such that in the first column there are terms from 5i and in the second the other terms. Now we notice that the terms in the first line should combine to give the term proportional to vµν and the other two lines should combine to give the term proportional to D on in δ1 λ. As a matter of fact the first line gives −

¡ ¢ ¢ 1 ¡ (2) F − F (2)† vˆ 0i ²1α σiαα˙ = −iI ²1 σ µν σ 0 α˙ vµν 2

(116)

and the first term in the second line ¡ ¢ 1 1 γ √ F (3) λβ (σ 0i )β ψγ ²1α σiαα˙ = √ F (3) ²1 ψλα σα0α˙ − ²1 λψ α σα0α˙ 2 2 2

(117)

combined to the second term in the same line give ¡ ¢ 1 1 − √ F (3) ²1 ψλα σα0α˙ + ²1 λψ α σα0α˙ = √ F (3) ψλ²1α σα0α˙ . 2 2 2 2

(118)

SUSY NOETHER CURRENTS AND SEIBERG–WITTEN THEORY

179

Finally we can write the first term in the third line as ¡ ¢ 1 √ F (3)† ²1 σ 0 ψ¯ λ¯ α˙ − ²1 σ 0 λ¯ ψ¯ α˙ 2 2

(119)

which combined with the second term in the same line gives 1 ¯ 1α σα0α˙ . √ F (3)† ψ¯ λ² 2 2

(120)

Collecting the terms in (116), (118), and (120) we have ¡ ¢ 1 1 000 ¯ 1α σα0α˙ iIσα0α˙ 11 λα = −iI ²1 σ µν σ 0 α˙ vµν + √ F (3) ψλ²1α σα0α˙ + √ F † ψ¯ λ² 2 2 2 2

(121)

which eventually gives the wanted expression. We want also to show here that the transformations of the dummy-fields on-shell can be obtained by the transformations of the fermions. After a lengthy computation we obtain δ1 E = 0 ¶ ·µ ¶ ¸ ¡ ¢ 1 1 2 1 i † α † i † ¯ α˙ † ¯ ¯ ∂/ αα˙ A ψ + λα g − f ψλ − √ f f ψ λ − √ δ1 E = ²1 − f 2 2 2i 2 i4 2 ¸ √ ¡ ¢ 1 + √ f 2ψα E † + (σ µν λ)α vµν + iλα D , 2 2 ·

and

µ

¸ · ·µ ¶ √ ¡√ ¡ ¢ ¢ √ 1 2 1 f ψλ − f f † ψ¯ λ¯ − 2 2δ1 D = ²1α − f † i 2 ∂/αα˙ A† λ¯ α˙ + 2ψα g − 2i 2i ¸ ¢ ¡√ + f 2λα E − (σ µν ψ)α vµν + iψα D ,

(122)

(123)

(124)

where f (A, A† ) ≡ F (3) /I and g(A, A† ) ≡ F (4) /I and we used δ f = δ A(g − 2i1 f 2 ) + δ A† 2i1 f f † . Comparing these expressions with the Euler–Lagrange equations for the fermions we have √ ¯ δ1 E † = i 2²1 ∂/ ψ, δ1 D = − ²1 ∂/ λ¯ (125) δ1 E = 0, in agreement with the given Susy transformations. D. SU(2) Effective Hamiltonian and Lagrangian In this appendix we want to show in some detail the computations leading to the Hamiltonian and Lagrangian for the SU(2) sector of the SW model. The Poisson brackets relevant to compute the Hamiltonian are given by ½ Z i ¯ 1 }− = {²1 Q 1 , ²¯1 Q d 3 x d 3 y 5ai δ1 via + δ1 ψ¯ a πψa¯ + F †ab ²1 σi λ¯ a v ∗0ib 2 1 − √ F †abc ²1 σ 0 ψ¯ a λ¯ b λ¯ c + iI ab ²1 σ 0 λ¯ b ² acd Ac Ad† , 2 2 ¡ ¢ 1 e + √ ²¯1 ψ¯ e F e f g† ψ¯ f σ¯ 0 ψ g + λ¯ f σ¯ 0 λg 5ej δ¯1 v ej + δ¯1 Ae† π A† 2

180

IORIO, O’RAIFEARTAIGH, AND WOLF

√ ef i 2I ²¯1 σ¯ j σ 0 ψ¯ f D j Ae + F e f ²¯1 σ¯ j λe v ∗0 j f 2 ¾ 1 ef g 0 e f g e e f g f g† + √ F ²¯1 σ¯ ψ λ λ − ²¯1 πλ¯ ² A A , 2 2 − +

(126)

√ √ a e i²1 σi λ¯ a , δ¯1 v ej = i ²¯1 σ¯ j λ√ , δ¯1 Ae† = 2¯²1 ψ¯ e , δ1 Ab = 2²1 ψ b , δ1 ψ¯ a = where √ δ1 vi = −i 2²1 /D Aα† , and δ1 ψ¯ a πψa¯ = δ1 Ab π Ab + 2I ab ²1 σ i σ¯ 0 ψ b Di Aa† . The computation is lengthy but straightforward. There are 44 non-zero contributions to H that have to be rearranged and manipulated according to the identities above given. Let us write the following useful formulae © a bª © ª ψ¯ α˙ , ψα + = λ¯ aα˙ , λaα + = −i(I ab )−1 σα0α˙ (127) ¡ ¢ª ¡ ¢ i π Aa , ψαb λbα − = − (I bc )−1 F cad ψαd λdα 2 ¡ ¢ª ¡ ¢ © a i π A† , ψαb λbα − = + (I bc )−1 F †cad ψαd λdα 2 ©

(128) (129)

and ª © ²1 σi λ¯ a , ²¯1 σ¯ j λe − = i(I ae )−1 ²1 σi σ¯ 0 σ j ²¯1 ª © ²1 σi λ¯ a , λ¯ f σ¯ 0 λg − = i(I ag )−1 ²1 σi λ¯ f ª © ai 5 , D j Ae − = ² aeh δ ij Ah ¢ ¡ y {5ai (x), v ∗0 j f (y)}− = −2² 0i jk δ a f ∂k + ² a f h vkh (y) δ (3) (Ex − Ey ) ª © ²1 σi λ¯ a , λ f λg − = −i(I a f )−1 ²1 σ1 σ¯ 0 λg + ( f ↔ g) ª © ²1 σi λ¯ a , ²¯1 πλ¯e − = δ ae ²¯1 σ¯ i ²1 © ª ¡ ¢ Dµ A†a (x), π Ae † (y) − = δ ae ∂ix + ² ade vid (x) δ (3) (Ex − Ey ) ª © a α˙ πψ¯ , ²¯1 ψ¯ e ψ¯ f σ¯ 0 ψ g − = ²¯1α˙ δ ae ψ¯ f σ¯ 0 ψ g + ²¯1 ψ¯ e (σ¯ 0 ψ g )α˙ δ a f ª © b c λ¯ λ¯ , ²¯1 σ¯ j λe − = i(I ec )−1 ²¯1 σ¯ j σ 0 λ¯ b + (b ↔ c) ª © ²1 σ 0 ψ¯ a , ψ¯ f σ¯ 0 λg − = i(I ag )−1 ²1 σ 0 ψ¯ f {λ¯ b λ¯ c , λ¯ f σ¯ 0 λg }− = i(I gc )−1 λ¯ f λ¯ b + (b ↔ c) c f −1 ¯ g

bg −1 ¯ c

f

g

0 b

(131) (132) (133) (134) (135) (136) (137) (138) (139) (140)

{λ λ , λ λ }− = i(I ) λ σ¯ λ − i(I ) λ σ¯ λ + ( f ↔ g). ¯b ¯c

(130)

0

f

(141)

¯ 1 }− = ² α ²¯ α˙ {Q 1α , Q ¯ 1α˙ }+ and we After commutation we get rid of ²1 and ²¯1 by using {²1 Q 1 , ²¯1 Q 1 1 ˙ obtaining (note that we have still to divide by a factor 4i): trace with σ¯ 0αα “Classical” terms. Kinetic terms for the e. m. field15 , i −2i(I ab )−1 5ai 5bi − 2i(I ab )−1 Rbc π ai v ∗0ic − (I ae )−1 F †ab F e f v ∗0ib v ∗0i f . 2

(142)

Classical test on the e.m. kinetic piece: I −1 52 + I −1 R5∗ + 14 I −1 (R2 + I 2 )v ∗2 = I(E 2 + B 2 ) where v ∗ = 2B and 5 = −(I E + RB). 15

SUSY NOETHER CURRENTS AND SEIBERG–WITTEN THEORY

181

Kinetic terms for the scalar fields, ¡ ¢† − 4i(I ab )−1 π Aa π Ab + 4iI ab (Di Aa )(Di A†b ).

(143)

Kinetic terms for the spinors, 2iF ab λa σ i Di λ¯ b − 2iF †ab λ¯ a σ¯ i Di λb

(144)

and £ ¤ −2I ab ψ a σ i Di ψ¯ b − 2I ab ψ¯ a σ¯ i Di ψ b + 2ψ¯ e σ¯ i ψ g ∂i I ab + ² ebc vib I gc + ² gbc vib I ec .

(145)

Integrating by parts we have ¡ ¢ ¡ ¢ ¡ ¢ −4I ab λa σ i Di λ¯ b + ψ a σ i Di ψ¯ b + 2i ∂i F †ab λ¯ a σ¯ i λb − ∂i 2iF †ab λ¯ a σ¯ f λb + 2I ab ψ¯ a σ¯ i ψ b . (146) Yukawa potential, √ √ 3 − i3 2² eah I e f Ah ψ¯ f λ¯ a − i 2² acd I ad Ac ψ¯ d λ¯ b − √ ² b f g F †abc A f A†g ψ¯ a λ¯ c . 2 √ aeh ab †h b e √ e f g ed †g f d 1 − 3i 2² I A ψ λ − i 2² I A ψ λ − √ ² acd F eag Ac A†d ψ e λg . 2 By using the properties of ² abc F ade listed in Appendix A.3 we can recast these terms into √ −i2 2² abc I ad (Ac ψ¯ d λ¯ b + A†c ψ d λb ).

(147)

(148)

Higgs potential, 2i² acd I ab ² b f g Ac A†d A f A†g .

(149)

Purely quantum corrections. Terms that contribute to F abc σµν v µν and to dummy-fields on-shell via the two fermions piece 5, √ √ 2 2(I ec )−1 F †abc 5ei ψ¯ a σ¯ i0 λ¯ b − 2 2(I a f )−1 F f eg 5ai ψ e σi0 λg √ √ + 2(I ag )−1 Rab F †e f g v ∗0ib ψ¯ e σ¯ i0 λ¯ f − 2(I a f )−1 Rab F e f g v ∗0ib ψ e σi0 λg √ √ + i 2F †abc v ∗0ia ψ¯ b σ¯ i0 λ¯ c + i 2F abc v ∗0ia ψ b σi0 λc . (150) Summing them up we obtain16 √ √ −2 2(I a f )−1 F f eg ψ e σi0 λg (5ia + F †ab B ib ) + 2 2(I ec )−1 F †abc ψ¯ a σ¯ i0 λ¯ b (5ie + F ed B id ). (151) Terms that contribute to the dummy-fields on-shell only, i †e f g cad gc −1 ¯ e ¯ f a c 3i bec e f g ab −1 a c f g F (I ) ψ ψ ψ ψ + F F (I ) ψ ψ λ λ F 4 4 3i i + F †bec F †e f g (I ab )−1 ψ¯ a ψ¯ c λ¯ f λ¯ g + F †abc F e f g (I ae )−1 λ¯ b λ¯ c λ f λg . 4 4 16

†

5 + 12 F (2) v ∗ = − 2i1 I vˆ † + 5F and 5 + 12 F (2) v ∗ = − 2i1 I vˆ + 5F . Also vˆ = E + i B and vˆ † = E−i B.

(152)

182

IORIO, O’RAIFEARTAIGH, AND WOLF

F abcd -type of terms, 1 ¯ a ψ¯ b λ¯ c λ¯ d ). − (F abcd ψ a ψ v λc λd − F †abcd ψ 2

(153)

Collecting all these terms and dividing by 4i we obtain the Hamiltonian given in the body of the paper. The Lagrangian. The Lagrangian obtained by Legendre transforming the Hamiltonian in (60) is given by L ≡ L1 + L2 ,

(154)

where L1 =

· ¸ · 1 1 1 b †b − F ab v aµν vˆ µν + F †ab v aµν vˆ µν / µ A†b + iψ a D / ψ¯ b + iλa D / λ¯ b − I ab Dµ Aa D 2i 4 4 ¸ 1 adc c ¯ b ¯ d 1 acd b f g c †d f †g †c b d − √ ² (A ψ λ + A ψ λ ) + ² ² A A A A 2 2 ¢ 1¡ 1 − (∂µ F †ab )λ¯ a σ¯ µ λb − ∂0 F †ab ψ¯ a σ¯ 0 ψ b 2 2

(155)

contains the “classical” terms, and · ¸ 1 1 abc a µν b c 1 †abc ¯ a µν ¯ b c L2 = λ σ¯ ψ vµν √ F λ σ ψ vµν − √ F 2i 2 2 · 3 + (I ab )−1 F acd F be f (ψ d ψ f λc λe − ψ d λe λc ψ f ) − F gbc F ge f ψ a ψ c λe λ f 16 + F †acd F †be f (ψ¯ d ψ¯ f λ¯ c λ¯ e − ψ¯ d λ¯ e λ¯ c ψ¯ f ) − F †gbc F †ge f ψ¯ a ψ¯ c λ¯ e λ¯ f ¸ + F †acd F †be f (λc σ 0 ψ¯ f ψ d σ 0 λ¯ e − λc σ 0 λ¯ e ψ d σ 0 ψ¯ f ) 1 ab −1 †acd be f ¯ c ¯ d e f (I ) F F (ψ ψ ψ ψ + λ¯ c λ¯ d λe λ f ) 16 µ ¶ 1 1 abcd a b c d 1 †abcd ¯ a ¯ b ¯ c ¯ d F ψ ψ λλ − F ψ ψ λλ + 2i 4 4 −

(156)

contains the purely quantum terms. Note that these expressions are in agreement with those obtained by superfield expansion [4] and correctly reduce to the U(1) Lagrangian given in (22) in the limit.

ACKNOWLEDGMENTS The authors thank M. Magro, I. Sachs, and F. Delduc for useful discussions. A. I. was partially supported by the fellowship “Theoretical Physics of the Fundamental Interactions, COFIN 1999”—University of Rome Tor Vergata. S.W. was supported by the Swiss National Science Foundation during his staying at the Ecole Normale Sup´erieure de Lyon.

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