Novel symmetries in an interacting N = 2 supersymmetric quantum mechanical model
aa r X i v : . [ h e p - t h ] J un Novel symmetries in an interacting N = 2 supersymmetric quantummechanical model S. Krishna ( a ) , D. Shukla ( a ) , R. P. Malik ( a,b )( a ) Physics Department, Centre of Advanced Studies,Banaras Hindu University, Varanasi - 221 005, (U.P.), India ( b ) DST Centre for Interdisciplinary Mathematical Sciences,Faculty of Science, Banaras Hindu University, Varanasi - 221 005, India e-mails: [email protected]; [email protected]; [email protected]
Abstract:
We demonstrate the existence of a set of novel discrete symmetry transforma-tions in the case of an interacting N = 2 supersymmetric quantum mechanical model ofa system of an electron moving on a sphere in the background of a magnetic monopoleand establish its interpretation in the language of differential geometry. These discretesymmetries are, over and above, the usual three continuous symmetries of the theory which together provide the physical realizations of the de Rham cohomological operators of differ-ential geometry. We derive the nilpotent N = 2 SUSY transformations by exploiting ouridea of supervariable approach and provide geometrical meaning to these transformationsin the language of Grassmannian translational generators on a (1, 2)-dimensional super-manifold on which our N = 2 SUSY quantum mechanical model is generalized. We expressthe conserved supercharges and the invariance of the Lagrangian in terms of the supervari-ables (obtained after the imposition of the SUSY invariant restrictions) and provide thegeometrical meaning to (i) the nilpotency property of the N = 2 supercharges, and (ii) theSUSY invariance of the Lagrangian of our N = 2 SUSY theory.PACS numbers: 11.30.Pb, 03.65.-w, 02.40.-k Keywords : An interacting N = 2 SUSY quantum mechanical model; continuous and dis-crete symmetries; de Rham cohomological operators; Hodge duality operation; Hodge the-ory; supervariable approach; nilpotency property; geometrical interpretations Introduction
It is a well-known fact that three (out of four) fundamental interactions of nature aregoverned by the gauge theories which are endowed with the first-class constraints in thelanguage of Dirac’s prescription for the classification scheme. These theories are character-ized by the existence of local gauge symmetries which are generated by the above first-classconstraints. There is a class of gauge theories that respect the dual-gauge symmetry trans-formations in addition to the above cited local gauge symmetry transformations. Suchgauge theories provide the physical models for the Hodge theory within the framework ofBecchi-Rouet-Stora-Tyutin (BRST) formalism where the local gauge symmetries are tradedwith the nilpotent (anti-)BRST symmetries and the dual-gauge symmetry transformationsare elevated to the (anti-) co-BRST symmetry transformations at the quantum level. In anearlier article (see, e.g. [1] and references therein), we have shown that such examples ofgauge theories are the Abelian p -form ( p = 1 , ,
3) gauge theories that are described withinthe framework of BRST formalism in D = 2 p dimensions of spacetime.In a recent set of papers [2-5], we have established that the N = 2 supersymmetric(SUSY) quantum mechanical models (QMMs) also provide a set of tractable physical ex-amples of Hodge theory because their continuous symmetries (and corresponding conservedcharges) provide the physical realizations of the de Rham cohomological operators ∗ of dif-ferential geometry [6-10] and discrete symmetry transformations correspond to the physicalanalogue of the Hodge duality operation. We have also established the exact similaritiesbetween the Hodge algebra obeyed by the cohomological operators and algebra respectedby the conserved SUSY charges and the Hamiltonian of the theory (which is nothing butone of the simplest N = 2 SUSY algebra (i.e. sl (1 / physically because exploiting the inputs from these kind ofstudies, we have been able to establish [11] that the two (1 + 1)-dimensional (2D) free (non-)Abelian 1-form gauge theories (without any interaction with matter fields) are the new models for the topological field theories (TFTs) which capture some salient featuresof Witten-type TFTs and a few key features of Schwarz-type TFTs (see, e.g. [11-13] fordetails). We have also shown that the above free gauge field theoretic models and the 2D U (1) gauge theory interacting with the Dirac fields [14,15] are the perfect models for theHodge theory within the framework of BRST formalism. Such a set of Hodge theoreticmodels have also been proven in the description of the 2D modified versions of the Abelian1-form anomalous gauge theory as well as Proca theory (where the gauge and matter fieldsare present [16,17] along with the mass term for the gauge fields).In a very recent paper [5], we have established that the free version of the widely-studied interacting N = 2 SUSY QMM of a charged particle (i.e. an electron) constrained to moveon a sphere, in the background of a magnetic monopole, is a model for the Hodge theory. ∗ On a compact manifold without a boundary, there exists a set of three differential operators ( d, δ, ∆)which are called as the de Rham cohomological operators of differential geometry. The (co-) exteriorderivatives ( δ ) d are nilpotent of order two and their anticommutator defines the Laplacian operator ∆.They follow the algebra: d = δ = 0 , ∆ = ( d + δ ) = { d, δ } , [∆ , d ] = [∆ , δ ] = 0. The (co-)exteriorderivatives are connected with each-other by the relationship: δ = ± ∗ d ∗ where the ( ∗ ) stands for theHodge duality operation on the given compact manifold. The Laplacian operator behaves like the Casimiroperator (see, e.g. [6-10]) for this algebra. interacting N = 2 SUSY QMM of the above system also provides a tractable physicalexample of Hodge theory where the conserved Noether charges of this theory follow exactlythe same algebraic structure as that of the de Rham cohomological operators of differentialgeometry. Furthermore, we apply the theoretical arsenal of supervariable approach † [18-21] to derive the nilpotent N = 2 SUSY transformations of this theory and provide thegeometrical interpretations for the nilpotency property of the N = 2 SUSY transformations(and their generators ) as well as the SUSY invariance of the Lagrangian of our theory.In the context of the statements made on the method of supervariable approach toderive the N = 2 SUSY symmetries, we would like to mention that we have to concentrateon the (anti-)chiral supervariables so that we could capture only the nilpotency propertyof the SUSY symmetries (and avoid the property of absolute anticommutativity). In ourpresent endeavor (and its predecessor [5]) we have accomplished this goal and we haveprovided the geometrical basis for the nilpotency and SUSY invariance of the Lagrangianfor our present theory. The choice of the (anti-)chiral supervariables should be contrastedwith the superfield approach to BRST formalism (see, e.g. [22-27]) where the superfields ‡ ,defined on the (D, 2)-dimensional supermanifold, are expanded along all the Grassmanniandirections of the supermanifold so that one could capture the nilpotency as well as theanticommutativity properties of the (anti-)BRST symmetries for a given D -dimensionalgauge theory which is generalized onto the above supermanifold.The main motivating factors behind our present endeavor are as follows. First andforemost, it is important for us to prove that the interacting N = 2 SUSY QMM of themotion of an electron on a sphere in the background of a magnetic monopole is also a modelfor the Hodge theory (as has been shown by us that its free version is a tractable modelfor the Hodge theory [5]). Second, the physical realization of the abstract mathematicalde Rham cohomological operators in the language of discrete and continuous symmetrytransformations of a physical theory is interesting in its own right . Third, the goal ofproving various SUSY models (e.g. N = 2 , , ... ) to be the models for Hodge theory istheoretically important as it might turn out to be useful in the study of such SUSY gaugetheories (in various dimensions of spacetime) which are important, at the moment, becauseof their connection with the superstring theories. Finally, our present endeavor (and itspredecessor [5]), are our modest steps towards our main goal of proving the interacting N = 4 SUSY QMM to provide a tractable SUSY model for the Hodge theory (becausethe generalization of our present interacting N = 2 SUSY QMM to its counterpart N = 4SUSY quantum theory has already been discussed in the literature [28]).The contents of our present investigation are organized as follows. In Sec. 2, we re-capitulate the bare essentials of the nilpotent N = 2 SUSY transformations and show † We christen our present approach as the supervariable approach because when we set the Grassmannianvariables equal to zero in a supervariable (defined on the (1, 2)-dimensional supermanifold), we obtain anordinary variable which is a function of t only in the realm of SUSY quantum mechanics. ‡ When we set the Grassmannian variables equal to zero in a superfield (defined on a (D, 2)-dimensionalsupermanifold), we obtain an ordinary field which is a function of the D -dimensional spacetime ordinarycoordinates. This is why, we christen the approach, adopted in the realm of BRST formalism, as thesuperfield approach because it is applied to the description of an ordinary D -dimensional gauge fieldtheory. N = 2SUSY transformations. In Sec. 4, we derive the algebraic structure of the symmetry trans-formations in their operator form and show the existence of one of the simplest form of N = 2 SUSY algebra amongst the conserved charges. Our Sec. 5 deals with the connectionbetween N = 2 SUSY algebra and the de Rham cohomological operators of differential ge-ometry thereby providing the proof that our present model is a SUSY quantum mechanicalexample of a Hodge theory. Sec. 6 deals with the derivation of N = 2 SUSY transforma-tions by exploiting the supervariable approach. Finally, we make some concluding remarksin our Sec. 7 and point out a few future directions for further investigation.In our Appendix A, for the readers’ convenience, we mention a few key points that areconnected with the superspace formalism where the (anti-)chiral supervariables are used[29]. Our Appendices B, C and D are devoted to clarify some of the expressions/equationsthat have been used in the main body of our text. Let us begin with the following Lagrangian for the N = 2 SUSY quantum mechanicalmodel of the motion an electron on a sphere in the background of the Dirac’s magneticmonopole based on the CP (1) -model approach (see, e.g. [29] for details) L = 2 ( D t ¯ z ) · ( D t z ) + i (cid:2) ¯ ψ · ( D t ψ ) − ( D t ¯ ψ ) · ψ (cid:3) − g a, (1)where t is the evolution parameter in the theory and ∂ t = ∂/∂t . Here the dot productbetween ¯ z and z is taken to be ¯ z · z = Σ i =1 | z i | which clearly demonstrates that z = (cid:18) z z (cid:19) , ¯ z = (cid:0) z ∗ z ∗ (cid:1) = ⇒ ¯ z · z = | z | + | z | . (2)Similarly, other dot products, defined in the Lagrangian (1), should be taken into account.We also have the “covariant” derivatives D t z = ( ∂ t − i a ) z, D t ¯ z = ( ∂ t + i a ) ¯ z, D t ψ =( ∂ t − i a ) ψ, D t ¯ ψ = ( ∂ t + i a ) ¯ ψ where a is the “gauge” variable and the complex variables z and ¯ z are bosonic in nature while their superpartners ψ and ¯ ψ are fermionic (i.e. ψ =¯ ψ = 0 , ψ ¯ ψ + ¯ ψ ψ = 0) at the classical level § . In the Lagrangian (1), the parameter g stands for the charge on the monopole which interacts with the electron via gauge variable a through the coupling ( − g a ). This is the last term in the Lagrangian (1). The chargeon the electron has been set equal to ( −
1) and its mass has been taken to be (+1). § We lay empashis on the fact that the absolute anticommutativity ( ψ ¯ ψ + ¯ ψ ψ = 0) is true only at the classical level. As is evident from equation (9) (see, below), at the quantum level, ψ and ¯ ψ would not absolutely anticommute because they are canonically conjugate to each-other. s = s = 0) N = 2 SUSY transformations ( s , s ): s z = ψ √ , s ψ = 0 , s ¯ ψ = 2 i D t ¯ z √ , s ¯ z = 0 ,s ( D t z ) = D t ψ √ , s ( D t ¯ z ) = 0 , s a = 0 ,s ¯ z = ¯ ψ √ , s ¯ ψ = 0 , s ψ = 2 i D t z √ , s z = 0 ,s ( D t ¯ z ) = D t ¯ ψ √ , s ( D t z ) = 0 , s a = 0 , (3)because the Lagrangian (1) transforms to the total time derivatives as s L = ddt (cid:20) ( D t ¯ z ) · ψ √ (cid:21) , s L = ddt (cid:20) ¯ ψ · ( D t z ) √ (cid:21) . (4)The above expression demonstrates that the N = 2 SUSY transformations s and s arethe symmetry transformations for the action integral S = R dt L . It is good to mentionthat the “gauge” variable, defined in the following fashion ¶ a = − i z · ˙ z − ˙¯ z · z ) −
12 ( ¯ ψ · ψ ) , (5)remains invariant k ( s a = s a = 0) under the N = 2 SUSY transformations s and s dueto the supersphere defined by the following constraints (see, e.g. [29])¯ z · z − , ¯ z · ψ = 0 , ¯ ψ · z = 0 , (6)which define the CP (1) model on a sphere ¯ z · z = 1 (that is supersymmetrized with theinclusion of the above fermionic variables ψ and ¯ ψ ). It is worthwhile to mention that theinvariance of the constraints (¯ z · z − N = 2 SUSY transformations s and s , leads to the constraints ¯ z · ψ = 0 and ¯ ψ · z = 0. Mathematically, these constraintshave also been determined by the superspace formalism in [29] which has been conciselymentioned in our Appendix A for the readers’ convenience.We obtain a bosonic symmetry ( s ω ) in the theory which is nothing but the anticom-mutator s ω = { s , s } of the N = 2 SUSY transformations s and s . The dynamicalvariables of our theory transform, under s ω , as follows: s ω z = D t z, s ω ¯ z = D t ¯ z, s ω ψ = D t ψ, s ω ¯ ψ = D t ¯ ψ, (7) ¶ Actual superspace formalism [29] yields the expression for this “gauge” variable (see, Appendix A) as: a = − (cid:0) [ i (¯ z · ˙ z − ˙¯ z · z ) + ( ¯ ψ · ψ )] / [2 ¯ z · z ] (cid:1) . However, the substitution of the constraint ¯ z · z = 1 leads tothe expression for a as quoted in (5) which avoids the presence of the variables in the denominatorthe aswell as singularity in our present theory. k The explicit application of s and s on a yields the results: s a = ddt [ − i √ (¯ z · ψ )] + a √ (¯ z · ψ )and s a = ddt [+ i √ ( ¯ ψ · z )] + a √ ( ¯ ψ · z ) which, finally, imply that s a = s a = 0 due to the constraintconditions: ¯ z · ψ = 0 and ¯ ψ · z = 0 which define the supersymmetrized version of the sphere (¯ z · z = 1). i . Thus, it is clear that the anticommutator of two SUSY transfor-mations generates the “covariant” time translation for the variables. This is one of thekey requirements of a consistent N = 2 interacting SUSY theory which is clear from thesuperspace and the (anti-)chiral supervariable formalism (see, Appendix A). Under thebosonic symmetry transformations (7), the Lagrangian transforms as: s ω L = ddt [ L + 2 ga ] ≡ ddt [2( D t ¯ z ) · ( D t z ) + i { ¯ ψ · ( D t ψ ) − ( D t ¯ ψ ) · ψ } ]. As a consequence, the action integral of ourtheory remains invariant under s ω for the physically well-defined variables.The existence of continuous symmetries in a theory leads to the derivation of conservedNoether charges which turn out to be the generators for the continuous symmetry transfor-mations. The continuous and nilpotent N = 2 SUSY transformations s and s and theiranticommutator s ω lead to the derivation of the following Noether conserved charges Q = Π z · ψ √ √ (cid:20) D t ¯ z + i ψ · ψ + 2 g ) ¯ z + 2 i a ¯ z (1 − ¯ z · z ) (cid:21) · ψ ≡ √ (cid:20) D t ¯ z + i ψ · ψ + 2 g ) ¯ z (cid:21) · ψ, ¯ Q = ¯ ψ · Π ¯ z √ √ ψ · (cid:20) D t z − i ψ · ψ + 2 g ) z − i a z (1 − ¯ z · z ) (cid:21) ≡ √ ψ · (cid:20) D t z − i ψ · ψ + 2 g ) z (cid:21) ,Q ω = H = 2 ( D t ¯ z ) · ( D t z ) −
12 ( ¯ ψ · ψ + 2 g ) ( ¯ ψ · ψ ) ≡
12 Π z · Π ¯ z − (cid:0) ¯ ψ · ψ + 2 g (cid:1) , (8)where H is the Hamiltonian of our present theory. It is evident that, we have used theconstraint ¯ z · z = 1 in the shorter version of Q and ¯ Q in the above equation (8).We note that the above charges have also been expressed in terms of canonical conjugatemomenta Π z and Π ¯ z w.r.t. the dynamical variables z and ¯ z . In fact, it is elementary tocheck that the canonical momenta (that emerge from Lagrangian (1)) corresponding to thedynamical variables z, ¯ z, ψ and ¯ ψ are:Π z = ∂L∂ ˙ z = 2 D t ¯ z + i ψ · ψ + 2 g ) ¯ z + 2 i a ¯ z (1 − ¯ z · z ) , ≡ D t ¯ z + i ψ · ψ + 2 g ) ¯ z, Π ¯ z = ∂L∂ ˙¯ z = 2 D t z − i ψ · ψ + 2 g ) z − i a z (1 − ¯ z · z ) , ≡ D t z − i ψ · ψ + 2 g ) z, Π ψ = ∂L∂ ˙ ψ ≡ − i ψ, Π ¯ ψ = ∂L∂ ˙¯ ψ ≡ − i ψ, (9)where we have adopted the convention of left derivative w.r.t. the fermionic superpart-ners ψ and ¯ ψ in the computation of Π ψ and Π ¯ ψ . The equivalent forms of the canonicalmomenta (Π z , Π ¯ z ) have been derived by imposing the constraint ¯ z · z = 1. We also note6hat the Hamiltonian H of our theory [cf. (8)] can also be computed by the Legendretransformations ∗∗ H = Π z · ˙ z + ˙¯ z · Π ¯ z − Π ψ · ˙ ψ + ˙¯ ψ · Π ¯ ψ − L by exploiting the defini-tion of canonical conjugate momenta (9) and the expression for the Lagrangian (1). Inthis derivation, we have to use the constraint ¯ z · z = 1 and definitions of D t z, D t ¯ z and a = − / i (¯ z · ˙ z − ˙¯ z · z ) + ¯ ψ · ψ ].The Noether charges ( Q, ¯ Q, Q ω ) are conserved and their conservation law (i.e. ˙ Q =˙¯ Q = ˙ Q ω = 0) can be proven by exploiting the following equations of motion (along withthe constraint ¯ z · z = 1) that emerge from the Lagrangian (1) of our theory, namely; d Π ¯ z dt − i (cid:20) a D t z + ˙ z ψ · ψ + 2 g ) (cid:21) = 0 ,d Π z dt + i (cid:20) a D t ¯ z + ˙¯ z ψ · ψ + 2 g ) (cid:21) = 0 ,D t ψ − i ψ · ψ + 2 g ) ψ = 0 , D t ¯ ψ + i ψ · ψ + 2 g ) ¯ ψ = 0 , (10)where the canonical conjugate momenta (Π z , Π ¯ z ) are defined in the equation (9). It is worthpointing out that the above charges are conserved on the constrained surface defined bythe constraints ¯ z · z = 1 , ¯ z · ψ = 0 , ¯ ψ · z = 0. Thus, in the explicit proof of the conservationlaw, the EOM and constraints both are exploited together (see, Appendix B). It is straightforward to note that the Lagrangian (1) remains invariant under the followingdiscrete symmetry transformations, namely; z → ∓ ¯ z, ¯ z → ∓ z, ψ → ∓ ¯ ψ, ¯ ψ → ± ψ,t → − t, a → + a, g → g, (11)where we note that there is a time-reversal (i.e. t → − t ) symmetry. In other words, weobserve that, in reality, the transformation z → ∓ ¯ z denotes that z ( t ) → z ( − t ) = ∓ ¯ z T ( t )where the superscript T on ¯ z denotes the transpose operation on ¯ z . We have suppressedthe transpose operations in the transformation (11). This operation, however, should betaken into account in the rest of the discrete symmetry transformations (e.g. ψ → ∓ ¯ ψ ⇒ ψ ( t ) → ψ ( − t ) = ∓ ¯ ψ T ( t ) , ¯ ψ → ψ ⇒ ¯ ψ ( t ) → ¯ ψ ( − t ) = ± ψ T ( t ) , a ( t ) → a ( − t ) = a ( t ), etc.).The above discrete symmetry transformations are useful and important because theyfacilitates a connection between the two nilpotent ( s = s = 0) SUSY transformations s and s in the following manner, namely; s = ± ∗ s ∗ , (12)where the notation ∗ has been exploited for the discrete symmetry transformations (11).The ( ± ) signs in the above equation are dictated by two successive operations of the discrete ∗∗ We note that the ordering and the proper signatures have been taken into account in our definitionof the Legendre transformation which leads to the derivation of the Hamiltonian H . ∗ ( ∗ Φ ) = + Φ , Φ = z, ¯ z, ∗ ( ∗ Φ ) = − Φ , Φ = ψ, ¯ ψ. (13)As a side remark, we observe that the bosonic variables z ( t ) and ¯ z ( t ) have the (+) signwhen they are acted upon by two successive discrete symmetry transformations. On thecontrary, the fermionic variables (i.e. ψ ( t ) and ¯ ψ ( t )) acquire a ( − ) sign under the abovetransformations. Thus, we have the following relationships: s Φ = + ∗ s ∗ Φ ⇒ s = + ∗ s ∗ , Φ = z, ¯ z,s Φ = − ∗ s ∗ Φ ⇒ s = − ∗ s ∗ , Φ = ψ, ¯ ψ, (14)where there is an elegant interplay between the discrete and continuous symmetries of ourtheory. We note that there is existence of a set of reciprocal relationships, namely; s Φ = − ∗ s ∗ Φ ⇒ s = − ∗ s ∗ , Φ = z, ¯ z,s Φ = + ∗ s ∗ Φ ⇒ s = + ∗ s ∗ , Φ = ψ, ¯ ψ, (15)which is also true for a duality invariant theory [30].We end this section with the remarks that the relationships (12) between the N = 2SUSY transformations s and s are reminiscent of the relationships that exist betweenthe exterior and co-exterior derivatives of differential geometry (i.e. δ = ± ∗ d ∗ ). Thus,it is very interesting to note that the relationship (12) provides a physical realization forthe above mathematical relationship in the language of the interplay between discrete andcontinuous symmetries of our present theory. We would like to lay emphasis on the factthat we have discussed only one set of discrete symmetry transformation in (11). However,there might exist many other useful discrete symmetry transformations (see, Appendix D)in our theory, too. We shall focus, however, for our rest of the discussions on (11) asthe discrete symmetry transformations of our theory. It will be noted that the conservedcharges ( Q, ¯ Q, Q ω ) transform, under the discrete symmetry transformations (11), as: ∗ Q = − ¯ Q, ∗ ¯ Q = Q, ∗ H = H. (16)Thus, we point out that the transformations of the SUSY charges Q and ¯ Q are exactly likethe duality transformation of Maxwell’s theory of electrodynamics where ~E → ~B, ~B → − ~E (under the Maxwell duality symmetry transformations). Furthermore, we observe that acouple of successive operations of the discrete symmetry transformation on the chargesyield the following transformations: ∗ ( ∗ Q ) = − Q, ∗ ( ∗ ¯ Q ) = − ¯ Q, ∗ ( ∗ H ) = H. (17)The above observation establishes the fermionic nature of Q and ¯ Q as we have seen earlierin the case of ψ and ¯ ψ (i.e. ∗ ( ∗ ψ ) = − ψ, ∗ ( ∗ ¯ ψ ) = − ¯ ψ ). It is worthwhile to point outthat the algebra: Q = ¯ Q = 0 , { Q, ¯ Q } = H, [ H, Q ] = [ H, ¯ Q ] = 0 remains invariant undera couple of successive discrete symmetry transformations.8 Operators and their algebra
We focus on the algebraic structure that is followed by the transformation operators( s , s , s ω ) and their corresponding Noether charges ( Q, ¯ Q, Q ω ). In this respect, as hasalready been discussed, we note that the following is true, namely; s = 0 , s = 0 , { s , s } = s ω ≡ ( s + s ) , [ s ω , s ] = 0 , [ s ω , s ] = 0 , { s , s } 6 = 0 . (18)Thus, we observe that the bosonic symmetry operator ( s ω ) commutes with all the three continuous symmetry operators ( s , s , s ω ) of our interacting N = 2 SUSY QMM. Hence,it behaves like the Casimir operator. In exactly similar fashion, it can be seen that (due tothe relationship between the continuous symmetries and generators) we have the following: s Q = i { Q, Q } = 0 , s ¯ Q = i { ¯ Q, ¯ Q } = 0 ,s ¯ Q = i { ¯ Q, Q } = i H, s Q = i { Q, ¯ Q } = i H,s ω Q = − i [ Q, H ] = 0 , s ω ¯ Q = − i (cid:2) ¯ Q, H (cid:3) = 0 . (19)The above relationships lead to the realization of one of the simplest form of the N = 2SUSY algebra amongst the generators of ( s , s , s ω ), namely; Q = 12 { Q, Q } = 0 , ¯ Q = 12 { ¯ Q, ¯ Q } = 0 , { Q, ¯ Q } = H, [ H, Q ] = (cid:2) H, ¯ Q (cid:3) = 0 , (20)where there is no central extension. Thus, the continuous symmetry transformations andtheir generators respect the celebrated sl (1 /
1) algebra (without any central extension).The proof of (19) is algebraically a bit involved. Thus, for the readers’ convenience, wehave explicitly derived these relations in our Appendix C.Physically, the above algebra can be understood as follows. The nilpotency ( Q = 0 , ¯ Q = 0) of the charges ( Q, ¯ Q ) shows that these charges are fermionic in nature. Thevanishing of the commutators (i.e. [ Q, H ] = [ ¯
Q, H ] = 0) demonstrates that the SUSY(i.e. fermionic) charges ( Q, ¯ Q ) are conserved. Finally, the algebraic structure { Q, ¯ Q } = H says that the operation of two consecutive operations of s and s on any arbitrary variablelead to the “covariant” time translation [cf. Eqn. (7) of Sec. 2] of the same variable.This physical statement is mathematically captured by the presence of bosonic symmetrytransformation ( s ω ) in our theory (i.e. s ω = { s , s } ).The algebraic structures (18) and (20) are reminiscent of the algebra obeyed by the deRham cohomological operators of differential geometry, namely; d = 0 , δ = 0 , ∆ = ( d + δ ) = { d, δ } , [∆ , d ] = 0 , [∆ , δ ] = 0 , (21)where the operators ( d, δ, ∆) from a set called the de Rham cohomological operators ofdifferential geometry. Thus, we note that the Laplacian operator ∆ behaves like a Casimiroperator †† exactly as s ω and H behave in our equations (18) and (20). We note that the †† This Laplacian operator is not like the precise Casimir operator of the Lie algebra (as the Hodgealgebra (21) is not a Lie algebra). Similarly, the algebras (18) and (20) are also not Lie algebra. s , s , s ω ) and their generators ( Q, ¯ Q, Q ω ) provide thephysical realization of the cohomological operators ( d, δ, ∆) of differential geometry in thelanguage of symmetries and conserved charges.We close this section with the remark that the well-known relation: δ = ± ∗ d ∗ (thatexists between the co-exterior derivative and exterior derivative) is realized in the languageof the interplay between the discrete and continuous symmetries of our present theory [cf.(12)]. Thus, as far as the algebraic structure is concerned, we note that there is completesimilarity amongst the relations (18), (20) and (21) and there is a mapping between ( d, δ, ∆)and the conserved charges ( Q, ¯ Q, Q ω ). This identification is, however, still not completebecause there are cohomological properties that are associated with ( d, δ, ∆) and we have tocapture these properties in the language of conserved charges ( Q, ¯ Q, Q ω ). We shall dwellon these aspects in our next section. It is a well-known fact that when the de Rham cohomological operators ( d, δ, ∆) act on agiven form (i.e. f ( p ) ) of degree p , the following consequences ensue. First, the degree of thisform increases by one when it is acted upon by the exterior derivative d (i.e. d f ( p ) ∼ f ( p +1) ).On the contrary, the operation of δ on the same form (i.e. f ( p ) ) lowers the degree of theform by one (i.e. δ f ( p ) ∼ f ( p − ). It turns out that the operation of the Laplacian operatordoes not change the degree of the form at all (i.e. ∆ f ( p ) ∼ f ( p ) ). These properties are theessential ingredients when we discuss the cohomological aspects of a differential form w.r.t.the de Rham cohomological operators. We have to find the analogues of these observationsin the language of the symmetry properties and conserved charges of the interacting N = 2SUSY QMM of our present endeavor and establish a proper relationship.We have noted that H is like the Casimir operator for the sl (1 /
1) algebra: Q = ¯ Q = 0, { Q, ¯ Q } = H, [ H, Q ] = (cid:2) H, ¯ Q (cid:3) = 0. As a consequence of the latter two relations, it is evidentthat H − Q = Q H − and H − ¯ Q = ¯ Q H − (where we assume that H − is properly definedfor the Hamiltonian of our theory). We note that the following relations turn out to betrue due to the validity of the sl (1 /
1) algebra, namely; (cid:20) Q ¯ QH , Q (cid:21) = + Q, (cid:20) Q ¯ QH , ¯ Q (cid:21) = − ¯ Q, (cid:20) ¯ Q QH , Q (cid:21) = − Q, (cid:20) ¯ Q QH , ¯ Q (cid:21) = + ¯ Q, (22)where it has been taken for granted that H − Q = Q H − and H − ¯ Q = ¯ Q H − are verymuch true. Let us define now a state | χ > p (in the total quantum Hilbert space of states),with an eigenvalue equation w.r.t. the Hermitian operator ( Q ¯ Q/H ), as: (cid:18) Q ¯ QH (cid:19) | χ > p = p | χ > p , (23)where p is the eigenvalue which is real because ( Q ¯ Q/H ) operator is Hermitian. Now, wecan discuss about the impact of the algebra (22) on the state | χ > P which could be useful10n the description of the cohomological aspects of states. In this context, we shall utilizethe beauty of equation (22) as it behaves like the ladder operators of quantum mechanics.For instance, using the top two relationships from (22), we find that (cid:18) Q ¯ QH (cid:19) Q | χ > p = ( p + 1) Q | χ > p , (cid:18) Q ¯ QH (cid:19) ¯ Q | χ > p = ( p −
1) ¯ Q | χ > p , (cid:18) Q ¯ QH (cid:19) H | χ > p = p H | χ > p . (24)Thus, it is crystal clear that the states Q | χ > p , ¯ Q | χ > p and H | χ > p have the eigenval-ues ( p + 1) , ( p −
1) and p , respectively, w.r.t. the Hermitian operator (cid:0) Q ¯ Q/H (cid:1) . Theseconsequences are exactly same as the results that emerge from the operations of the coho-mological operators ( d, δ, ∆) on a form f ( p ) of degree p (because we do obtain the forms ofdegree ( p + 1) , ( p −
1) and p , respectively) due to their operations.Now we are in a position to exploit the mathematical beauty and power of the lower two entries of equation (22). If we define an eigenstate | ξ > q (in the total quantum Hilbertspace of states) w.r.t. the Hermitian operator (cid:0) ¯ Q Q/H (cid:1) , as (cid:18) ¯ Q QH (cid:19) | ξ > q = q | ξ > q , (25)it is evident that the following consequences ensue, namely; (cid:18) ¯ Q QH (cid:19) ¯ Q | ξ > q = ( q + 1) ¯ Q | ξ > q , (cid:18) ¯ Q QH (cid:19) Q | ξ > q = ( q − Q | ξ > q , (cid:18) ¯ Q QH (cid:19) H | ξ > q = q H | ξ > q , (26)which shows that the eigenvalues of states ¯ Q | χ > q , Q | χ > q and H | χ > q are ( q +1) , ( q − q , respectively, w.r.t. the Hermitian operator (cid:0) ¯ Q Q/H (cid:1) . Once again, we note that theseconsequences are exactly same as the operations of the cohomological operators ( d, δ, ∆)on the differential form of degree q . Hence, these mathematical operators can be realizedin the language of conserved charges of our present theory and explained below.Due to the beauty of equations (24) and (26), we observe that if an eigenstate | χ > p has an eigenvalue p w.r.t. the Hermitian operator ( Q ¯ Q/H ), the states Q | χ > p , ¯ Q | χ > p and H | χ > p would have the eigenvalues ( p + 1) , ( p −
1) and p , respectively. Now, if weidentify a corresponding differential form f ( p ) of degree p , we know that d f ( p ) , δ f ( p ) and∆ f ( p ) would have the degrees ( p + 1) , ( p −
1) and p , respectively. Thus, we conclude thatwe have a mapping: ( Q, ¯ Q, H ) ⇐⇒ ( d, δ, ∆) , (27)11etween the two kinds of operators. On the other hand, if we identify the eigenstate | ξ > q with eigenvalue q , w.r.t. the Hermitian operator ( ¯ Q Q/H ), we observe that the quantumstates ¯ Q | ξ > q , Q | ξ > q and H | χ > q would have their eigenvalues ( q + 1) , ( q − q , respectively. Similarly, the forms d f ( q ) , δ f ( q ) and ∆ f ( q ) would carry the degrees( q + 1) , ( q −
1) and q , respectively, if we start with a form f ( q ) of degree q . Thus, we havethe mapping of the operators: ( ¯ Q, Q, H ) ⇐⇒ ( d, δ, ∆) , (28)which are defined in two different spaces. Whereas, the operators ( ¯ Q, Q, H ) are definedin the quantum Hilbert space of states, the operators ( d, δ, ∆) are defined in the space offorms on a given manifold. All the above discussions imply that there are two realizationsof ( d, δ, ∆) in the language of conserved charges of our present N = 2 SUSY theory.Depending on the starting quantum state w.r.t. a given Hermitian operator, we have themappings: ( Q, ¯ Q, H ) ⇐⇒ ( d, δ, ∆) and/or ( ¯ Q, Q, H ) ⇐⇒ ( d, δ, ∆). N = 2 SUSY transformations: Supervari-able approach and geometrical interpretations
First of all, we focus on the derivation of s within the framework of the supervariable ap-proach. In this regards, we generalize the ordinary variables (cid:0) z ( t ) , ¯ z ( t ) , ψ ( t ) , ¯ ψ ( t ) (cid:1) to theircounterparts supervariables on a (1, 1)-dimensional chiral super-submanifold parametrizedby ( t, θ ). This super submanifold is a part of the general (1, 2)-dimensional supermanifoldon which our theory is generalized. Thus, chiral supervariables and their expansions are: z ( t ) −→ Z ( t, θ ) = z ( t ) + θ f ( t ) , ¯ z ( t ) −→ ¯ Z ( t, θ ) = ¯ z ( t ) + θ f ( t ) ,ψ ( t ) −→ Ψ( t, θ ) = ψ ( t ) + i θ b ( t ) , ¯ ψ ( t ) −→ ¯Ψ( t, θ ) = ¯ ψ ( t ) + i θ b ( t ) , (29)where secondary variables ( f ( t ) , f ( t )) and ( b ( t ) , b ( t )) are fermionic and bosonic in natureas is evident from the fermionic nature (i.e. Q = 0) of Grassmannian variable θ .To determine these secondary variables in terms of the basic variables, we have to invokeSUSY invariant restrictions (SUSYIRs). In this connection, we note that the followinguseful quantities remain invariant under s , namely; s ( ψ ) = 0 , s (¯ z ) = 0 , s ( z T · ψ ) = 0 , s (cid:2) D t ¯ z · z + i ¯ ψ · ψ (cid:3) = 0 , (30)where z T ( t ) · ψ ( t ) = z ψ + z ψ . The SUSYIRs demand that all the quantities that arepresent in the above brackets should remain independent of the “soul” coordinate ‡‡ θ whenthey are generalized onto the (1, 1)-dimensional chiral super submanifold. In other words, ‡‡ In the old literature (see, e.g. [27]), the Grassmannian variables have been christened as the “soul”coordinates and the ordinary spacetime variables have been called as the “body” coordinates.
12e have the following equalities as the SUSYIRs, namely;Ψ( t, θ ) = ψ ( t ) = ⇒ b ( t ) = 0 , ¯ Z ( t, θ ) = ¯ z ( t ) = ⇒ f ( t ) = 0 ,Z T ( t, θ ) · Ψ( t, θ ) = z T ( t ) · ψ ( t ) , D t ¯ Z ( t, θ ) · Z ( t, θ ) + i ¯Ψ( t, θ ) · Ψ( t, θ ) = 2 D t ¯ z ( t ) · z ( t ) + i ¯ ψ ( t ) · ψ ( t ) . (31)From the relationship number three, we obtain the relationship that f ( t ) ∝ ψ ( t ). Inthis regards, we make a choice and take f ( t ) = ψ ( t ) / √
2. This choice, entails upon b ( t ) = 2 D t ¯ z ( t ) / √
2. Plugging in these values into the expansions, we obtain the following Z (1) ( t, θ ) = z ( t ) + θ (cid:18) ψ ( t ) √ (cid:19) ≡ z ( t ) + θ ( s z ( t )) , ¯ Z (1) ( t, θ ) = ¯ z ( t ) + θ (0) ≡ ¯ z ( t ) + θ ( s ¯ z ( t )) , Ψ (1) ( t, θ ) = ψ ( t ) + θ (0) ≡ ψ ( t ) + θ ( s ψ ( t )) , ¯Ψ (1) ( t, θ ) = ¯ ψ ( t ) + θ (cid:18) i D t ¯ z ( t ) √ (cid:19) ≡ ¯ ψ ( t ) + θ (cid:0) s ¯ ψ ( t ) (cid:1) , (32)where the superscript (1) on the supervariables stand for the chiral supervariables thathave been obtained after the application of SUSYIRs in (31). A close look at (32) showsthat we have already obtained the SUSY transformations s .The above chiral expansions of the supervariables provide the geometrical meaning tothe SUSY transformations s because we note that ∂∂θ Ω (1) ( t, θ ) = s ω ( t ) ≡ ± i [ ω ( t ) , Q ] ± , (33)where Ω (1) ( t, θ ) is the generic chiral supervariable that has been obtained after the appli-cation of SUSYIRs in (31) and ω ( t ) is the generic ordinary variable. The subscripts ( ± ),on the square bracket, denote the existence of (anti)commutator for the given variable ω ( t ) being (fermionic)bosonic in nature. Thus, we observe that the operators ( ∂ θ , s , Q )are inter-connected. Geometrically, the SUSY transformation s on an ordinary variable isequivalent to the translation of the corresponding supervariable (obtained after SUSYIRs)along the Grassmannian θ -direction of the (1, 1)-dimensional chiral super-submanifold.Furthermore, two successive translations along this Grassmannian direction captures thenilpotency ( s = 0 , Q = 0) of s and Q which is nothing other than the nilpotency ( ∂ θ = 0)property of the translational generator ∂ θ along the θ -direction.To derive the SUSY transformations s , we generalize the ordinary variables ( z ( t ) , ¯ z ( t ), ψ ( t ) , ¯ ψ ( t )) onto (1, 1)-dimensional anti-chiral super-submanifold that is characterized by( t, ¯ θ ). In other words, we have the following: z ( t ) −→ Z ( t, ¯ θ ) = z ( t ) + ¯ θ f ( t ) , ¯ z ( t ) −→ ¯ Z ( t, ¯ θ ) = ¯ z ( t ) + ¯ θ f ( t ) ,ψ ( t ) −→ Ψ( t, ¯ θ ) = ψ ( t ) + i ¯ θ b ( t ) , ¯ ψ ( t ) −→ ¯Ψ( t, ¯ θ ) = ¯ ψ ( t ) + i ¯ θ b ( t ) , (34)13here ( f ( t ) , f ( t )) and ( b ( t ) , b ( t )) are the pair of fermionic and bosonic secondary vari-ables that have to be determined by exploiting the SUSYIRs on the anti-chiral supervari-ables. In this connection, it can be checked that the following is true, namely; s ( ¯ ψ ) = 0 , s ( z ) = 0 , s (¯ z · ¯ ψ T ) = 0 , s (cid:2) z · D t z − i ¯ ψ · ψ (cid:3) = 0 . (35)Thus, the above invariant quantities would remain independent of the “soul” coordinate ¯ θ when they are generalized onto the anti-chiral super sub-manifold. In other words, we havethe following SUSYIRs in the form of the equalities, namely; Z ( t, ¯ θ ) = z ( t ) , ¯Ψ( t, ¯ θ ) = ¯ ψ ( t ) , ¯ Z ( t, ¯ θ ) · ¯Ψ T ( t, ¯ θ ) = ¯ z ( t ) · ¯ ψ T ( t ) , Z ( t, ¯ θ ) · D t Z ( t, ¯ θ ) − i ¯Ψ( t, ¯ θ ) · Ψ( t, ¯ θ ) = 2 ¯ z ( t ) · D t z ( t ) − i ¯ ψ ( t ) · ψ ( t ) , (36)which lead to the determination of the secondary variables as f ( t ) = 0 , b ( t ) = 0 , f ( t ) = ¯ ψ ( t ) √ , b ( t ) = 2 D t z ( t ) √ . (37)The substitution of these secondary variables into the expansions (34) lead to the following Z (2) ( t, ¯ θ ) = z ( t ) + ¯ θ (0) ≡ z ( t ) + ¯ θ ( s z ) , ¯ Z (2) ( t, ¯ θ ) = ¯ z ( t ) + ¯ θ (cid:18) ¯ ψ √ (cid:19) ≡ ¯ z ( t ) + ¯ θ ( s ¯ z ) , Ψ (2) ( t, ¯ θ ) = ψ ( t ) + ¯ θ (cid:18) i D t z √ (cid:19) ≡ ψ ( t ) + ¯ θ ( s ψ ) , ¯Ψ (2) ( t, ¯ θ ) = ¯ ψ ( t ) + ¯ θ (0) ≡ ¯ ψ ( t ) + ¯ θ ( s ¯ ψ ) , (38)where the superscript (2) denotes the expansions of the supervariables after the applicationof SUSYIRs in (36). The geometrical meaning of the transformations s and its nilpotencycan be given in terms of the translational generator ∂ ¯ θ along ¯ θ -direction of the (1, 1)-dimensional anti-chiral super submanifold along exactly similar lines as the ones that havebeen given for s after equation (32) and (33).The conserved charges Q and ¯ Q can also be generalized onto the (anti-)chiral supersub-manifolds where the supervariables would be taken after the application of SUSYIRs.14hese expressions in terms of the ∂ θ , ∂ ¯ θ , dθ, d ¯ θ and (anti-)chiral supervariables are Q = ∂∂θ h D t ¯ Z (1) ( t, θ ) · Z (1) ( t, θ ) i ≡ ∂∂θ h D t ¯ z ( t ) · Z (1) ( t, θ ) i , = Z dθ h D t ¯ Z (1) ( t, θ ) · Z (1) ( t, θ ) i ≡ Z dθ h D t ¯ z ( t ) · Z (1) ( t, θ ) i ,Q = ∂∂θ h − i ¯Ψ (1) ( t, θ ) · Ψ (1) ( t, θ ) i ≡ ∂∂θ h − i ¯Ψ (1) ( t, θ ) · ψ ( t ) i , = Z dθ h − i ¯Ψ (1) ( t, θ ) · Ψ (1) ( t, θ ) i ≡ Z dθ h − i ¯Ψ (1) ( t, θ ) · ψ ( t ) i , ¯ Q = ∂∂ ¯ θ h Z (2) ( t, ¯ θ ) · D t Z (2) ( t, ¯ θ ) i ≡ ∂∂ ¯ θ h Z (2) ( t, ¯ θ ) · D t z ( t ) i , = Z d ¯ θ h Z (2) ( t, ¯ θ ) · D t Z (2) ( t, ¯ θ ) i ≡ Z d ¯ θ h Z (2) ( t, ¯ θ ) · D t z ( t ) i , ¯ Q = ∂∂ ¯ θ h + i ¯Ψ (2) ( t, ¯ θ ) · Ψ (2) ( t, ¯ θ ) i ≡ ∂∂ ¯ θ h + i ¯ ψ ( t ) · Ψ (2) ( t, ¯ θ ) i , = Z d ¯ θ h + i ¯Ψ (2) ( t, ¯ θ ) · Ψ (2) ( t, ¯ θ ) i ≡ Z d ¯ θ h + i ¯ ψ ( t ) · Ψ (2) ( t, ¯ θ ) i . (39)The nilpotency of ∂ θ and ∂ ¯ θ (i.e. ∂ θ = 0 , ∂ θ = 0) implies that the nilpotency of charges Q and ¯ Q (ı.e. ∂ θ Q = 0 , ∂ ¯ θ ¯ Q = 0). Furthermore, when we express the above expressions forthe charges Q and ¯ Q in terms of the symmetry transformations s and s and dynamicalvariables, we observe that Q = s (cid:16) D t ¯ z · z (cid:17) ≡ s (cid:16) − i ¯ ψ · ψ (cid:17) , ¯ Q = s (cid:16) z · D t z (cid:17) ≡ s (cid:16) + i ¯ ψ · ψ (cid:17) . (40)The nilpotency of charges Q and ¯ Q can be readily proven by using the constraints ¯ ψ · z = 0and ¯ z · ψ = 0 that have been taken into considerations in our present endeavor.We can also capture the invariance of the Lagrangian (1) in terms of the (anti-) chiralsupervariables that are obtained after the impositions of the SUSYIRs as given below L → ˜ L ( ac ) = 2 D t ¯ Z (2) · D t Z (2) + i (cid:2) ¯Ψ (2) · D t Ψ (2) − D t ¯Ψ (2) · Ψ (2) (cid:3) − g a,L → ˜ L ( c ) = 2 D t ¯ Z (1) · D t Z (1) + i (cid:2) ¯Ψ (1) · D t Ψ (1) − D t ¯Ψ (1) · Ψ (1) (cid:3) − g a, (41)where the superscripts ( c ) and ( ac ) denote the chiral and anti-chiral nature of the La-grangians ˜ L ( c )0 and ˜ L ( ac )0 , respectively. The superscripts (1) and (2) on the supervariablescorrespond to the expansions (32) and (38). Mathematically, we observe that ∂∂θ h ˜ L ( c ) i = ddt (cid:16) D t ¯ z · ψ √ (cid:17) ≡ s L,∂∂ ¯ θ h ˜ L ( ac ) i = ddt (cid:16) ¯ ψ · D t z √ (cid:17) ≡ s L. (42)The relation (42) provides the geometrical interpretation for the SUSY invariances of theLagrangian (1). This can be stated that in the following manner. The (anti-)chiral super15agrangians (41) are the sum of composite supervariables that have been obtained afterthe application of the SUSYIRs. The translations of these super Lagrangians along the¯ θ and θ -directions of the (anti-)chiral super-submanifolds are such that they produce theordinary total time derivatives [cf. (42)] in the ordinary space thereby leaving the actionintegral ( S = R dt L ) invariant as the physical variables vanish off at t = ±∞ on physicalgrounds. The central theme of our present investigation has been to establish that the interacting N = 2 SUSY QMM of the motion of an electron on a sphere, in the background of amagnetic monopole, provides a tractable SUSY model for the Hodge theory. We haveaccomplished this goal in our present endeavor because we have shown that the discreteand continuous symmetries of our present theory are such that they together provide thephysical realizations of the de Rham cohomological operators of differential geometry. Afew of the specific properties of these cohomological operators are also captured in thelanguage of the conserved charges of our present theory in the quantum Hilbert space.Some of the subtle issues of our present model are the observations that the constraints(e.g. ¯ z · z = 1 , ¯ ψ · z = 0 , ¯ z · ψ = 0) and the equations of motion (10) together play veryimportant roles in the proof of the conservation laws as well as in the determination ofthe sl (1 /
1) algebra from the symmetry principles and conserved charges. In particular, asdiscussed in Appendix C, the proof of { Q, ¯ Q } = H , from the symmetry transformations s ¯ Q and s Q , requires the mathematical beauty and power of the constraints as well as theequations of motion. It is gratifying to observe that the interacting version of our earlierwork on the free N = 2 SUSY QMM [5] also turns out to be the model for the Hodgetheory as the symmetries of the theory provide the physical realizations of ( d, δ, ∆).We have derived the N = 2 SUSY transformations s and s by exploiting the supervari-able approach where we have been theoretically compelled to consider only the (anti-)chiralsupervariables. This is due to the fact that N = 2 SUSY transformations are nilpotent( s = s = 0) but they are not absolutely anticommuting (i.e. s s + s s = 0). Thus,eventhough our theory is generalized onto a specific (1, 2)-dimensional supermanifold, wehave not taken the full expansions of the supervariables along (1 , θ, ¯ θ, θ ¯ θ )-directions of thesupermanifold (because the full expansions would automatically imply the validity of ab-solute anticommutativity). We have also provided the geometrical basis for nilpotency andinvariance of the Lagrangian within the framework of our supervariable approach [5,18-21].Besides our present work on the N = 2 SUSY case of the charge-monopole system,there are many interesting works (see, e.g. [31-40]) on this particular system as well as thatof the charge-dyon system (and their very beautiful and diverse super-extensions). Forinstance, it has been shown, in a couple of very interesting papers [31,32], that our presentsystem exhibits a hidden conical dynamics. The elaborate discussions on the integrabilityproperties of this system, its systematic SUSY generalizations, its connection with thereflectionless potentials, etc., have also attracted a great deal of interest amongst theoreticalphysicists of very different backgrounds (see, e.g. [31-40] for more details). We would liketo lay emphasis on the fact that, in our present investigation, we have concentrated only
16n the algebraic structures of the symmetries and corresponding conserved charges for thissystem. However, a whole range of other directions remain to be explored and these remainprecisely an open set of interesting problems for the future investigations [41].Our present N = 2 SUSY QMM is special in the sense that there are no singularitiesin our theory mainly because the constraints ¯ z · z = 1 , ¯ ψ · z = 0 , ¯ z · ψ = 0 (and their timederivatives) do play important role in avoiding them. The other good feature of our presentmodel is the observation that it can be generalized to its counterpart N = 4 version [28].Thus, it would be a very nice idea to look for the physical realizations of the cohomologicaloperators in the case of N = 4 and N = 8 versions of our present model. We speculatethat this understanding might be useful for us in the study of N = 2 , , new class of topological field theories. These new topological field theories have been shownto possess the Lagrangian density that look like the Witten-type topological field theories.However, their (anti-)BRST and (anti-)co-BRST symmetries are just like the Schwarz-typetopological field theories as they do not incorporate the shift symmetries which are thehallmark of a typical Witten-type topological field theories. We are presently busy withthese theoretical ideas and we shall report about our progress in our future publications.
Acknowledgements:
Two of us (SK and DS) would like to gratefully acknowledge finan-cial support from UGC, Government of India, New Delhi, under their SRF-schemes.
Appendix A: On Superspace Formalism
We invoke here some of the essential ingredients of the superspace formalism to clarify someof the expressions and equations that have been used in our main body of the text. In thisconnection, we define the supercovariant derivatives ( D, ¯ D ) for the interacting N = 2SUSY quantum mechanical model as follows (see, e.g. [29]) D = ∂ θ − i ¯ θ D t , ¯ D = ∂ ¯ θ − i θ D t . ( A. t, θ, ¯ θ ))and anti-chiral ( ¯Φ( t, θ, ¯ θ )) supervariables as:Φ( t, θ, ¯ θ ) = z ( t ) + θ ψ ( t ) − i θ ¯ θ D t z ( t ) , ( D t z = ˙ z − i a z ) , ¯Φ( t, θ, ¯ θ ) = ¯ z ( t ) − ¯ θ ¯ ψ ( t ) + i θ ¯ θ D t ¯ z ( t ) , ( D t ¯ z = ˙¯ z + i a ¯ z ) , ( A. D Φ( t, θ, ¯ θ ) = 0 and D ¯Φ( t, θ, ¯ θ ) = 0. In the CP (1) model approach, we havethe following supersymmetric constraint in terms of Φ( t, θ, ¯ θ ) and ¯Φ( t, θ, ¯ θ ), namely;¯Φ( t, θ, ¯ θ ) · Φ( t, θ, ¯ θ ) − , ( A. z · z = 1) under backgroundof a monopole. Setting the coefficients of θ, ¯ θ, θ ¯ θ and constant term equal to zero in (A.3),17e obtain the following expressions for the constraints and a , namely;¯ z · z = 1 , ¯ z · ψ = 0 , ¯ ψ · z = 0 , a = − [ i (¯ z · ˙ z − ˙¯ z · z ) + ( ¯ ψ · ψ )]2 (¯ z · z ) , ( A. N = 2 SUSYgenerators for our N = 2 interacting quantum mechanical model as: Q = 1 √ (cid:0) ∂ θ + i ¯ θ D t (cid:1) , ¯ Q = 1 √ ∂ ¯ θ + i θ D t ) . ( A. N = 2 SUSY algebra, as { Q, Q } ≡ Q = 0 , { ¯ Q, ¯ Q } ≡ ¯ Q = 0 , { Q, ¯ Q } = i D t . ( A. N = 2 SUSY transformations (corresponding to s and s ) on a variable leads to the covariant “time” translation on the same variablewhich has been demonstrated to be true in our Sec. 2. In fact, from equation (7), it isclear that the bosonic symmetry ( s ω = { s s } ) is such that it transforms the dynamicalvariable of our present theory to their “covariant” time derivative modulo a factor of i . Appendix B: Conservation law for the charges
In this Appendix, we perform some explicit computations that are connected with theproof of the conservations of charges Q and ¯ Q . In this connection, first of all, we take thestraightforward time derivative on the charge Q as follows dQdt = 1 √ (cid:20) d Π z dt · ψ + Π z · dψdt (cid:21) . ( B. Q :˙ Q = 1 √ h − i (cid:26) a D t ¯ z + 12 ( ¯ ψ · ψ + 2 g ) ˙¯ z (cid:27) · ψ + (cid:26) D t ¯ z + i ψ · ψ + 2 g ) ¯ z (cid:27) · ˙ ψ i . ( B. ψ variable which leads to˙ ψ = i ψ · ψ + 2 g ) ψ + i a ψ. ( B. z in the above equation demonstrate that ¯ z · ˙ ψ = 0 dueto the constraint equation ¯ z · ψ = 0. This, in turn, implies that ˙¯ z · ψ = 0 (due to theobservation that d/dt (¯ z · ψ ) = 0 ⇒ ¯ z · ˙ ψ + ˙¯ z · ψ = 0). It is now trivial to prove that ˙ Q = 0.We lay emphasis on the fact that it is an elegant combination of the equations of motion(10) and the constraints (¯ z · z = 1 ¯ z · ψ = 0 , ¯ ψ · z = 0) which are to be invoked for theproof of ˙ Q = 0. In exactly similar fashion, it can be seen that ˙¯ Q = 0 due to the equationsof motion (10) and the constraints ¯ ψ · z = 0 , ˙¯ ψ · z = 0 , ¯ ψ · ˙ z = 0. The proof of ˙ H ≡ ˙ Q ω = 018s elementary at the classical as well as the quantum levels because the Poisson bracketand/or commutator of H with itself is zero. Appendix C: Symmetries and Algebraic Structure
We exploit here the ideas of continuous symmetries and their generators to derive one of thesimplest form of sl (1 /
1) algebra that is satisfied amongst the conserved charges ( Q, ¯ Q, Q ω )of our theory. By exploiting the concept of generators, it is trivial to note that s Q = i { Q, Q } = 0 , s ¯ Q = i { ¯ Q, ¯ Q } = 0 , ( C. Q, ¯ Q ) from (8) and symmetry transformations from (3), we observe that s Q = −
12 ( D t ¯ z · ψ )(¯ z · ψ ) , s ¯ Q = 12 ( ¯ ψ · D t z )( ¯ ψ · z ) , ( C. zero on the constrained surface defined by the constraint conditions¯ z · ψ = 0 and ¯ ψ · z = 0, respectively. In similar fashion, we compute the l.h.s. of thefollowing relationships: s ¯ Q = i { ¯ Q, Q } = i H, s Q = i { Q, ¯ Q } = i H, ( C. s Q = 2 i D t ¯ z · D t z + D t ¯ ψ · ψ + 12 ( ¯ ψ · D t z )(¯ z · ψ )+ i ψ · ψ + 2 g ) ( ¯ ψ · ψ ) −
12 ( ¯ ψ · ψ + 2 g ) (¯ z · D t z ) ,s ¯ Q = 2 i D t ¯ z · D t z − ( ¯ ψ · D t ψ ) −
12 ( D t ¯ z · ψ )( ¯ ψ · z )+ i ψ · ψ + 2 g ) ( ¯ ψ · ψ ) + 12 ( D t ¯ z · z ) ( ¯ ψ · ψ + 2 g ) . ( C. D t z, D t ¯ z and constraints ¯ ψ · z = 0 , ¯ z · ψ = 0 plus the equationsof motion for ψ and ¯ ψ from (10), we obtain the following: s ¯ Q = 2 i D t ¯ z · D t z + 12 ( ˙¯ z · z + i a ¯ z · z ) ( ¯ ψ · ψ + 2 g ) − i ψ · ψ + 2 g ) ( ¯ ψ · ψ ) ,s Q = 2 i D t ¯ z · D t z −
12 ( ¯ ψ · ψ + 2 g ) (¯ z · ˙ z − i a ¯ z · z ) − i ψ · ψ + 2 g ) ( ¯ ψ · ψ ) . ( C. a and constraint ¯ z · z = 1 and ddt (¯ z · z −
1) = 0, in theabove equation, to obtain the following s ¯ Q = i (cid:20) D t ¯ z · D t z −
12 ( ¯ ψ · ψ + 2 g ) ( ¯ ψ · ψ ) (cid:21) ≡ iH, Q = i (cid:20) D t ¯ z · D t z −
12 ( ¯ ψ · ψ + 2 g ) ( ¯ ψ · ψ ) (cid:21) ≡ iH. ( C. s ¯ Q = i { ¯ Q, Q } = i H , we have used the constraintconditions ¯ ψ · z = 0 , ¯ z · z = 1 , ddt (¯ z · z −
1) = 0 and the definitions of a, D t ψ, D t ¯ z . On theother hand, in the explicit composition of s Q = i { Q, ¯ Q } = i H , we have exploited theconstraint conditions ¯ z · ψ = 0 , ¯ z · z = 1 , ddt (¯ z · z −
1) = 0 and the definitions of a, D t ¯ ψ, D t z .Thus, we note that the constraints as well as the equations of motion are to be exploitedjudiciously to prove that s ¯ Q = s Q = i H which, ultimately, implies that { Q, ¯ Q } = H . Appendix D: On Discrete Symmetries
In addition to the discrete symmetry transformations (11), we have the following usefuldiscrete symmetry transformations for the Lagrangian (1), namely; z → ± i ¯ z, ¯ z → ∓ i z, ψ → ± i ¯ ψ, ¯ ψ → ± i ψ,t → − t, a → + a, g → g. ( D. ∗ L = L, ∗ H = H, ∗ Q = − ¯ Q, ∗ ¯ Q = Q are true under (D.1).We dwell a bit on a couple of discrete symmetry transformations for the Lagrangian (1)that are not acceptable to us because they do not comply with the strictures laid down bythe duality invariant theories [30]. These transformations, for instance, are z → ± ¯ z, ¯ z → ± z, ψ → ± ¯ ψ, ¯ ψ → ± ψ,t → + t, a → − a, g → − g, ( D. ∗ ( ∗ z ) = z, ∗ ( ∗ ¯ z ) = ¯ z, ∗ ( ∗ ψ ) = ψ, ∗ ( ∗ ¯ ψ ) = ¯ ψ. ( D. s Φ = + ∗ s ∗ Φ , Φ = z, ¯ z, ψ, ¯ ψ. ( D. s Φ = − ∗ s ∗ Φ , Φ = z, ¯ z, ψ, ¯ ψ, ( D. not satisfied at all. Thus, the discrete symmetry transformations (D.2) of the Lagrangian(1) are not acceptable because they do not comply with the conditions (e.g. reciprocal re-lationship) laid down by the duality invariant theories [30].20 eferences [1] R. Kumar, S. Krishna, A. Shukla and R. P. Malik, Int. J. Mod. Phys. A , 1450135 (2014)[2] R. Kumar and R. P. Malik, Europhys. Lett. , 11002 (2012)[3] R. P. Malik and Avinash Khare, Ann. Phys. bf 334, 142 (2013)[4] R. Kumar and R. P. Malik,
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