Twisted Superalgebras and Cohomologies of the N=2 Superconformal Quantum Mechanics
aa r X i v : . [ h e p - t h ] N ov CERN-PH-TH/2010CBPF-NF-005/11
Twisted Superalgebras and Cohomologies of the N = 2 Superconformal Quantum Mechanics
Laurent Baulieu †‡ and Francesco Toppan ∗ † Theoretical Division CERN ‡ LPTHE Universit´e Pierre et Marie Curie ∗ CBPF, Rio de Janeiro Abstract
We prove that the invariance of the N = 2 superconformal quantum mechanics is con-trolled by subalgebras of a given twisted superalgebra made of 6 fermionic (nilpotent) gen-erators and 6 bosonic generators (including a central charge). The superconformal quantummechanics actions are invariant under subalgebras of this quite large twisted superalgebra.They are in fact fully determined by a subalgebra with only 2 fermionic and 2 bosonic (thecentral charge and the ghost number) generators. The invariant actions are Q i -exact, witha Q i ′ -exact ( i ′ = i ) antecedent for some of its 6 fermionic generators. It follows that the su-perconformal quantum mechanics actions with Calogero potentials are uniquely determinedeven if, in its bosonic sector, the twisted superalgebra does not contain the one-dimensionalconformal algebra sl (2), but only its Borel subalgebra. The general coordinate covariance ofthe non-linear sigma-model for the N = 2 supersymmetric quantum mechanics in a curvedtarget space is fully implied only by its worldline invariance under a pair of the 6 twistedsupersymmetries. The transformation connecting the ordinary and twisted formulations ofthe N = 2 superconformal quantum mechanics is explicitly presented. email address: [email protected] email address: [email protected] CH-1211 Gen`eve, 23, Switzerland Rua Dr. Xavier Sigaud 150, cep 22290-180, Rio de Janeiro, RJ, Brazil
Introduction.
Both the ordinary and the conformal N = 2 supersymmetric quantum mechanical models de-scribe interesting dynamics. The N = 2 supersymmetry has applications to solvable potentialsand the motion of particles in certain gravitational backgrounds [1]–[9]. It is an appropri-ate framework to describe topological invariants of various target-spaces for the propagatingbosons [10]. The worldline N = 2 global supersymmetry has a natural extension as a super-conformal algebra ( N = 2 SCA) [11].The interpretation of the N = 2 supersymmetric quantum mechanics as a topological quan-tum field theory derives from the invariance under a twisted superalgebra, whose nilpotentfermionic generators define a BRST-type cohomology [12, 13, 14, 15].We construct here a twisted superalgebra, called “twisted N = 2 SCA”, acting on a setof (1 , ,
1) supermultiplets (cid:0) X µ ( t ); Ψ µ ( t ) , ¯Ψ µ ( t ); b µ ( t ) (cid:1) . For any given µ , 1 ≤ µ ≤ d , X µ ( t ) isa propagating boson, Ψ µ ( t ), ¯Ψ µ ( t ) are its two anticommuting supersymmetric partners and b µ ( t ) is a commuting auxiliary field. Such multiplets are called balanced because they containan equal number of bosonic and fermionic component fields. For us the expression “twistedsuperalgebra” just means that its fermionic generators are all nilpotent; the relation betweenthe twisted N = 2 SCA and the ordinary N = 2 SCA is that the invariance of an actionunder the twisted N = 2 SCA implies the invariance under the ordinary N = 2 SCA andvice-versa. In fact, the construction extends that of [15] from global supersymmetries to thecase of supersymmetries whose generators carry an explicit dependence on the t coordinate.The ordinary one-dimensional N = 2 superconformal algebra of the (1 , ,
1) supermultiplet isthe simple Lie superalgebra sl (2 |
1) [16], with 4 bosonic and 4 fermionic generators. It containsas a subalgebra, in its bosonic sector, the sl (2) algebra which defines the one-dimensionalConformal Quantum Mechanics [17].The existence of the twisted N = 2 SCA is interesting for at least two reasons. It selectsinvariants as local cocycles and, furthermore, it contains quite a small subalgebra which issufficient to determine the full superconformal invariance.To build the twisted N = 2 SCA, we can construct at first a global twisted N = 2 super-algebra with 4 fermionic generators, acting on the (1 , ,
1) supermultiplet, the two worldline“twisted scalar supersymmetries” Q and ¯ Q and the two “twisted vector supersymmetries” Q V and ¯ Q V . Then, one can complement these 4 fermionic nilpotent generators by two extra nilpo-tent ones which carry the explicit t -dependence, the “twisted conformal supersymmetries” Q C and ¯ Q C .The Z -graded anticommutators of 6 fermionic generators close on a set of 6 bosonic gener-ators, one of them being a central charge. Due to the presence of the central charge, the twisted N = 2 SCA is not a simple Lie superalgebra as the ordinary N = 2 SCA.The generators of the twisted N = 2 SCA can be realized as 4 × × t -dependent differential operators. This provides a so-called D -module representation.The twisted N = 2 SCA induces the same invariant actions as the ordinary N = 2 SCA.However, it does not contains the conventional conformal sl (2) symmetry as a bosonic subalge-bra. Rather, only a Borel subalgebra of the conformal sl (2) belongs to the twisted N = 2 SCA.We found that imposing the invariance under a small subalgebra is sufficient to guaranteethe invariance under the full twisted N = 2 SCA and, consequently, the ordinary N = 2 SCA.The minimal superalgebra which enforces the N = 2 superconformal invariance is madeof only 2 fermionic generators (either Q and ¯ Q C or ¯ Q and Q C ) and 2 bosonic generators (thecentral charge c with a fixed coefficient and the ghost number N gh ). Therefore, the whole super-conformal invariance is obtained “for free”, with the extra generators regarded as “accidental”symmetries of this minimal set-up.This intriguing result offers a new perspective for investigating the conformal properties ofthe supersymmetric models, since it suggests that one can replace the demand of the conformalsymmetry by that of a much smaller symmetry .Conformally invariant topological theories such as the topological quantum field theory toymodel [19] and the Calogero models are thus very economically defined by simply demandingthe invariance under the 2 supercharges Q and ¯ Q C , instead of imposing the complete world-line superconformal symmetry. New Lagrangians with higher-order interactions among Fermifields, involving fields that are one-dimensional analogous of the Ramond fields, can also beconstructed.Further results in this paper can be summarized as follows. The invariant actions I are Q i -exact, with a local Q i ′ -exact ( i ′ = i ) antecedent for some of the 6 fermionic generators ofthe twisted N = 2 SCA ( I = R dtQ i Q i ′ Z ii ′ ).If one relaxes the condition of conformal invariance, the general coordinate covariance of thenon-linear sigma-model for the N = 2 supersymmetric quantum mechanics in a curved targetspace is fully implied by its worldline invariance under the action of only two of the abovementioned 6 supercharges, either Q and ¯ Q V or ¯ Q and Q V . (The invariance under the 2 extrasupercharges is automatically obtained). One can search for target metrics such that the actionis superconformally invariant [8].An invertible complex transformation relates the component fields and the free actions ofthe ordinary and the twisted N = 2 superconformal algebra.The paper is organized as follows. In Section we introduce the twisted N = 2 supercon-formal algebra and discuss the relevant subalgebras. In section we investigate the differentcohomologies which induce superconformally invariant actions (both free and in the presence These considerations have analogies in the N = 4, d = 4 super-Yang–Mills theory, where the superconformalYang–Mills supersymmetry with its 32 supersymmetric generators is implied by a much smaller superalgebra,with only 4 scalar twisted nilpotent supercharges [18].
2f an interacting potential) in the case of a flat target-space. In Section we discuss the im-plications of the cohomologies for the covariance of a curved target manifold. In Section the construction of higher-order Fermi interactions is pointed out. The explicit transformationrelating the ordinary and the twisted formulations of the N = 2 SCA is presented in Section .In the Conclusion we make a comparison between our observations concerning these simple su-persymmetric quantum mechanical systems and the intriguing results that have been recentlyobserved in higher dimensional supersymmetric quantum field theories. We also discuss thefuture perspectives of our work. N = 2 superconformal algebra. We recall at first some known facts. Let M d be a d -dimensional manifold, locally parametrizedby the 1 ≤ µ ≤ d coordinates X µ , with metric g µν ( X ), Christoffel symbols Γ µν,ρ and Riemanntensor R µν,ρσ . The target-space reparametrization covariant action with worldline N = 2supersymmetry is expressed by the Lagrangian L = − g µν ˙ X ν ˙ X ν + ¯Ψ µ ( g µν ˙Ψ ν + Γ µ,ρσ ˙ X ρ Ψ ν ) + 14 R µνρσ ¯Ψ ρ Ψ σ ¯Ψ ν Ψ ν . (1) t parametrizes the worldline and ˙Φ ≡ d Φ dt . The possibility of choosing any given parametrizationis obvious since we are working with a one-dimensional parametrization.The t -dependent coordinates X µ ( t ) are bosons, while Ψ µ ( t ) and ¯Ψ µ ( t ) are fermions. Usingan auxiliary field b µ ( t ), one can express L as [14] L = g µν b ν b µ + b µ ( − g µν ˙ X ν + Γ [ µ,ρ ] σ ¯Ψ ρ Ψ σ ) + ∂ ρ g µν ¯Ψ µ Ψ ρ X ν + ¯Ψ µ ( g µν ˙Ψ ν + Γ µ,ρσ ˙ X ρ Ψ ν ) . (2)The general covariance in the curved target-space with coordinates X µ is explicit for theaction (1). However, such an important invariance is only enforced after the eliminationfrom the action (2) of the auxiliary fields b µ via their algebraic equations of motion b µ = g µν ˙ X ν − Γ [ µ,ρ ] σ ¯Ψ ρ Ψ σ . These equations show that the “on-shell” b µ -replaced fields are not vec-tors. The action (2) is not invariant under target-space general coordinates transformations dueto the fact that it inherently involves the Christoffel symbols Γ [ µ,ρ ] σ and that it is not possible toredefine the b µ fields in order to absorb this dependence. On the other hand, when the auxiliaryfields are present, all (twisted or untwisted) supersymmetry transformations close “off-shell”.After the elimination of the b µ fields via their equations of motion, the supersymmetries onlyclose modulo some fermionic equations of motion. All this boils down to the fact that X, Ψ , ¯Ψ , b is a balanced multiplet (for simplicity, from now on we drop, when not necessary, the µ suffix),while X, Ψ , ¯Ψ is an unbalanced multiplet, that is it contains an unequal number of bosonic andfermionic component fields. These intriguing facts about what is happening in the presence orafter eliminating the auxiliary fields are however not troublesome when the fermionic twisted3enerators are realized as nilpotent generators [14]. From the point of view of studying theworld-line supersymmetry, the action (2) is more suitable.The balanced quantum mechanical supersymmetric multiplet is thus made of d independent(1 , ,
1) supermultiplets, whose fields are target-space vectors X µ ( t ) , Ψ µ ( t ) , ¯Ψ µ ( t ) , b µ ( t ) , ≤ µ ≤ d. (3)In the flat case the metric is g µν ( X ) = η µν and the Lagrangian is simply given by L = b µ η µν b ν − b µ η µν ˙ X ν + ¯Ψ µ η µν ˙Ψ ν ∼ − η µν ˙ X µ ˙ X ν + ¯Ψ µ η µν ˙Ψ ν . (4)Two important bosonic charges are conserved and compatible with all fermionic transforma-tions, the field dimension (also known as “engineering dimension”) and the ghost number.Their values for the components of the balanced multiplet are, respectively, ( − , , , ) and(0 , , − , t to possess the engineering dimension − N = 2 SCA.
The 6 nilpotent fermionic generators can be divided into one pair of worldline scalar twistedsupersymmetry operators Q, ¯ Q , one pair of worldline vector twisted supersymmetry operators Q V , ¯ Q V and one pair of worldline scalar special twisted supersymmetry operators Q C , ¯ Q C . Q, ¯ Q, Q V , ¯ Q V are constructed with the prescription of [15], while Q C , ¯ Q C are determined bydemanding explicit t -dependence and compatibility with ghost number and engineering dimen-sion. 4hese 6 operators act on the component fields according to the transformations QX = Ψ , ¯ QX = ¯Ψ ,Q Ψ = 0 , ¯ Q Ψ = − b,Q ¯Ψ = b, ¯ Q ¯Ψ = 0 ,Qb = 0 , ¯ Qb = 0 .Q V X = ¯Ψ , ¯ Q V X = Ψ ,Q V Ψ = − b + ˙ X, ¯ Q V Ψ = 0 ,Q V ¯Ψ = 0 , ¯ Q V ¯Ψ = b − ˙ X,Q V b = ˙¯Ψ , ¯ Q V b = ˙Ψ .Q C X = t Ψ , ¯ Q C X = t ¯Ψ ,Q C Ψ = 0 , ¯ Q C Ψ = − tb + ¯ λX,Q C ¯Ψ = tb − λX, ¯ Q C ¯Ψ = 0 ,Q C b = λ Ψ , ¯ Q C b = ¯ λ ¯Ψ . (5)Until now the real parameters λ, ¯ λ are arbitrary. We are however forced to set ¯ λ = λ in orderto eliminate unwanted anticommutation relations (the presence of a t -multiplication operator)arising from { Q C , ¯ Q C } . This setting guarantees that { Q C , ¯ Q C } = 0.Therefore, the only non-vanishing anticommutators are { Q, Q V } = H, { ¯ Q, ¯ Q V } = H, { Q, ¯ Q C } = c, { ¯ Q, Q C } = − c, { Q C , Q V } = S, { ¯ Q C , ¯ Q V } = ¯ S, { Q V , ¯ Q C } = Z, { ¯ Q V , Q C } = ¯ Z. (6)A central charge c , c = λ , (7)has arisen from the anticommutators of Q with ¯ Q C and ¯ Q with Q C .The action on the component fields of the bosonic operators H, S, ¯ S, Z, ¯ Z is as follows H = ddt ,S = t ddt + ∆ , ¯ S = − t ddt + ¯∆ , (8)and ZX = Z ¯Ψ = Zb = 0 , Z Ψ = ¯Ψ , ¯ ZX = ¯ Z Ψ = ¯ Zb = 0 , ¯ Z ¯Ψ = − Ψ , (9)5here ∆, ¯∆ in (8) act as diagonal operators:∆ X = − λX, ¯∆ X = λX, ∆ b = (1 − λ ) b, ¯∆ b = ( λ − b, ∆Ψ = (1 − λ )Ψ , ¯∆Ψ = λ Ψ , ∆ ¯Ψ = − λ ¯Ψ , ¯∆ ¯Ψ = ( λ −
1) ¯Ψ . (10)The 12 operators entering (6) are closed under (anti)commutation relations, so that G ♯ ≡ { Q, ¯ Q, Q V , ¯ Q V , Q C , ¯ Q C , H, c, S, ¯ S, Z, ¯ Z } (11)is a Lie superalgebra.The non-vanishing commutators involving the even operators of G ♯ are[ H, S ] = [ ¯
S, H ] = H, [ S, Z ] = [ ¯
S, Z ] = Z, [ ¯ Z, S ] = [ ¯ Z, ¯ S ] = ¯ Z, [ ¯ Z, Z ] = S + ¯ S. (12)The non-vanishing commutators between even and odd generators of G ♯ are[ H, Q C ] = Q, [ H, ¯ Q C ] = ¯ Q, [ S, Q ] = − Q, [ S, ¯ Q V ] = − ¯ Q V , [ S, ¯ Q C ] = ¯ Q C , [ ¯ S, ¯ Q ] = ¯ Q, [ ¯ S, Q V ] = Q V , [ ¯ S, Q C ] = − Q C , [ Z, Q ] = − ¯ Q, [ Z, ¯ Q V ] = − Q V , [ Z, Q C ] = − ¯ Q C , [ ¯ Z, ¯ Q ] = Q, [ ¯ Z, Q V ] = ¯ Q V , [ ¯ Z, ¯ Q C ] = Q C . (13)From the action of the G ♯ operators on the component fields one can immediately writedown a D -module representation of G ♯ in terms of 4 × G ♯ is compatible with the following assignment for the scaling dimen-sions of the component fields and of the generators (we set for the worldline coordinate t thedimension [ t ] = − x + z, [ ¯Ψ] = x + 1 − z, [ b ] = x + 1 , (14)where x = [ X ] is an arbitrary parameter.For the fermionic generators we have[ ¯ Q ] = [ Q V ] = − [ Q C ] = 1 − z, [ ¯ Q V ] = − [ ¯ Q C ] = z, (15)with z = [ Q ] an arbitrary parameter. 6o far the parameters λ, x, z are arbitrary. On the other hand λ and x have to be fixed bythe requirement of scale and conformal invariance. Indeed, x has to be set x = −
12 (16)in order to make dimensionless the free kinetic action. The parameter λ has to be fixed λ = 12 (17)in order to guarantee the invariance under Q C of the free kinetic action.Without loss of generality, the parameter z can be fixed to be z = to allow Ψ , ¯Ψ havingthe same dimension.The combinations S ± ¯ S are particularly important. S + ¯ S is the ghost number operator N gh := S + ¯ S, (18)while D := 12 ( S − ¯ S ) = t ddt + d s (19)contains the diagonal matrix d s with the engineering or scaling dimension of the componentfields. The ghost number and the scale dimensions are given by N gh d s X − b Ψ 1 0¯Ψ − sl (2) conformal algebra of the one-dimensional conformal quantum mechanics acts with thefollowing D -module unidimensional transformations on an arbitrary s -dimensional field Φ s ( t )(in our case s = − for X , s = 0 for Ψ and ¯Ψ, s = for b ): L − = ddt ,L = t ddt + s,L = − t ddt − st. (21)The non-vanishing commutators are [ L , L ± ] = ± L ± , [ L , L − ] = 2 L . (22)7he conformal sl (2) is not a bosonic subalgebra of G ♯ . G ♯ possesses an sl (2) subalgebra givenby N gh , Z, ¯ Z . This sl (2) subalgebra does not generate the conformal transformations on thecomponent fields. On the other hand, G ♯ possesses the Borel subalgebra of sl (2). Indeed, thesubalgebra { D, H } , with D introduced in (19), is identified with { L , L − } , so that we canidentify D ≡ L and H ≡ L − . One should note that D acts on the component fields with thecorrect assignment of their scale dimensions.It is quite remarkable, as we will discuss later, that the invariance under this subalgebra issufficient to determine the conformally invariant actions in quantum mechanics, both for thebosonic and the fermionic sectors.Even more remarkable, the invariance under just 2 twisted fermionic generators, togetherwith the requirement of the vanishing of the ghost number, is sufficient to determine the super-conformally invariant actions.We denote as “ G ♯min ” the minimal G ♯ subalgebra which, imposed as invariance of the action,determines the full set of N = 2 superconformal invariances. G ♯min is given by G ♯min = { Q, Q C , c, N gh } . (23)Imposing the G ♯min invariance is a very economical way to impose the full N = 2 superconformalinvariance.It is convenient to present here a list of subalgebras for G ♯ , which can be relevant for differentpurposes. We have, for instance, { Q, ¯ Q C , c } , { Q, ¯ Q C , c, S } , { Q, ¯ Q, ¯ Q C , c, S, H } , { Q, ¯ Q, Q V , ¯ Q C , c, S, H, Z } , { Q, ¯ Q, Q C , ¯ Q C , c, S, H, Z } , { Q, ¯ Q, Q c , ¯ Q V , Q V , c, S, H, ¯ S } . (24)As we will see in the next Section, Q - and Q V -invariant actions are not necessarily Q C -invariant,while Q - and Q C -invariant actions are necessarily Q V -invariant.An important subalgebra of G ♯ is denoted by B . Its 5 generators are B ≡
12 ( S − ¯ S ) , H, Z + ¯ Z, Q + Q V − ¯ Q, ¯ Q − ¯ Q V − Q. (25)As discussed in Section , B coincides with the Borel subalgebra of the sl (2 |
1) superalgebra.After having defined the twisted superalgebra G ♯ with its set of nilpotent fermionic gen-erators, we are now looking for actions which are invariant under the full G ♯ or some of itssubalgebras. We will restrict ourselves to the case of a standard kinetic term for the bosons X µ , namely with a Lagrangian of the form L ∼ R dtg µν ˙ X µ ˙ X ν + . . . .8 Invariant actions in the flat target-space. Q and Q V . Let us enforce the Q and Q V invariance of the action S = R dt L and let us assume the freeaction to be non-dimensional. It will therefore be uniquely defined, with the dimensionality of X µ fixed to be x = − .We have S = QQ V Z dt (cid:0) ¯Ψ µ η µν Ψ ν (cid:1) = Z dtQ (cid:16) ¯Ψ µ η µν ( b ν − ˙ X ν ) (cid:17) = Z dt (cid:16) b µ η µν b ν − b µ η µν ˙ X ν + ¯Ψ µ η µν ˙Ψ ν (cid:17) . (26)By eliminating the b µ fields through their algebraic equations of motion one gets the supersym-metric Lagrangian L ∼ − ˙ X µ ˙ X ν + ¯Ψ µ η µν ˙Ψ ν .The QQ V -exact term QQ V ( b µ η µν X ν ) has the appropriate dimension. It is however, moduloa pure time-derivative, equal to QQ V ( ¯Ψ µ η µν Ψ ν ) and, therefore, it is not independent. Indeed,both terms b µ η µν b ν and − b µ η µν ˙ X ν + ¯Ψ µ ˙Ψ ν are separately Q -exact and thus Q -invariant; the Q V invariance on the other hand fixes the relative coefficients of these terms. The action R dt ( b µ η µν b ν − b µ η µν ˙ X ν + ¯Ψ µ ˙Ψ ν ) is thus completely determined by requiring the invarianceunder both Q and Q V . Q C invariance. We leave for the time being arbitrary the parameter λ entering (5) and we check under whichcondition the QQ V -invariant action QQ V R dt ( ¯Ψ µ η µν Ψ ν ) is also Q C -invariant. We obtain Q C L = Q C QQ V (cid:0) ¯Ψ µ η µν Ψ ν (cid:1) = ddt ( b µ η µν X ν ) + (2 λ − b µ η µν Ψ ν . (27)Therefore the Q C -invariance is ensured provided that λ = (see the formula (17)).Modulo a time derivative one gets, for the Lagrangian, L = Q C ¯ Q (cid:16) ¯Ψ µ η µν ˙Ψ ν (cid:17) . (28)Therefore L is also, modulo a time derivative, Q C Q V -exact, L = Q C Q V (cid:18) b µ η µν b ν (cid:19) . (29)Therefore the N = 2 free Lagrangian is Q, Q V , Q C -invariant provided that λ = .9he action admits the following quite remarkable set of equalities Z dt L = Z dt (cid:16) b µ η µν b ν − b µ η µν ˙ X ν + ¯Ψ µ ˙Ψ ν (cid:17) = Z dtQ (cid:16) ¯Ψ µ η µν ( b ν − ˙ X ν ) (cid:17) = Z dtQQ V (cid:0) ¯Ψ µ η µν Ψ ν (cid:1) == Z dtQ C ¯ Q (cid:16) ¯Ψ µ η µν ˙Ψ ν (cid:17) = Z dtQ C Q V (cid:18) b µ η µν b ν (cid:19) . (30)One can also check the Q C invariance as follows. Q C Q V Q (cid:0) ¯Ψ µ η µν Ψ ν (cid:1) = { Q C , Q V } Q (cid:0) ¯Ψ µ η µν Ψ ν (cid:1) + Q V (cid:16) QQ C ( ¯Ψ µ η µν ˙Ψ ν ) (cid:17) . (31)For λ = the action of the bosonic symmetry { Q C , Q V } on Q ( ¯Ψ µ η µν Ψ ν ) gives zero, modulo atime-derivative. Furthermore, for what concerns the second term in the r.h.s. of (31), we have QQ C (cid:0) ¯Ψ µ η µν Ψ ν (cid:1) = Q (cid:18) ( tb µ − X µ ) η µν Ψ ν (cid:19) = Q (cid:18) −
12 Ψ µ η µν Ψ ν (cid:19) = 0 . (32) The supersymmetric interaction is introduced in term of the “prepotential” W [ X µ ].A manifest Q -invariant term can be added to the action by setting L int = Q (cid:18) ¯Ψ µ δWδX µ (cid:19) = b µ δWδX µ − ¯Ψ µ δ WδX µ δX ν Ψ ν . (33)The full Q -invariant action is thus S = Z dt (cid:18) b µ η µν b ν − b µ η µν ( ˙ X ν + δWδX µ ) + ¯Ψ µ η µν ( ˙Ψ ν + δ WδX µ δX ν Ψ ν ) (cid:19) ∼ Z dt (cid:18) −
14 ˙ X µ ˙ X ν − δWδX µ η µν δWδX ν + ¯Ψ µ η µν ( ˙Ψ ν + δ WδX µ δX ν Ψ ν ) (cid:19) . (34)The Q , Q V , ¯ Q and ¯ Q V invariances, modulo a time derivative, of L int are warranted because L int = Q ¯ Q ( W ) = QQ V ( W ) = − ¯ Q ¯ Q V ( W ) . (35) Q C invariance. The Q C , ¯ Q C invariances, modulo a time derivative, of L int imply the following condition onthe prepotential Ψ µ ∂W∂X ν − X ν ∂ W∂X µ ∂X ν Ψ ν = 0 ⇒ ∂∂X ρ (cid:16) X µ ∂W∂X µ (cid:17) = 0 . (36)10herefore, the condition for having a Q C -invariance is X µ ∂W∂X µ = C, (37)whose general solution is W = C ln R + f (cid:18) X µ R (cid:19) . (38) C is an arbitrary constant and f is an arbitrary function of the non-dimensional quantities X µ R ,where R ≡ X µ η µν X ν .This gives us a so-called conformal twisted supersymmetric quantum mechanics, often withpossible topological observables. Its action can be untwisted to a Lagrangian that has ordinaryconformal supersymmetry. For instance, for one particle on a plane with coordinates X i , i = 1 , f = 0, one can select the conformal potential (cid:16) ∂W∂X i (cid:17) = C R . (39)It defines an interesting topological solvable quantum mechanics on the R ∗ plane with theorigin excluded. The topological gauge function is then ˙ X i = Cǫ ij X j X , since (cid:18) ˙ X i + Cǫ ij X j X (cid:19) = (cid:16) ˙ X i (cid:17) + C X + 2 C ˙ θ. (40)The corresponding N = 2 supersymmetric quantum mechanics mimics as an elementary modelthe topological Yang–Mills theory, which uses the selfduality equations as quantum field theorytopological gauge functions [19].Still keeping i = 1 ,
2, with X ≡ X and Y ≡ X , the Q C “topological” invariance alsoproduces the superconformal quantum mechanics of a pair of particles evolving on a line withcoordinates X ( t ) and Y ( t ) and interacting via a Calogero potential.The latter is (cid:16) ∂W∂X (cid:17) + (cid:16) ∂W∂Y (cid:17) = 2( X − Y ) (41)and depends on the relative distance.The prepotential W is indeed W = ln | X − X | = ln R + ln | X − X R | . (42)It satisfies the general condition (37) for Q C invariance with C = 1 everywhere apart from theunphysical (infinite energy) coincidence line X = X in R .11 Curved target-space.
Let us come back to the action (2), built in [14]. The value of the coefficient in front ofthe Christoffel coefficient in the “topological gauge function” g µν ( ˙ X ν ) − Γ µ,ρσ ¯Ψ ρ Ψ σ must befined-tuned in order to get target-space covariance. For topological observables, defined fromthe cohomology of Q , this is not a critical issue, since they do not depend on the addition of Q -exact terms to the action.The choice of this coefficient however, and thus the target-space covariance of the action,is implied by the additional requirement of the Q V invariance. Indeed, one can easily check(by using the chain rule for the Q V operator) that the above action can be expressed as astraightforward generalization of the flat space formula, this time with a coordinate-dependentmetric g µν . We get Z dtL = QQ V Z dt ( ¯Ψ µ g µν Ψ ν ) . (43)Therefore, the requirement of invariance under both Q and Q V implies the target-space covari-ance, which is an intriguing new result.The Q C invariance, on the other hand, is generally broken by terms which are proportional toderivatives of the metric. Thus, the compatibility between conformal invariance of the worldlineand the target-space covariance is broken in the presence of a curved target-space metrics (see[7, 8] for a discussion of the conformal invariance with non-trivial backgrounds). A series of other of Q and Q V invariant actions can be constructed by simply using the B µ ! ...µ p ( X ) forms of various degrees. They can be written as Z dtL = QQ V Z dt ( ¯Ψ µ Ψ ν )( g µν ( X ) + B µν ( X ) + X p> B µνµ ν ...µ p ν p ( X ) ¯Ψ µ Ψ ν . . . ¯Ψ µ p ¯Ψ ν p ) . (44)Such actions are by construction Q and Q V invariant. When one expands the above QQ V -exactterm, one finds a b dependence, such that higher Fermi field interactions do occur. Since theengineering dimensions of Ψ and ¯Ψ adds to zero, there is no further limitation on the value of p , apart the relation 2 p ≤ d , due to the fact that Ψ µ and ¯Ψ ν are anticommuting fields. Theforms B µν are analogous to the Kalb–Ramond fields. We discuss now the relation between the twisted N = 2 SCA G ♯ and the ordinary N = 2 SCA(the sl (2 |
1) superalgebra). Their D -module representations are given in Appendix A and B ,12espectively.Both superalgebras possess a maximal common subalgebra B , which is made of 5 (3 evenand 2 odd) generators, defined as follows (in the right hand side of the equations we presentthe combinations in terms of the G ♯ generators), B = { D = 12 ( S − ¯ S ) , H, W = Z + ¯ Z, Q = Q + Q V − ¯ Q, Q = ¯ Q − ¯ Q V − Q } . (45)The superalgebra B , G ♯ ⊃ B ⊂ sl (2 | sl (2 |
1) given by the Cartanand the negative-root generators.One should note that no “conformal generator” (i.e., carrying an explicit dependence on t )belongs to both G ♯ and sl (2 | N = 2 superconformal algebra (ending up with an sl (2 | N = 2 twistedsuperconformal algebra G ♯ ).Let us denote respectively as L and L ♯ both invariant free Lagrangian for the ordinary N = 2 SCA and the twisted N = 2 SCA. It is convenient to introduce different notationsfor the component fields entering the ordinary and the twisted (1 , ,
1) supermultiplets. Forthe ordinary N = 2 SCA let us have the ( Y ( t ); ξ ( t ) , ξ ( t ); g ( t )) component fields and for thetwisted N = 2 SCA the ( X ( t ); Ψ( t ) , ¯Ψ( t ); b ( t )) component fields.With a convenient normalization we can present the free Lagrangians as L = 12 (cid:16) ˙ Y + g − ξ ˙ ξ − ξ ˙ ξ (cid:17) , L ♯ = b − b ˙ X + ¯Ψ ˙Ψ . (46)In order to identify them ( L ♯ = L ), we have to provide the invertible “twist transformation” T ,which link both sets of fields. We have X = √ iY,b = 1 √ g + i ˙ Y ) , Ψ = 1 √ ξ − ξ ) , ¯Ψ = i √ ξ + ξ ) , or, in matrix form, T = √ i i∂ t −
10 1 i i , T − = √ − i − ∂ t − i − − i . (47)13he above twist transformation T is intrinsically complex. No solution for T can be found withinthe real numbers. Therefore T only makes sense if the component fields X, Y, b, g, Ψ , ¯Ψ , ξ , ξ are taken as complex fields. We can introduce a reality condition on the twisted fields X, b, Ψ , ¯Ψor a reality condition on the ordinary fields Y, g, ξ , ξ . Both reality conditions, however, aremutually incompatible.This feature of the twist transformation does not make it less useful. Indeed, to recover,via path integral, the ordinary correlation functions from the twisted correlation functions (orvice-versa), the only needed tool is an analytical extension. This allows to perform the twisttransformation (or its inverse).In the Conclusions we comment more on the implications of the relation between ordinaryand twisted formulations for extended supersymmetric theories.Let us mention here that the superalgebra G ♯ could be further enlarged by adding the evenconformal generator K (which, together with H and D , closes the sl (2) conformal algebra) andall the extra generators which are required to close the (anti)commutation relations. Such anenlarged superalgebra is of little significance since it does not produce any further informationand constraint on the superconformal invariance, besides those obtained from G ♯ and its relevantsubalgebras. In this work we proved that the N = 2 superconformal quantum mechanics based on the (1 , , G ♯ contains 6 nilpotent odd generators and 6 even generators (includ-ing a central charge). The fermionic generators can be used to define BRST-type cohomologies.The invariance under different subalgebras determine different types of models. For some ofthem, the invariance under just a pair of the 6 generators implies the invariance under some ofthe other ones. For a curved target-space, the request of both Q and Q V invariance determines,after solving the equations of motion of the auxiliary fields, a supersymmetric action which iscovariant, being expressed in terms of the metric, the vectors X , Ψ, ¯Ψ, the covariant derivativesand the Riemann curvature.A striking observation concerns the fact that the invariance under Q and Q C , whose an-ticommutator closes on a central charge, together with the property that one is consideringLagrangians with ghost number zero and standard dimension, completely determines the su-perconformally actions with all their extra symmetries.These features share many features and might be at the root of recent results in higherdimensions, for both twisted super-Yang–Mills theories [18] and supergravity [20]. These quan-tum field cases are much more constrained, due to the presence of additional symmetries suchas the Lorentz symmetry, the R-symmetry, etc. The key point is that the twisted formulation14educes in a controllable way the size of the symmetries, keeping however the symmetry alge-bra large enough to uniquely determine the theory . Therefore, and quite remarkably, once thetheory has been defined by this smaller set of generators, one discovers that it possesses moresymmetries, which appear “for free”. The presence of these extra symmetries eventually allowsto untwist the fermions, so that one is able to recover the physical spin-statistic relation.For the twisted N = 4 maximal superYang-Mills theory [21, 22, 15] one can for instancecovariantly select, among its 16 supersymmetry generators, n ≤ N = 4, d = 4 as well as in the N = 1, d = 10 SYM cases.Furthermore, one can enlarge the twist from the super-Poincar´e algebra to the super-conformalcase [26]. Within this framework one finds that the invariance under 4 twisted fermionic gen-erators mixing sectors of both Poincar´e and conformal supersymmetry fully determines thetheory [18].In the setting of the N = 2 supersymmetric mechanics we proved that the two fermionicgenerators Q and ¯ Q C , together with a central charge, contain enough information to determinethe N = 2 superconformal action with its full set of fermionic and bosonic symmetries. We mayalso mention the recently found example of the N = 1, d = 4 supergravity, which is very well-known in the untwisted formalism, and can be therefore used as a safe playground for exploringnon-trivial properties of the twist procedure. It can be obtained by using only a U (2) ⊂ SO (4)symmetry, in analogy with the N = 1, d = 4 super-Yang–Mills theory. Only 3 = 1 + 2 globalsupersymmetries are needed to define, modulo boundary terms, the supergravity. The fourthsusy generator is implied by the 3 former ones and the full SO (4) symmetry is finally foundafter the untwisting.As a consequence of the seemingly general existence of the twist, burdening quantum fieldtheory questions, such as the existence of supercharges which only close modulo the use ofsome equations of motion may well just become irrelevant since, in the twisted formulation,one discovers that the theory is determined and controlled by a subset of supercharges whichclose off-shell. The Poincar´e (as well as the conformal) supersymmetry might thus appear asa kind of physical (and over-determining) effective symmetry, which sometimes emerges afteruntwisting a TQFT, whenever it is possible.As a closing remark we wish to point out that we made explicit the connection betweentwisted and ordinary N = 2 superconformal quantum mechanics, determining the twistedsuperalgebra (and its D -module representation) associated with the twisted invariance. A The twisted theories possess normal local Feynmann propagators. Therefore, their simplification cannot becompared with that obtained by choosing a light-cone gauge. In the latter case some simplicity is gained, but thelocal properties of the theory are harmed, due to a propagator with cut singularities which cannot be handledin a satisfactory way. Twisted theories are better behaved. N =4 superconformal quantum mechanics. In its “ordinary” side [27, 28, 29, 30], the N = 4superconformal algebra (which plays the role of the sl (2 | N = 2 SCA) is the exceptionalsuperalgebra D (2 , α ). An open question is to determine its twisted superalgebra counterpart.A natural motivation to investigate twisted superconformal quantum mechanics with large N comes from the geometric Langlands program which can be described [31] as a twisted N = 4Super-Yang–Mills theory compactified in 2 D . It is expected that a world-line method basedon the one-dimensional twisted superconformal mechanics can be employed to reconstruct thetwo-dimensional theory. Acknowledgments.
L.B. is grateful to CBPF, where this work was completed, for hospitality and for the financialsupport through a PCI-BEV grant. F.T. is grateful to the LPTHE, Paris VI, for both hospitalityand financial support. He acknowledges a CNPq research grant.16 ppendix A: the D -module representation of the twisted N = 2 superconformal algebra. For completeness we present here the D -module representation of the twisted N = 2 su-perconformal algebra G ♯ . It is a Z -graded Lie algebra G ♯ = G ♯ ⊕ G ♯ , with the even sec-tor G ♯ given by the generators H, c, S, ¯ S, Z, ¯ Z and the odd sector G ♯ given by the generators Q, ¯ Q, Q V , ¯ Q V , Q C , ¯ Q C .The D -module representation consists of 4 × X ( t ) b ( t )Ψ( t )¯Ψ( t ) , whose two upper component fields are bosonic and the two lower component fieldsare fermionic. It is convenient to leave arbitrary the real parameter λ . The closure of the(anti)commutation relations is not affected by it. As discussed in the main text, see formula(17), the applicability of the G ♯ D -module representation to the superconformal invarianceforces us to set λ = .In the expressions below we set for convenience ∂ t := ddt . We have Q = , ¯ Q = − ,Q V = ∂ t ∂ t − , ¯ Q V = ∂ t
00 0 0 0 − ∂ t ,Q C = t
00 0 λ
00 0 0 0 − λ t , ¯ Q C = t λλ − t ,H = ∂ t ∂ t ∂ t
00 0 0 ∂ t , c = λ λ λ
00 0 0 λ , (48)17 = t∂ t − λ t∂ t + 1 − λ t∂ t + 1 − λ
00 0 0 t∂ t − λ , ¯ S = − t∂ t + λ − t∂ t + λ − − t∂ t + λ
00 0 0 − t∂ t + λ − ,Z = , ¯ Z = − . The (anti)-commutation relations are presented in Section (formulas (6),(12) and (13)). Appendix B: the D -module representation of sl (2 | . In order to allow the comparison with the twisted case we present here the D -modulerepresentation of the N = 2 superconformal algebra sl (2 |
1) acting on a (1 , ,
1) supermultipletwhose component fields have the same engineering dimensions as in the twisted case. Asbefore, a free parameter λ is allowed. The application of the transformations to superconformalinvariant actions forces us to set λ = .The D -module representation can be obtained by closing the superalgebra recovered byapplying the sl (2) D -module representation to the D -module representation, given in [32, 33],of the global N = 2 supercharges.We end up with the following set of even ( H, W, D, K ) and odd ( Q , Q , e Q , e Q ) generatorsclosing the (anti)commutation relations[ D, H ] = − H, [ D, K ] = K, [ K, H ] = 2 D, (49)together with { Q , Q } = { Q , Q } = 2 H, { e Q , e Q } = { e Q , e Q } = − K, { Q , e Q } = { Q , e Q } = 2 D, { Q , e Q } = { e Q , Q } = W (50)and [ H, e Q i ] = Q i , [ K, Q i ] = e Q i , [ D, Q i ] = − Q i , [ D, e Q i ] = e Q i , [ W, Q i ] = − ǫ ij Q j , [ W, e Q i ] = − ǫ ij e Q j , (51)18or i = 1 , ǫ = − ǫ = 1).Their explicit expression is given by H = ∂ t ∂ t ∂ t
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00 0 0 t∂ t + − λ ,K = − t ∂ t + 2 λt − t ∂ t + (2 λ − t − t ∂ t + (2 λ − t
00 0 0 − t ∂ t + (2 λ − t ,Q = ∂ t ∂ t ,Q = − ∂ t − ∂ t , e Q = t
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