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Minimal unitary representation of D(2,1;λ) and its SU(2) deformations and d=1 , N=4 superconformal models
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Nuclear Physics B 869 [PM] (2013) 111–130 www.elsevier.com/locate/nuclphysb
Minimal unitary representation of D(2, 1; λ) and its SU(2) deformations and d = 1, N = 4 superconformal models Karan Govil, Murat Gunaydin ∗ Physics Department, Pennsylvania State University, University Park, PA 16802, USA Received 28 September 2012; received in revised form 16 November 2012; accepted 11 December 2012 Available online 14 December 2012
Abstract Quantization of the geometric quasiconformal realizations of noncompact groups and supergroups leads directly to their minimal unitary representations (minreps). Using quasiconformal methods massless unitary supermultiplets of superconformal groups SU(2, 2|N ) and OSp(8∗ |2n) in four and six dimensions were constructed as minreps and their U (1) and SU(2) deformations, respectively. In this paper we extend these results to SU(2) deformations of the minrep of N = 4 superconformal algebra D(2, 1; λ) in one dimension. We find that SU(2) deformations can be achieved using n pair of bosons and m pairs of fermions simultaneously. The generators of deformed minimal representations of D(2, 1; λ) commute with the generators of a dual superalgebra OSp(2n∗ |2m) realized in terms of these bosons and fermions. We show that there exists a precise mapping between symmetry generators of N = 4 superconformal models in harmonic superspace studied recently and minimal unitary supermultiplets of D(2, 1; λ) deformed by a pair of bosons. This can be understood as a particular case of a general mapping between the spectra of quantum mechanical quaternionic Kähler sigma models with eight super symmetries and minreps of their isometry groups that descends from the precise mapping established between the 4d, N = 2 sigma models coupled to supergravity and minreps of their isometry groups. © 2012 Elsevier B.V. All rights reserved.
* Corresponding author.
E-mail addresses: [email protected] (K. Govil), [email protected] (M. Gunaydin). 0550-3213/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nuclphysb.2012.12.006
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1. Introduction This paper belongs to a series of papers on the construction of the minimal unitary representations of noncompact groups and supergroups and their deformations using quasiconformal methods. For physical motivations and references to earlier work on the subject we refer to [1,2]. In this paper we study the deformations of the minimal unitary supermultiplet of D(2, 1; λ) with the even subgroup SU(1, 1) × SU(2) × SU(2). D(2, 1; λ) represents a one parameter family of N = 4 superconformal algebras in one dimension and is relevant to AdS2 /CFT 1 dualities. It is also important for AdS3 /CFT 2 dualities since the AdS3 group SO(2, 2) factorizes as SU(1, 1) × SU(1, 1). Supersymmetric extensions factorize as well and each factor can be extended to D(2, 1; λ) [3]. Another motivation for our work is to establish the connection between the spectra of N = 4 superconformal quantum mechanical models that have been studied in recent years [4–23] and the deformations of minimal unitary supermultiplets of corresponding conformal superalgebras. This provides a quantum mechanical example of the deep connection between N = 2 sigma models that couple to 4d supergravity and minimal unitary representations of their isometry groups established in [24,25]. The plan of the paper is as follows. In Section 2 after reviewing the minimal unitary realization of D(2, 1; λ) [26] we construct its SU(2) deformations using bosonic oscillators in the noncompact 5-graded basis. In Section 3 we reformulate the results of Section 2 in the compact 3-graded basis and show that the deformations of the minrep of D(2, 1; λ) are positive “energy” (unitary lowest weight) representations. We then present the corresponding unitary supermultiplets. Section 4 discusses deformations of the minrep using both bosons and fermions and how the deformed D(2, 1; λ) commutes with a noncompact super algebra OSp(2n∗ |2m) with the even subgroup SO∗ (2m) × USp(2n) constructed using “deformation” bosons and fermions. In Section 5 we review some of the results of work on N = 4 superconformal quantum mechanics and show how its symmetry generators and spectrum map into the generators of D(2, 1; λ) deformed by a pair of bosonic oscillators and the resulting unitary supermultiplets. We conclude with a brief discussion of our results. 2. The minimal unitary representation of D(2, 1; λ) Of all the noncompact real forms of the one parameter family of supergroups D(2, 1; λ) only the real form with the even subgroup SU(2) × SU(2) × SU(1, 1) admits unitary lowest weight (positive energy) representations. In this paper we shall study the minimal unitary representations of this real form, which we shall denote as D(2, 1; λ), and their deformations. We shall label the even subgroup as SU(2)A × SU(2)T × SU(1, 1)K with the odd generators transforming in the (1/2, 1/2, 1/2) representation with respect to it. The Lie super algebra of D(2, 1; λ) can be given a 5-graded decomposition of the form D(2, 1; λ) = g(−2) ⊕ g(−1) ⊕ g(0) ⊕ g(+1) ⊕ g(+2)
(2.1)
where the grade ±2 subspaces are one-dimensional and the grade zero subalgebra is g(0) = su(2)A ⊕ su(2)T ⊕ so(1, 1)
(2.2)
The grade ±1 subspaces contain 4 odd generators transforming in the (1/2, 1/2) representation of SU(2)A × SU(2)T subgroup and the generators belonging to g(−2) and g(+2) together with the SO(1, 1) generator of grade zero subspace form the su(1, 1)K subalgebra.
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For λ = −1/2 the Lie superalgebra D(2, 1; λ) is isomorphic to OSp(4∗ |2) = OSp(4|2, R). The minrep of D(2, 1; λ) was obtained in [26] using quasiconformal techniques. In this paper we shall use a different label λ for the one parameter superalgebras D(2, 1; λ) which is related to the label σ (used in [26]) as: 2λ + 1 3 With this labeling we have σ=
(2.3)
D(2, 1; λ = −1/2) = OSp(4|2, R)
(2.4)
2.1. Minimal unitary realization of D(2, 1; λ) in 5-grading Before studying the deformations of the minrep of D(2, 1; λ), we shall first rewrite the results of [26] in a split basis in which the U (1) generators of the two SU(2) subgroups are diagonalized and the generators of SU(2)T and SU(2)A , denoted by T±,0 and A±,0 respectively, are given by the following bilinears of a pair of fermionic oscillators: T+ = α † β,
T− = β † α,
A+ = α † β † ,
A− = βα,
1 T0 = (Nα − Nβ ) 2 1 A0 = (Nα + Nβ − 1) 2
(2.5) (2.6)
where Nα = α † α and Nβ = β † β are the respective number operators. The fermionic oscillators satisfy the following anti-commutation relations: α, β † = 0 = β, α † (2.7) α, α † = 1 = β, β † , These generators satisfy the commutation relations: [T+ , T− ] = 2T0 ,
[T0 , T± ] = ±T±
(2.8)
with the Casimir 1 T 2 = T02 + (T+ T− + T− T+ ) 2
(2.9)
The quadratic Casimir A2 of SU(2)A is related to T 2 as follows 3 4 The single bosonic coordinate and its canonical momentum (x, p) satisfy T 2 + A2 =
(2.10)
[x, p] = i
(2.11)
and the generator that gives the 5-grading, is realized as: 1 (2.12) = (xp + px) 2 The grade −1 generators can then be written as bilinears of the coordinate x with the fermionic oscillators: Q = xα,
Q† = xα † ,
S = xβ,
S † = xβ †
(2.13)
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They close into K− under anti-commutation: Q, Q† = 2K− The grade zero generators in the 5-grading determined by are the generators T+ , T− and T0 of SU(2)T , A+ , A− and A0 of SU(2)A and . The grade +2 generator with respect to is given by: 1 2 1 2 3 1 2 2 K+ = p + 2 2λT + (λ − 1)A + + λ(λ − 1) 2 3 8 2 x 2 1 1 2 λ (2λ + 1)T 2 + +1 (2.14) K+ = p 2 + 2 2 2 x 3 where we used T 2 + A2 = 34 . 2.2. Deformations of minreps In order to obtain unitary irreducible representations that are “deformations” of the minrep of D(2, 1; λ) we introduce bosonic oscillators am , bm and their hermitian conjugates a m = (am )† , bm = (bm )† (m, n, . . . = 1, 2) that satisfy the commutation relations: n a m , a n = b m , b n = δm , [am , an ] = [am , bn ] = [bm , bn ] = 0 (2.15) and introduce an SU(2)S Lie algebra whose generators are as follows: 1 S0 = (Na − Nb ) 2 where Na = a m am and Nb = bm bm are the respective number operators. They satisfy: S+ = a m bm ,
S− = (S+ )† = am bm ,
[S+ , S− ] = 2S0 ,
[S0 , S± ] = ±S±
(2.16)
(2.17)
The quadratic Casimir of su(2)S is 1 C2 su(2)S = S 2 = S02 + (S+ S− + S− S+ ) 2 1 1 = (Na + Nb ) (Na + Nb ) + 1 − 2a [m bn] a[m bn] 2 2
(2.18)
where square bracketing a[m bn] = 12 (am bn − an bm ) represents antisymmetrization of weight one. The bilinears a[m bn] and a [m bn] close into U (P ) generated by the bilinears U m n = a m an + b m bn
(2.19)
under commutation and all together they form the Lie algebra of noncompact group SO∗ (2P ) with the maximal compact subgroup U (P ). The group SO∗ (2P ) thus generated commutes with SU(2)S as well as with D(2, 1; λ). To obtain the SU(2) deformed D(2, 1; λ) superalgebra we replace the generators of su(2)T subalgebra with the generators of the diagonal subgroup of SU(2)T and SU(2)S su(2)T
⇒
su(2)S ⊕ su(2)T
(2.20)
and denote the diagonal subgroup as SU(2)T and its Lie algebra as su(2)T . Its generators are simply:
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T+ = S+ + T+ = a m bm + α † β T− = S− + T− = bm am + β † α 1 (2.21) T0 = S0 + T0 = (Na − Nb + Nα − Nβ ) 2 The generator and the negative grade generators defined by it remain unchanged in going over to the deformed minreps. The grade +1 generators are then given by the commutators:
† = (Q)
† = i Q† , K+
= i[Q, K+ ], Q Q
(2.22) S † = (
S)† = i S † , K+ S = i[S, K+ ], Thus we make an ansatz for grade +2 generator K+ of the form 1 1 K+ = p 2 + 2 c1 T 2 + c2 S 2 + c3 A2 + c4 (2.23) 2 x where c1 , . . . , c4 are some constant parameters. Using the closure of the algebra, we determine these four unknown constants in terms of λ and obtain: 1 1 8 3 K+ = p 2 + 2 8λT 2 + (λ − 1)A2 + + 8λ(λ − 1)S 2 + 2λ(λ − 1) (2.24) 2 3 2 4x The +1 grade generators then take the form
= −pα + 2i λ T0 + 3 α + T− β − λ − 1 A0 − 3 α − 2A− β † Q x 4 3 4
† = −pα † − 2i λ T0 − 3 α † + T+ β † − λ − 1 A0 + 3 α † − 2A+ β Q x 4 3 4
3 λ−1 3 2i
λ T0 − β − T+ α − A0 + β − A− α † S = −pβ − x 4 3 4
λ−1 2i 3 † 3 † † † †
λ T0 + β − T− α − A0 − β − A− α S = −pβ + (2.25) x 4 3 4 The anti-commutators of grade +1 and grade −1 generators close into grade zero subalgebra
† = − − 2iλT0 + i(λ + 1)A0 Q, Q †
= − + 2iλT0 − i(λ + 1)A0 Q ,Q S,
S † = − + 2iλT0 + i(λ + 1)A0 † S ,
S = − − 2iλT0 − i(λ + 1)A0 (2.26) Q,
S † = −2iλT− (2.27) {Q,
S} = +2i(λ + 1)A− , † † †
Q , S = +2iλT+ (2.28) Q , S = −2i(λ + 1)A+ , †
= −2iλT+
= −2i(λ + 1)A− , S, Q (2.29) {S, Q} † † †
= +2i(λ + 1)A+ ,
= +2iλT− S ,Q (2.30) S ,Q † † † †
= {S,
= Q ,Q S} = S ,
(2.31) S =0 {Q, Q} † † † †
[Q, K− ] = iQ, Q , K− = iQ , [S, K− ] = iS, S , K− = iS (2.32)
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The quadratic Casimir of su(1, 1)K generated by K±2 and is 1 1 C2 su(1, 1)K = K2 = (K+ K− + K− K+ ) − 2 2 4 λ−1 2 λ(λ − 1) 2 2 = λT + A + λ(λ − 1)S + (2.33) 3 4 There exists a one parameter family of quadratic Casimir elements C2 (μ) that commute with all the generators of D(2, 1; λ). μ 2 λ 1 i K − (μ − 8)T 2 − 16 + 8λ + μ(λ − 1) A2 + F(Q, S) 4 4 12 4 2 1 λ = 8 + μ(λ − 1) S + 4 4
C2 (μ) =
(2.34)
where
† + Q† , Q
+ S,
F(Q, S) = Q, Q S† + S†,
S
(2.35)
is the contribution from the odd generators. Since the eigenvalues of the quadratic Casimir depends on the eigenvalues s(s + 1) of the Casimir operator S 2 of SU(2)S the corresponding deformed unitary supermultiplets will be uniquely labeled by spin s of SU(2)S . 3. SU(2) deformed minimal unitary representations as positive energy unitary supermultiplets of D(2, 1; λ) 3.1. Compact 3-grading The 5-graded decomposition of the Lie superalgebra D(2, 1; λ) is determined by the noncompact generator : D(2, 1; λ) = g(−2) ⊕ g(−1) ⊕ su(2)T ⊕ su(2)A ⊕ so(1, 1) ⊕ g(+1) ⊕ g(+2)
Q
† ,
= K− ⊕ Q, Q† , S, S † ⊕ [A±,0 , T±,0 , ] ⊕ Q, S,
S † ⊕ K+ The Lie superalgebra D(2, 1; λ) can also be given a 3-graded decomposition with respect to its compact subsuperalgebra osp(2|2) ⊕ u(1) = su(2|1) ⊕ u(1), which we shall refer to as compact 3-grading: D(2, 1; λ) = C− ⊕ C0 ⊕ C+ D(2, 1; λ) = (A− , B− , Q− , S− ) ⊕
(3.1)
T±,0 , J , H, Q0 , S0 , Q†0 , S†0 ⊕ (A+ , B+ , Q+ , S+ )
(3.2) The expressions for the generators in the 3-grading in terms of the generators in 5-grading is given in Table 1. The generator H that determines the compact 3-grading is given by 1 1 H = (K+ + K− ) + A0 = B0 + (Nα + Nβ − 1) 2 2
(3.3)
The other generators in C0 form the compact superalgebra su(2|1) whose even subalgebra is generated by T±,0 and J .
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Table 1 Generators in compact 3-grading. C−
C0
C+
A−
T±,0 H = 12 (K+ + K− ) + A0
A+
B− = 2i [ + i(K+ − K− )]
Q− = 12 (Q − i Q)
J = (λ + 1)A0 + 12 (K+ + K− )
Q0 = 1 (Q + i Q) 2
† ) Q†0 = 12 (Q† − i Q 1 S0 = 2 (S + i
S) † 1 † S†) S0 = 2 (S − i
S) S− = 12 (S − i
B+ = − 2i [ − i(K+ − K− )]
† ) Q+ = (Q− )† = 12 (Q† + i Q S+ = (S− )† = 12 (S † + i
S†)
3.2. Unitary supermultiplets of D(2, 1; λ) The generators B− and B+ defined above close into B0 under commutation and generate the distinguished su(1, 1)K subalgebra [B− , B+ ] = 2B0 ,
[B0 , B+ ] = +B+ ,
[B0 , B− ] = −B−
(3.4)
where B0 =
(L3 + 3/16) 1 2 x + p2 + 4 x2
where L2 = λT 2 + 13 (λ − 1)A2 + λ(λ − 1)S 2 + 14 λ(λ − 1). The generator B0 can be interpreted as 12 the Hamiltonian, HConf , of conformal quantum mechanics [27] or of a singular oscillator [28] HConf = 2B0 =
g2 1 2 x + p2 + 2 2 x
(3.5)
with g 2 = (2L2 + 38 ) playing the role of coupling constant. A unitary lowest weight (positive energy) irreducible representation of SU(1, 1)K is uniquely determined by the state |ψ0α with the lowest eigenvalue of B0 that is annihilated by B− . In the coordinate (x) representation its wave function is given by ψ0ω (x) = C0 x ω e−x
2 /2
where C0 is the normalization constant, ω is given by 1/2 1 1 2 ω= + + 2gˆ 2 4 and gˆ 2 is the eigenvalue of (2L2 + 38 ) 3 ω 2L2 + ψ0 = gˆ 2 ψ0ω 8
(3.6)
(3.7)
(3.8)
We shall denote the functions obtained by the repeated action of differential operators B+ on ψ0ω (x) in the coordinate representation as ψnω (x) and the corresponding states in the Hilbert space as |ψnω :
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ψnω (x) = cn (B+ )n ψ0ω (x)
(3.9)
where the normalization constant is given as √ Γ (ω + 1/2) (−1) cn = n √ 2 n!Γ (n + ω + 1/2)
(3.10)
The wave functions ψnω (x) can be written as 2(n!) (ω−1/2) 2 ω −x 2 /2 ω x x e ψn (x) = Ln Γ (n + ω + 1/2)
(3.11)
(ω−1/2)
(x 2 ) is the generalized Laguerre polynomial. where Ln Irreducible unitary lowest weight representations of D(2, 1; λ) are uniquely labeled by a set of states, which we shall simply denote as {|Ω}, that transform irreducibly under the grade zero compact subsupergroup OSp(2/2) × U (1) and are annihilated by the grade −1 generators B− , A− , Q− and S− in C− . In a unitary lowest weight (positive energy) representation of D(2, 1; λ) the spectrum of H is bounded from below. We shall refer to H as the (total) Hamiltonian and its eigenvalues as total energy. Each state in the set |Ω is a lowest (conformal) energy state of a positive energy irrep of SU(1, 1)K , since they are all annihilated by B− . The conditions B− |Ω = 0,
A− |Ω = 0,
Q− |Ω = 0,
S− |Ω = 0
(3.12)
imply that the states |Ω must be linear combinations of the tensor product states of the form (3.13) |F × |B × ψ0ω where the state |F in (3.13) is either the fermionic Fock vacuum |0F = |mt = 0; ma = −1/2F = |0, ↓
(3.14)
or one of the following SU(2)T doublet of states: α † |0F ≡ |mt = 1/2; ma = 0F = |↑, 0
(3.15)
β |0F ≡ |mt = −1/2; ma = 0F = |↓, 0
(3.16)
†
and the state |B in (3.13) is any one of the states a m1 · · · a mk bmk+1 · · · bm2s |0B
(3.17)
where |0B is the bosonic Fock vacuum annihilated by the bosonic oscillators am and bm (m = 1, 2, . . . , P ). For fixed n the states of the form (3.17) transform in the spin s representation of SU(2)S . They also form representations of SO∗ (2P ) generated by the bilinears U m n = a m an + bm bn , (am bn − an bm ) and (a m bn − a n bm ) and which commutes with D(2, 1; λ). Therefore as far as the SU(2) deformations of the minrep of D(2, 1; λ) are concerned we can restrict our analysis to P = 1. Then for P = 1 we simply have S+ = a † b,
S− = b † a
and 1 † a a − b† b 2 Then the states |B belong to the set S0 =
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k 2s−k |s, ms B ≡ a † b† |0B
119
(3.18)
where ms = k − s (k = 0, 1, . . . , 2s) and transform in spin s representation of SU(2)S : S 2 |s, ms B = s(s + 1)|s, ms B
(3.19)
S0 |s, ms B = ms |s, ms B
(3.20)
and lowering operator S− = The action of raising operator S+ = given as S+ |s, ms = k − sB = (k + 1)(2s − k)|s, k − s + 1B S− |s, ms = k − sB = k(2s − k + 1)|s, k − s − 1B a†b
(1/4 + 2gˆ 2 )
b† a
on this state is then (3.21) (3.22)
(4L2 + 1)
of on the above states determine the values of ω labeling The eigenvalues the eigenstates |ψ0ω of B0 annihilated by B− : 2 4L + 1 |mt = 0; ma = −1/2F × |s, ms B = λ2 (2s + 1)2 |0; −1/2F × |s, ms B (3.23) 2 2 4L + 1 |mt = ±1/2; ma = 0F × |s, ms B = λ(2s + 1) + 1 |±1/2; 0F × |s, ms B (3.24) The SU(2) subalgebra of SU(1|2) is the diagonal subalgebra SU(2)T of SU(2)S and SU(2)T . Therefore we shall work in a basis where T 2 and T0 are diagonalized and denote the simultaneous eigenstates of B0 , T 2 , T0 , A2 and A0 as1 |ω; t, mt ; a, ma
(3.25)
where ω is the eigenvalue of B0 . The set of states |Ω must be linear combinations of the tensor product states of the form |F × |B × |ψoω where the state |F could be either |0F = |0, ↓F or one of the SU(2)T doublet of states |↑, 0F or |↓, 0F . We will now study the unitary representations for these two cases. 3.2.1. |F = |0F For states with t = 0, we can use Eq. (3.7) to write 1 ± λ(2s + 1) (3.26) 2 where the sign of the square root is determined by the sign of λ and the range of λ is determined by the square integrability of the states and the positivity of gˆ 2 . This leads to the following restriction on λ 1 |λ| > (3.27) 2(2s + 1) Let us first consider the case s = 0 and with the positive square root taken in the above equaλ+1/2 , annihilated by all the generators tions. Then the lowest energy state |0, ↓F × |0B × |ψ0 −1 in C , is a singlet of the grade zero super algebra SU(1|2) since it is also annihilated by all the supersymmetry generators in C0 . ω=
1 We introduce this notation for the states because the tensor product states of the form |F × |B × |ψ ω are not always 0 definite eigenstates of T 2 and T0 but it is easier to understand the structure of supermultiplets in terms of these tensor product states. Thus we will use both notations for states and hope that the meaning would be clear from the context.
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States generated by action of grade +1 generators C+ on this lowest weight state |ψ0 λ+1/2 λ+1/2 B + |0, ↓F × |0B × ψ0 = |0, ↓F × |0B × ψ1 λ+1/2 λ+1/2 = |0, ↑F × |0B × ψ0 A+ |0, ↓F × |0B × ψ0 λ+1/2 λ+3/2 = |↑, 0F × |0B × ψ0 Q+ |0, ↓F × |0B × ψ0 λ+1/2 λ+3/2 S+ |0, ↓F × |0B × ψ0 = |↓, 0F × |0B × ψ0 (3.28) (λ+1/2)
form a supermultiplet transforming in the representation with super tableau λ+1/2
of SU(2|1).
λ+1/2
and S+ |0, ↓F × |0B × |ψ0 are both lowest The states Q+ |0, ↓F × |0B × |ψ0 weight vectors of SU(1, 1)K transforming as a doublet of SU(2)T . The state A+ |0, ↓F × |0B × λ+1/2 λ+1/2 is a lowest weight vector of SU(1, 1)K and together with |0, ↓F × |0B × |ψ0 |ψ0 + form a doublet of SU(2)A . The commutator of two susy generators in C satisfies λ+1/2 λ+1/2 [Q+ , S+ ]|0, ↓F × |0B × ψ0 ∝ α † β † B + |0F × |0B × ψ0 Hence one does not generate any new lowest weight vectors of SU(1, 1)K by further actions of grade +1 supersymmetry generators. Every positive energy unitary representation of the conformal group SO(d, 2) corresponds to a conformal field in d-dimensional Minkowskian spacetime. The eigenvalues of the SO(2) generator determine the conformal dimension of the field. In one dimension the positive energy unitary representations of SO(2, 1) are identified with conformal wave functions. We shall denote the conformal wave function associated with a positive energy unitary representation of SO(2, 1) (ω) with lowest weight vector |ψ0 as Ψ ω (x). The conformal wave functions transforming in the (t, a) representation of SU(2)T × SU(2)A will then be denoted as ω Ψ(t,a) (x)
Thus the unitary supermultiplet of D(2, 1; λ) with the lowest weight vector |0F × |0B × λ+1/2 decomposes as: |ψ0 (λ+1)/2
(λ+2)/2
Ψ(0,1/2) ⊕ Ψ(1/2,0)
(3.29)
This is simply the minimal unitary supermultiplet of D(2, 1; λ) and for λ = −1/2 coincides with the singleton supermultiplet of OSp(4|2, R) = D(2, 1; −1/2). Next we consider the representations for the case s = 0 with λ > 0. In this case the lowest energy states |0, ↓F × s, ms = (k − s) B × ψ0ω (3.30) where ω = 1/2 + λ(2s + 1), and k = 0, . . . , 2s, are not annihilated by all supersymmetry generators of SU(2|1): Q0 |0, ↓F × |s, k − sB × ψ0ω = 0 S0 |0, ↓F × |s, k − sB × ψ ω = 0 Q†0 |0, ↓F
× |s, k
0 − sB × |ψ0ω
= λ(2s − k)|↑, 0F × |s, k − sB × ψ0ω−1
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− λ (k + 1)(2s − k)|↓, 0F × |s, k − s + 1B × ψ0ω−1 S†0 |0, ↓F × |s, k − sB × ψ0ω = λk|↓, 0F × |s, k − sB × ψ0ω−1 − λ k(2s − k + 1)|↑, 0F × |s, k − s − 1B × ψ0ω−1 † † Q , S |0, ↓F × |s, k − sB × ψ ω = 2s|0, ↑F × |s, k − sB × ψ ω−2 0
0
0
(3.31)
0
Since the supersymmetry generators Q†0 and S†0 transform in the spin 1/2 representation of SU(2)T acting on the states with spin t = s one would expect to obtain states with both t = s ± 1/2. However setting k = 2s in the above formulas we find Q†0 |0, ↓F × |s, sB × ψ0ω = 0 S†0 |0, ↓F × |s, sB × ψ0ω √ (3.32) = 2λs|↓, 0F × |s, sB × ψ0ω−1 − λ 2s|↑, 0F × |s, s − 1B × ψ0ω−1 which implies that we only get states with t = s − 1/2. Therefore the lowest energy supermultiplets of SU(2|1) that uniquely determine the deformed minimal unitary supermultiplets of D(2, 1; λ) transform in the representation with the super tableau which 2s
decomposes under the even subgroup SU(2)T × U (1)J as = ,0 ⊕ 2s
2s
λ , 2
(3.33)
(2s−1)
By acting with grade +1 generators of the compact 3-grading on these states with t = s and t = s − 1/2 one obtains states with t = s ± 1/2 and t = s: B + |0, ↓F × |s, k − sB × ψ0ω = |0, ↓F × |s, k − sB × ψ1ω A+ |0, ↓F × |s, k − sB × ψ0ω = |0, ↑F × |s, k − sB × ψ0ω Q+ |0, ↓F × |s, k − sB × ψ0ω = |↑, 0F × |s, k − sB × ψ0ω+1 − λ(2s − k)|↑, 0F × |s, k − sB × ψ0ω−1 + λ (k + 1)(2s − k)|↓, 0F × |s, k − s + 1B × ψ0ω−1 S+ |0, ↓F × |s, k − sB × ψ0ω = |↓, 0F × |s, k − sB × ψ0ω+1 − λ(2s − k)|↓, 0F × |s, k − sB × ψ0ω−1 + λ k(2s − k + 1)|↑, 0F × |s, k − s − 1B × ψ ω−1 (3.34) 0
The commutator of two supersymmetry generators does not generate any new lowest weight vector of SU(1, 1)K : [Q+ , S+ ]|0, ↓F × |s, k − sB × ψ0ω (3.35) ∝ |0, ↑F × |s, k − sB × ψ1ω = α † β † B + |0, ↓F × |s, k − sB × ψ0ω Thus the complete supermultiplet is simply p
p+1/2
p+1
Ψ(s−1/2,0) ⊕ Ψ(s,1/2) ⊕ Ψ(s+1/2,0)
(3.36)
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Table 2 Decomposition of SU(2) deformed minimal unitary lowest energy supermultiplets of D(2, 1; λ) with respect to SU(2)T × SU(2)A × SU(1, 1)K . The conformal wave functions transforming in the (t, a) representation of SU(2)T × ω . The first column shows the super tableaux of the lowest energy SU(2)A with conformal energy ω are denoted as Ψ(t,a) SU(2|1) supermultiplet, the second column gives the eigenvalue of the U (1) generator H. The allowed range of λ in this case is λ > 1/(4s + 2). SU(2|1)
H
SU(1, 1)K × SU(2)T × SU(2)A
1
λ/2
Ψ(0,1/2) ⊕ Ψ(1/2,0)
λ
λ+1 λ Ψ(0,0) ⊕ Ψ(1/2,1/2) ⊕ Ψ(1,0)
3λ/2 . ..
Ψ(1/2,0) ⊕ Ψ(1,1/2) . ..
(2s + 1)λ/2
Ψ(s−1/2,0) ⊕ Ψ(s,1/2) ⊕ Ψ(s+1/2,0)
. ..
(λ+1)/2
(λ+2)/2
λ+1/2
3λ/2
p
(3λ+1)/2
(3λ/2)+1
⊕ Ψ(3/2,0)
p+1/2
p+1
p = (2s + 1)λ/2
2s
where p = λ(2s + 1)/2. We have summarized the deformed supermultiplets for lowest weight states with t = s and λ > 1/(4s + 2) in Table 2. So far we have considered the representations for λ > 0 when the lowest weight state has t = s. Now we take a look at the case when λ < 0 and ω is then given as ω=
1 − λ(2s + 1) 2
(3.37)
The action of grade 0 supersymmetry generators on these states produces states with t = s + 1/2. This is different from the case with λ > 0 where we obtained states with t = s − 1/2 by the action of grade 0 supersymmetry generators. Thus the lowest energy supermultiplets of SU(2|1) that uniquely determine the deformed minimal unitary supermultiplets of D(2, 1; λ) transform which decomposes under the in the representation with the super tableau 2s+1
even subgroup SU(2)T × U (1)J as
=
2s+1
2s+1
,0 ⊕
λ , 2
(3.38)
2s
By acting with the grade +1 supersymmetry generators on these states, we complete the D(2, 1; λ) supermultiplet given as: p
p+1/2
p+1
Ψ(s+1/2,0) ⊕ Ψ(s,1/2) ⊕ Ψ(s−1/2,0)
(3.39)
where p = |λ|(2s + 1)/2. We have summarized the deformed supermultiplets for lowest weight states with t = s and λ < −1/(4s + 2) in Table 3. F 3.2.2. |F = |↑,0 |↓,0F If we choose the doublet of states |F = |↑, 0F and |F = |↓, 0F as part of the lowest energy supermultiplet, the states |ω, t, mt , a, ma satisfying the conditions given in (3.12) will have
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Table 3 Decomposition of SU(2) deformed minimal unitary lowest energy supermultiplets of D(2, 1; λ) with respect to SU(2)T × SU(2)A × SU(1, 1)K . The conformal wave functions transforming in the (t, a) representation of SU(2)T × ω . The first column shows the super tableaux of the lowest energy SU(2)A with conformal energy ω are denoted as Ψ(t,a) SU(2|1) supermultiplet, the second column gives the eigenvalue of the U (1) generator H. The allowed range of λ in this case is λ < −1/(4s + 2). SU(2|1)
.. .
H
SU(1, 1)K × SU(2)T × SU(2)A
|λ|/2
Ψ(1/2,0) ⊕ Ψ(0,1/2)
|λ|
Ψ(1,0) ⊕ Ψ(1/2,1/2) ⊕ Ψ(0,0)
3|λ|/2 .. .
Ψ(3/2,0) ⊕ Ψ(1,1/2) .. .
(2s + 1)|λ|/2
Ψ(s+1/2,0) ⊕ Ψ(s,1/2) ⊕ Ψ(s−1/2,0) p = (2s + 1)|λ|/2
2s+1
|λ|/2 |λ|
3|λ|/2
p
(|λ|+1)/2
|λ|+1/2
(3|λ|+1)/2
p+1/2
|λ|+1 3|λ|/2+1
⊕ Ψ(1/2,0)
p+1
t = s ± 1/2 and can be written as linear combinations of |↑(↓), 0 and |s, ms with appropriate Clebsch–Gordon coefficients for adding spin 1/2 with spin s. Square integrability of states and positivity of gˆ 2 imposes restrictions on range of λ and (3.12) determines the value of ω. For states with t = s + 1/2, we have ω = −1/2 − λ(2s + 1) 3 with λ < − 2(2s+1) . On the other hand for t = s − 1/2, we have ω = −1/2 + λ(2s + 1) with 3 λ > 2(2s+1) . As was done in previous section, we now proceed to study the supermultiplets but the action of supersymmetry generators in C0 and C+ on states with t = s + 1/2. The lowest energy supermultiplet for t = s + 1/2 corresponds to the following SU(2|1) supertableau (3.40) = ,1 ⊕ , 2s+1
2s+1
2s
and the resulting unitary supermultiplet of D(2, 1; λ) decomposes as p
p+1/2
p+1
Ψ(s+1/2,0) ⊕ Ψ(s,1/2) ⊕ Ψs−1/2,0
(3.41)
where p = (2s + 1)|λ|/2 with λ < 0. We have summarized the deformed supermultiplets for lowest weight states with t = s + 1/2 in Table 4. Note that these occur only for λ < −3/(4s + 2). Next we look at the states with t = s − 1/2. The lowest energy super multiplet for t = s − 1/2 corresponds to the following SU(2|1) supertableau = ,1 ⊕ , 2s 2s 2s (3.42) and leads to the supermultiplet p
p+1/2
p+1
Ψ(s−1/2,0) ⊕ Ψ(s,1/2) ⊕ Ψ(s+1/2,0)
(3.43)
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Table 4 Decomposition of SU(2) deformed minimal unitary lowest energy supermultiplets of D(2, 1; λ) with respect to SU(2)T × SU(2)A × SU(1, 1)K . The conformal wave functions transforming in the (t, a) representation of SU(2)T × ω . The first column shows the super tableaux of the lowest energy SU(2)A with conformal energy ω are denoted as Ψ(t,a) SU(2|1) supermultiplet, the second column gives the eigenvalue of the U (1) generator H. The allowed range of λ in this case is λ < −3/(4s + 2). SU(2|1)
.. .
H
SU(1, 1)K × SU(2)T × SU(2)A
|λ|/2
Ψ(1/2,0) ⊕ Ψ(0,1/2)
|λ|
Ψ(1,0) ⊕ Ψ(1/2,1/2) ⊕ Ψ(0,0)
3|λ|/2 .. .
Ψ(3/2,0) ⊕ Ψ(1,1/2) .. .
(2s + 1)|λ|/2
Ψ(s+1/2,0) ⊕ Ψ(s,1/2) ⊕ Ψ(s−1/2,0) p = (2s + 1)|λ|/2
2s+1
|λ|/2 |λ|
3|λ|/2
p
(|λ|+1)/2
|λ|+1/2
(3|λ|+1)/2
|λ|+1 3|λ|/2+1
⊕ Ψ(1/2,0)
p+1/2
p+1
Table 5 Decomposition of SU(2) deformed minimal unitary lowest energy supermultiplets of D(2, 1; λ) with respect to SU(2)T × SU(2)A × SU(1, 1)K . The conformal wave functions transforming in the (t, a) representation of SU(2)T × ω . The first column shows the super tableaux of the lowest energy SU(2)A with conformal energy ω are denoted as Ψ(t,a) SU(2|1) supermultiplet, the second column gives the eigenvalue of the U (1) generator H. The allowed range of λ in this case is λ > 3/(4s + 2). SU(2|1)
.. .
H
SU(1, 1)K × SU(2)T × SU(2)A
λ
λ+1 λ Ψ(0,0) ⊕ Ψ(1/2,1/2) ⊕ Ψ(1,0)
3λ/2
Ψ(1/2,0) ⊕ Ψ(1,1/2)
2λ
2λ+1 2λ ⊕ Ψ Ψ(1,0) (3/2,1/2) ⊕ Ψ(2,0)
.. .
.. .
(2s + 1)λ/2
λ+1 Ψ(s−1/2,0) ⊕ Ψ(s,1/2) ⊕ Ψ(s+1/2) p = (2s + 1)λ/2
λ+1/2
3λ/2
(3λ+1)/2
3λ+1 ⊕ Ψ(3/2,0)
2λ+1/2
p
p+1/2
2s
where p = (2s + 1)λ/2 with λ > 0. We have summarized the deformed supermultiplets for lowest weight states with t = s − 1/2 in Table 5. Note that these occur only for s > 1/2 and λ > 3/(4s + 2). We note the similarities of Table 2 with Table 5, and that of Table 3 with Table 4. This shows that the supermultiplets obtained for lowest weight states with t = s and t = s − 1/2 and λ > 0 are the same and the supermultiplets for lowest weight states with t = s and t = s + 1/2 and λ < 0 are same. The difference between these two types of supermultiplets is that the SU(1, 1)K spin (labeled as p) gets interchanged between states with t = s + 1/2 and t = s − 1/2 as we change the sign of λ.
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4. SU(2) deformations of the minimal unitary representation of D(2, 1; λ) using both bosons and fermions and OSp(2n∗ |2m) superalgebras Above we obtained unitary supermultiplets of D(2, 1; λ) which are SU(2) deformations of the minimal unitary representation. This was achieved by introducing bosonic oscillators an and bn and extending the SU(2)T generators to the generators of the diagonal subgroup of SU(2)T and SU(2)S realized as bilinears of the bosonic oscillators 1 S0 = (Na − Nb ) 2 ∗ As stated above, the noncompact group SO (2n) generated by the bilinears S+ = a n bn ,
S − = b n an ,
Amn = am bn − an bm ,
Amn = a m bn − a n bm ,
Unm = a m an + bn bm
(4.1)
(4.2)
commutes with the generators of D(2, 1; λ). One can similarly obtain SU(2) deformations of the minimal unitary supermultiplet of D(2, 1; λ) by introducing fermionic oscillators ρr and σs (r, s, . . . = 1, . . . , n) satisfying ρr , ρ s = σr , σ s = δrs {ρr , ρs } = {ρr , σs }{σr , σs } = 0 (4.3) and extend the generators of SU(2)T to the generators of the diagonal subgroup of SU(2)T and SU(2)F generated by 1 r ρ ρr − σ r σr 2 In this case the compact USp(2n) generated by the fermion bilinears F+ = ρ r σ r ,
F− = σ r ρ r ,
Srs = ρr σs + ρs σr ,
F0 =
S rs = σ r ρ s + σ s ρ r ,
Ssr = ρ r ρs − σs σ r
(4.4)
(4.5)
commute with the generators of D(2, 1; λ). One can deform the minimal unitary representation of D(2, 1; λ) using fermions and bosons simultaneously. This is achieved by replacing the SU(2)T generators by the diagonal SU(2)D generators of SU(2)T × SU(2)S × SU(2)F , which we shall denote as U+ , U− and U0 U + = T + + S + + F+ ,
U− = T − + S − + F − ,
U 0 = T 0 + S 0 + F0
(4.6)
and substituting the quadratic Casimir of SU(2)T in the Ansatz for K− with the quadratic Casimir of SU(2)D . Remarkably, in this case the resulting generators of D(2, 1; λ) commute with the generators of the noncompact superalgebra OSp(2n∗ |2m) generated by the generators of SO∗ (2n) and USp(2m) given above and the supersymmetry generators [29]: Πmr = am σr − bm ρr , r Σm
= a m ρ + bm σ , r
r
Π¯ mr = (Πmr )† = a m σ r − bm ρ r r † Σ¯ rm = Σm = a m ρr + b m σ r
The (anti-)commutation relations for the OSp(2n∗ |2m) algebra are given below: i k r t Aj , Al = δjk Ail − δli Akj , Ss , Su = δst Sur − δur Sst Aij , Akl = δjk Ail − δik Aj l , Srs , Sut = δst Sru + δrt Sus ij k rs t j A , Al = δli Aj k − δl Aik , S , Su = −δus S rt − δur S ts Aij , Akl = −δjk Ali + δjl Aki − δil Akj + δik Alj
(4.7) (4.8)
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Srs , S tu = −δst Sru − δrt Ssu − δsu Srt − δru Sst r n s n r Πmr , Π¯ ns = δrs Anm − δm Sr , Σm , Σ¯ sn = δsr Anm + δm Ss s s n n Πmr , Σn = δr Amn , Πmr , Σ¯ s = −δm Srs m m r An , Πkr = −δkm Πnr , An , Σk = −δkm Σnr mn mn r A , Πkr = −δkm Σ¯ rn + δkn Σ¯ rm , A , Σk = −δkn Π¯ rm + δkm Π¯ rn [Amn , Πkr ] = 0, Amn , Σkr = 0 r r t r Ss , Πmt = −δtr Πms , Ss , Σm = δst Σm rs r s t S , Πmt = δts Σm =0 + δtr Σm , S rs , Σm t t [Srs , Πmt ] = 0, Srs , Σm = −δr Πms − δst Πmr 5. SU(2) deformed minimal unitary supermultiplets of D(2, 1; α) and N = 4 superconformal mechanics 5.1. N = 4 superconformal quantum mechanical models A new class of N = 4 supersymmetric Calogero-type models have been studied by various authors in recent years [4,10] which are invariant under the superconformal algebra D(2, 1; α). The construction of D(2, 1; α) mechanics and quantization was performed in [10]. In this section we will review that construction following [10]. The on shell component action was shown to take the form [10] (5.1) S = S b + Sf 2 k 2 α (¯zk z ) i − A z¯ k zk − c (5.2) Sb = dt x˙ x˙ + z¯ k z˙ k − z˙¯ k zk − 2 4x 2 ψ i ψ¯ k z(i z¯ k) 2 ψ i ψ¯ k ψ(i ψ¯ k) + (1 + 2α) dt Sf = −i dt ψ¯ k ψ˙ k − ψ˙¯ k ψ k + 2α dt 2 3 x x2 (5.3) Here x, zi and ψ j (i, j = 1, 2) are d = 1 bosonic and fermionic “fields”, respectively. The fields zi form a complex doublet of the R-symmetry group SU(2). The last term in (5.2) represents the constraint z¯ k zk = c
(5.4)
and A is the Lagrange multiplier. Upon quantization of the action given in (5.1), the dynamical variables were promoted to quantum mechanical operators with following commutators: [X, P ] = i,
Z i , Z¯ j = δji ,
1 Ψ i , Ψ¯ j = − δji 2
(i, j = 1, 2)
(5.5)
As the action is invariant under the group D(2, 1; α), the corresponding symmetry generators can be obtained by the Noether procedure. The results as given in [10] are:
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Z (i Z¯ k) Ψk
Ψk Ψ k Ψ¯ i + i(1 + 2α) X X k kΨ ¯ ¯ ¯ ¯ k Ψi Z
Ψ Z Ψ ¯ i = P Ψ¯ i − 2iα (i k) + i(1 + 2α) Q X X ¯i S¯ i = −2X Ψ¯ i + t Q Si = −2XΨ i + tQi ,
Qi = P Ψ i + 2iα
(5.6) (5.7) (5.8)
(Z¯ k Z k )2 + 2Z¯ k Z k Z (i Z¯ k) Ψ(i Ψ¯ k) 1 − 2α H = P 2 + α2 4 4X 2 X2 i k 2 ¯ ¯
Ψi Ψ Ψ Ψk (1 + 2α) − (1 + 2α) + 2X 2 16X 2 1 1 K = X 2 − t {X, P } + t 2 H, D = − {X, P } + tH 2 4 Jik = i Z (i Z¯ k) + 2Ψ (i Ψ¯ k)
I2 2 = i Ψ¯ k Ψ¯ k ,
i
(5.9) (5.10) k
Ψk , Ψ¯ 2 where t is time variable and the symbol · · · denotes Weyl ordering: I1 1 = −iΨk Ψ k ,
I1 2 = −
127
(5.11)
k 1 1 Ψk Ψ k Ψ¯ i = Ψk Ψ k Ψ¯ i + Ψ i , Ψ¯ Ψ¯ k Ψi = Ψ¯ k Ψ¯ k Ψi + Ψ¯ i 2 2 1 1 1 i ¯k ¯ i ¯k ¯ i ¯k ¯ i Ψi Ψ Ψ Ψk = Ψi Ψ , Ψ Ψk − = Ψi Ψ Ψ Ψk − Ψi Ψ¯ + 2 4 4 i + i + ¯ ¯ and Qi = −(Q ) , Si = −(S ) . In the above set of generators Qi and Si are supertranslation and superconformal boost generators respectively. The generators H, K and D are the Hamiltonian, special conformal transformations and dilatation generators respectively and they form an su(1, 1) algebra. The remaining generators Jik and Ii k are the generators of two su(2) algebras.
5.2. Mapping between the harmonic superspace generators of N = 4 superconformal mechanics and generators of deformed minimal unitary representations of D(2, 1; λ) A precise correspondence between the Killing potentials of the isometry groups G of N = 2 sigma models that couple to 4d supergravity in harmonic superspace and the generators of the minimal unitary representations of G was first shown in [24]. It was then suggested that the correspondence could be made concrete and precise by reducing the four-dimensional N = 2 sigma models to one dimension with eight supercharges and subsequently quantize them to get supersymmetric quantum mechanical models [24,25]. The results presented in this paper and the results of [10] on D(2, 1; α) superconformal quantum mechanics reviewed in previous section provide an opportunity to test this proposal. Remarkably one finds that the generators of quantized N = 4 superconformal mechanics in harmonic superspace go over to the generators of minimal unitary realization of D(2, 1; λ) deformed by a pair of bosonic oscillators if we make the simple substitutions listed in Table 6 and setting α = λ. Noting that the SU(2) indices i, j, . . . of [10] are raised and lowered by the Levi-Civita tensor ij (with 12 = 21 = 1) we find a oneto-one correspondence between the symmetry generators of D(2, 1; α) superconformal quantum mechanics and the generators of the minimal unitary representations of D(2, 1; α) deformed by a pair of bosons, which we present in Table 6. We also find that the quadratic Casimir obtained in Eq. (4.26) of [10] is the same as the one obtained by our construction given in (2.34) for μ = 4.
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Table 6 Mapping of the operators, quantum numbers and symmetry generators between N = 4 superconformal mechanics in harmonic superspace (denoted as HSS) and the minimal unitary representation of D(2, 1; λ) deformed by a pair of bosons (denoted as Minrep). HSS
Minrep
Operators: (ψ 1 , ψ 2 ) (ψ¯1 , ψ¯2 ) (Z 1 , Z 2 ) (Z¯ 1 , Z¯ 2 )
Minrep
Quantum numbers: − √i (α † , β † )
2 − √i (α, β) 2 −i(a † , b† )
i(a, b)
Odd generators: − √i (Q1 , Q2 ) 2 ¯ 2, Q ¯ 2) − √i (Q 2 i 1 2 − √ (S , S ) 2 − √i (S1 , S2 ) 2
HSS r0
ω
j
t
i c
a 2s
Even generators:
1 ,
(Q S1)
i(I 1 1 , −I 2 2 , I 1 2 )
(A+ , A− , A0 )
1 ,
(Q S1 )
i(−J 11 , J 22 , J 12 )
(T+ , T− , T0 )
(Q1 , S 1 )
(2H, 12 K)
(K+ , K− )
(Q1 , S1 )
−2D
The spectrum of N = 4 superconformal quantum mechanics was also studied in [10]. To compare its quantum spectrum with the deformed minimal unitary supermultiplets we need only to compare the SU(1, 1), SU(2)R , SU(2)L decomposition of the spectrum of [10] with the SU(1, 1)K , SU(2)T , SU(2)A decomposition of the minimal unitary supermultiplets constructed in this paper. We find that the superfield content of the quantum spectrum described in Table 2 of [10] is exactly the same as supermultiplets described in Tables 2, 3, 4 and 5 above. 6. Conclusions Above we constructed the SU(2) deformed minimal unitary supermultiplets of D(2, 1; λ) using quasiconformal methods as was done for the 4d and 6d superconformal algebras in [1,2]. We showed that for deformations obtained by a pair of bosons there exists a precise mapping to the generators and the quantum spectra of N = 4 superconformal mechanical models studied recently. This raises the question what kind of N = 4 superconformal quantum mechanical models correspond to more general deformations of the minimal unitary representations involving an arbitrary numbers of pairs of bosons and/or pairs of fermions as formulated above. We expect some of these more general deformations to describe the spectra of supersymmetric quantum mechanical models with quaternionic Kähler target spaces. One would also like to understand the precise connection between the above results and the N = 4 supersymmetric gauged WZW models in d = 2 studied in [30,31] that extends the results of [32] on N = 2 supersymmetric gauged WZW models. These gauged WZW models correspond to realization over spaces of the form Gc × SU(2) × U (1) H × SU(2) Gc is a compact quaternionic symmetric space. On the other hand the quaternionic where H ×SU(2) Kähler manifolds that couple to 4d, N = 2 supergravity are noncompact [24]. We hope to address these problems in a future study.
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Acknowledgements This work was supported in part by the National Science Foundation under grants numbered PHY-1213183 and PHY-0855356. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. References [1] S. Fernando, M. Gunaydin, Minimal unitary representation of SU(2, 2) and its deformations as massless conformal fields and their supersymmetric extensions, J. Math. Phys. 51 (2010) 082301, arXiv:0908.3624 [hep-th]. [2] S. Fernando, M. Gunaydin, SU(2) deformations of the minimal unitary representation of OSp(8∗ |2N ) as massless 6D conformal supermultiplets, Nucl. Phys. B 843 (2011) 784–815, arXiv:1008.0702 [hep-th]. [3] M. Gunaydin, G. Sierra, P. Townsend, The unitary supermultiplets of d = 3 anti-de Sitter and d = 2 conformal superalgebras, Nucl. Phys. B 274 (1986) 429. [4] S. Fedoruk, E. Ivanov, O. Lechtenfeld, Superconformal mechanics, J. Phys. A 45 (2012) 173001, arXiv:1112.1947 [hep-th]. [5] E. Ivanov, O. Lechtenfeld, N = 4 supersymmetric mechanics in harmonic superspace, JHEP 0309 (2003) 073, arXiv:hep-th/0307111. [6] E. Ivanov, S. Krivonos, O. Lechtenfeld, New variant of N = 4 superconformal mechanics, JHEP 0303 (2003) 014, arXiv:hep-th/0212303. [7] S. Krivonos, O. Lechtenfeld, Many-particle mechanics with D(2, 1 : α) superconformal symmetry, JHEP 1102 (2011) 042, arXiv:1012.4639 [hep-th]. [8] O. Lechtenfeld, K. Schwerdtfeger, J. Thurigen, N = 4 multi-particle mechanics, WDVV equation and roots, SIGMA 7 (2011) 023, arXiv:1011.2207 [hep-th]. [9] S. Fedoruk, E. Ivanov, O. Lechtenfeld, New super Calogero models and OSp(4|2) superconformal mechanics, arXiv:1001.2536 [hep-th]. [10] S. Fedoruk, E. Ivanov, O. Lechtenfeld, New D(2, 1, α) mechanics with spin variables, JHEP 1004 (2010) 129, arXiv:0912.3508 [hep-th]. [11] S. Fedoruk, E. Ivanov, O. Lechtenfeld, OSp(4|2) superconformal mechanics, JHEP 0908 (2009) 081, arXiv: 0905.4951 [hep-th]. [12] F. Delduc, E. Ivanov, N = 4 mechanics of general (4, 4, 0) multiplets, Nucl. Phys. B 855 (2012) 815–853, arXiv: 1107.1429 [hep-th]. [13] E.A. Ivanov, Harmonic superfields in N = 4 supersymmetric quantum mechanics, SIGMA 7 (2011) 015, arXiv: 1102.2288 [hep-th]. [14] E. Ivanov, J. Niederle, Bi-harmonic superspace for N = 4 mechanics, Phys. Rev. D 80 (2009) 065027, arXiv: 0905.3770 [hep-th]. [15] S. Fedoruk, E. Ivanov, O. Lechtenfeld, Supersymmetric Calogero models by gauging, Phys. Rev. D 79 (2009) 105015, arXiv:0812.4276 [hep-th]. [16] F. Delduc, E. Ivanov, Gauging isometries in N = 4 supersymmetric mechanics, arXiv:0801.4521 [hep-th]. [17] F. Delduc, E. Ivanov, Gauging N = 4 supersymmetric mechanics, Nucl. Phys. B 753 (2006) 211–241, arXiv:hep-th/ 0605211. [18] F. Delduc, E. Ivanov, Gauging N = 4 supersymmetric mechanics II: (1, 4, 3) models from the (4, 4, 0) ones, Nucl. Phys. B 770 (2007) 179–205, arXiv:hep-th/0611247. [19] S. Bellucci, S. Krivonos, A. Sutulin, N = 4, d = 1 supersymmetric hyper-Kaehler sigma models with isospin variables, JHEP 1102 (2011) 038, arXiv:1012.2250 [hep-th]. [20] S. Bellucci, S. Krivonos, A. Sutulin, Dual multiplets in N = 4 superconformal mechanics, arXiv:1012.2325 [hepth]. [21] S. Bellucci, S. Krivonos, Potentials in N = 4 superconformal mechanics, Phys. Rev. D 80 (2009) 065022, arXiv: 0905.4633 [hep-th]. [22] S. Bellucci, S. Krivonos, A. Sutulin, N = 4 supersymmetric 3-particles Calogero model, Nucl. Phys. B 805 (2008) 24–39, arXiv:0805.3480 [hep-th]. [23] S. Bellucci, S. Krivonos, Supersymmetric mechanics in superspace, Lect. Notes Phys. 698 (2006) 49–96, arXiv: hep-th/0602199.
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Nuclear Physics B 869 [PM] (2013) 111–130 www.elsevier.com/locate/nuclphysb
Minimal unitary representation of D(2, 1; λ) and its SU(2) deformations and d = 1, N = 4 superconformal models Karan Govil, Murat Gunaydin ∗ Physics Department, Pennsylvania State University, University Park, PA 16802, USA Received 28 September 2012; received in revised form 16 November 2012; accepted 11 December 2012 Available online 14 December 2012
Abstract Quantization of the geometric quasiconformal realizations of noncompact groups and supergroups leads directly to their minimal unitary representations (minreps). Using quasiconformal methods massless unitary supermultiplets of superconformal groups SU(2, 2|N ) and OSp(8∗ |2n) in four and six dimensions were constructed as minreps and their U (1) and SU(2) deformations, respectively. In this paper we extend these results to SU(2) deformations of the minrep of N = 4 superconformal algebra D(2, 1; λ) in one dimension. We find that SU(2) deformations can be achieved using n pair of bosons and m pairs of fermions simultaneously. The generators of deformed minimal representations of D(2, 1; λ) commute with the generators of a dual superalgebra OSp(2n∗ |2m) realized in terms of these bosons and fermions. We show that there exists a precise mapping between symmetry generators of N = 4 superconformal models in harmonic superspace studied recently and minimal unitary supermultiplets of D(2, 1; λ) deformed by a pair of bosons. This can be understood as a particular case of a general mapping between the spectra of quantum mechanical quaternionic Kähler sigma models with eight super symmetries and minreps of their isometry groups that descends from the precise mapping established between the 4d, N = 2 sigma models coupled to supergravity and minreps of their isometry groups. © 2012 Elsevier B.V. All rights reserved.
* Corresponding author.
E-mail addresses: [email protected] (K. Govil), [email protected] (M. Gunaydin). 0550-3213/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nuclphysb.2012.12.006
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1. Introduction This paper belongs to a series of papers on the construction of the minimal unitary representations of noncompact groups and supergroups and their deformations using quasiconformal methods. For physical motivations and references to earlier work on the subject we refer to [1,2]. In this paper we study the deformations of the minimal unitary supermultiplet of D(2, 1; λ) with the even subgroup SU(1, 1) × SU(2) × SU(2). D(2, 1; λ) represents a one parameter family of N = 4 superconformal algebras in one dimension and is relevant to AdS2 /CFT 1 dualities. It is also important for AdS3 /CFT 2 dualities since the AdS3 group SO(2, 2) factorizes as SU(1, 1) × SU(1, 1). Supersymmetric extensions factorize as well and each factor can be extended to D(2, 1; λ) [3]. Another motivation for our work is to establish the connection between the spectra of N = 4 superconformal quantum mechanical models that have been studied in recent years [4–23] and the deformations of minimal unitary supermultiplets of corresponding conformal superalgebras. This provides a quantum mechanical example of the deep connection between N = 2 sigma models that couple to 4d supergravity and minimal unitary representations of their isometry groups established in [24,25]. The plan of the paper is as follows. In Section 2 after reviewing the minimal unitary realization of D(2, 1; λ) [26] we construct its SU(2) deformations using bosonic oscillators in the noncompact 5-graded basis. In Section 3 we reformulate the results of Section 2 in the compact 3-graded basis and show that the deformations of the minrep of D(2, 1; λ) are positive “energy” (unitary lowest weight) representations. We then present the corresponding unitary supermultiplets. Section 4 discusses deformations of the minrep using both bosons and fermions and how the deformed D(2, 1; λ) commutes with a noncompact super algebra OSp(2n∗ |2m) with the even subgroup SO∗ (2m) × USp(2n) constructed using “deformation” bosons and fermions. In Section 5 we review some of the results of work on N = 4 superconformal quantum mechanics and show how its symmetry generators and spectrum map into the generators of D(2, 1; λ) deformed by a pair of bosonic oscillators and the resulting unitary supermultiplets. We conclude with a brief discussion of our results. 2. The minimal unitary representation of D(2, 1; λ) Of all the noncompact real forms of the one parameter family of supergroups D(2, 1; λ) only the real form with the even subgroup SU(2) × SU(2) × SU(1, 1) admits unitary lowest weight (positive energy) representations. In this paper we shall study the minimal unitary representations of this real form, which we shall denote as D(2, 1; λ), and their deformations. We shall label the even subgroup as SU(2)A × SU(2)T × SU(1, 1)K with the odd generators transforming in the (1/2, 1/2, 1/2) representation with respect to it. The Lie super algebra of D(2, 1; λ) can be given a 5-graded decomposition of the form D(2, 1; λ) = g(−2) ⊕ g(−1) ⊕ g(0) ⊕ g(+1) ⊕ g(+2)
(2.1)
where the grade ±2 subspaces are one-dimensional and the grade zero subalgebra is g(0) = su(2)A ⊕ su(2)T ⊕ so(1, 1)
(2.2)
The grade ±1 subspaces contain 4 odd generators transforming in the (1/2, 1/2) representation of SU(2)A × SU(2)T subgroup and the generators belonging to g(−2) and g(+2) together with the SO(1, 1) generator of grade zero subspace form the su(1, 1)K subalgebra.
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For λ = −1/2 the Lie superalgebra D(2, 1; λ) is isomorphic to OSp(4∗ |2) = OSp(4|2, R). The minrep of D(2, 1; λ) was obtained in [26] using quasiconformal techniques. In this paper we shall use a different label λ for the one parameter superalgebras D(2, 1; λ) which is related to the label σ (used in [26]) as: 2λ + 1 3 With this labeling we have σ=
(2.3)
D(2, 1; λ = −1/2) = OSp(4|2, R)
(2.4)
2.1. Minimal unitary realization of D(2, 1; λ) in 5-grading Before studying the deformations of the minrep of D(2, 1; λ), we shall first rewrite the results of [26] in a split basis in which the U (1) generators of the two SU(2) subgroups are diagonalized and the generators of SU(2)T and SU(2)A , denoted by T±,0 and A±,0 respectively, are given by the following bilinears of a pair of fermionic oscillators: T+ = α † β,
T− = β † α,
A+ = α † β † ,
A− = βα,
1 T0 = (Nα − Nβ ) 2 1 A0 = (Nα + Nβ − 1) 2
(2.5) (2.6)
where Nα = α † α and Nβ = β † β are the respective number operators. The fermionic oscillators satisfy the following anti-commutation relations: α, β † = 0 = β, α † (2.7) α, α † = 1 = β, β † , These generators satisfy the commutation relations: [T+ , T− ] = 2T0 ,
[T0 , T± ] = ±T±
(2.8)
with the Casimir 1 T 2 = T02 + (T+ T− + T− T+ ) 2
(2.9)
The quadratic Casimir A2 of SU(2)A is related to T 2 as follows 3 4 The single bosonic coordinate and its canonical momentum (x, p) satisfy T 2 + A2 =
(2.10)
[x, p] = i
(2.11)
and the generator that gives the 5-grading, is realized as: 1 (2.12) = (xp + px) 2 The grade −1 generators can then be written as bilinears of the coordinate x with the fermionic oscillators: Q = xα,
Q† = xα † ,
S = xβ,
S † = xβ †
(2.13)
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They close into K− under anti-commutation: Q, Q† = 2K− The grade zero generators in the 5-grading determined by are the generators T+ , T− and T0 of SU(2)T , A+ , A− and A0 of SU(2)A and . The grade +2 generator with respect to is given by: 1 2 1 2 3 1 2 2 K+ = p + 2 2λT + (λ − 1)A + + λ(λ − 1) 2 3 8 2 x 2 1 1 2 λ (2λ + 1)T 2 + +1 (2.14) K+ = p 2 + 2 2 2 x 3 where we used T 2 + A2 = 34 . 2.2. Deformations of minreps In order to obtain unitary irreducible representations that are “deformations” of the minrep of D(2, 1; λ) we introduce bosonic oscillators am , bm and their hermitian conjugates a m = (am )† , bm = (bm )† (m, n, . . . = 1, 2) that satisfy the commutation relations: n a m , a n = b m , b n = δm , [am , an ] = [am , bn ] = [bm , bn ] = 0 (2.15) and introduce an SU(2)S Lie algebra whose generators are as follows: 1 S0 = (Na − Nb ) 2 where Na = a m am and Nb = bm bm are the respective number operators. They satisfy: S+ = a m bm ,
S− = (S+ )† = am bm ,
[S+ , S− ] = 2S0 ,
[S0 , S± ] = ±S±
(2.16)
(2.17)
The quadratic Casimir of su(2)S is 1 C2 su(2)S = S 2 = S02 + (S+ S− + S− S+ ) 2 1 1 = (Na + Nb ) (Na + Nb ) + 1 − 2a [m bn] a[m bn] 2 2
(2.18)
where square bracketing a[m bn] = 12 (am bn − an bm ) represents antisymmetrization of weight one. The bilinears a[m bn] and a [m bn] close into U (P ) generated by the bilinears U m n = a m an + b m bn
(2.19)
under commutation and all together they form the Lie algebra of noncompact group SO∗ (2P ) with the maximal compact subgroup U (P ). The group SO∗ (2P ) thus generated commutes with SU(2)S as well as with D(2, 1; λ). To obtain the SU(2) deformed D(2, 1; λ) superalgebra we replace the generators of su(2)T subalgebra with the generators of the diagonal subgroup of SU(2)T and SU(2)S su(2)T
⇒
su(2)S ⊕ su(2)T
(2.20)
and denote the diagonal subgroup as SU(2)T and its Lie algebra as su(2)T . Its generators are simply:
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T+ = S+ + T+ = a m bm + α † β T− = S− + T− = bm am + β † α 1 (2.21) T0 = S0 + T0 = (Na − Nb + Nα − Nβ ) 2 The generator and the negative grade generators defined by it remain unchanged in going over to the deformed minreps. The grade +1 generators are then given by the commutators:
† = (Q)
† = i Q† , K+
= i[Q, K+ ], Q Q
(2.22) S † = (
S)† = i S † , K+ S = i[S, K+ ], Thus we make an ansatz for grade +2 generator K+ of the form 1 1 K+ = p 2 + 2 c1 T 2 + c2 S 2 + c3 A2 + c4 (2.23) 2 x where c1 , . . . , c4 are some constant parameters. Using the closure of the algebra, we determine these four unknown constants in terms of λ and obtain: 1 1 8 3 K+ = p 2 + 2 8λT 2 + (λ − 1)A2 + + 8λ(λ − 1)S 2 + 2λ(λ − 1) (2.24) 2 3 2 4x The +1 grade generators then take the form
= −pα + 2i λ T0 + 3 α + T− β − λ − 1 A0 − 3 α − 2A− β † Q x 4 3 4
† = −pα † − 2i λ T0 − 3 α † + T+ β † − λ − 1 A0 + 3 α † − 2A+ β Q x 4 3 4
3 λ−1 3 2i
λ T0 − β − T+ α − A0 + β − A− α † S = −pβ − x 4 3 4
λ−1 2i 3 † 3 † † † †
λ T0 + β − T− α − A0 − β − A− α S = −pβ + (2.25) x 4 3 4 The anti-commutators of grade +1 and grade −1 generators close into grade zero subalgebra
† = − − 2iλT0 + i(λ + 1)A0 Q, Q †
= − + 2iλT0 − i(λ + 1)A0 Q ,Q S,
S † = − + 2iλT0 + i(λ + 1)A0 † S ,
S = − − 2iλT0 − i(λ + 1)A0 (2.26) Q,
S † = −2iλT− (2.27) {Q,
S} = +2i(λ + 1)A− , † † †
Q , S = +2iλT+ (2.28) Q , S = −2i(λ + 1)A+ , †
= −2iλT+
= −2i(λ + 1)A− , S, Q (2.29) {S, Q} † † †
= +2i(λ + 1)A+ ,
= +2iλT− S ,Q (2.30) S ,Q † † † †
= {S,
= Q ,Q S} = S ,
(2.31) S =0 {Q, Q} † † † †
[Q, K− ] = iQ, Q , K− = iQ , [S, K− ] = iS, S , K− = iS (2.32)
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The quadratic Casimir of su(1, 1)K generated by K±2 and is 1 1 C2 su(1, 1)K = K2 = (K+ K− + K− K+ ) − 2 2 4 λ−1 2 λ(λ − 1) 2 2 = λT + A + λ(λ − 1)S + (2.33) 3 4 There exists a one parameter family of quadratic Casimir elements C2 (μ) that commute with all the generators of D(2, 1; λ). μ 2 λ 1 i K − (μ − 8)T 2 − 16 + 8λ + μ(λ − 1) A2 + F(Q, S) 4 4 12 4 2 1 λ = 8 + μ(λ − 1) S + 4 4
C2 (μ) =
(2.34)
where
† + Q† , Q
+ S,
F(Q, S) = Q, Q S† + S†,
S
(2.35)
is the contribution from the odd generators. Since the eigenvalues of the quadratic Casimir depends on the eigenvalues s(s + 1) of the Casimir operator S 2 of SU(2)S the corresponding deformed unitary supermultiplets will be uniquely labeled by spin s of SU(2)S . 3. SU(2) deformed minimal unitary representations as positive energy unitary supermultiplets of D(2, 1; λ) 3.1. Compact 3-grading The 5-graded decomposition of the Lie superalgebra D(2, 1; λ) is determined by the noncompact generator : D(2, 1; λ) = g(−2) ⊕ g(−1) ⊕ su(2)T ⊕ su(2)A ⊕ so(1, 1) ⊕ g(+1) ⊕ g(+2)
Q
† ,
= K− ⊕ Q, Q† , S, S † ⊕ [A±,0 , T±,0 , ] ⊕ Q, S,
S † ⊕ K+ The Lie superalgebra D(2, 1; λ) can also be given a 3-graded decomposition with respect to its compact subsuperalgebra osp(2|2) ⊕ u(1) = su(2|1) ⊕ u(1), which we shall refer to as compact 3-grading: D(2, 1; λ) = C− ⊕ C0 ⊕ C+ D(2, 1; λ) = (A− , B− , Q− , S− ) ⊕
(3.1)
T±,0 , J , H, Q0 , S0 , Q†0 , S†0 ⊕ (A+ , B+ , Q+ , S+ )
(3.2) The expressions for the generators in the 3-grading in terms of the generators in 5-grading is given in Table 1. The generator H that determines the compact 3-grading is given by 1 1 H = (K+ + K− ) + A0 = B0 + (Nα + Nβ − 1) 2 2
(3.3)
The other generators in C0 form the compact superalgebra su(2|1) whose even subalgebra is generated by T±,0 and J .
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Table 1 Generators in compact 3-grading. C−
C0
C+
A−
T±,0 H = 12 (K+ + K− ) + A0
A+
B− = 2i [ + i(K+ − K− )]
Q− = 12 (Q − i Q)
J = (λ + 1)A0 + 12 (K+ + K− )
Q0 = 1 (Q + i Q) 2
† ) Q†0 = 12 (Q† − i Q 1 S0 = 2 (S + i
S) † 1 † S†) S0 = 2 (S − i
S) S− = 12 (S − i
B+ = − 2i [ − i(K+ − K− )]
† ) Q+ = (Q− )† = 12 (Q† + i Q S+ = (S− )† = 12 (S † + i
S†)
3.2. Unitary supermultiplets of D(2, 1; λ) The generators B− and B+ defined above close into B0 under commutation and generate the distinguished su(1, 1)K subalgebra [B− , B+ ] = 2B0 ,
[B0 , B+ ] = +B+ ,
[B0 , B− ] = −B−
(3.4)
where B0 =
(L3 + 3/16) 1 2 x + p2 + 4 x2
where L2 = λT 2 + 13 (λ − 1)A2 + λ(λ − 1)S 2 + 14 λ(λ − 1). The generator B0 can be interpreted as 12 the Hamiltonian, HConf , of conformal quantum mechanics [27] or of a singular oscillator [28] HConf = 2B0 =
g2 1 2 x + p2 + 2 2 x
(3.5)
with g 2 = (2L2 + 38 ) playing the role of coupling constant. A unitary lowest weight (positive energy) irreducible representation of SU(1, 1)K is uniquely determined by the state |ψ0α with the lowest eigenvalue of B0 that is annihilated by B− . In the coordinate (x) representation its wave function is given by ψ0ω (x) = C0 x ω e−x
2 /2
where C0 is the normalization constant, ω is given by 1/2 1 1 2 ω= + + 2gˆ 2 4 and gˆ 2 is the eigenvalue of (2L2 + 38 ) 3 ω 2L2 + ψ0 = gˆ 2 ψ0ω 8
(3.6)
(3.7)
(3.8)
We shall denote the functions obtained by the repeated action of differential operators B+ on ψ0ω (x) in the coordinate representation as ψnω (x) and the corresponding states in the Hilbert space as |ψnω :
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ψnω (x) = cn (B+ )n ψ0ω (x)
(3.9)
where the normalization constant is given as √ Γ (ω + 1/2) (−1) cn = n √ 2 n!Γ (n + ω + 1/2)
(3.10)
The wave functions ψnω (x) can be written as 2(n!) (ω−1/2) 2 ω −x 2 /2 ω x x e ψn (x) = Ln Γ (n + ω + 1/2)
(3.11)
(ω−1/2)
(x 2 ) is the generalized Laguerre polynomial. where Ln Irreducible unitary lowest weight representations of D(2, 1; λ) are uniquely labeled by a set of states, which we shall simply denote as {|Ω}, that transform irreducibly under the grade zero compact subsupergroup OSp(2/2) × U (1) and are annihilated by the grade −1 generators B− , A− , Q− and S− in C− . In a unitary lowest weight (positive energy) representation of D(2, 1; λ) the spectrum of H is bounded from below. We shall refer to H as the (total) Hamiltonian and its eigenvalues as total energy. Each state in the set |Ω is a lowest (conformal) energy state of a positive energy irrep of SU(1, 1)K , since they are all annihilated by B− . The conditions B− |Ω = 0,
A− |Ω = 0,
Q− |Ω = 0,
S− |Ω = 0
(3.12)
imply that the states |Ω must be linear combinations of the tensor product states of the form (3.13) |F × |B × ψ0ω where the state |F in (3.13) is either the fermionic Fock vacuum |0F = |mt = 0; ma = −1/2F = |0, ↓
(3.14)
or one of the following SU(2)T doublet of states: α † |0F ≡ |mt = 1/2; ma = 0F = |↑, 0
(3.15)
β |0F ≡ |mt = −1/2; ma = 0F = |↓, 0
(3.16)
†
and the state |B in (3.13) is any one of the states a m1 · · · a mk bmk+1 · · · bm2s |0B
(3.17)
where |0B is the bosonic Fock vacuum annihilated by the bosonic oscillators am and bm (m = 1, 2, . . . , P ). For fixed n the states of the form (3.17) transform in the spin s representation of SU(2)S . They also form representations of SO∗ (2P ) generated by the bilinears U m n = a m an + bm bn , (am bn − an bm ) and (a m bn − a n bm ) and which commutes with D(2, 1; λ). Therefore as far as the SU(2) deformations of the minrep of D(2, 1; λ) are concerned we can restrict our analysis to P = 1. Then for P = 1 we simply have S+ = a † b,
S− = b † a
and 1 † a a − b† b 2 Then the states |B belong to the set S0 =
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k 2s−k |s, ms B ≡ a † b† |0B
119
(3.18)
where ms = k − s (k = 0, 1, . . . , 2s) and transform in spin s representation of SU(2)S : S 2 |s, ms B = s(s + 1)|s, ms B
(3.19)
S0 |s, ms B = ms |s, ms B
(3.20)
and lowering operator S− = The action of raising operator S+ = given as S+ |s, ms = k − sB = (k + 1)(2s − k)|s, k − s + 1B S− |s, ms = k − sB = k(2s − k + 1)|s, k − s − 1B a†b
(1/4 + 2gˆ 2 )
b† a
on this state is then (3.21) (3.22)
(4L2 + 1)
of on the above states determine the values of ω labeling The eigenvalues the eigenstates |ψ0ω of B0 annihilated by B− : 2 4L + 1 |mt = 0; ma = −1/2F × |s, ms B = λ2 (2s + 1)2 |0; −1/2F × |s, ms B (3.23) 2 2 4L + 1 |mt = ±1/2; ma = 0F × |s, ms B = λ(2s + 1) + 1 |±1/2; 0F × |s, ms B (3.24) The SU(2) subalgebra of SU(1|2) is the diagonal subalgebra SU(2)T of SU(2)S and SU(2)T . Therefore we shall work in a basis where T 2 and T0 are diagonalized and denote the simultaneous eigenstates of B0 , T 2 , T0 , A2 and A0 as1 |ω; t, mt ; a, ma
(3.25)
where ω is the eigenvalue of B0 . The set of states |Ω must be linear combinations of the tensor product states of the form |F × |B × |ψoω where the state |F could be either |0F = |0, ↓F or one of the SU(2)T doublet of states |↑, 0F or |↓, 0F . We will now study the unitary representations for these two cases. 3.2.1. |F = |0F For states with t = 0, we can use Eq. (3.7) to write 1 ± λ(2s + 1) (3.26) 2 where the sign of the square root is determined by the sign of λ and the range of λ is determined by the square integrability of the states and the positivity of gˆ 2 . This leads to the following restriction on λ 1 |λ| > (3.27) 2(2s + 1) Let us first consider the case s = 0 and with the positive square root taken in the above equaλ+1/2 , annihilated by all the generators tions. Then the lowest energy state |0, ↓F × |0B × |ψ0 −1 in C , is a singlet of the grade zero super algebra SU(1|2) since it is also annihilated by all the supersymmetry generators in C0 . ω=
1 We introduce this notation for the states because the tensor product states of the form |F × |B × |ψ ω are not always 0 definite eigenstates of T 2 and T0 but it is easier to understand the structure of supermultiplets in terms of these tensor product states. Thus we will use both notations for states and hope that the meaning would be clear from the context.
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States generated by action of grade +1 generators C+ on this lowest weight state |ψ0 λ+1/2 λ+1/2 B + |0, ↓F × |0B × ψ0 = |0, ↓F × |0B × ψ1 λ+1/2 λ+1/2 = |0, ↑F × |0B × ψ0 A+ |0, ↓F × |0B × ψ0 λ+1/2 λ+3/2 = |↑, 0F × |0B × ψ0 Q+ |0, ↓F × |0B × ψ0 λ+1/2 λ+3/2 S+ |0, ↓F × |0B × ψ0 = |↓, 0F × |0B × ψ0 (3.28) (λ+1/2)
form a supermultiplet transforming in the representation with super tableau λ+1/2
of SU(2|1).
λ+1/2
and S+ |0, ↓F × |0B × |ψ0 are both lowest The states Q+ |0, ↓F × |0B × |ψ0 weight vectors of SU(1, 1)K transforming as a doublet of SU(2)T . The state A+ |0, ↓F × |0B × λ+1/2 λ+1/2 is a lowest weight vector of SU(1, 1)K and together with |0, ↓F × |0B × |ψ0 |ψ0 + form a doublet of SU(2)A . The commutator of two susy generators in C satisfies λ+1/2 λ+1/2 [Q+ , S+ ]|0, ↓F × |0B × ψ0 ∝ α † β † B + |0F × |0B × ψ0 Hence one does not generate any new lowest weight vectors of SU(1, 1)K by further actions of grade +1 supersymmetry generators. Every positive energy unitary representation of the conformal group SO(d, 2) corresponds to a conformal field in d-dimensional Minkowskian spacetime. The eigenvalues of the SO(2) generator determine the conformal dimension of the field. In one dimension the positive energy unitary representations of SO(2, 1) are identified with conformal wave functions. We shall denote the conformal wave function associated with a positive energy unitary representation of SO(2, 1) (ω) with lowest weight vector |ψ0 as Ψ ω (x). The conformal wave functions transforming in the (t, a) representation of SU(2)T × SU(2)A will then be denoted as ω Ψ(t,a) (x)
Thus the unitary supermultiplet of D(2, 1; λ) with the lowest weight vector |0F × |0B × λ+1/2 decomposes as: |ψ0 (λ+1)/2
(λ+2)/2
Ψ(0,1/2) ⊕ Ψ(1/2,0)
(3.29)
This is simply the minimal unitary supermultiplet of D(2, 1; λ) and for λ = −1/2 coincides with the singleton supermultiplet of OSp(4|2, R) = D(2, 1; −1/2). Next we consider the representations for the case s = 0 with λ > 0. In this case the lowest energy states |0, ↓F × s, ms = (k − s) B × ψ0ω (3.30) where ω = 1/2 + λ(2s + 1), and k = 0, . . . , 2s, are not annihilated by all supersymmetry generators of SU(2|1): Q0 |0, ↓F × |s, k − sB × ψ0ω = 0 S0 |0, ↓F × |s, k − sB × ψ ω = 0 Q†0 |0, ↓F
× |s, k
0 − sB × |ψ0ω
= λ(2s − k)|↑, 0F × |s, k − sB × ψ0ω−1
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− λ (k + 1)(2s − k)|↓, 0F × |s, k − s + 1B × ψ0ω−1 S†0 |0, ↓F × |s, k − sB × ψ0ω = λk|↓, 0F × |s, k − sB × ψ0ω−1 − λ k(2s − k + 1)|↑, 0F × |s, k − s − 1B × ψ0ω−1 † † Q , S |0, ↓F × |s, k − sB × ψ ω = 2s|0, ↑F × |s, k − sB × ψ ω−2 0
0
0
(3.31)
0
Since the supersymmetry generators Q†0 and S†0 transform in the spin 1/2 representation of SU(2)T acting on the states with spin t = s one would expect to obtain states with both t = s ± 1/2. However setting k = 2s in the above formulas we find Q†0 |0, ↓F × |s, sB × ψ0ω = 0 S†0 |0, ↓F × |s, sB × ψ0ω √ (3.32) = 2λs|↓, 0F × |s, sB × ψ0ω−1 − λ 2s|↑, 0F × |s, s − 1B × ψ0ω−1 which implies that we only get states with t = s − 1/2. Therefore the lowest energy supermultiplets of SU(2|1) that uniquely determine the deformed minimal unitary supermultiplets of D(2, 1; λ) transform in the representation with the super tableau which 2s
decomposes under the even subgroup SU(2)T × U (1)J as = ,0 ⊕ 2s
2s
λ , 2
(3.33)
(2s−1)
By acting with grade +1 generators of the compact 3-grading on these states with t = s and t = s − 1/2 one obtains states with t = s ± 1/2 and t = s: B + |0, ↓F × |s, k − sB × ψ0ω = |0, ↓F × |s, k − sB × ψ1ω A+ |0, ↓F × |s, k − sB × ψ0ω = |0, ↑F × |s, k − sB × ψ0ω Q+ |0, ↓F × |s, k − sB × ψ0ω = |↑, 0F × |s, k − sB × ψ0ω+1 − λ(2s − k)|↑, 0F × |s, k − sB × ψ0ω−1 + λ (k + 1)(2s − k)|↓, 0F × |s, k − s + 1B × ψ0ω−1 S+ |0, ↓F × |s, k − sB × ψ0ω = |↓, 0F × |s, k − sB × ψ0ω+1 − λ(2s − k)|↓, 0F × |s, k − sB × ψ0ω−1 + λ k(2s − k + 1)|↑, 0F × |s, k − s − 1B × ψ ω−1 (3.34) 0
The commutator of two supersymmetry generators does not generate any new lowest weight vector of SU(1, 1)K : [Q+ , S+ ]|0, ↓F × |s, k − sB × ψ0ω (3.35) ∝ |0, ↑F × |s, k − sB × ψ1ω = α † β † B + |0, ↓F × |s, k − sB × ψ0ω Thus the complete supermultiplet is simply p
p+1/2
p+1
Ψ(s−1/2,0) ⊕ Ψ(s,1/2) ⊕ Ψ(s+1/2,0)
(3.36)
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Table 2 Decomposition of SU(2) deformed minimal unitary lowest energy supermultiplets of D(2, 1; λ) with respect to SU(2)T × SU(2)A × SU(1, 1)K . The conformal wave functions transforming in the (t, a) representation of SU(2)T × ω . The first column shows the super tableaux of the lowest energy SU(2)A with conformal energy ω are denoted as Ψ(t,a) SU(2|1) supermultiplet, the second column gives the eigenvalue of the U (1) generator H. The allowed range of λ in this case is λ > 1/(4s + 2). SU(2|1)
H
SU(1, 1)K × SU(2)T × SU(2)A
1
λ/2
Ψ(0,1/2) ⊕ Ψ(1/2,0)
λ
λ+1 λ Ψ(0,0) ⊕ Ψ(1/2,1/2) ⊕ Ψ(1,0)
3λ/2 . ..
Ψ(1/2,0) ⊕ Ψ(1,1/2) . ..
(2s + 1)λ/2
Ψ(s−1/2,0) ⊕ Ψ(s,1/2) ⊕ Ψ(s+1/2,0)
. ..
(λ+1)/2
(λ+2)/2
λ+1/2
3λ/2
p
(3λ+1)/2
(3λ/2)+1
⊕ Ψ(3/2,0)
p+1/2
p+1
p = (2s + 1)λ/2
2s
where p = λ(2s + 1)/2. We have summarized the deformed supermultiplets for lowest weight states with t = s and λ > 1/(4s + 2) in Table 2. So far we have considered the representations for λ > 0 when the lowest weight state has t = s. Now we take a look at the case when λ < 0 and ω is then given as ω=
1 − λ(2s + 1) 2
(3.37)
The action of grade 0 supersymmetry generators on these states produces states with t = s + 1/2. This is different from the case with λ > 0 where we obtained states with t = s − 1/2 by the action of grade 0 supersymmetry generators. Thus the lowest energy supermultiplets of SU(2|1) that uniquely determine the deformed minimal unitary supermultiplets of D(2, 1; λ) transform which decomposes under the in the representation with the super tableau 2s+1
even subgroup SU(2)T × U (1)J as
=
2s+1
2s+1
,0 ⊕
λ , 2
(3.38)
2s
By acting with the grade +1 supersymmetry generators on these states, we complete the D(2, 1; λ) supermultiplet given as: p
p+1/2
p+1
Ψ(s+1/2,0) ⊕ Ψ(s,1/2) ⊕ Ψ(s−1/2,0)
(3.39)
where p = |λ|(2s + 1)/2. We have summarized the deformed supermultiplets for lowest weight states with t = s and λ < −1/(4s + 2) in Table 3. F 3.2.2. |F = |↑,0 |↓,0F If we choose the doublet of states |F = |↑, 0F and |F = |↓, 0F as part of the lowest energy supermultiplet, the states |ω, t, mt , a, ma satisfying the conditions given in (3.12) will have
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Table 3 Decomposition of SU(2) deformed minimal unitary lowest energy supermultiplets of D(2, 1; λ) with respect to SU(2)T × SU(2)A × SU(1, 1)K . The conformal wave functions transforming in the (t, a) representation of SU(2)T × ω . The first column shows the super tableaux of the lowest energy SU(2)A with conformal energy ω are denoted as Ψ(t,a) SU(2|1) supermultiplet, the second column gives the eigenvalue of the U (1) generator H. The allowed range of λ in this case is λ < −1/(4s + 2). SU(2|1)
.. .
H
SU(1, 1)K × SU(2)T × SU(2)A
|λ|/2
Ψ(1/2,0) ⊕ Ψ(0,1/2)
|λ|
Ψ(1,0) ⊕ Ψ(1/2,1/2) ⊕ Ψ(0,0)
3|λ|/2 .. .
Ψ(3/2,0) ⊕ Ψ(1,1/2) .. .
(2s + 1)|λ|/2
Ψ(s+1/2,0) ⊕ Ψ(s,1/2) ⊕ Ψ(s−1/2,0) p = (2s + 1)|λ|/2
2s+1
|λ|/2 |λ|
3|λ|/2
p
(|λ|+1)/2
|λ|+1/2
(3|λ|+1)/2
p+1/2
|λ|+1 3|λ|/2+1
⊕ Ψ(1/2,0)
p+1
t = s ± 1/2 and can be written as linear combinations of |↑(↓), 0 and |s, ms with appropriate Clebsch–Gordon coefficients for adding spin 1/2 with spin s. Square integrability of states and positivity of gˆ 2 imposes restrictions on range of λ and (3.12) determines the value of ω. For states with t = s + 1/2, we have ω = −1/2 − λ(2s + 1) 3 with λ < − 2(2s+1) . On the other hand for t = s − 1/2, we have ω = −1/2 + λ(2s + 1) with 3 λ > 2(2s+1) . As was done in previous section, we now proceed to study the supermultiplets but the action of supersymmetry generators in C0 and C+ on states with t = s + 1/2. The lowest energy supermultiplet for t = s + 1/2 corresponds to the following SU(2|1) supertableau (3.40) = ,1 ⊕ , 2s+1
2s+1
2s
and the resulting unitary supermultiplet of D(2, 1; λ) decomposes as p
p+1/2
p+1
Ψ(s+1/2,0) ⊕ Ψ(s,1/2) ⊕ Ψs−1/2,0
(3.41)
where p = (2s + 1)|λ|/2 with λ < 0. We have summarized the deformed supermultiplets for lowest weight states with t = s + 1/2 in Table 4. Note that these occur only for λ < −3/(4s + 2). Next we look at the states with t = s − 1/2. The lowest energy super multiplet for t = s − 1/2 corresponds to the following SU(2|1) supertableau = ,1 ⊕ , 2s 2s 2s (3.42) and leads to the supermultiplet p
p+1/2
p+1
Ψ(s−1/2,0) ⊕ Ψ(s,1/2) ⊕ Ψ(s+1/2,0)
(3.43)
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Table 4 Decomposition of SU(2) deformed minimal unitary lowest energy supermultiplets of D(2, 1; λ) with respect to SU(2)T × SU(2)A × SU(1, 1)K . The conformal wave functions transforming in the (t, a) representation of SU(2)T × ω . The first column shows the super tableaux of the lowest energy SU(2)A with conformal energy ω are denoted as Ψ(t,a) SU(2|1) supermultiplet, the second column gives the eigenvalue of the U (1) generator H. The allowed range of λ in this case is λ < −3/(4s + 2). SU(2|1)
.. .
H
SU(1, 1)K × SU(2)T × SU(2)A
|λ|/2
Ψ(1/2,0) ⊕ Ψ(0,1/2)
|λ|
Ψ(1,0) ⊕ Ψ(1/2,1/2) ⊕ Ψ(0,0)
3|λ|/2 .. .
Ψ(3/2,0) ⊕ Ψ(1,1/2) .. .
(2s + 1)|λ|/2
Ψ(s+1/2,0) ⊕ Ψ(s,1/2) ⊕ Ψ(s−1/2,0) p = (2s + 1)|λ|/2
2s+1
|λ|/2 |λ|
3|λ|/2
p
(|λ|+1)/2
|λ|+1/2
(3|λ|+1)/2
|λ|+1 3|λ|/2+1
⊕ Ψ(1/2,0)
p+1/2
p+1
Table 5 Decomposition of SU(2) deformed minimal unitary lowest energy supermultiplets of D(2, 1; λ) with respect to SU(2)T × SU(2)A × SU(1, 1)K . The conformal wave functions transforming in the (t, a) representation of SU(2)T × ω . The first column shows the super tableaux of the lowest energy SU(2)A with conformal energy ω are denoted as Ψ(t,a) SU(2|1) supermultiplet, the second column gives the eigenvalue of the U (1) generator H. The allowed range of λ in this case is λ > 3/(4s + 2). SU(2|1)
.. .
H
SU(1, 1)K × SU(2)T × SU(2)A
λ
λ+1 λ Ψ(0,0) ⊕ Ψ(1/2,1/2) ⊕ Ψ(1,0)
3λ/2
Ψ(1/2,0) ⊕ Ψ(1,1/2)
2λ
2λ+1 2λ ⊕ Ψ Ψ(1,0) (3/2,1/2) ⊕ Ψ(2,0)
.. .
.. .
(2s + 1)λ/2
λ+1 Ψ(s−1/2,0) ⊕ Ψ(s,1/2) ⊕ Ψ(s+1/2) p = (2s + 1)λ/2
λ+1/2
3λ/2
(3λ+1)/2
3λ+1 ⊕ Ψ(3/2,0)
2λ+1/2
p
p+1/2
2s
where p = (2s + 1)λ/2 with λ > 0. We have summarized the deformed supermultiplets for lowest weight states with t = s − 1/2 in Table 5. Note that these occur only for s > 1/2 and λ > 3/(4s + 2). We note the similarities of Table 2 with Table 5, and that of Table 3 with Table 4. This shows that the supermultiplets obtained for lowest weight states with t = s and t = s − 1/2 and λ > 0 are the same and the supermultiplets for lowest weight states with t = s and t = s + 1/2 and λ < 0 are same. The difference between these two types of supermultiplets is that the SU(1, 1)K spin (labeled as p) gets interchanged between states with t = s + 1/2 and t = s − 1/2 as we change the sign of λ.
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4. SU(2) deformations of the minimal unitary representation of D(2, 1; λ) using both bosons and fermions and OSp(2n∗ |2m) superalgebras Above we obtained unitary supermultiplets of D(2, 1; λ) which are SU(2) deformations of the minimal unitary representation. This was achieved by introducing bosonic oscillators an and bn and extending the SU(2)T generators to the generators of the diagonal subgroup of SU(2)T and SU(2)S realized as bilinears of the bosonic oscillators 1 S0 = (Na − Nb ) 2 ∗ As stated above, the noncompact group SO (2n) generated by the bilinears S+ = a n bn ,
S − = b n an ,
Amn = am bn − an bm ,
Amn = a m bn − a n bm ,
Unm = a m an + bn bm
(4.1)
(4.2)
commutes with the generators of D(2, 1; λ). One can similarly obtain SU(2) deformations of the minimal unitary supermultiplet of D(2, 1; λ) by introducing fermionic oscillators ρr and σs (r, s, . . . = 1, . . . , n) satisfying ρr , ρ s = σr , σ s = δrs {ρr , ρs } = {ρr , σs }{σr , σs } = 0 (4.3) and extend the generators of SU(2)T to the generators of the diagonal subgroup of SU(2)T and SU(2)F generated by 1 r ρ ρr − σ r σr 2 In this case the compact USp(2n) generated by the fermion bilinears F+ = ρ r σ r ,
F− = σ r ρ r ,
Srs = ρr σs + ρs σr ,
F0 =
S rs = σ r ρ s + σ s ρ r ,
Ssr = ρ r ρs − σs σ r
(4.4)
(4.5)
commute with the generators of D(2, 1; λ). One can deform the minimal unitary representation of D(2, 1; λ) using fermions and bosons simultaneously. This is achieved by replacing the SU(2)T generators by the diagonal SU(2)D generators of SU(2)T × SU(2)S × SU(2)F , which we shall denote as U+ , U− and U0 U + = T + + S + + F+ ,
U− = T − + S − + F − ,
U 0 = T 0 + S 0 + F0
(4.6)
and substituting the quadratic Casimir of SU(2)T in the Ansatz for K− with the quadratic Casimir of SU(2)D . Remarkably, in this case the resulting generators of D(2, 1; λ) commute with the generators of the noncompact superalgebra OSp(2n∗ |2m) generated by the generators of SO∗ (2n) and USp(2m) given above and the supersymmetry generators [29]: Πmr = am σr − bm ρr , r Σm
= a m ρ + bm σ , r
r
Π¯ mr = (Πmr )† = a m σ r − bm ρ r r † Σ¯ rm = Σm = a m ρr + b m σ r
The (anti-)commutation relations for the OSp(2n∗ |2m) algebra are given below: i k r t Aj , Al = δjk Ail − δli Akj , Ss , Su = δst Sur − δur Sst Aij , Akl = δjk Ail − δik Aj l , Srs , Sut = δst Sru + δrt Sus ij k rs t j A , Al = δli Aj k − δl Aik , S , Su = −δus S rt − δur S ts Aij , Akl = −δjk Ali + δjl Aki − δil Akj + δik Alj
(4.7) (4.8)
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Srs , S tu = −δst Sru − δrt Ssu − δsu Srt − δru Sst r n s n r Πmr , Π¯ ns = δrs Anm − δm Sr , Σm , Σ¯ sn = δsr Anm + δm Ss s s n n Πmr , Σn = δr Amn , Πmr , Σ¯ s = −δm Srs m m r An , Πkr = −δkm Πnr , An , Σk = −δkm Σnr mn mn r A , Πkr = −δkm Σ¯ rn + δkn Σ¯ rm , A , Σk = −δkn Π¯ rm + δkm Π¯ rn [Amn , Πkr ] = 0, Amn , Σkr = 0 r r t r Ss , Πmt = −δtr Πms , Ss , Σm = δst Σm rs r s t S , Πmt = δts Σm =0 + δtr Σm , S rs , Σm t t [Srs , Πmt ] = 0, Srs , Σm = −δr Πms − δst Πmr 5. SU(2) deformed minimal unitary supermultiplets of D(2, 1; α) and N = 4 superconformal mechanics 5.1. N = 4 superconformal quantum mechanical models A new class of N = 4 supersymmetric Calogero-type models have been studied by various authors in recent years [4,10] which are invariant under the superconformal algebra D(2, 1; α). The construction of D(2, 1; α) mechanics and quantization was performed in [10]. In this section we will review that construction following [10]. The on shell component action was shown to take the form [10] (5.1) S = S b + Sf 2 k 2 α (¯zk z ) i − A z¯ k zk − c (5.2) Sb = dt x˙ x˙ + z¯ k z˙ k − z˙¯ k zk − 2 4x 2 ψ i ψ¯ k z(i z¯ k) 2 ψ i ψ¯ k ψ(i ψ¯ k) + (1 + 2α) dt Sf = −i dt ψ¯ k ψ˙ k − ψ˙¯ k ψ k + 2α dt 2 3 x x2 (5.3) Here x, zi and ψ j (i, j = 1, 2) are d = 1 bosonic and fermionic “fields”, respectively. The fields zi form a complex doublet of the R-symmetry group SU(2). The last term in (5.2) represents the constraint z¯ k zk = c
(5.4)
and A is the Lagrange multiplier. Upon quantization of the action given in (5.1), the dynamical variables were promoted to quantum mechanical operators with following commutators: [X, P ] = i,
Z i , Z¯ j = δji ,
1 Ψ i , Ψ¯ j = − δji 2
(i, j = 1, 2)
(5.5)
As the action is invariant under the group D(2, 1; α), the corresponding symmetry generators can be obtained by the Noether procedure. The results as given in [10] are:
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Z (i Z¯ k) Ψk
Ψk Ψ k Ψ¯ i + i(1 + 2α) X X k kΨ ¯ ¯ ¯ ¯ k Ψi Z
Ψ Z Ψ ¯ i = P Ψ¯ i − 2iα (i k) + i(1 + 2α) Q X X ¯i S¯ i = −2X Ψ¯ i + t Q Si = −2XΨ i + tQi ,
Qi = P Ψ i + 2iα
(5.6) (5.7) (5.8)
(Z¯ k Z k )2 + 2Z¯ k Z k Z (i Z¯ k) Ψ(i Ψ¯ k) 1 − 2α H = P 2 + α2 4 4X 2 X2 i k 2 ¯ ¯
Ψi Ψ Ψ Ψk (1 + 2α) − (1 + 2α) + 2X 2 16X 2 1 1 K = X 2 − t {X, P } + t 2 H, D = − {X, P } + tH 2 4 Jik = i Z (i Z¯ k) + 2Ψ (i Ψ¯ k)
I2 2 = i Ψ¯ k Ψ¯ k ,
i
(5.9) (5.10) k
Ψk , Ψ¯ 2 where t is time variable and the symbol · · · denotes Weyl ordering: I1 1 = −iΨk Ψ k ,
I1 2 = −
127
(5.11)
k 1 1 Ψk Ψ k Ψ¯ i = Ψk Ψ k Ψ¯ i + Ψ i , Ψ¯ Ψ¯ k Ψi = Ψ¯ k Ψ¯ k Ψi + Ψ¯ i 2 2 1 1 1 i ¯k ¯ i ¯k ¯ i ¯k ¯ i Ψi Ψ Ψ Ψk = Ψi Ψ , Ψ Ψk − = Ψi Ψ Ψ Ψk − Ψi Ψ¯ + 2 4 4 i + i + ¯ ¯ and Qi = −(Q ) , Si = −(S ) . In the above set of generators Qi and Si are supertranslation and superconformal boost generators respectively. The generators H, K and D are the Hamiltonian, special conformal transformations and dilatation generators respectively and they form an su(1, 1) algebra. The remaining generators Jik and Ii k are the generators of two su(2) algebras.
5.2. Mapping between the harmonic superspace generators of N = 4 superconformal mechanics and generators of deformed minimal unitary representations of D(2, 1; λ) A precise correspondence between the Killing potentials of the isometry groups G of N = 2 sigma models that couple to 4d supergravity in harmonic superspace and the generators of the minimal unitary representations of G was first shown in [24]. It was then suggested that the correspondence could be made concrete and precise by reducing the four-dimensional N = 2 sigma models to one dimension with eight supercharges and subsequently quantize them to get supersymmetric quantum mechanical models [24,25]. The results presented in this paper and the results of [10] on D(2, 1; α) superconformal quantum mechanics reviewed in previous section provide an opportunity to test this proposal. Remarkably one finds that the generators of quantized N = 4 superconformal mechanics in harmonic superspace go over to the generators of minimal unitary realization of D(2, 1; λ) deformed by a pair of bosonic oscillators if we make the simple substitutions listed in Table 6 and setting α = λ. Noting that the SU(2) indices i, j, . . . of [10] are raised and lowered by the Levi-Civita tensor ij (with 12 = 21 = 1) we find a oneto-one correspondence between the symmetry generators of D(2, 1; α) superconformal quantum mechanics and the generators of the minimal unitary representations of D(2, 1; α) deformed by a pair of bosons, which we present in Table 6. We also find that the quadratic Casimir obtained in Eq. (4.26) of [10] is the same as the one obtained by our construction given in (2.34) for μ = 4.
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Table 6 Mapping of the operators, quantum numbers and symmetry generators between N = 4 superconformal mechanics in harmonic superspace (denoted as HSS) and the minimal unitary representation of D(2, 1; λ) deformed by a pair of bosons (denoted as Minrep). HSS
Minrep
Operators: (ψ 1 , ψ 2 ) (ψ¯1 , ψ¯2 ) (Z 1 , Z 2 ) (Z¯ 1 , Z¯ 2 )
Minrep
Quantum numbers: − √i (α † , β † )
2 − √i (α, β) 2 −i(a † , b† )
i(a, b)
Odd generators: − √i (Q1 , Q2 ) 2 ¯ 2, Q ¯ 2) − √i (Q 2 i 1 2 − √ (S , S ) 2 − √i (S1 , S2 ) 2
HSS r0
ω
j
t
i c
a 2s
Even generators:
1 ,
(Q S1)
i(I 1 1 , −I 2 2 , I 1 2 )
(A+ , A− , A0 )
1 ,
(Q S1 )
i(−J 11 , J 22 , J 12 )
(T+ , T− , T0 )
(Q1 , S 1 )
(2H, 12 K)
(K+ , K− )
(Q1 , S1 )
−2D
The spectrum of N = 4 superconformal quantum mechanics was also studied in [10]. To compare its quantum spectrum with the deformed minimal unitary supermultiplets we need only to compare the SU(1, 1), SU(2)R , SU(2)L decomposition of the spectrum of [10] with the SU(1, 1)K , SU(2)T , SU(2)A decomposition of the minimal unitary supermultiplets constructed in this paper. We find that the superfield content of the quantum spectrum described in Table 2 of [10] is exactly the same as supermultiplets described in Tables 2, 3, 4 and 5 above. 6. Conclusions Above we constructed the SU(2) deformed minimal unitary supermultiplets of D(2, 1; λ) using quasiconformal methods as was done for the 4d and 6d superconformal algebras in [1,2]. We showed that for deformations obtained by a pair of bosons there exists a precise mapping to the generators and the quantum spectra of N = 4 superconformal mechanical models studied recently. This raises the question what kind of N = 4 superconformal quantum mechanical models correspond to more general deformations of the minimal unitary representations involving an arbitrary numbers of pairs of bosons and/or pairs of fermions as formulated above. We expect some of these more general deformations to describe the spectra of supersymmetric quantum mechanical models with quaternionic Kähler target spaces. One would also like to understand the precise connection between the above results and the N = 4 supersymmetric gauged WZW models in d = 2 studied in [30,31] that extends the results of [32] on N = 2 supersymmetric gauged WZW models. These gauged WZW models correspond to realization over spaces of the form Gc × SU(2) × U (1) H × SU(2) Gc is a compact quaternionic symmetric space. On the other hand the quaternionic where H ×SU(2) Kähler manifolds that couple to 4d, N = 2 supergravity are noncompact [24]. We hope to address these problems in a future study.
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