ЯДЕРНАЯ ФИЗИКА, 2010, том 7S, № S, с. 5в5-5в9
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
MINIMAL REPRESENTATIONS OF SUPERSYMMETRY AND 1D N-EXTENDED a MODELS
©2010 F. Toppan*
Centra Brasileiro de Pesquisa Fisicas, Rio de Janeiro, Brazil Received April 17,2009
We discuss the minimal representations of the 1D N-Extended Supersymmetry algebra (the Z2-graded symmetry algebra of the Supersymmetric Quantum Mechanics) linearly realized on a finite number of fields depending on a real parameter t, the time. Their knowledge allows to construct one-dimensional sigma-models with extended off-shell supersymmetries without using superfields.
1. INTRODUCTION
The superalgebra of the Supersymmetric Quantum Mechanics (1^ N-Extended Supersymmetry Algebra) [1] is a Z2-graded algebra expressed by N odd generators Qi (i = 1,...,N) and a single even generator H (the Hamiltonian). It is defined by the (anti)commutation relations
{Qi, Qj} = 25ijH, [Qi,H] = 0. (1)
The structure of its minimal linear representations realized on a finite number of fields depending on a single real parameter (t, the time) has been substantially elucidated in recent years. Several results have been obtained [2—7]. They are based on the Atiyah—Bott— Shapiro [8] classification of the irreducible Clifford algebras. In this paper we discuss several results on the classification of the (1) minimal representations and their use in constructing one-dimensional sigma-models with extended number of off-shell supersymmetries.
The problem we addressing can be stated as follows: to construct and classify, for any given integer N, the linear representations of (1) acting on a finite, minimal, number of fields, even and odd (bosonic and fermionic), depending on t. The generator H has to be represented by a time-derivative, while the Qi's generators must be realized by finite-dimensional matrices whose entries are either c numbers or timederivatives up to a certain power. The representation space we are considering is infinite-dimensional, being given by the set of fundamental fields and their time-derivatives of any order. In the physical literature these representations are called "finite" since they are obtained by a finite number of fundamental fields (the situation parallels here the representation theory
of chiral algebras [9], given by the generating set of primary fields and their descendants; the timederivatives of the fundamental fields play, for (1) representations, the role of the descendants in chiral algebra representations). For the same reason, the notion of "minimal representations" is expressed, in the physical literature, as "irreducible representations".
The program of classifying the (1) minimal representations starts with [2], with the recognition that formulating an eigenvalue problem for the Hamilto-nian H (for an eigenvalue different from zero) reduces the Qi's anticommutators to, up to normalization, the basic relation for Euclidean Clifford algebra generators. The [2] main result can be stated as follows. The minimal representations of (1), for a given N, are obtained by applying a dressing transformation to a fundamental representation (nowadays called in the literature the "root multiplet"), with equal number of bosonic and fermionic fields. The root multiplet is specified by an associated Euclidean Clifford algebra. As a main corollary, the total number n of bosonic fundamental fields entering a minimal representation equals the total number of fermionic fundamental fields and is expressed, for any given N, by the following relation [2]:
(2)
N = 8l + m, n = 2 G(m), where l = 0,1,2,... and m = 1—8.
G(m) appearing in (2) is the Radon-Hurwitz function
E-mail: toppan@sbpf.br
m 12345678 G(m) 1 2 4 4 8 8 8 8 Note the mod 8 Bott's periodicity.
An integral Z-grading, compatible with the Z2-grading of the superalgebra, can be assigned to the fundamental fields and their time-derivatives. In the physical literature, the grading is referred as "massdimension". The integral grading will be denoted by z. For convenience, the mass-dimension d will be expressed as d = z/2. The Hamiltonian H has mass-dimension d = 1 (its fermionic roots, the Qi's operators, have mass-dimension d = 1/2). Bosonic (fermionic) fields have integer (respectively, halfinteger) mass-dimension. Each linear representation admitting a finite number of fundamental fields is characterized by its "fields content", i.e., the set of integers (nl,n2,..., nl) specifying the number ni of fundamental fields of dimension di (di = dl + ( — — 1)/2, with dl an arbitrary constant) entering the representation. Physically, the nl fields of highest dimension are the auxiliary fields which transform as a time-derivative under any supersymmetry generator. The maximal value l (corresponding to the maximal dimensionality di) is defined to be the length of the representation (a root representation has length l = 2). Either nl,n3,... correspond to the bosonic fields (therefore, n2,n4,... specify the fermionic fields) or vice versa. In both cases the equality nl + n3 + ... = n2 + n4 + ... = n is guaranteed.
The representation theory does not discriminate the overall bosonic or fermionic nature of the representation.
According to [2], if (nl,n2,...,nl) specifies the fields content of an irreducible representation, (nl,nl-l,...,nl) specifies the fields content of a dual irreducible representation. Representations such that nl = nl,n2 = nl-l,... are called "self-dual representations". In [3] it was shown how to extract from the associated Clifford algebras the admissible fields content of the (1) linear finite irreducible representations. We discuss these results in the next section.
2. SUPERSYMMETRIC QUANTUM MECHANICS AND CLIFFORD ALGEBRAS
In this section we give a more detailed account of the connection between representations of the Supersymmetric Quantum Mechanics and Clifford algebras.
According to [2] the length-2 minimal representations of the (1) supersymmetry algebra are uniquely determined by a representation of an associated Clifford algebra. The connection goes as follows. The supersymmetry generators acting on a length-2 irreducible multiplet can be expressed as
ài
à H 0
(4)
where ai and ai are matrices entering a Weyl type (i.e., block antidiagonal) irreducible representation of the Clifford algebra relation
0 ài ài 0
, {ri, r } = 2n
ij
(5)
The Qi's in (4) are supermatrices with vanishing bosonic and nonvanishing fermionic blocks, acting on a multiplet m (thought of as a column vector) which can be either bosonic or fermionic (we conventionally consider a length-2 irreducible multiplet as bosonic if its upper-half part of component fields is bosonic and its lower half is fermionic; it is fermionic in the converse case). The connection between Clifford algebra irreps of the Weyl type and minimal representations with minimal length of the N-extended one-dimensional supersymmetry is such that D, the dimensionality of the (Euclidean, in the present case) space—time of the Clifford algebra (5) coincides with the number N of the extended supersymmetries, according to
of space—time dim. (Weyl—Clifford) & § of extended su.sies (in 1-dim.) D = N
(6)
The matrix size of the associated Clifford algebra (equal to 2n, with n given in (2)) corresponds to the number of (bosonic plus fermionic) fields entering the one-dimensional N-extended supersymmetry irrep.
The classification of Weyl-type Clifford irreps, furnished in [2], can be easily recovered from the well-
known classification of Clifford irreps, given in [8] (see also [10] and [11]).
The (4) Qi,s matrices realizing the N-extended supersymmetry algebra (1) on length-2 minimal representations have entries which are either c numbers or are proportional to the Hamiltonian H. Minimal
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representations of higher length (l > 3) are systematically produced [2] through repeated applications of the dressing transformations
Qi » Q(k) = 5(k)QiS(k)
-i
realized by diagonal matrices S(k)'s (k = 1, with entries sj given by
j = 5ij (1 - 5jk + 5jkH). Some remarks are in order [2].
(7)
,2n)
(8)
(i) The dressed supersymmetry operators Q'i (for a given set of dressing transformations) have entries which are integral powers of H. A subclass of the Qi's dressed operators is given by the local dressed operators, whose entries are non-negative integral powers of H (their entries have no 1/H poles). A local representation (minimal representations fall into this class) of an extended supersymmetry is realized by local dressed operators. The number of the extension, given by N' (N' < N), corresponds to the number of local dressed operators.
(ii) The local dressed representation is not necessarily a minimal representation. Since the total number of fields (n bosons and n fermions) is unchanged under dressing, the local dressed representation is a minimal representation if n and N' satisfy the (2) requirement (with N' in place of N).
(iii) The dressing changes the dimension of the fields of the original multiplet m. Under the S(k) dressing transformation (7), m ^ S(k)m, all fields entering m are unchanged apart from the kth one (denoted, e.g., as pk and mapped to pk). Its dimension is changed from [k] ^ [k] + 1. This is why the dressing changes the length of a minimal representation. As an example, if the original length-2 multiplet m is a bosonic multiplet with d 0-mass-dimension bosonic fields and d ^-mass-dimension fermionic fields (in the following such a multiplet will be denoted as (xi;^j) = (d,d)s=0, for i,j = 1,... , d), then S(k)m, for k < d, corresponds to a length-3 multiplet with d — 1 bosonic fields of 0-mass-dimension, d fermionic fields of ^-mass-dimension, and a single bosonic field of mass-dimension 1 (in the following we employ the notation (d — 1, d, 1)s=0 for such a multiplet of fields).
When looking purely at the representation properties of a given multiplet the assignment of an overall mass
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