Embedding of bases: from the M(2,2k+1) to the M(3,4k+2-delta) models

A new quasi-particle basis of states is presented for all the irreducible modules of the M(3,p) models. It is formulated in terms of a combination of Virasoro modes and the modes of the field phi_{2,1}. This leads to a fermionic expression for particular combinations of irreducible M(3,p) characters, which turns out to be identical with the previously known formula. Quite remarkably, this new quasi-particle basis embodies a sort of embedding, at the level of bases, of the minimal models M(2,2k+1) into the M(3,4k+2-delta) ones, with 0 \leq delta \leq 3.

The M(3, p) models have been reformulated recently [1] in terms of the extended algebra defined by the OPEs φ(z)φ(w) = 1 (z − w) 2h I + (z − w) 2 2h c T (w) + · · · S , and S = (−1) pF where F counts the number of φ modes. The highest-weight states |σ ℓ are completely characterized by an integer ℓ such that 0 ≤ ℓ ≤ (p − 2)/2 and satisfy The highest-weight modules are described by the successive action of the lowering φ-modes subject to exclusion-type constraints. In the N -particle sector, with strings of lowering modes written in the form [1] 1 (see also [5,6] for ℓ = 0): these constraints are with s N −2i ∈ Z + h + ℓ 2 and The complete module of |σ ℓ is obtained by summing over all these states (4) satisfying (5) and all values of N . The enumeration of these states lead to the standard form of the fermionic character for the sum of the two Virasoro modules |φ 1,ℓ+1 and |φ 1,p−ℓ−1 of the M(3, p) models [2,3,4] when 0 ≤ ℓ ≤ [p/3] (the closed form expression of the generating functions has not been obtained for the remaining cases).
Here we display a new form of the basis of states of the M(3, p) models, still viewed form the point of view of the extended algebra (1). This basis is written in terms of combined sequences of Virasoro and φ modes, as The module over |σ ℓ is again the direct sum of the two Virasoro modules |φ 1,ℓ+1 and |φ 1,p−ℓ−1 . In order to specify the constraints on the mode indices, we first define two integers κ and δ through the decomposition of p as The conditions take the form 1 In [1], the conditions are formulated in terms of the indices n i defined by: The relation between s i and n i is These are supplemented by the boundary conditions: (10) The n i are always integers but the range of the indices m i is defined as follows. Given that h = −δ/4 mod 1, we have which are actually equivalent to the conditions on the s i in (6).
The conditions (9) indicates that the Virasoro modes are ordered and further subject to a difference 2 condition at distance κ − 1. The φ modes are also ordered, being in fact all distinct if δ = 0. In addition, they are subject to a difference κ condition at distance 2 (which is almost the 'dual' of the conditions on the n i ).
The different inequalities in the boundary conditions (10) have the following interpretation. At first, m M ≥ h− ℓ 2 is simply the highest-weight condition (3). The condition on m M−1 partially specifies the different descendant states according to the value of ℓ. It is analogous to the third condition in (5). The inequality n N −ℓ ≥ 2 means that the maximal number of L −1 modes that can appear in the descendants of the |σ ℓ module is ℓ. Actually, this number is also bounded by the difference 2 condition at distance κ − 1, so that this maximal number is actually min (ℓ, κ − 1).
The most interesting condition is the remaining one in (10), which, for the vacuum module (ℓ = 0) reads n N ≥ M + 2. For M = 0, this takes into account the Virasoro highest-weight condition on the vacuum. But if there are M φ-modes already acting on the highest-weight state, the condition implies that all the modes L −n with 2 ≤ n ≤ M + 1 have to be excluded. This can be interpreted as a sort of repulsion between the T and φ 'quasi-particles'. For any other module (ℓ = 0), the bound on n N reads n N ≥ M + 1.
If κ = 1, the difference condition on the Virasoro modes becomes n i ≥ n i + 2, which is impossible. This means that when κ = 1, there can be no Virasoro modes; the basis is solely described by the φ modes. Let us check that it reduces then to basis (5). When κ = 1, p = 6 − δ, but in order for p to be relatively prime with 3, we require δ = 1 or 2. For δ = 2, so that p = 4, the conditions (9) reduce to m i ≥ m i+1 + 1, in agreement with (5) (note that the condition m i ≥ m i+2 + 1 is thus automatically satisfied). In that case h = 1/2 and this indeed describes the free-fermionic basis of the Ising model. For p = 5, these conditions take the form m i ≥ m i+1 + 1/2, which again implies the condition at distance 2. This agrees with (5) and the known quasi-particle basis formulated in terms of the graded parafermion of dimension h = 3/4 (cf. [8], end of section 5, and [1] section 1.4).
Let us stress a remarkable feature of the new basis. The conditions (9) for the Virasoro modes are precisely the one pertaining to the quasi-particle basis of the M(2, p) models, with p = 2κ + 1 [7]. Moreover, the boundary condition on n N −ℓ , which specifies the maximal number of L −1 factors, thereby distinguishing the different modules, is also the very one that occurs in these models. Therefore, in absence of φ modes, the above M(3, 4κ + 2 − δ) basis reduces to the M(2, 2κ + 1) one. It thus appears that the above basis describes a sort of embedding of the M(2, 2κ + 1) models within the M(3, 4κ + 2 − δ) ones.
Let us consider the expression for the characters associated to this new basis. Constructing these characters amounts to finding the generating function for the composition of the two partitions (n 1 , · · · , n N ) and (m 1 , · · · , m M ) satisfying (9) and (10). This is essentially built from the composition of two corresponding generating functions, both of which being known (up to a restriction on ℓ to be specified).
The generating functions for partitions (n 1 , · · · , n N ) is obtained as follows. First, delete M from each parts n i and introduce q N M to correct for this. The resulting restricted partitions are enumerated by the Andrews multiple-sum [9,10]: where N i = s i + · · · + s κ−1 .
Similarly, the generating function for partitions (m 1 , · · · , m M ) can be extracted from [5] up to simple modifications. The latter generating function enumerates the partitions (λ 1 , · · · , λ M ) satisfying for 0 ≤l ≤ [(2r + 5)/3] where [x] stands for the integer part of x (the boundary condition of λ M−1 induces a correcting term in the generating function that has been introduced in [1].) To connect the two problems, let us redefine m i as: The conditions (9)-(10) become then We thus recover the counting problem of [5,1] but with 2r → κ − δ andl − 1 → κ − ℓ. (Note that the generating function of [5] does not hold for those cases where 2r + 5 is divisible by 3. But this is not restrictive since if κ − δ + 5 were a multiple of 3, say 3n, then p would be 12n + 3δ − 18, which is divisible by 3 and that would not corresponds to a minimal model.) The correcting factor q M(M−1) δ 4 +M(h− ℓ 2 −1) will keep track of the shifted staircase that must be added to adjust the weight when passing from the partitions (λ 1 · · · , λ M ) to our original partitions (m 1 , · · · , m M ). From [5,1], we see that the generating function is written as a g-multiple sum, where g is given by and it takes the form (with the understanding that tBt = g i,j=1 t i B ij t j and Ct = g i=1 C i t i ), and the g × g symmetric matrix B reads · · · · · · · · · · · · · · · κ − δ κ − δ + 1 · · · κ − δ + g − 2 κ−δ while the entries of the row matrix C are C j = −κ + δ + j + 1 + max (0, ℓ − κ) for j < g and C g = −g + 2 . (24) We stress that this result holds only for 0 ≤ ℓ ≤ κ + g − 1 (and this range is identical to the previous one 0 ≤ ℓ ≤ [p/3] since 3κ + 3g − 3 = p). For the remaining values of ℓ, that is, for κ + g ≤ ℓ ≤ p/2 − 1, we stress that although the generating function has not been found in closed form, the validity of the basis has been verified to high order in q.