The ${\cal N}=4$ Supersymmetric Linear $W_{\infty}[λ]$ Algebra

Abstract

From the recently known N = 2 supersymmetric linear W K,K ∞ [λ] algebra where K is the dimension of fundamental (or antifundamental) representation of bifundamental β γ and b c ghost system, we determine its N = 4 supersymmetric enhancement at K = 2. We construct the N = 4 stress energy tensor, the first N = 4 multiplet and their operator product expansions (OPEs) in terms of above bifundamentals. We show that the OPEs between the first N = 4 multiplet and itself are the same as the corresponding ones in the N = 4 coset SU (N +2) SU (N ) model under the large (N, k) 't Hooft-like limit with fixed λ co ≡ (N +1) (k+N +2) , up to two central terms. The two parameters are related to each other λ = 1 2 λ co . We also provide other OPEs by considering the second, the third and the fourth N = 4 multiplets in the N = 4 supersymmetric linear W ∞ [λ] algebra. 3.5.2 The OPE between the weight-3 2 operators and the weight-1 operator . . 3.5.3 The OPE between the weight-2 operators and the weight-1 operator . . 3.5.4 The OPE between the weight-5 2 operators and the weight-1 operator . .

The Structure of the ${\cal N}=4$ Supersymmetric Linear $W_{\infty}[λ]$ Algebra

“…For the weights h = 1, 2, 3, 4, we can observe that the corresponding numerical values appearing in the relative coefficients of (2.7) can be seen from the ones in [9]. The normalization in (2.7) is different from the one in [9] where there appear the additional numerical factors 16, 8, 12, 24 for the weights h = 1, 2, 3, 4 respectively. At the moment, it is not easy to figure out the exact h dependence from these values.…”

“…The third components contain the various bosonic currents and their relative coefficients are equal to each other for any weights h = 1, 2, 3, 4 in the analysis of [9]. Then we obtain the following results by replacing them with the corresponding expressions for arbitrary weight h with the above overall factors in previous section…”

“…From the observation of [9] for the weights h = 1, 2, 3, 4, the second components contain the various fermionic currents and their relative coefficients are common for any weights h = 1, 2, 3, 4. This implies that it is natural to take these relative coefficients for any h and the question is how we obtain the overall numerical factor.…”

“…Recently [9], by considering the particular number K = 2 of free fields above, the N = 4 supersymmetric linear W ∞ [λ] algebra is studied. For low weights, the explicit OPEs between the currents of N = 4 multiplet are obtained.…”

“…In this paper, we continue to study the structure of the N = 4 supersymmetric linear W ∞ [λ] algebra found in [9]. We try to determine the N = 4 multiplet for any weight h in terms of above free field operators.…”