A deformed supersymmetric $$w_{1+\infty }$$ symmetry in the celestial conformal field theory

Abstract

By using the K-free complex bosons and the K-free complex fermions, we construct the $$\mathcal {N}\,{=}\,2$$ N = 2 supersymmetric $$W_{\infty }^{K,K}$$ W ∞ K , K algebra which is the matrix generalization of previous $${{\mathcal {N}}}\,{=}\,2$$ N = 2 supersymmetric $$W_{\infty }$$ W ∞ algebra. By twisting this $${{\mathcal {N}}}\,{=}\,2$$ N = 2 supersymmetric $$W_{\infty }^{K,K}$$ W ∞ K , K algebra, we obtain the $${{\mathcal {N}}}\,{=}\,1$$ N = 1 supersymmetric $$W_{\infty }^{K}$$ W ∞ K algebra which is the matrix generalization of known $${{\mathcal {N}}}\,{=}\,1$$ N = 1 supersymmetric topological $$W_{\infty }$$ W ∞ algebra. From this two-dimensional symmetry algebra, we propose the operator product expansion (OPE) between the soft graviton and gravitino (as a first example), at nonzero deformation parameter, in the supersymmetric Einstein–Yang–Mills theory explicitly. Other six OPEs between the graviton, gravitino, gluon and gluino can be determined completely. At vanishing deformation parameter, we reproduce the known result of Fotopoulos, Stieberger, Taylor and Zhu on the above OPEs via celestial holography.

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

“…The final anticommutator relation between the fermionic currents is summarized by (m, r, λ), where 4 is realized by (h 1 −1), from the above, we obtain − 2 525 (λ−1)(λ+1)(λ+2)(2λ−3)(2λ+1)(2λ+3). Note that the additional term in the q 4, 7 2 ,5 F (m, r, λ) contains the factor λ. By considering the numerical factor 46080 when we move from the modes in the commutator to the differential operators in the OPE and multiplying this into the above factor, we obtain the previous structure constant in the current Q λ h1+h2− 3 2 −h= 3 2 (w).…”

“…This can be obtained by adding the extra contribution from q 4, 7 2 ,5 B (m, r, λ) in addition to the one from q 5, 7 2 ,5 B (m, r, λ) as in previous paragraph. Note that the Qλ (w) term on the right hand side of the OPE can be determined by the contribution from the structure constant q 5, 7 2 ,6 B (m, r, λ) only. Therefore, the last equation of (3.12) should be modified for nonzero λ with more general h 1 and h 2 where the weight h 2 is outside of the above allowed region.…”

“…where h 1 = h 2 − 2 can be determined and the fourth order pole contains the structure constant 128 5 (λ − 1)(2λ − 3)(2λ − 1) appearing in the current W λ F,h1+h2−h=1 (w). On the other hand, the structure constant o 3 2 , 7 2 ,3 F (r, s, λ) contains the λ dependent factor 8 15 (2λ − 1)(4λ 2 + 2λ + 3). We can take the sum (due to the odd h) of these with proper numerical value −48 for the latter and obtain − 128 5 λ(2λ − 1)(2λ + 7) which vanishes at λ = 0 and this implies that this extra contribution should appear.…”

“…Furthermore, the fourth order pole contains the structure constant 256 5 λ(λ + 1)(2λ + 1) appearing in the current W λ B,h1+h2−h=1 (w). On the other hand, the structure constant o 3 2 , 7 2 ,3 B (r, s, λ) contains 16 15 λ(4λ 2 − 6λ + 5). Obviously they are different from each other and the extra term − 256 5 (λ − 4)λ(2λ − 1) should appear from similar analysis.…”

“…Then it is straightforward to write down the corresponding algebra because the defining operator product expansions(OPEs) between these multiple free fields satisfy independently. See also the relevant work in [7] without a deformation parameter we mentioned above.…”