The N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 supersymmetric w1+∞ symmetry in the two-dimensional SYK models

We identify the rank (qsyk + 1) of the interaction of the two-dimensional N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = (2, 2) SYK model with the deformation parameter λ in the Bergshoeff, de Wit and Vasiliev (in 1991)’s linear W∞[λ] algebra via λ=12qsyk+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \lambda =\frac{1}{2\left({q}_{\mathrm{syk}}+1\right)} $$\end{document} by using a matrix generalization. At the vanishing λ (or the infinity limit of qsyk), the N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 supersymmetric linear W∞N,N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {W}_{\infty}^{N,N} $$\end{document}[λ = 0] algebra contains the matrix version of known N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 W∞ algebra, as a subalgebra, by realizing that the N-chiral multiplets and the N-Fermi multiplets in the above SYK models play the role of the same number of βγ and bc ghost systems in the linear W∞N,N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {W}_{\infty}^{N,N} $$\end{document}[λ = 0] algebra. For the nonzero λ, we determine the complete N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 supersymmetric linear W∞N,N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {W}_{\infty}^{N,N} $$\end{document}[λ] algebra where the structure constants are given by the linear combinations of two different generalized hypergeometric functions having the λ dependence. The weight-1,12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{1}{2} $$\end{document} currents occur in the right hand sides of this algebra and their structure constants have the λ factors. We also describe the λ = 14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{1}{4} $$\end{document} (or qsyk = 1) case in the truncated subalgebras by calculating the vanishing structure constants.


Introduction
The celestial holography [1] connects the gravitational scattering in asymptotically flat spacetimes with the conformal field theory which lives on the celestial sphere. By using the low energy scattering problems, the symmetry algebra of the conformal field theory for flat space has been found in [2]. Furthermore, in [3], the group of symmetries on the celestial sphere plays the role of the wedge subalgebra of w 1+∞ algebra [4]. We should understand the unknown structures behind these findings in order to convince the above duality. In [5,6], the supersymmetric w 1+∞ algebra has been identified with the corresponding soft current algebra in the supersymmetric Einstein-Yang-Mills theory. Recently, in [7], the holographic map from two-dimensional SYK models to the conformally soft sector of gravity in four-dimensional asymptotically flat spacetimes is studied. One of the motivations in this paper is to consider other types of SYK models and to check whether we have similar w 1+∞ symmetry or not. See the review papers [1,8,9,10] on the celestial holography 1 .
In N = (2, 2) SYK models [19,20,21], the two U(1) symmetries of the N = (0, 2) SYK models can be combined with U(1) R symmetry and the chiral and Fermi multiplets are also combined into N = (2, 2) chiral multiplet. This implies that their charges are related to each other. It turns out that the stress energy tensor takes simple form and the coefficients of the stress energy tensor are related to the rank of the interaction of the SYK models.
The standard N = 2 superconformal algebra is realized by the chiral multiplets and Fermi multiplets in quadratic form with various powers of (antiholomorphic) derivatives.
In the N = 2 supersymmetric linear W ∞ [λ] algebra [22,23], by so-called β γ and b c ghost systems, the higher spin currents with one-parameter are determined by the quadratic forms of these bosonic and fermionic operators 2 . In this case, the standard N = 2 superconformal algebra can be written in terms of the currents of low weights. Moreover, the so-called N = 2 scalar multiplet can be described by the lowest bosonic and fermionic currents. As a subalgebra, the bosonic algebra contains W ∞ [λ] algebra and W ∞ [λ + 1 2 ] algebra. They claim that the N = 2 supersymmetric linear W ∞ [λ] algebra is isomorphic to the N = 2 supersymmetric linear W ∞ [ 1 2 − λ] algebra because there exist some transformations between the above β γ and b c ghost systems by introducing two real anticommuting parameters 3 .
In this paper, by realizing that the above two models have their own one parameter, i) the rank (q syk + 1) of the interaction of the SYK models, and ii) the λ parameter and the fundamental building blocks are characterized by chiral and Fermi multiplets on the one hand and by β γ and b c ghost systems on the other hand, we would like to study the precise relation between the N = (2, 2) SYK models and the N = 2 β γ and b c ghost systems.
At first, we make a generalization of [22,23] by introducing the multiple β γ and b c ghost systems. Then we can compare with each stress energy tensor (or the generators of N = 2 superconformal algebra) described above. This will provide the exact correspondence between the two parameters mentioned before. At λ = 0, we identify the free field realization in [6] with the ones from β γ and b c ghost systems. This implies that the realization of N = 2 supersymmetric linear W N,N ∞ [λ = 0] algebra is described by the above N = (2, 2) SYK models together with the infinity limit of the rank of the interaction.
At nonzero λ, by using the higher spin currents of the matrix generalized β γ and b c ghost systems [22,23], we determine the complete N = 2 supersymmetric linear W N,N ∞ [λ] algebra in terms of various (anti)commutator relations. The structure constants originate from the oscillator construction in the AdS 3 Vasiliev higher spin theory [25]. At λ = 1 4 (corresponding to the rank (q syk + 1) = 2 of the interaction of the N = (2, 2) SYK models), we show how the truncated subalgebra arises by calculating the vanishing structure constants. Finally, we also describe the relation with celestial holography briefly.
In section 3, by matrix generalization of the N = 2 supersymmetric linear W ∞ [λ] algebra, the realization of N = 2 supersymmetric linear W N,N ∞ [λ = 0] algebra in the N = (2, 2) SYK models is described. For nonzero λ, starting from the λ dependent higher spin currents, we construct the (anti)commutator relations by checking the structure constants explicitly. The realization of N = 2 supersymmetric linear W N,N ∞ [λ = 1 4 ] algebra in the N = (2, 2) SYK models is studied. The relation with celestial holography is obtained.
In section 4, we summarize what we have obtained in this paper and further directions are also described.
In Appendices, some detailed calculations in section 3 are explained.
We are using the Thielemans package [26] with a mathematica [27].

Review
2.1 Two-dimensional N = (2, 2) SYK models In the two-dimensional SYK model [19], there are N chiral multiplets Φ a (a = 1, 2, · · · , N) and M Fermi multiplets Λ i (i = 1, 2, · · · , M) with a random coupling. This random coupling of the interaction of the SYK model has a rank of (q syk + 1). The model with N = M has an enhanced N = (2, 2) supersymmetry and reduces to the one studied in [19,20] 4 . The lowest components of these superfields satisfy the following operator product expansions (OPEs) For the fermions in the second equation of (2.1), the proper normalization is performed, compared to the one in [21]. The conformal weights for φ a ,∂φ a , λ a andλ a in the antiholomorphic sector are given by 1 2(q syk +1) , 1 − 1 2(q syk +1) , 1 2 + 1 2(q syk +1) and 1 2 − 1 2(q syk +1) . The lowest supermultiplet contains the weight-1 operator, two supercharges and the stress energy tensor. Then the standard N = 2 superconformal algebra is realized by [33,21] J = q syk 2(q syk + 1)λ a λ a − 1 (q syk + 1) φ a∂φa , (2. 2) The central charge in (2.2), where the fourth order pole in the OPE T (z) T (w) is c 2 , is given by Each independent term in the stress energy tensor contributes to its own central term and the overall factor N appears in (2.3). Some typo in [21] is corrected in (2.2). Note that we can multiply any (pure imaginary) numerical number into the G + and its inverse into the G − without changing the definition of the N = 2 superconformal algebra. The central terms of the OPEs, J(z) J(w) and G + (z) G − (w), are given by c 3 and c 3 respectively 5 .

The
In [22,23], the N = 2 supersymmetric linear W ∞ [λ] algebra 6 is realized by the following β γ and b c ghost systems which satisfy the OPEs The conformal weights for β, γ, b and c are given by λ, 1 − λ, 1 2 + λ and 1 2 − λ respectively 7 . Note that the normalizations in the right hand sides of (2.4) are given by +1.
Then the higher spin currents are given by [22,23] 5 Note that in the N = (0, 2) SYK model, the stress energy tensor takes the more general form [21] and the condition for the N = 2 supersymmetry enables us to have simpler form for the stress energy tensor. 6 In this paper, we are considering the linear algebra where the corresponding OPE does not have any quadratic or higher order terms in the currents of the right hand side although the currents are quadratic in the operators. See also the relevant work in [34] where the nonlinear structures occur in the context of AdS 3 higher spin theory. 7 In terms of the parameter λ hs of the higher spin algebra hs[λ hs ], there exists a relation λ = 1 2 λ hs . 8 We thank M. Vasiliev for pointing out that these are quasiprimary operators under the stress energy tensor V (2)+ λ ten years ago.
The λ-dependent coefficients appearing in (2.5) are given by The λ-dependent coefficients (2.6) are not independent. Some properties of these coefficients are given by Appendix A of [22]. The binomial coefficients for parentheses are used and the rising Pochhammer symbol (a) n ≡ a(a + 1) · · · (a + n − 1) is also used here. We can check that the N = 2 superconformal generators are given by 9 The lowest s value for the bosonic currents V (s)± λ is given by s = 1. One of them plays the role of the weight-1 current of the N = 2 superconformal algebra in (2.7). The lowest s value for the fermionic currents Q (s)± λ is given by s = 1 also. In [22,23], the N = 2 scalar multiplet is denoted by (Q . We can easily see that the weights for the composite operators β γ, b c, β c and b γ are given by 1, 1, 1 2 and 3 2 respectively and all the λ dependence is gone. This means that their weights for the bosonic currents V (s)± λ are given by s while the weights for the fermionic currents Q (s)± λ are given by (s − 1 2 ) 10 .
9 In next section we will present their explicit forms in terms of the composite operators in the ghost systems. When we take N = 1 over there, then we obtain the exact results [22,23]. 10 In terms of the ghost systems, we have V In order to calculate some OPEs between the ghost systems and the currents in (2.5), some partial results on the highest order poles between them in [22,35]  In order to describe the multiple number of the chiral multiplets (or the Fermi multiplets), we need to introduce the multiple number of β γ and b c systems [36] satisfying the following defining OPEs The fundamental indices a, b, · · · of SU(N) in (3.1) runs over a, b, · · · = 1, 2, · · · , N and the antifundamental indicesā,b, · · · of SU(N) runs overā,b, · · · = 1, 2, · · · , N. Similarly, we can associate the indices i, j, · · · and the indicesī,j, · · · with the corresponding fundamental and antifundamentals of SU(L) 11 .
By multiplying the generators of SU(N) into the previous relations (2.5), we obtain the following matrix generalization of the work in [22,23] 11 We consider SU (L)-singlet currents in this paper.
The central charge of the stress energy tensor is given by 13 By comparing (2.3) with (3.3), the deformation parameter in [22,23] plays the role of the rank of the random coupling of the SYK model and it is given by 14 λ = 1 2(q syk + 1) .
We can write down the generators of the N = 2 superconformal algebra for matrix generalization from (2.7) as follows: For N = 1, we observe that the above relations (3.5) are reduced to the ones in [22]. By realizing the following relations with (3.4) 15 , we observe that the relations (3.5) can be identified with the ones in (2.2) together with a factor √ 2i in G + and a factor − i does not change with these factors). The conformal weights for both sides in (3.6) are consistent with each other. We expect that there is a one-to one correspondence between the N = 2 SYK model and the β γ and b c ghost systems in the N = 2 supersymmetric linear W N,N ∞ [λ] algebra. 12 Note that in [36], there appear the extra factors ±(−1) s or ±(−1) s− 1 2 in various places in the coefficients of (3.2). 13 We consider the L = 1 case. If we consider the general L, then this L factor appears in the central charge. 14 It has been conjectured in [37] from the N = (0, 2) SYK models that the parameter λ hs (See also the footnote 7) is related to the q syk and is given by λ hs = 1 M N q syk . 15 In [21], we used the different terminology for the bosons. Note that when we change the ordering in the first OPE of (3.1), there is a minus sign in the right hand side while there is no minus sign in the second OPE of (3.1) after this change. We should also make sure that they have the correct weights in terms of deformation parameter.

3.2
The N = 2 supersymmetric linear W N,N ∞ [λ = 0] algebra From the exact correspondence between the chiral multiplets and the Fermi multiplets of the N = 2 (2, 2) SYK model and the β γ and b c ghost systems in (3.6), we expect that there exist the precise relations for the higher spin currents between them. By linear combinations among the higher spin currents in (3.2) we can write down the higher spin currents of [6] in terms of (3.2) at λ = 0 and it turns out that for SU(N)-singlet currents we have 16 .

(3.7)
For h = 1 with λ = 0, the coefficient of the first term of W F,h=1 in (3.7) vanishes and the See also (2.7) and (3.5). On the other hand, the coefficient of the first term of W B,h=1 in (3.7) does not vanish and the W B,h=1 with λ = 0 is proportional to −γ a β a (which holds for nonzero λ). Then the current W B,h=1 arises only in the β γ and b c ghost systems. For h = 0, the Q1 2 vanishes and theQ1 2 is proportional to −β a c a which does not occur in the construction of [6]. See also the footnote 21. Therefore, we expect that there appears the presence of the current W B,h=1 and the currentQ1 2 in the N = 2 supersymmetric linear W N,N ∞ [λ = 0] algebra 17 . Furthermore, we can compare with each coefficient appearing in the free field realization in [6] and the one in (3.2) at λ = 0. In order to do this, we should act the antiholomorphic partial derivatives on the composite operators fully. Then the binomial coefficients appear.
It turns out that there appear the following identities It is rather nontrivial to check these relations for generic h and k, but we can try to do this for several values for these quantities. Note that in the right hand sides of (3.8), the additional binomial coefficients occur by expanding the antiholomorphic partial derivatives fully as described before 18 . Therefore, the N = 2 SYK model has N = 2 supersymmetric linear W N,N ∞ [λ = 0] algebra where the higher spin currents are given by (3.2) and by using the relations (3.7), the explicit (anti)commutator relations can be read off from the previous results in [6](See also [38]).
The relation between the parameters is given by (3.4). Of course, as explained before, in the (anti)commutators relations, we observe that there appear the currents W B,h=1 andQ1 2 .
Moreover, their OPEs with other higher spin currents W F,h≥1 , W B,h≥1 , Q h+ 1 2 ≥ 1 2 andQ h+ 1 2 ≥ 1 2 will appear in general. In next section, we will present the (anti)commutator relations for nonzero λ. Therefore, once we put the λ to be zero in these equations, we obtain the final results. (3.10) The algebra between the currents in the first line of (3.10) is closed and the explicit form is given by the ones in [6]. In the right hand sides of these (anti)commutator relations we can see only the operators in the first line of (3.10). Due to the presence of the operators in the second line of (3.10), we should calculate the OPEs between these weight-1, 1 2 currents and the remaining ones in the first line of (3.10) as well as their own OPEs in order to describe the full algebra if we do not decouple these currents 19 . As we will see next section, once the λ becomes nonzero value (a deviation from (3.9)), then this does not hold any more because the right hand sides of these (anti)commutator relations possess the operators in the second line of (3.10).

The
The higher spin currents for nonzero λ Let us consider the nonzero λ case. We take the previous expressions (3.7) by considering the λ dependence explicitly. Then we have the following SU(N)-singlet currents 20 In particular, V (1)− λ has γ a β a term also for nonzero λ. See also the weight-1 current in (3.5). We would like to obtain the algebra generated by these currents in (3.11).
We present the currents for low weights as follows: 19 We will see that we have their explicit forms as W B,1 = − 1 4 γ a β a andQ 1 2 = − 1 √ 2 β a c a . Even they do not depend on the λ parameter from the footnote 21. By construction of [6], there is no φī ,a dependence and its derivative∂ φī ,a appears only. 20 The SU (N )-adjoint currents are given by Appendix A. Each λ independent coefficient of bosonic current is the same and each λ independent coefficient of fermionic current is the same: At λ = 0, the weight-1 current J in (3.13) of the N = 2 superconformal algebra does not depend on the bosonic β γ operators 21 .

The structure constants for nonzero λ
Let us introduce the generalized hypergeometric function In general, the sum of upper four elements plus 1 (= 5 2 + a − r) is not equal to the sum of lower three elements (= 7 2 − r) for generic a = 1 22 . Furthermore, we introduce the mode By using the notation of (3.14), we havê φ h1,h2 r−1−a (0, The last of lower elements contains the additional (a − 1) which is nonzero for a = 1. We check that for a = 1, the expression of (3.14) reduces to the one in [39] where their s, r, i and j in (3.14) correspond to our The falling Pochhammer symbol [a] n ≡ a(a − 1) · · · (a − n + 1) in (3.15) is used.
We have found three different kinds of structure constants in the context of the matrix generalization of AdS 3 Vasiliev higher spin theory as follows [25]: where the relations (3.14) and (3.15) are needed 23 .
From the lesson of [25] where the mode dependent structure constants for vanishing λ can be written in terms of the linear combinations of (3.16), we do expect that for nonzero λ, they satisfy as follows: 23 We have the following symmetry between the structure constants [25] under the transformation µ ↔ 1−µ BB h1,h2 r, ± (m, n; µ) = ±BB h1,h2 r, ± (m, n; 1 − µ), BF This implies that the half of the structure constants vanishes at µ = 1 2 (λ = 1 4 or q syk = 1).
We have checked that the above relations for several h 1 , h 2 and h are satisfied in the specific OPE examples. That is, the structure constants are indeed the right hand sides of (3.17).

3.3.3
The example of the explicit OPE W λ F,4 (z) W λ F,4 (w) for nonzero λ For example, for h 1 = 4 and h 2 = 4, we can calculate the OPE W λ F,4 (z) W λ F,4 (w) by using (3.11), (3.2) and (3.1) and reexpressing each pole in terms of W λ F,h (w) with h = 2, 4, 6 and their derivatives as follows 24 : In the second relation of (3.18), we reexpress the structure constants in terms of the differential operators p 4,4 F,6−h (∂z,∂w, λ) with h = 2, 4, 6. By using the first equation of (3.17) for fixed h 1 = h 2 = 4, we obtain From this (3.19), we can read off the corresponding differential operator by taking the terms having a degree (h + 1) = 5 as follows: Then we can calculate the quantity −p 4,4 F,4 (∂z,∂w, λ) by acting (3.20) on the operator and this will lead to the corresponding terms in (3.18). Similarly, we can calculate . Then we can determine w and this leads to the current W λ F,4 (w) and its derivatives in (3.18) by performing −p 4,4 F,2 (∂z,∂w, λ) provides the corresponding terms in (3.18) 25 .

The complete (anti)commutator relations between the SU(N)-singlet currents for nonzero λ
From the analysis of previous section together with the similar descriptions in Appendix B, we conclude that the final complete (anti)commutator relations between the SU(N)-singlet currents for nonzero λ with the insertion of q dependence appropriately can be summarized by 26 In the right hand sides of (3.21), we emphasize that the additional weights 1 2 , 1 currents appear by taking the square brackets 27 . Of course we assume that the possible lowest weights for are given by h = 1, h = 2, h + 1 2 = 3 2 and h + 1 2 = 3 2 respectively. In other words, among the field contents in [42], the above weights 1 2 , 1 currents occur in the right hand sides of the (anti)commutator relations at nonzero λ. In order to fully describe the complete structure of the N = 2 supersymmetric linear W N,N ∞ [λ] algebra, we need to calculate the OPEs between the additional weights 1 2 , 1 currents and the remaining currents. Also their own OPEs should be calculated.
In Appendix C, we present some OPEs containing the additional currentsQ λ 1 2 or W λ B,1 for fixed h 1 and h 2 . It turns out that the OPEs containing the weight-1 2 currentQ λ 1 2 have the previous known structure constants while the OPEs containing the weight-1 current W λ B,1 , at first sight, do not have their structure constants which can be written in terms of the known expressions appearing in (3.21), although there are explicit λ-dependent terms in their OPEs.
In particular, we can check the following relations 28 Then according to (3.22), the square brackets in the above (anti)commutator relations (3.21) vanish at λ = 0 and we reproduce the subalgebra of the N = 2 supersymmetric linear [6]. The bosonic subalgebra is given by W N 1+∞ [λ = 0] generated by 27 We can still use the (anti)commutator relations for λ=0 by allowing the corresponding upper limits in the summation over the dummy variable h properly at the four places. Each single term can combine with each summation term because the λ and mode dependent structure constants in each single term can be written in terms of the same structure constants in each summation term. 28 Similar relations can be checked as follows: p h1,h2,h=h1+h2−3 (w) and removing the unwanted currentQ λ 1 2 . In this way we determine the new currents, (3.23), (3.24), (3.25) and (3.26). We expect that we can continue to perform this procedure and remove the above weight-1, 1 2 currents 30 . Therefore, in principle, eventually we obtain the complete (anti)commutator relations with modified λ-dependent known structure constants, as a subalgebra, where the unwanted weight-1, 1 2 currents disappear completely 31 . 29 We need to admit that the new currents are not quasiprimary operators. 30 As described before, by using the partial results in [35], we can obtain the new currents for any weight h by calculating the highest order pole in any OPE in order to remove the weigh-1 2 current. 31 We can decouple the W λ F,1 andQ λ  , all the second terms in the right hand sides of the structure constants (3.17) vanish. In the N = 2 SYK models, this is equivalent to take the following limit q syk → 1. (3.27) The interaction is quadratic. As observed in [22], there exists a subalgebra generated by 33 There is no supersymmetry in the first case (3.28) [22] 34 . We calculate some OPEs for fixed h 1 and h 2 in order to see this behavior explicitly in Appendix E. In the basis of [22], we obtain the following relations from (3.11) 32 At this point, N = 2 supersymmetric linear W N,N ∞ [λ = 1 4 ] algebra is self isomorphic because the solution of λ = 1 2 − λ provides the λ = 1 4 , as in the introduction. 33 We thank M. Vasiliev for discussion on this matter further. 34 It would be interesting to study this case in the context of the SYK models because there is no supersymmetry.

Then we can calculate the commutator relation [(V
By substituting the first equation of (3.30) into this commutator, we obtain the following coefficient function  (3.27) in the N = 2 SYK models) implying that we can decouple the currents V ) n ] can be obtained by substituting the first and second equations of (3.30) into this commutator, we obtain the following coefficient function of (V (h 1 +h 2 −2−h),+ λ ) m+n as follows: The above coefficient (3.32) vanishes at λ = 1 4 and we can decouple the currents V (h),+ λ with odd h and therefore, we do not have these currents in (3.28) or (3.29). For even h 1 and odd h 2 , the combination (h 2 + h 2 − 2) is odd. In Appendix E, we will see more details on this matter.

The relation with celestial holography
We have found the matrix generalization of the N = 2 supersymmetric W ∞ algebra [42] by adding the additional parameter λ. Then we can follow the procedure of [43] by using the topological twisting [44,45]. The bosonic SU(N)-singlet current of weight h can be given The corresponding SU(N)-adjoint current can be constructed by multiplying the SU(N) generators into the above four kinds of operators.
For the fermionic currents, we take Q λ h+ 1 2 and Q λ,Â h+ 1 2 . Then the seven OPEs between these currents (or the corresponding (anti)commutator relations) can be determined explicitly. The structure constants found in [43,6] can be generalized to λ dependent ones where the explicit expressions are given by (3.17). When we apply the two-dimensional algebra to the N = 1 supersymmetric Einstein-Yang-Mills theory, it is crucial to realize that the mode dependent function (3.15) is obtained by performing the nontrivial contour integrals [14]. Then the OPEs between the graviton, the gravitino, the gluon and the gluino can be obtained and the corresponding structure constants are given in (3.17) with λ dependence.

Conclusions and outlook
We derived that the parameter of N = 2 SYK models can be realized by It is an open problem to compare the present results with the ones in [22,23] and to observe how they coincide with each other analytically. So far, we have considered the N = 2 SYK models and it is interesting to study the N = (0, 2) SYK models and check whether there exists a higher spin realization or not. There is a partial work in [21] on the limit of q syk → N M in this direction. It is also interesting problem to generalize the work of [46] to the case having the above N = 2 supersymmetric linear W N,N ∞ [λ] algebra.

A
The SU (N )-adjoint higher spin currents As done in (3.7), we can check that the SU(N)-adjoint higher spin currents at vanishing λ can be obtained as follows: Similarly, for nonzero λ, we take the following higher spin currents together with (3.2) In the normalization of Appendix (A.2), we take the same normalization of Appendix (A.1). That is, the overall factor does not depend on the λ explicitly.
It is straightforward to calculate this result because we are considering the linear algebra and collect each pole in terms of the various descendant terms and new quasiprimary operator inside the Thielemans package [26]. Then all the structure constants can be determined and depend on the λ explicitly as above. In Appendix (B.1), we also present the structure constants in terms of (3.17) after inserting the derivatives. As we expect, up to minus sign, we observe that the above OPE behaves as the first and third terms of the second equation in (3.21) in the sense that there are three terms with correct mode dependent structure constants. Due to the (−1) h−1 factor when we change from the second equation of (3.21) to the above OPE and there are h = 0, 2, 4 cases which are even, we have all the minus signs in the above OPE. Then how we can see the existence of the second term of the second equation of (3.21)?
We can read off the weight from the condition h = h 1 + h 2 − 3 with h 1 + h 2 − 2 − h = 1 and we take h 1 = 4 and h 2 = 3. Then we observe the weight for the W λ B,h 1 +h 2 −2−h is equal to 1. Then we can calculate the following OPE (B.2) Therefore, in this example, we observe the three terms of the second equation of (3.21). Note that all the structure constants associated with the weight-1 current W λ B,1 (w) in Appendix (B.2) contain the λ factor explicitly. This implies that when we take the vanishing limit for the λ, the weight-1 current and its descendant terms disappear.
Let us consider the third equation of (3.21) with h 1 = 4 and h 2 = 3. Then we obtain the following result as done before

(w)
Let us move on the fourth equation of (3.21) and we can calculate the corresponding OPE for h 1 = 4 and h 2 = 3 in our notation and this leads to the following OPE (w) Let us continue to calculate the OPE associated with the fifth equation of (3.21) and it turns out that (w) Note that in the fifth equation of (3.21), there exists (−1) h factor. So when we write down the OPE as above, this factor combines with the previous factor (−1) h−1 . This implies that there are no h dependence in the (−1) factor. We are left with the final (−1) factor which appears in Appendix (B.5). As described before, the structure constants associated with the weight-1 2 current contain the λ factor and this leads to the fact that this weight-1 2 current disappear when we take the vanishing λ limit 35 .
For the sixth equation of (3.21), we can calculate the following OPE There is a factor (2λ − 1) which vanishes at λ = 1 2 . See also the footnote 28.
In Appendix (B.6), there are overall minus signs as mentioned before because the sixth equation of (3.21) has the (−1) h factor. Again, the presence of the weight-1 2 current appears at the nonzero λ 36 .
Now let us look at the final equation of (3.21) and we consider the following OPE for h 1 = h 2 = 3 in our notation (B.7) There appear four different kinds of terms in Appendix (B.7). In this case, for even h, there are plus signs. Note that there is a factor (4λ − 1) which vanishes at λ = 1 4 . Moreover, the λ factor appears in the weight-1 current W λ B,1 (w) (and its descendant terms) 37 .
C Some OPEs containing theQ λ 1 2 or the W λ B,1 We consider the OPEs corresponding to the (anti)commutator relations in (3.21) where the left hand sides containQ λ 1 2 or the W λ B,1 . For the first four cases, we haveQ λ 1 2 and for the remaining ones we have W λ B,1 .
Let us consider the fifth equation of (3.21). We calculate the following OPE by taking the second current asQ λ 37 There is a factor (2λ − 1) which vanishes at λ = 1 2 in the weight-1 current W λ F,1 (w) (and its descendant terms). See also the footnote 28.

−2q
It turns out that we can express the OPE in Appendix (C.1) as the one in Appendix (B.5) except the numerical values −2 rather than −1. If we rescaleQ λ 1 2 (w) by 1 2 , then there appears 2 in the first term of Appendix (B.5) and the above 2's in Appendix (C.1) disappear. Even at λ = 0, this OPE arises.
Let us consider the sixth equation of (3.21). We calculate the following OPE by taking the second current asQ λ λ(λ + 1)(2λ + 1)∂ 3Qλ The OPE in Appendix (C.2) looks similar to the one in Appendix (B.6). Again, by the rescaling of theQ λ (w), we observe that this term (and its descendant terms) vanishes at λ = 0.
Let us consider the final equation of (3.21), where the corresponding second current is given byQ λ 1 2 . It turns out that (6λ 2 − 3λ + 1)

(w)
When we take the first current as W B,1 further corresponding to the sixth equation of (3.21) then we obtain Now we see that the similar behavior in Appendix (C.4) arises, compared to the ones in Appendix (B.6) or Appendix (C.2). From now on, we consider that the first current is given by W λ B,1 .
In Appendices (C.5), (C.6), (C.7), and (C.8), we cannot express the structure constants in terms of (3.17  (r, λ) δÂB δ h 1 h 2 q 2(h 1 + 1 2 −1) δ r+s . (D.1) As in the section 3, we intentionally make the square brackets in the Appendix (D.1) in order to emphasize that the currentQ λ 1 2 or the current W λ B,1 occurs inside of the square brackets when we restrict to the operators in the left hand sides which do not have these weight-1 2 , 1 currents. E Some OPEs for λ = 1 4 We present some OPEs for the particular value of λ = 1 4 for fixed h 1 and h 2 as follows. We keep the λ dependence without substituting this value in order to see the factor (1 − 4λ). It turns out that