The supersymmetric affine Yangian

Journal of High Energy Physics, May 2018

Abstract The affine Yangian of gl1 is known to be isomorphic to \( {\mathcal{W}}_{1+\infty } \), the W-algebra that characterizes the bosonic higher spin — CFT duality. In this paper we propose some of the defining relations of the Yangian that are relevant for the \( \mathcal{N}=2 \) superconformal version of \( {\mathcal{W}}_{1+\infty } \). Our construction is based on the observation that the \( \mathcal{N}=2 \) superconformal \( {\mathcal{W}}_{1+\infty } \) algebra contains two commuting bosonic W1+∞ algebras, and that the additional generators transform in bi-minimal representations with respect to these two algebras. The corresponding affine Yangian can therefore be built up from two affine Yangians of \( \mathfrak{g}{\mathfrak{l}}_1 \) by adding in generators that transform appropriately.

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The supersymmetric affine Yangian

Revised: May The supersymmetric a ne Yangian Matthias R. Gaberdiel 0 1 4 Wei Li 0 1 2 Cheng Peng 0 1 3 Hong Zhang 0 1 2 Zurich 0 1 Switzerland 0 1 0 182 Hope Street, Providence , RI 02912 , U.S.A 1 100190 Beijing , P.R. China 2 Institute of Theoretical Physics, Chinese Academy of Sciences 3 Department of Physics, Brown University 4 Institut fur Theoretische Physik, ETH Zurich The a ne Yangian of gl1 is known to be isomorphic to W1+1, the W -algebra that characterizes the bosonic higher spin | CFT duality. In this paper we propose some of the de ning relations of the Yangian that are relevant for the N = 2 superconformal version of W1+1. Our construction is based on the observation that the N = 2 superconformal W1+1 algebra contains two commuting bosonic W1+1 algebras, and that the additional generators transform in bi-minimal representations with respect to these two algebras. The corresponding a ne Yangian can therefore be built up from two a ne Yangians of gl1 by adding in generators that transform appropriately. Conformal and W Symmetry; Higher Spin Symmetry; Quantum Groups - 1 Introduction 2 3 4 5 2.1 2.2 3.1 3.2 3.3 5.1 5.2 5.3 5.4 The minimal and conjugate minimal representation 4.2 Identifying the representations The Yangian at generic parameters The generators in minimal representations 5.1.1 Other minimal generators The generators in conjugate minimal representations The OPEs with e and f The N = 2 algebra 6 Conclusions A Additional relations of the free eld theory B The de ning relations B.1 The OPE like description B.2 The mode relations B.3 The initial conditions C Supercharge constraints free) conformal eld theory [1{3], where string theory is expected to contain a higher spin theory [4]. At this point in moduli space the large symmetry algebra underlying string { 1 { theory (see [5{7] for indirect evidence) is expected to become visible. This is also the place where the integrability of the theory should be most easily discerned. In the context of AdS3, the emergence of a higher spin symmetry at the tensionless point was recently seen quite explicitly in [8], see also [9, 10] for attempts to observe this directly from a world-sheet perspective. In that case, the dual 2d conformal eld theory of string theory on AdS3 S 3 T4, the symmetric orbifold of T4, was shown to contain a W1 symmetry algebra. This is the hallmark of the duality between higher spin theories on AdS3 and 2d CFT's [11{13], see [14] for a review. On the other hand, there has also been progress in understanding the integrable structure of string theory on AdS3 [15{18], and it would be very interesting to relate the higher spin and integrable symmetries. Integrable theories are usually distinguished by having a Yangian symmetry, and one may therefore try to identify the relevant Yangian in the explicit higher spin description. This was recently done [19, 20] for the bosonic toy model of [13], where the generators of W1 [ ], the symmetry algebra of the higher spin theory, were explicitly identi ed with those of the a ne Yangian of gl1. (The underlying isomorphism was rst noted in [21, 22], generalizing the construction of [23], and independently by [24] and [25{27], see also [28] for further generalizations. The a ne Yangian of gl1 is also isomorphic to the spherical degenerate double a ne Hecke algebra SHc of [29], and was also constructed independently in [30].) In this paper we show how to construct the Yangian algebra corresponding to the N = 2 superconformal generalisation of W1 . Our approach is partially inspired by the fact that the underlying higher spin algebra shs[ ] contains two commuting bosonic higher spin subalgebras hs[ ] hs[1 ]. Subsequently, it was suggested in [31, section 11.1] that this relation may also be true for the full W1 W1(N =2)[ ] actually has this structure, i.e. that it contains two decoupled W1 (N =2)[ ] algebra. We begin by showing that [ ] algebras. (This analysis relies on the precise form of the de ning structure constant of the W1 algebra that was identi ed in [32].) We then show that the additional generators that have to be added to the two bosonic W1 algebras in order to generate the full W1 algebra transform in what one may call bi-minimal representations with respect to the two W1 algebras. (This is to say, they transform as a minimal representation with respect to one, and as an anti-minimal representation with respect to the other; here \anti-minimal" means that it is the conjugate representation to the minimal representation.) The basic idea of our construction is then to add generators to the two a ne Yangians of gl1 that have these transformation properties. The main technical di culty of this approach comes from the fact that the description of conjugate minimal representations in terms of the a ne Yangian was not known. The a ne Yangian viewpoint gives rise to an elegant description of representations in terms of plane partitions [19], see also [33], but this language only applies to the \box"representations, but not to those made of \anti-boxes". However, the bi-minimal representations that are relevant for the above extension always involve also anti-box representations. We propose a general formula for the description of anti-box representations in terms of plane partitions, see section 3.2. With this insight we can then propose some of the commutation relations of the two sets of a ne Yangian generators of gl1 with the { 2 { additional modes, and thus undertake the rst steps towards de ning the supersymmetric generalisation of the a ne Yangian. The paper is organized as follows. In section 2 we show that the W1(N =2)[ ] algebra contains (and can be built up from) two commuting bosonic W1 algebras. We identify the additional generators that need to be added, and in particular, their representation properties with respect to the two bosonic W1 algebras. In section 3 we review the relevant minimal representation of the a ne Yangian, and explain how to describe the conjugate representation. In section 4 we analyze the N where the W1(N =2)[ ] algebra has a free eld realization, in terms of which also the two bosonic W1 algebras can be identi ed. In particular, we can make an explicit ansatz for the additional generators that need to be added and compute their commutation and = 2 construction for = 0, anti-commutation relations for = 0. In section 5, we then deform these relations away from the free eld point ( = 0), using as a guiding principle our insight into the correct description of the minimal and conjugate minimal representations. We furthermore test our ansatz by comparing to the free eld limit, and by showing that the additional generators lead to states in the correct representations. Our conclusions and avenues for future work are outlined in section 6. There are two appendices: in appendix A, we have spelled out some of the free eld relations that we did not want to put in the main part of the text, and in appendix B we have summarized the de ning relations of the supersymmtric a ne Yangian we have found. Note added: as we were in the nal stages of this work we were made aware of [39] which contains some overlap with section 2 of our paper. 2 Building up the N = 2 W1 algebra In this section we explain that the W1(N =2)[ ] algebra contains two bosonic W1 as mutually commuting subalgebras, one at = and one at = 1 [ ] algebras . Note that it is known, see e.g. eq. (202) in [14], that the N = 2 higher spin algebra can be written in this manner shs[ ](bos) = hs[ ] hs[1 However, it is not obvious whether this will also be true for the full quantum W1(N =2)[ ] algebra.1 This viewpoint will be important below because it will allow us to construct the full W1(N =2)[ ] algebra starting with these bosonic subalgebras. 2.1 Decoupling the bosonic subalgebras algebra as WN(N;k=2), i.e. to express both In order to understand how this comes about, it is convenient to parametrise the W1(N =2)[ ] and c in terms of N and k as c c(NN;k=2) = 3N k 1For the case of the N = 2 W3 algebra this was already noted in [34, 35]; however, for general this is not known. (2.1) (2.2) { 3 { see e.g. [32] for our conventions. As is also explained there, the WN;k spin h = 2 elds: the stress energy tensor T , and the primary spin h = 2 eld W W is also primary with respect to the u(1) current J , but T is not; however, we can de ne the decoupled spin 2 eld via T~ = T : J J : : 3 2c The modes of these two elds then satisfy the commutation relations [T~m; T~n] = (m n)T~m+n + m(m2 1) m; n (c 1) 12 [T~m; Wn] = (m n)Wm+n [Wm; Wn] = (m n) 2n2 (c 1) T~m+n + 2 where we are using the same conventions as in [32], and the last identity is directly read o from eq. (2.14) of that paper. To nd the two commuting Virasoro algebras, we make the ansatz ~ Tm = a+L+m + a Lm ; Wm = b+L+m + b Lm ; and demand that L commute with one another and each lead to a Virasoro algebra with central charge c where c+ c . In particular, it follows that (c 1) = c+ + c ; 0 = b+c+ + b c : These two conditions x the coe cients a and b uniquely, and one nds that the two solutions are a = 1 ; b = c22;2 4 (c22;2)2 + 32 n2 (c 1) : The corresponding central charges are then c+ = (c b 1)b b+ ; (c b 1)b+ b+ : cN;k = (N h from [32], one nds that c equals cN;k and ck;N , respectively, where is the central charge of the bosonic WN;k algebra (without the additional u(1) current). Note that the full (decoupled) stress energy tensor T~, de ned in eq. (2.3), is indeed the sum of L+ and L , ~ Tm = L+m + Lm ; as also follows from (2.7). In particular, this implies that the total central charge of the N = 2 algebra must equal | the \+1" comes from the u(1) factor we have divided out | cN;k + ck;N + 1 = (3k 1)N k 1 (N =2)) algebra contains at least two Virasoro primary elds. It seems plausible that among these spin s elds, we can always nd two elds W (s) such that2 [Lm; Wn(s) ] = (s 1)m n) W m(s+)n ; [Lm; Wn(s) ] = 0 : (2.12) Since all W (s)+ elds commute with L , the same must be true for their commutator, and hence the VOA generated by L+ and the W (s)+ elds must close. (Obviously, a similar statement also holds for L and the W (s) elds.) Furthermore, the commutator of W (s)+ with W (t) must vanish since, with respect to L+, say, W (t) behaves like the identity eld and hence does not give rise to a non-trivial commutator. Thus, if for each spin s there are two elds such that (2.12) holds, it follows that the W1 two commuting bosonic W1 and the other generated by L [ ] algebras, one generated by the elds L+ and the W (s)+, and the W (s) . Given that their central charges equal cN;k (N =2)[ ] algebra contains and ck;N , it is very plausible that the relevant W1 algebras are just the bosonic WN;k and Wk;N algebras, respectively, i.e. that WN(N;k=2) WN;k Wk;N : In order to con rm this we would have to construct the relevant elds and determine their C334 structure constants, but we have not attempted to do so here. Note that this structure also nicely re ects the Z2 Z2 realized by N $ k. 2.2 Character analysis Next we want to understand the additional generators that need to be added to the two bosonic W1 venient to ad[d] aalgseinbgrlaes firneeorbdoesrotno geelndertaoteWW1(N1=2)[ ]. Then the corresponding vaccum (N =2)[ ]. For the following it will be concharacter equals Z2 symmetry of the WN(N;k=2) algebra that is 0(q) = Y1 (1 + qn+ 12 )2n n=1 (1 qn)2n ; 00(q) = Y 1 1 Y s=1 n=s (1 (1 + qn+ 12 )2 qn)(1 qn+1) ; fb = 1 Y n=1 (1 1 qn) : pp = 1 Y n=1 (1 1 qn)n ; 2We have checked explicitly that this is the case for spin s = 3. { 5 { since the vacuum character of the W1(N =2) algebra is and a single free boson contributes We want to organise this character in terms of W1+1[ ] W1+1[1 character of each W1+1[ ] algebra is described by a plane partition, see e.g. [19, 20] ]. The vacuum (2.13) (2.14) (2.15) (2.16) (2.17) where R runs over all representations that appear in nite tensor powers of the two biminimal representations, and RT is the conjugate representation to RT , with T denoting the transpose of R. Since R involves in general box and anti-box representations, and since the wedge character of such a mixed representation is simply the product of the wedge character of the box representation and that of the anti-box representation, the above identity follows from S 1 n=1 Y (1 + yqn+ 12 )n = X yjSj (Swedge) [ ](q) ST (wedge) [1 ](q) ; where S runs over all Young diagrams (labelling say box-representations), ST is the transpose Young diagram (labelling now anti-box representations), and jSj denotes the number of boxes in S. The rst few cases are explicitly (see [36] for the general method for how to derive them) 1 n=1 and hence the two algebras account precisely for the denominator of 0(q) in eq. (2.14). The numerator of (2.14) corresponds to the fermionic excitations, and is accounted for in terms of bi-minimal representations of the two algebras3 together with their tensor powers. Since for each representation R of W1+1[ ] the character is of the form this amounts to the condition that (wedge)(q) pp(q) ; R (wedge)(q) = (wedge)(q) = (wedge)(q) = (wedge)(q) = (wedge)(q) = (wedge)(q) = q h 1 (1 (1 (1 (1 (1 q q2h q)(1 q2h+1 q)(1 q)(1 q3h+1 q)2(1 q2) q2) q3h q2)(1 q3) q3) q3h+3 q)(1 q2)(1 q3) : { 6 { 3The bi-minimal representations that are relevant here are \minimal" with respect to one factor, and \conjugate-minimal" with respect to the other. There are therefore two such representations, namely \minimal"{\conjugate-minimal" and \conjugate-minimal"{\minimal". 1 2 h + h^ = 1 2 h^ = 1 2 1 2 (1 + ) + It is then straightforward to check (2.20) explicitly,4 provided that we take the box repre ] to be the ones with conformal dimensions respectively, so that the total conformal dimension equals thus reproducing the conformal dimension of the supercharge. HJEP05(218) 3 The minimal and conjugate minimal representation As we have seen in the previous section, the additional generators that need to be added to the two bosonic W1+1 algebras transform in bi-minimal representations with respect to these two algebras. For the following it will be important to describe these representations from the viewpoint of the a ne Yangian. Recall from [19, 20] that the de ning relations of the a ne Yangian can be written as (2.27) (2.28) (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) and e(z) f (w) f (w) e(z) = 1 3 (z) z w (w) ; e(z) e(w) f (z) f (w) (z) e(w) (z) f (w) '3(z '3 1(z '3(z '3 1(z w) e(w) e(z) w) f (w) f (z) w) e(w) (z) w) f (w) (z) ; where ` ' means equality up to terms that are regular at z = 0 or w = 0. Here the elds are expanded in terms of modes as e(z) = 1 X j=0 zje+j1 ; f (z) = 1 X j=0 zfj+j1 ; (z) = 1 + 3 1 X j=0 zj+j1 ; and the function '3(z) is de ned by '3(z) = (z + h1)(z + h2)(z + h3) (z h1)(z h2)(z h3) = z3 + 2z z3 + 2z + 3 : 3 The hi parameters satisfy h1 + h2 + h3 = 0, and we have de ned 2 = h1h2 + h2h3 + h1h3 ; 3 = h1h2h3 : The structure of these OPEs can be summarised by the diagram of gure 1. 4We have done this up to q10. It should not be too hard to prove this analytically, but we have not attempted to do so. { 7 { ϕ3(Δ) e ϕ3(Δ) ψ f In terms of the conformal eld theory language, the hi parameters and As is explained in [19, 20], the di erent states of the vacuum representation are described by plane partitions where the eigenvalues of i on the con guration are given by (z) = 1 + '3(z h( )) ; corresponding states are still given by (3.11), except that now the in nite product (over the in nitely many boxes de ning the asymptotic con guration) must be suitably regularized. 3.1 The minimal representation The simplest non-trivial representations are the minimal representations whose asymptotic box con guration consists of a single row of boxes extending along either x1, x2 or x3. For our analysis above, the minimal representation corresponding to an asymptotic single box in the x2 direction will play a central role.5 Its ground state has the charges (u) = 1 + = 1 + Expanding out in inverse powers of u, this is of the form (u) = 1 + 5It is the one whose conformal dimension equals (2.27) in the classical limit. Note that selecting out x2 breaks the S3 symmetry of the a ne Yangian to Z2. { 8 { Comparing with (3.6) we read o 1 = 1 h2 ; As will become clear from the free eld analysis of the next section, the above minimal representations will not su ce. In fact, the bi-minimal representations that appear in (2.19) have the property that they are minimal with respect to one W1 algebra, but conjugate minimal (or anti-minimal) with respect to the other. Thus in order to describe these generators we also need to understand how to describe the conjugate minimal representation from the a ne Yangian perspective. In the following we shall make a general proposal for how this works. charges given by Given a box representation described by '(u) (where again '(u) does not include the vacuum factor 0(u)), we claim that the corresponding conjugate representation has The sign cance of this shift is that it turns the vacuum factor 0(u) = (1 + 0u 3 ) into Thus the full eigenvalue function of the anti-box representation is 0(u) ' 1( u 0 3) = 0(v) '(v) 1 ; where v = u 0 3 , and therefore indeed just the inverse of (u) = 0(u)'(u). (Note that the shift (and sign) transformation from u to v is just the spectral ow (and scaling) automorphism of the a ne Yangian, see e.g. sections 2.2 and 2.3 of [19].) We can check this proposal explicitly by checking whether the rst few W0s charges change correctly | recall that for the conjugate representation the charges of the odd spin W0s generators must have the opposite sign, while the even spin generators are the same. Suppose then that '(u) describes a given representation with eigenvalues j, i.e. we have the expansion sentation is (1 + 0u 3 ) '(u) = 1 + u Then, according to the above proposal, the power series expansion of the conjugate repreThis predicts that the conjugate representation has the charges 0 = 0 ; 1 = 1 ; 2 = 2 ; 3 = { 9 { (3.16) (3.17) (3.18) (3.19) (3.20) (3.21) HJEP05(218) Together with the form of the spin s = 3 charge from [20] it follows that the value of W03 on the conjugate representation equals as expected. This is a fairly non-trivial consistency check of this proposal. In particular, for the conjugate minimal representation from above we nd while for s 3 we nd the simple closed form expression s = h 2 s 2 (1 As an aside, the above analysis now allows us to check the nite N and k corrections to the conformal dimension (2.28). According to [19, 20], the conformal dimensions with respect to the coupled theory (where the u(1) generator has not been removed) equals, see eq. (3.15) h = 1 2 2 = 1 2 1 0h1h3 = 1 + 1 2 N N + k where we have used the dictionary (3.10). Note that this is true both for the minimal representation, as well as the anti-minimal representation, see eq. (3.24). We should also mention in passing that the decoupled conformal dimension is then, see eq. (5.67) of [19] hdec = 2 1 2 2 1 = and hence agrees indeed with the conformal dimension of h(f; 0) in the coset, see, e.g., eq. (2.13) of [13]. For the problem at hand, however, we should work with the coupled conformal dimension (since we are dealing with W1+1 rather than just W1). Furthermore, for the N = 2 construction, we consider two plane partitions that correspond to and 1 , i.e., for the second W1 algebra we should exchange the roles of N and k. Note that, according to the dictionary of eq. (3.10), this only a ects 0, but not the values of hi. Thus we shall work with the same values of hi for both a ne Yangians, but distinguish the two a ne Yangians by setting 0 = N ; ^0 = k : ; ; (3.22) (3.23) (3.24) (3.25) (3.26) (3.27) (3.28) Note that, using (3.10), this relation can be written as or equivalently as If we then add (3.26) to the corresponding expression with N $ k, we nd altogether 1 2 1 2 htot = 1 as desired. This is another highly non-trivial consistency check for this construction to work also at the quantum level. It may also be worth noting that the Z2 Z2 symmetry of the W1(N =2) algebra [32] has a nice geometric interpretation in this setting. First of all, the S3 symmetry of each of the two a ne Yangians is broken down to a common Z2 symmetry that exchanges the x1 and x3 direction. In addition, there is the symmetry exchanging the roles of the two a ne Yangians, which corresponds to the N $ k transformation. 3.3.1 Representations One can similarly understand how the two minimal N = 2 representations appear from the above bosonic viewpoint. The relevant representations have conformal dimensions N 2(N + k + 1) k 2(N + k + 1) h(f; 0; N ) = ; and h 0; f; (N + 1) = ; (3.32) see, e.g. eq. (3.9) of [32]. From the above perspective, these representations correspond to the representation that has an in nite row of boxes along the x1 direction for either of the two plane partitions. Indeed, it follows from [19, 20] that the relevant conformal dimensions are (cf. eq. (3.26)) and h = 1 2(N + k + 1) : Note that these conformal dimensions are higher than those in (3.32), with the di erence in both cases being equal to This is the contribution of the overall u(1) generator that was added in (2.16). In general, the representations of the N = 2 a ne Yangian will therefore be described by in nite box con gurations extending in the x1 and x3 direction for both plane partitions. (The representations where the boxes extend along the x3 direction are again the non-perturbative representations, in analogy to what happens in the bosonic case, see [37].) (3.29) (3.30) (3.31) For the following it will sometimes be important to compare our ansatz with an explicit free eld realization. Recall that at = 0 (or = 1), the W1 (N =2) algebra has a free eld construction in terms of free complex fermions and bosons. More explicitly, the neutral bilinears in the fermions, i.e. the elds of the form generate W1+1[0], while the neutral bilinears in the bosons, i.e. the elds of the form j j (4.1) (4.2) (4.3) (4.4) HJEP05(218) give rise to W1[1]. (Here j and j are the complex fermions, and j and j are the complex bosons.) On the other hand, the fermionic generators are linear combinations of the form and j j From the viewpoint of the two bosonic W1 algebras, i.e. W1+1[0] and W1[1], these generators transform in the `bi-minimal' representation. Indeed, the fermion eld j corresponds son (with respect to W1+1[0]) to the representation (f; 0) in the coset language, while the boeld @ j describes (with respect to W1[1]) the representation (f; 0). Thus the two fermionic elds above correspond to the states in (f; 0) (f; 0) and (f; 0) (f; 0) ; respectively. (Here we have used that the complex conjugate fermion transforms in (f; 0), and correspondingly for the complex conjugate boson.) In terms of the description in terms of plane partitions, this means that the rst fermionic generators act as an addition of an in nite row of boxes with respect to the rst plane partition | the one corresponding to W1+1[0] | while it acts as an addition of an in nite row of anti-boxes with respect to the second plane partition | the one corresponding to W1[1]. 4.1 The a ne Yangian generators at = 0 We can use this free eld realisation to construct the relevant a ne Yangian generators for this special case, and work out their commutation and anti-commutation relations. In the next section we shall explain how to modify these relations as we move away from = 0. We recall from [20] (see also [19]) that for the bosonic W1+1[0] algebra, the corresponding a ne Yangian generators can be de ned as r = er = fr = X X m+1 im : ; and m2Z i m2Z i xs = X X( m where the free fermion modes are denoted by im and i . As was shown there, these m generators satisfy the a ne Yangian algebra of [19] for 2 = 1 and 3 = 0 with 3 0 = 0. In terms of the hi parameters, this corresponds to the case see eq. (3.10) above. We also need a description for the a ne Yangian generators associated to W1[1], and they are given as ^r = e^r = f^r = X X m2Z i X m2Z i X X m2Z i X( m)r 1 : i 1 m mi : m + 1 r 1 : 1i m mi : : (m + 1)( m)r 2 + m + 1 r 1 : i This leads to the a ne Yangian with hi being given by (4.8); the only di erence to the sense for e^r, f^r with r case of W1[0] above is that now 1 and ^r with r 2. However, we can at least formally extend these de nitions to include also e^0 and ^1 by setting i 0 with m = 0 from all of these expressions. (Similarly, we could de ne f^0 by setting 0i The generator e^0 is then the 1 mode of a non-local eld with spin 1. One checks by an explicit calculation that it satis es the correct commutation relation with the ^r modes, 0, i.e., by dropping the term in particular (see eq. (4.13) of [20]) [ ^1; e^0] = 0 ; [ ^2; e^0] = 2e^0 ; [ ^3; e^0] = 6e^1 2e^0 : For the fermionic generators we now make the ansatz (4.5) (4.6) (4.7) (4.8) (4.9) (4.10) (4.11) (4.12) (4.13) (4.14) where s = 12 ; 32 ; : : :, and we de ne the generating functions by i.e. the term with s = 12 corresponds to the supercharge, etc. Obviously, the algebra also contains the corresponding +3=2 modes, which we may de ne via conjugation as and and where the corresponding generating functions are x(z) = 1 X xs z s 1=2 ; s=1=2 x(z) = 1 X xs z s 1=2 : s=1=2 xs W (s3+=21)+ ; and xs and xys = ys : Their treatment is similar to that of the xr and xr generators, and is therefore relegated to appendix A. It is now straightforward to work out the commutation and anti-commutation relations of these generators. For example, one nds (4.15) (4.16) (4.17) (4.18) (4.19) (4.20) (4.21) (4.22) (4.23) (4.24) (4.25) (4.26) (4.28) (4.29) as well as 0 = [ r+2; xs] 2[ r+1; xs+1] + [ r; xs+2] + [ r+1; xs] [ r; xs+1] 0 = [er+1; xs] [er; xs+1] + [er; xs] 0 = [fr+1; xs] [fr; xs+1] ; 0 = [ r+2; xs] 2[ r+1; xs+1] + [ r; xs+2] [ r+1; xs] + [ r; xs+1] 0 = [er+1; xs] [er; xs+1] [er; xs] 0 = [fr+1; xs] [fr; xs+1] : On the other hand, the commutation relations with the hatted modes are 0 = [ ^r+2; xs] 2[ ^r+1; xs+1] + [ ^r; xs+2] 3[ ^r+1; xs] + 3[ ^r; xs+1] + 2[ ^r; xs] (4.27) 0 = [e^r+1; xs] [e^r; xs+1] 2[e^r; xs] 0 = [f^r+1; xs] [f^r; xs+1] [f^r; xs] and In addition, there are the initial conditions Note that the hatted generators only start with ^2, i.e. the modes ^0 and ^1 are initially not de ned in (4.9). We have added the mode ^1 by hand | the result also agrees with what one obtains upon extending the de nition of ^r in (4.9) to r = 1, see the comments above | and de ned it so that it satis es (4.27) and (4.30) for r 1. However, (4.27) and (4.30) are then not compatible with [ ^0; xs] = 0 and [ ^0; xs] = 0. The reason for this will become clear below: the deformed relations, see appendix B.2, contain an additional contribution that survives (for r = 0) since 1. With this correction term the ^ above results are then also compatible with the recursion relations of appendix B.2 for r = 0. For the unhatted modes, these problems do not arise, and in fact the commutator with 2 is determined using (4.21) and (4.24) with r = 0 as [ 2; xs] = xs 2xs+1 ; [ 2; xs] = xs + 2xs+1 : (4.34) Finally, for the anti-commutator of the xs and xr we nd 0 = fxi+2; xjg 2fxi+1; xj+1g + fxi; xj+2g + fxi+1; xjg fxi; xj+1g 2fxi; xjg : (4.35) It is also convenient to de ne with the initial condition that fxr; ysg = Pr+s ; P1 = 1 2 where N is the number of complex free bosons and fermions. The Pr modes satisfy a number of relations that are also spelled out in appendix A. 4.2 Identifying the representations The discussion around eq. (4.4) suggests that the fermionic generators transform in a minimal representation with respect to one W1 algebra, but in the conjugate minimal with respect to the other. We can now verify this also more explicitly. Let us rst analyse the generators described by x. The eigenvalues of r on the state i 1=2j0i | this is the relevant state for the description of x1=2j0i | equals 1 = 1 ; 2 = 1 ; 3 = 1 ; (4.38) (4.30) (4.31) (4.32) (4.33) (4.36) (4.37) where the rst few r generators are explicitly, see eq. (4.5) 1 = 1, h2 = 1 and h3 = 0 (with 3 0 = 0) this agrees then with the charges of the minimal representation, see (3.15). On the other hand, the charges of the state i of xs) are 1(x) = 1 ; 2(x) = 1 ; 3(x) = 1 : These are not the charges of the minimal representation, but rather that of the conjugate minimal representation. Indeed, evaluating (3.24) for h1 = 1, h2 = 1, h3 = 0 with 3 0 = 0 we nd 1 = 1, 2 = 1 and 3 = 1, which reproduces indeed (4.42). Incidentally, the situation is precisely reverse with respect to the hatted modes. In that case, we need to evaluate the charges 1=2j0i (that is relevant for the description ^2 = 2 X X : i First consider the state i 1j0i (that is relevant for the xs modes), for which we nd This is then of the form (3.15) with h1 = 1, h2 = 1, h3 = 0 and 3 ^0 = 1; thus the xr generators transform in the minimal representation with respect to the hatted modes. On the other hand, on the state see eq. (4.13), the charges equal i 1j0i that is relevant for the description of the xs modes, ^2(x) = 2 ; ^3(x) = 4 : Now, this does not correspond to the minimal representation, i.e. it does not match (3.15) with h1 = 1, h2 = 1, h3 = 0 and 3 ^0 = 1, but rather corresponds to the conjugate minimal representation, i.e., it agrees with (3.24) for h1 = 1, h2 = 1, h3 = 0 with 1. The situation for the y and y generators is similar; we have summarized the representation properties of these generators in the table 1. 5 The Yangian at generic parameters Next we want to make a proposal for how the algebra should be deformed away from the special point = 0, see eq. (4.8). Our guiding principle is that, with respect to the two bosonic a ne algebras, denoted by Y and Y^ respectively in the following, the fermionic generators sit in \bi-minimal" representations. generator unhatted modes Y x x y y minimal conj. minimal conj. minimal minimal hatted modes Y^ conj minimal minimal minimal conj. minimal The generators in minimal representations Let us begin with studying the generators that transform in minimal (rather than conjugate minimal) representations. As we have seen above, the generator xs transforms in the minimal representation of Y. By analogy with the construction of the bosonic a ne Yangian, the operation of adding xs should therefore change the eigenvalue of the modes by | this is (u) in eq. (3.13) without the \vacuum" factor 0(u) = (1 + 0u 3 ) HJEP05(218) This then suggests that (4.21) should become '2(u) = whose modes | this can be deduced as in [20], see the discussion around eq. (2.12) there | then satisfy 2[ r+1; xs+1] + [ r; xs+2] + h2 [ r+1; xs] [ r; xs+1] + h1h3 rxs = 0 : (5.3) Note that this reduces to (4.21) for h2 = 1 and h1h3 = 0. Before we proceed further, we can test this proposal by working out the charges of the state that is created by x1=2 from the vacuum. Recall that on the vacuum state the bosonic and fermionic modes satisfy eij0i = i;0e0j0i ; xij0i = i;1=2 x1=2j0i ; xij0i = i;1=2 x1=2j0i : (5.4) We want to con rm that the state x1=2j0i has the charges of the minimal representation of Y. We postulate that the initial condition (4.33) is now modi ed to Then it follows that 0 x 12 j0i = N x 12 j0i 1 x 12 j0i = 1 h2 x 12 j0i : (5.1) (5.2) (5.5) (5.6) (5.7) In order to determine the higher charges we deduce from (5.3) that (h1 + h3), the last equation can now be rewritten as These charges then agree precisely with eq. (3.15), thus con rming that our ansatz (5.2) leads to states with the correct charges. We also make the ansatz where here and in the following e(z) x(w) f (z) x(w) G( ) and H( ) are functions that will be constrained further below, see section 5.3. Other minimal generators The construction works similarly for the generators yr, which behave like the conjugate operators to xr, i.e. they are like the fr modes relative to er in the bosonic a ne Yangian. Because of that we expect them to satisfy the inverse OPE, cf. eq. (3.4) and (3.5) Given the simple relation between (5.2) and (5.14), we postulate that also the y-analogues of (5.11) and (5.12) only involve simple inverses. The structure of the OPEs of x and y with respect to the unhatted elds can then be summarized by the diagram of gure 2. The analysis is completely analogous for the two elds x and y with respect to the hatted elds, and the structure of the corresponding OPEs can thus be similarly realized, see gure 3. 5.2 The generators in conjugate minimal representations A more interesting case are the OPEs of the unhatted elds with x and y, or equivalently, that of the hatted elds with x and y. For concreteness, let us describe the former case in G(Δ) ϕ2(Δ) x ϕ3(Δ) ψ ϕ3−1(Δ) f H(Δ) H(Δ) H−1(Δ) G(Δ) ϕ2(Δ) eˆ ϕ3(Δ) ϕ3(Δ) ˆ ψ ϕ3−1(Δ) ˆ f detail. Since the charges of the conjugate minimal representation are described by (3.16), the natural ansatz for the OPE is thus By the usual arguments, this then leads to the commutation relations [ r+2; xs] 2[ r+1; xs+1] + [ r; xs+2] + ( h2 + 2 0 3) [ r+1; xs] [ r; xs+1] which reduces indeed to (4.24) in the free eld limit. Again, before proceeding further, we should check that this gives the correct charges on the corresponding states. In analogy to (5.5) we now postulate (5.15) (5.16) (5.17) HJEP05(218) ϕ3(Δ) ϕ3−1(Δ) G−1(−Δ − σx3ψ0) ϕ2−1(−Δ − σ3ψ0) H−1(−Δ − σ3ψ0) ψ ˆ ψ ϕ3(Δ) ϕ3−1(Δ) f ϕ2−1(Δ) f y¯ y x¯ G(Δ) gures 2 and 3 for comparison. Then we nd, using (5.16) = (1 h1h3 0)x1=2j0i 1 x1=2j0i = x1=2j0i 1 h2 2 x1=2j0i = [ 2; x1=2] j0i = (h2 2 0 3)[ 1; x1=2] j0i + h1h3 0x1=2 j0i 3 x1=2j0i = (h2 thus giving the correct charges of the anti-minimal representation, see eq. (3.24). The structure of the corresponding OPEs can therefore be summarized as in gure 4. The situation for the hatted elds with respect to x and y is completely analogous and summarized in gure 5. 5.3 The OPEs with e and f Since the eld appears in the OPE of the e and the f eld, see eq. (3.1), we can also deduce constraints on the OPE of the e and f eld with x from that with . To this end, { 20 { G(Δ) x ϕ3(Δ) ϕ3−1(Δ) ψ ˆ ψ f y G−1(−Δ − σx¯3ψˆ0) ϕ2−1(−Δ − σ3ψˆ0) H−1(−Δ − σ3ψˆ0) H(−Δ − σ3ψˆ0) ϕ2(−Δ − σ3ψˆ0) G(−Δ − σ3ψˆ0) ϕ3(Δ) ϕ3−1(Δ) gures 2 and 3 for comparison. we recall the ansatz from eqs. (5.11) and (5.12) Note that, just like the identities of eqs. (3.2) { (3.5), these relations cannot be exactly correct, but are only true up to terms that are regular at either z = 0 or w = 0, see the discussion around eq. (5.15) in [20]. Applying this identity twice we nd that e(z1) f (z2) x(w) f (z2) e(z1) x(w) G(z1 G(z1 w) H(z2 w) H(z2 w) x(w) e(z1)f (z2) w) x(w) f (z2) e(z1) : Subtracting the two equations from one another and using (3.1) we thus deduce that (z1)x(w) (z2)x(w) z1 z2 G(z1 w) H(z2 w) x(w) (z1) z1 z2 x(w) (z2) : Next we apply (5.2) to the left-hand-side, from which we deduce that this equals Thus it follows that '2(z1 w)x(w) (z1) z1 '2(z2 z2 w)x(w) (z2) : '2(z1 z1 w) z2 '2(z2 z1 w) z2 G(z1 w) H(z2 w) z1 z2 : (5.21) (5.22) (5.23) (5.24) (5.25) (5.26) (5.27) HJEP05(218) Because these identities are only true up to regular terms, this implies that the functions G( ) and H( ) have to satisfy the identity The most natural ansatz that is compatible with the free eld limit (see below) is G( ) H( ) = '2( ) : G( ) = + h2 h1 ; H( ) = h3 ; which reduces indeed to the correct free eld answers, see eqs. (4.22) and (4.23). However, this ansatz cannot be right since there are two box-descendants of the state generated by x1=2j0i | this follows from the bosonic structure of the minimal representation | and hence the function G( ) must have two poles [38]. In fact, one can use the representation theory to constrain the function G( ) (and hence H( )) further, but this goes beyond the scope of the present paper and will be described elsewhere [38]. Our starting point in section 2 was the W1(N =2)[ ] algebra, and we can now try to identify its generators with those of the supersymmetric a ne Yangian. The vacuum character, see eq. (2.14), contains two spin s = 1 elds: the u(1) generator of the N = 2 superconformal algebra, as well as the extra bosonic generator that we added by hand to the W1 vacuum character (2.15). This free boson should be completely decoupled, and its zero mode be identi ed with the central generator U0 1 + ^1 : Obviously, U0 commutes with all er, fr, e^r and f^r generators, and because of the relations we have imposed, it also commutes with the xs and xs generators. We also know that the total Mobius generators correspond to L 1 = e1 + e^1 ; L1 = f1 f^1 ; L0 = 2 + ^2 ; 1 2 and it is thus natural to assume that the 1 modes of the decoupled boson are The u(1) generator of the N = 2 algebra, on the other hand, should be identi ed with U 1 = e0 + e^0 ; U1 = f0 + f^0 : J0 = ( 3 ^0) 1 ( 3 0) ^1 ; In terms of commutators this leads to HJEP05(218) and J 1 = ( 3 ^0) e0 ( 3 0) e^0 ; J1 = ( 3 ^0) f0 + ( 3 0) f^0 : Then [Jm; Un] = 0 for m; n 2 f0; 1g, and we nd c(N =2) 3 m m; n ; with c(N =2) = 3 ( 3)2 0 ^0 ( 0 + ^0) : In terms of the dictionary of [20], this central charge then equals c(N =2) = 3N k ; (5.37) (5.38) (5.39) (5.40) (5.41) in agreement with the N = 2 central charge, see eq. (2.11). The other bosonic generators can be similarly identi ed: for each integer s, there are two decoupled bosonic elds, see the discussion below (2.12), and they can be identi ed with the a ne Yangian generators of the two bosonic a ne Yangians, using the dictionary of [20]. This leaves us with the fermionic generators. The lowest fermionic generators are the supercharge generators Gr , which, at the free point, can be identi ed with It would be natural to postulate this identi cation also for generic hi. However, there is a problem with this proposal. The N = 2 generators should commute with the decoupled free boson described by Un. But even at the free point one nds The origin of this problem is that the e^0 generator corresponds to the see also the discussion below eq. (4.11). The fact that e^0 is non-local at the free point is an artefact of the free limit | for generic , both e0 and e^0 describe the 1 mode of local elds, as follows from the discussion in [20]. One may therefore suspect that the fact that [U0; x1=2] 6= 0 is purely a free- eld artefact, but this is not the case: as we show in appendix C, with the above identi cations, this problem persists for generic . We believe that the resolution of this problem is that we need to correct the identi cation (5.39) by (non-local) correction terms. The fact that such non-local correction terms appear is maybe not surprising in view of the fact that also in the bosonic setting non-local correction terms were required for the identi cation of the spin 3 and 4 elds, see [20]. The relevant analysis is, however, rather cumbersome, and we leave it to future work. 6 In this paper we have found some of the de ning relations of the Yangian algebra that is expected to be isomorphic to W1 [ ], the N = 2 superconformal version of W1 We have extensively used the fact that W1(N =2)[ ] contains two commuting bosonic W1 algebras, each of which in turn is isomorphic to an a ne Yangian of gl1. The additional generators transform in bi-minimal representations with respect to these two W1 gebras. We have shown how this translates into explicit commutation relations for the additional Yangian generators | our main technical advance is the description of the conjugate representations, see section 4.2. This has allowed us to make a proposal for at least some of the de ning relations of the N = 2 superconformal a ne Yangian. We have also checked | in fact this was an important guiding principle | that these relations reduce to the expected identities in the free eld case ( = 0). There are many open questions which we hope to address in the future. First of all, it would be nice to construct the representation theory of this Yangian algebra, see [38] for rst steps in this direction; this will involve two plane partitions on which the various generators should have some natural action. (The two a ne Yangians of gl1 act separately on each, while the additional bi-minimal generators generate in nite rows of boxes (and anti-boxes), connecting the two plane partitions.) Among other things, this would allow us to prove the consistency of our construction and to nd the remaining relations. It would also be interesting to establish the dictionary to the W1 generalizing the construction of [20] to the current context, and to explore the various duality symmetries this picture suggests. Note that the construction selects out one of the three directions of each plane partition, thereby breaking the S3 symmetry [37] of each a ne Yangian of gl1 to a Z2 symmetry that exchanges the remaining two directions. Together with the exchange symmetry of the two a ne Yangians, the N = 2 a ne Yangian therefore has a Z2 Z2 symmetry. We hope to come back to these questions in the near future. (N =2)[ ] algebra in more detail, Acknowledgments We thank Tomas Prochazka and Miroslav Rapcak for sending us [39] prior to publication, and Tomas Prochazka for comments on the draft. We also thank Yang Lei for help with drawing the gures. The work of MRG is (partly) supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation. WL is grateful for support from the \Thousand talents grant" and from the \Max Planck Partnergruppe". The work of CP is supported by the US Department of Energy under contract DE-SC0010010 Task A. HZ is partially supported by the General Financial Grant from the China Postdoctoral Science Foundation, with Grant No. 2017M611009. MRG thanks Beijing University for hospitality during the very nal stages of this work. We gratefully acknowledge the hospitality of the Galileo Galilei Institute for Theoretical Physics (GGI) for hospitality, and INFN for partial nancial support during the program \New Developments in AdS3/CFT2 Holography". A Additional relations of the free eld theory In addition to the free eld relations that were given in the main body of the text, see eqs. (4.21){(4.33), the commutation relations of ys and ys are 0 = [ r+2; ys] 2[ r+1; ys+1] + [ r; ys+2] + [ r+1; ys] [ r; ys+1] 0 = [er+1; ys] [er; ys+1] (A.1) (A.2) 0 = [ r+2; ys] 2[ r+1; ys+1] + [ r; ys+2] [ r+1; ys] + [ r; ys+1] are given explicitly by as well as (for r 2) Pr = Pr = X X The Pr modes that were de ned in eq. (4.36), and their corresponding barred version 0 = [Pi+2; xj] 2[Pi+1; xj+1] + [Pi; xj+2] + [Pi+1; xj] [Pi; xj+1] 2[Pi; xj] 0 = [Pi+2; yj] 2[Pi+1; yj+1] + [Pi; yj+2] + [Pi+1; yj] [Pi; yj+1] 2[Pi; yj] 0 = [Pi+2; ej] 2[Pi+1; ej+1] + [Pi; ej+2] ([Pi+1; ej] [Pi; ej+1]) 0 = [Pi+2; fj] 2[Pi+1; fj+1] + [Pi; fj+2] ([Pi+1; fj] [Pi; fj+1]) 0 = [Pi+2; e^j] 2[Pi+1; e^j+1] + [Pi; e^j+2] + 3([Pi+1; e^j] [Pi; e^j+1]) + 2[Pi; e^j] 0 = [Pi+2; f^j] 2[Pi+1; f^j+1] + [Pi; f^j+2] + 3([Pi+1; f^j] [Pi; f^j+1]) + 2[Pi; f^j] 0 = [Pi+2; xj] 2[Pi+1; xj+1] + [Pi; xj+2] [Pi+1; xj] + [Pi; xj+1] 2[Pi; xj] 0 = [Pi+2; yj] 2[Pi+1; yj+1] + [Pi; yj+2] [Pi+1; yj] + [Pi; yj+1] 2[Pi; yj] (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) (A.10) (A.11) (A.12) (A.13) (A.14) (A.15) (A.16) (A.17) (A.18) (A.19) (A.20) (A.21) HJEP05(218) 0 = [Pi+2; f^j] 2[Pi+1; f^j+1] + [Pi; f^j+2] ([Pi+1; f^j] [Pi; f^j+1]) 0 = [Pi+2; ej] 2[Pi+1; ej+1] + [Pi; ej+2] + ([Pi+1; ej] [Pi; ej+1]) 0 = [Pi+2; fj] 2[Pi+1; fj+1] + [Pi; fj+2] + ([Pi+1; fj] [Pi; fj+1]) : B The de ning relations Yangian. B.1 The OPE like description In this appendix we collect some of the de ning relations of the supersymmetric a ne HJEP05(218) (A.22) (A.23) (A.24) (A.25) (B.1) (B.2) (B.3) (B.4) (B.5) (B.6) (B.7) (B.8) where '2(u) is de ned in eq. (5.1). For the y elds we nd In terms of modes, these identities are [ r+2; xs] 2[ r+1; xs+1] + [ r; xs+2] + h2 [ r+1; xs] [ r; xs+1] + h1h3 rxs = 0 [ ^r+2; xs] 2[ ^r+1; xs+1] + [ ^r; xs+2] (h2 2 ^0 3) [ ^r+1; xs] [ ^r; xs+1] + (h1 + ^0 3)(h3 + ^0 3)[ ^r; xs] h1h3 ^rxs = 0 [ r+2; xs] 2[ r+1; xs+1] + [ r; xs+2] + ( h2 + 2 0 3) [ r+1; xs] [ r; xs+1] [ r+2; ys] 2[ r+1; ys+1] + [ r; ys+2] + h2 [ r+1; ys] [ r; ys+1] + ^0 3( ^0 3 h2)[ ^r; ys] + h1h3 ^rys = 0 [ ^r+2; ys] 2[ ^r+1; ys+1] + [ ^r; ys+2] (h2 2 ^0 3) [ ^r+1; ys] [ ^r; ys+1] [ r+2; ys] 2[ r+1; ys+1] + [ r; ys+2] (h2 2 0 3) [ r+1; ys] [ r; ys+1] For completeness, we also give the mode relations of the bosonic generators [19, 20], which we write only for the unhatted generators (but which hold similarly also for the hatted generators). 3fej; ekg = [ej+3; ek] 3[ej+2; ek+1] + 3[ej+1; ek+2] [ej; ek+3] 3ffj; fkg = [fj+3; fk] 3[fj+2; fk+1] + 3[fj+1; fk+2] [fj; fk+3] 3f j; ekg = [ j+3; ek] 3[ j+2; ek+1] + 3[ j+1; ek+2] [ j; ek+3] 3f j; fkg = [ j+3; fk] 3[ j+2; fk+1] + 3[ j+1; fk+2] [ j; fk+3] In addition, they satisfy the Serre relations B.3 The initial conditions The modi ed initial conditions, generalizing (4.33) are Sym(j1;j2;j3)[ej1; [ej2; ej3+1]] = 0 ; Sym(j1;j2;j3)[fj1; [fj2; fj3+1]] = 0 : Furthermore, and Finally, the initial relations of the bosonic generators are (B.9) (B.10) (B.11) (B.12) (B.13) (B.14) (B.15) (B.16) (B.17) (B.18) (B.19) Supercharge constraints We start by assuming that the decoupled u(1) generator commutes with the x1=2 generator, which we would like to identify with the supercharge, see eq. (5.39), We will deduce two identities from this assumption, using the commutation relations we have postulated. First we consider For the left-hand side we then use from which it follows, using again (C.1), that tator with 3 + ^3, = 2[(e0 e^0); x1=2] = 4[e0; x1=2] : The other identity can be derived similarly, except that now we consider the commu[[( 3 + ^3); x1=2]; (e0 + e^0)] = [[( 3 + ^3); (e0 + e^0)]; x1=2] + [[(e0 + e^0)]; x1=2]; ( 3 + ^3)] = [[( 3 + ^3); e0 + e^0]; x1=2] : Now the relevant charge relations are where d is some constant, and This then leads to the identity [ 3 + ^3; x1=2] = (10 + 2h1h3 0)x3=2 + d x1=2 (10 + 2h1h3 0)[x3=2; e0 + e^0] = 6[e1 + e^1; x1=2] + 2 3 0[e0; x1=2] + 2 3 ^0[e^0; x1=2] = 6[ 3 ^0 e0 3 0 e^0; x1=2] + 2 3 0[e0; x1=2] +2 3 ^0[e^0; x1=2] where we have also used that [e1 + e^1; x1=2] = [L 1; x1=2] = [J 1; x1=2] = [ 3 ^0 e0 3 0 e^0; x1=2] ; (C.1) (C.2) (C.3) (C.4) (C.5) (C.6) (C.7) (C.8) N N + k 1 ; 7! 1 (C.10) one may HJEP05(218) as follows from the fact that x1=2 G+3=2 is the mode of a primary eld of spin 3=2 with charge +1, i.e., from the relations of the N = 2 superconformal algebra Gm+r ; r Gm+r : (C.9) n 2 The two identities (C.4) and (C.7) are only compatible provided that (10 + 2h1h3 0) = (4 4h1h3 0) ; i.e. h1h3 0 = = 0, the analogue of (5.40) vanishes, [e0 + e^0; x1=2] = 0. is then at which corresponds to the case = 1. 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Matthias R. Gaberdiel, Wei Li, Cheng Peng, Hong Zhang. The supersymmetric affine Yangian, Journal of High Energy Physics, 2018, 200, DOI: 10.1007/JHEP05(2018)200