Higher-rank fields and currents

Sp(2M) invariant field equations in the space ℳM with symmetric matrix coordinates are classified. Analogous results are obtained for Minkowski-like subspaces of ℳM which include usual 4d Minkowski space as a particular case. The constructed equations are associated with the tensor products of the Fock (singleton) representation of Sp(2M) of any rank r. The infinite set of higher-spin conserved currents multilinear in rank-one fields in ℳM is found. The associated conserved charges are supported by rM−rr−12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{r}M-\frac{\mathrm{r}\left(\mathrm{r}-1\right)}{2} $$\end{document}-dimensional differential forms in ℳM , that are closed by virtue of the rank-2r field equations. The cohomology groups Hp(σ−r) with all p and r, which determine the form of appropriate gauge fields and their field equations, are found both for ℳM and for its Minkowski-like subspace.


Introduction
As originally observed by Fronsdal [1], infinite towers of massless fields in four dimensions admit an extension of the conformal algebra su (2,2) to sp(4) allowing a description in the generalized matrix space M 4 with symmetric matrix coordinates X AB = X BA (A, B = 1, . . . , 4). This observation was farther elaborated in [2][3][4]. The rank-one Sp(8)-invariant unfolded equation describing massless fields of all spins is [4] where Y A are auxiliary commuting variables that will be referred to as twistor variables.
To simplify formulae we use notation ξ AB for anticommuting differentials dX AB . Note that eq. (1.1) admits an interesting interpretation in terms of world-like particle models of [2,3], providing also a field-theoretical realization of the observation of

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Fronsdal [1] that the infinite tower of all 4d massless fields enjoys Sp (8) symmetry. More generally M M is the space with local coordinates X AB which are symmetric M × M matrices. Sp(2M ) invariant unfolded field equations corresponding to rank-r tensor products of the Fock (singleton) representation of Sp(2M ) were introduced in [5] where these equations were argued to describe "branes" of different dimensions in the Sp(2M ) invariant generalized space-time.
In [5], the case of rank-two equations where was considered in detail. In particular, all rank-two dynamical (primary) fields and field equations were found and it was shown that dynamical equations for most of the rank-two fields have the form of conservation conditions for conserved currents found in [6], which give rise to the full set of bilinear conserved charges in the rank-one theory. Rank-r unfolded equations are δ kj (k, j = 1, . . . , r) . (1.5) In this paper the analysis of [6] is extended to the fields and equations of arbitrary rank. Namely we find all dynamical fields, which are primary fields from the conformal field theory perspective, along with the explicit form of their field equations. It is shown that, similarly to the rank-two case, some of these fields give rise to differential forms that are closed by virtue of their field equations, thus generating conserved currents.
The sp(2M ) invariant field equations are appropriate for description of infinite towers of massless fields that appear in higher-spin theories. For M = 2, 4, 8, 16, the Sp(2M ) invariant field equations describe towers of conformal massless fields in usual Minkowski spaces of dimensions d = 3, 4, 6 and 10, respectively [3,7,8]. Pattern of Sp(2M ) invariant field equations for other values M so far has not been analysed including the case of M = 32 which is most interesting in the context of M theory.

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notations, and ξ αα ′ = dx αα ′ are anticommuting differentials. Equations (1.6) decompose into an infinite set of subsystems for fields of different helicities h C(µy, µ −1ȳ |x) = µ 2h C(y,ȳ|x) . (1.7) The space with coordinates x αβ ′ with α = 1, . . . , K and α ′ = 1, . . . , K for any K we call generalized Minkowski space M M nk 2K . The rank-r generalization of (1.6) is for any Hermitian form η kj , k, j = 1, . . . , r , α, α ′ = 1, . . . , K. Though interpretation of these equations for higher K from the perspective of usual Minkowski space embedded into M M nk 2K demands more detailed analysis which is beyond the scope of this paper, we briefly comment on the cases of K = 2, 4 and 8 being reductions of the Sp(2M ) systems with M = 4, 8 and 16, respectively. 1 The case of K = 2 gives the genuine 4d Minkowski space. The case of K = 4 results from the reduction of the Sp(16) invariant system. The latter was shown in [7,8] to describe conformal massless fields in the 6d Minkowski space carrying spinning degrees of freedom valued in SU (2). As emphasized in [8] this implies that the original Sp(16) invariant system describes an infinite tower of 6d massless fields of all spins such that the multiplicity of a spin s is 2s + 1 coinciding with the dimension of the spin-s representation of the spinning SU (2). Coordinates x αβ ′ can be interpreted as the part of the coordinates X AB that are invariant under the action of one of the generators H in the spinning su (2), that acts on the primed and unprimed indices by the conjugated phase transformations. As a result, the irreducible subsystems in M M nk 8 are characterized by sl(2) helicities h. Each of these systems contains an infinite tower of conformal fields in which every spin s ≥ |h| appears once (s is related to the length of the rectangular Young diagram associated with a 6d conformal field in Minkowski space).
The case of K = 8 results from the reduction of the Sp(32) invariant system. The latter describes a tower of conformal massless self-dual fields of all spins in the 10d Minkowski space [8] such that each spin appears once. Now the reduction of the Sp(32) invariant system to M M nk 2K has different interpretation. Since indices of the coordinates X AB of M 16 are associated with 10d chiral spinors, the coordinates x αβ ′ cannot be obtained by a 10d Lorentz covariant projection from X AB . In other words, the helicity-like operator distinguishing between primed and unprimed indices cannot be 10d Lorentz covariant. Hence, the M M nk 16 setup must break manifest 10d Lorentz covariance describing the system in a Minkowski space of some lower dimension d < 10. The simplest option is to identify the helicity-like generator with one of the 10d rotations, considering the system in eight dimensions. An interesting alternative possibility to be explored is to embed a 9d Minkowski space in M M nk 16 using another helicity operator that would partially act in the spinning space. In both of these cases the 10d conformal invariance will be a kind of hidden and the JHEP10(2016)067 same time the self-dual 10d fields will be traded for lower-dimensional fields not restricted by the self-duality conditions. We hope to elaborate details of this analysis elsewhere.
In this paper we analyse the pattern of Minkowski-like equations with general r and K from the perspective of the generalized space M M nk 2K with no reference to the underlying physical space. However, since in the particular case of K = 2 the space M M nk 4 is the usual 4d Minkowsky space, this allows us to derive all conformal primary currents in the four-dimensional Minkowski space that are built from 4d massless fields of all spins. (These results have been already announced and used in [10] for the analysis of the operator algebra and correlators of conserved currents in four dimensions.) We expect that results of the present paper may have applications in the context of AdS/CF T holography [11][12][13] and especially, higher-spin holography (see, e.g., [14][15][16][17][18][19][20][21][22][23][24][25] and references therein) because, as emphasized in [26], the duality between fields in higher dimensions and currents in lower dimensions to large extent amounts in the language of this paper to the duality between lower-rank fields in M M with higher M and higher-rank fields in M M with lower M .
Another interesting application is to the analysis of multiparticle amplitudes (see e.g. [27] and references therein). The key fact here is that the product of r rank-one solutions gives a solution to the rank-r equation. In these terms an r-particle amplitude represents a solution to the rank-r system. From this perspective the constructed conserved multiparticle charges may be of most interest since amplitudes supported by such charges can be represented as multiple integrals independent of local variations of the integration cycle. In this language the nontrivial dynamics of a model in question should be hidden in the so-called parameters which in our formalism are arbitrary functions of certain twistor variables, which parameterize different amplitudes. We hope to come back to a more detailed analysis of this issue in a future publication. The analysis of the pattern of dynamical fields and their field equations is performed in the σ − -cohomology language which is analogous (and in many cases equivalent) to the search of singular vectors of conformal modules. Specifically, zero-form fields and their field equations are classified by H 0 (σ − ) and H 1 (σ − ), respectively. In this paper, we find all cohomology groups H p (σ − ) with p ≥ 0. These results determine the form of appropriate gauge fields and their field equations both for M M and for its Minkowski-like subspace M M nk M . The rest of the paper is organized as follows. Section 2 contains our Young diagram conventions and some their properties. In particular, the structure of differential forms in M M is analyzed here. Section 3 presents the full list of dynamical fields and field equations of any rank-r in M M . Details of derivation of the equations of motion as well as the form of multilinear conserved currents in M M are also given here. Section 4 contains summary of our results for conformal fields and their field equations both in the usual four-dimensional

Young diagrams
. . ≥ l m ) obeys symmetry properties of the Young diagram (YD) Y(l 1 , . . . , l m ) with manifest symmetrization provided that A is symmetric with respect to permutations of l k indices B k 1 , . . . , B k l k of any k−th row, while the symmetrization over l k + 1 indices B k 1 , . . . , B k l k , B p j yields zero for any p > k. Such tensors are conventionally denoted A B 1 (l 1 ) ,...,B m (lm) .
Analogously, a tensor A height of the k−th column of Y is denoted h k (Y) or simply h k , while a length of the k−th row of Y is denoted l k (Y) or l k . Let us stress that components of a tensor with symmetry properties of any Young diagram with manifest symmetrization are linear combinations of the components of a tensor with symmetry properties of the same Young diagram with manifest antisymmetrization and vise versa. Weight |Y| equals to the number of cells of a YD Y[h 1 , . . . , h p ], i.e., (2.1) Any Young diagram can be represented as the unification of its elementary cells S(i , j) on the intersection of its i − th row and j − th column, i.e., Y = S(i,j)∈Y S(i, j). For any cell S(i , j) and parameter a ∈ R we introduce the characteristic function χ a (S(i , j)) = (j − i + a) . (2.2) For any set of cells A we introduce its characteristic function 2 Considering a diagram Y[h 1 , . . . , h k ] as the unification of either its columns H j (h j ) or rows L j (l j ) we obtain Hence, the right-hand sides of (2.4) and (2.5) are equal for any YD Y. Cells of a Young diagram can be ordered as follows: for any two cells S 1 and S 2 , . (Note that this definition is insensitive to a.) Definition (2.3) implies that, with respect to this ordering, for any two cells the smaller is situated South-West to the larger.
Thus the partial ordering can be introduced in particular for Young diagrams containing equal numbers of cells.
The symmetrized tensor product of a number of rank-two symmetric tensors δ ij described by the YD Y[1, 1] decomposes into a linear combination of tensors with the symmetry properties (2.6) This is because antisymmetrization of indices i 1 , i 2 , . . . , i d , say, in δ i 1 j 1 . . . δ i d j d implies antisymmetrization of indices j 1 , j 2 , . . . , j d . Components of the symmetrized tensor product of a number of Y[1, 1] will be referred to as Kronecker diagrams.
On the other hand, as shown below, the antisymmetrized tensor product of tensors with the symmetry properties of Y [1,1] contains various tensors with the symmetry properties of almost symmetric Young diagrams Y A defined as follows. For any Young diagram invariant under reflection with respect to the diagonal, one cell should be added to each row that intersects the diagonal. Pictorially, shading the added cells, The simplest one is the almost symmetric hook of height h
Any almost symmetric diagram Y A (2.8) admits a nested hook realization, namely Y A is a unification of a set of shifted almost symmetric hooks Y(k, a k ) where Y(k, a k ) is the almost symmetric hook shifted by k −1 cells down along the diagonal. More precisely, (2.12) Note that by definition h n+1 (Y nest {a 1 , . . . a n }) = n := ♯(Y nest {a 1 , . . . a n }) . (2.14)

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Indeed, by definition (2.2) any pair of cells symmetric with respect to the diagonal does not contribute to χ 0 , hence for an almost symmetric shifted hook Y(k, h k ) (2.11) Summation over all shifted almost symmetric hooks gives (2.14).
It is convenient to introduce block hooks of the form Y nest {m, m − 1, . . . , m − n n+1 } with arbitrary integer m > n. Pictorially, A general almost symmetric diagram can be treated as nested block hooks consisting of p block hooks where n i is the number of hooks in the i th block and n is the total number of hooks in Y A . Let ξ AB be anticommuting differentials The coefficients of differential forms belong to the space of antisymmetric tensor products of Y [1,1]. To show that they are described by almost symmetric diagrams we observe that anticommutativity of the differentials implies that the projection of ξ AB ξ CD to the window diagram is zero From here it follows that the product of n variables ξ antisymmetrized over n indices has the symmetry of the hook Y[n, 1, . . . , (2.20) However the one containing window is zero by (2.19). Due to antisymmetrization in every column, this implies that any Young diagram associated with a differential form has the nested hook structure, i.e., is almost symmetric.
are reconstructed in terms of (X-derivatives of) the primary fields that satisfy In the rank-one case, the primary fields are [4] C(X) := C(0|X) , The symmetry properties of C and C A are represented by the Young diagrams • (empty diagram) and , respectively. The following equations hold as a consequence of unfolded equations (1.1): and, in particular, The symmetry properties of the left-hand sides of equations (3.5) and (3.6) are represented by the Young diagrams and , respectively. In the language of σ − -cohomology [29] (see also [30]) convenient for the analysis of the pattern of zero-form higher-rank fields, JHEP10(2016)067 primary fields and their field equations are represented by the cohomology groups H 0 (σ 1 − ) and H 1 (σ 1 − ), respectively. Hence, the structure of rank-one fields and field equations is represented by the following diagrams Note that eqs. (3.5), (3.6) are the only independent equations obeyed by the primary (=dynamical) fields as a consequence of (1.1). Such equations will be referred to as dynamical.
Rank-two unfolded equations (1.2) are most conveniently analyzed in terms of variables (3.9) The following homogeneous differential equations hold as a consequence of (3.9) where arbitrary rank-m totally antisymmetric tensors ε A 1 ,...,Am are introduced to impose appropriate antisymmetrizations. As shown in [5], the rank-two primary fields are Dynamical equations for the primary fields are [5]
One observes that in the examples of ranks one and two, dynamical fields associated with H 0 (σ r − ) are such that the total number of indices in the first two columns of the respective Young diagrams does not exceed r.
Another property illustrated by these examples is that all columns starting from the third one of the Young diagrams Y 0 and Y 1 associated with H 0 (σ r − ) and H 1 (σ r − ) are equal, while the first two columns are such that together they form a rectangular two-column block of height r + 1, i.e., We say that a pair of Young diagrams are rank-r two-column dual if they obey (3.14). The rank-one and two examples suggest that dynamical fields and their field equations are described by rank-r two-column dual Young diagrams. As we explain now this is indeed true for fields of any rank.

Any rank
Rank-r unfolded equations (1.4) are sp(2M ) symmetric by the general argument of [5]. Indeed, it is well known (for more details see e.g. [4]) that any system of equations of the form where C(X) is some set of p-forms taking values in a g-module V and the one-form ω(X) = dX A ω A (X) is some fixed connection of g obeying the flatness condition

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Note that the oscillator representation provides a standard tool for the study of representations of sp(2M ) [31,32] (and references therein). Rank-r equations (1.4) have the form (3.15). The operator σ r − (1.5) obeys which implies that the corresponding connection is flat. System (1.4) as well as generators (3.17) are invariant under the action of o(r) on the color indices i, j, . . . = 1, . . . r, that leaves invariant δ ij . Generators of o(r) obey the standard commutation relations Being mutually commuting, o(r) and sp(2M ) form a Howe-dual pair [33]. As in the lower-rank cases, rank-r primary fields obey the condition In terms of the expansion

22) implies tracelessness of the component fields with respect to color indices
Since Y A i commute, the tensors C i 1 ;...; in A 1 ;...;An (X) are symmetric with respect to permutation of any pair of upper and lower indices. Hence C i 1 ;...; in A 1 ;...;An (X) can be decomposed into a direct sum of tensors forming irreducible representations of o(r) as well as of gl M with the same symmetry properties described by some YD Y(l 1 , . . . , l m ) (l 1 + . . . + l m = n). Abusing terminology, a gl(M ) YD associated with a traceless tensor with respect to color indices will be referred to as traceless YD.
Recall that if a traceless tensor with respect to color indices taking r values has symmetry of Y[h 1 , . . . , h m ], it can be nonzero only if As shown in section 5.2, for any rank-r primary field C ∈ H 0 (σ r − ) associated with a YD Y 0 (n 1 , . . . , n m , 1, . . . , 1 q ) that obeys (3.26), the left-hand side of the dynamical equation

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has the symmetry properties of the rank-r two-column dual YD Y 1 (n 1 , . . . , n m , 2, . . . , 2 Note that for M < r some of them may be zero which would mean either that the corresponding primary field is absent (Y 0 = 0) or that it does not obey any dynamical equations (Y 1 = 0), i.e., the system is off-shell.
Dynamical equations are most conveniently described in terms of Young diagrams with manifest antisymmetrization. For nonnegative integers in the upper ones. So defined Y 0 and Y 1 obey two-column duality condition (3.14). Let E, which plays a role of projector to the respective tensor representations, be traceless with respect to the color indices. It is not difficult to see that, for any E (3.28), the following equations hold as a consequence of (1.4) where

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Indeed, by virtue of (1.4), eq. (3.29) is equivalent to (3.31) resulting from the total antisymmetrization of indices i and j. Then The proof follows from the determinant formula applied to the double dual tensorG Expressing the product of two totally antisymmetric symbols in terms of Kronecker symbols one observes that for m+n > r−p at least one pair of indices of the traceless tensor F i[m] ,j[n] is contracted hence giving zero. Note that eq. (3.33) at p = 0 yields eq. (3.25).
Since the differential operator E Y 0 (3.30) is homogeneous, the primary fields also satisfy (3.29). The parameter (3.28) is designed in such a way that eq. (3.29) is nontrivial only for primaries C(Y |X) with the symmetry properties of Y 0 [h 1 , . . . , h n ]. The nontrivial part of the story is to prove that the presented list of dynamical equations is complete. This follows from the analysis of the cohomology group H 1 (σ r − ) in section 5. As anticipated, there is precise matching between the primaries and field equations. This means that the respective subspaces of H 0 (σ r − ) and H 1 (σ r − ) form isomorphic o(r)modules. Here one should not be confused by the fact that the two-column dual diagram Y 1 does not respect condition (3.25). The point is that, as an o(r)-tensor, Y 1 is not JHEP10(2016)067 traceless, containing explicitly a number of o(r) metric tensors δ ij which add additional o(r) indices to Y 1 compared to Y 0 .
The particular case of (3.29) which plays the key role in the construction of conserved currents is that with r = 2κ and a traceless o (3.36) Then operator E Y 2κcur 0 (3.30) is of the first order in ∂ ∂X . Hence, eq. (3.29) acquires the form of conservation condition For instance, in the particular case of two-column diagrams this yields a straightforward generalization of the conservation condition of [6] 3.2 Higher σ r − -cohomology in M M As shown in [29] (for more detail see [5,30]), when the fields C(Y |X) satisfying (1.4) are pforms, dynamical fields and their field equations are associated with H p (σ − ) and H p+1 (σ − ), respectively. Recall that H p (σ r − ) = ker(σ r − )/Im(σ r − ) p , where p denotes the restriction to p-forms. Hence, we are interested in σ r − -closed p-forms P (ξ, Y |X), that are not σ r − -exact. In the expansion the coefficients C i 1 ;...;in A 1 ;...;An | C 1 ;...;C 2p (X) form almost symmetric diagrams (2.8) with respect the indices C 1 ; . . . ; C 2p contracted with the differentials ξ AB and are symmetric with respect to permutation of any pair of upper and lower indices A k , i k . Hence, with respect to the latter indices, C i 1 ;...;in A 1 ;...;An | C 1 ;...;C 2p can be decomposed into a direct sum of tensors forming irreducible representations of o(r) as well as of gl M with the same symmetry properties described by some YD Y ′ .

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Differently from the zero-form case, the coefficients C i 1 ;...;in A 1 ;...;An |C 1 ;...;C 2p (X) in eq. (3.40) are not necessarily traceless with respect to color indices. In addition to the indices of a o(r)-traceless YD, the diagram Y ′ can contain indices carried by products of δ ij . The respective gl(M ) tensor contains products of the "tracefull" combinations that, as mentioned in section 2, are described by the Kronecker diagrams (2.6). As a result, P (ξ, Y |X) satisfying (2.6) has symmetry properties that described by direct sum of the following tensor products of YD for different m, n such that |Y A | = 2p. Analysis of the higher cohomology with p > 1 is also interesting in many respects. For instance, it characterizes further relations on the left-hand sides of field equations as well as gauge fields in higher-spin gauge theories, described by higher differential forms. In this section we present the final results for H p (σ r − ) leaving details of their derivation to section 5 and appendix A.
H p (σ r − ) is realized by a space of homogeneous polynomials P (ξ, Y ) of degree p in ξ, forming an o(r)-module with respect to the color indices i carried by Y A i and a gl Mmodule with respect to the spinor indices A, B, . . . carried by Y A i and ξ AB . To avoid the restriction that the maximal height of gl M -Young diagrams is M , we will assume that M is large enough. In the final result one should simply take into account that such diagrams are zero. On the other hand, the rank r is not supposed to be large because nontrivial cohomology is just associated with special values of r.

Main results
It turns out that the cohomology groups H p (σ r − ) are characterized by various o(r)-modules and gl M -modules formed by the differentials ξ AB . The former can be described by a traceless o(r)-diagram Y 0 while the latter by an almost symmetric gl M -diagram Y A . Given Y 0 and Y A , there exists some tensor with the symmetry of a Kronecker YD Y δ (2.6), composed of the Kronecker symbols δ ij , and some component To identify Y ′ it is convenient to introduce the infinite almost symmetric matrix S = S(r|Y 0 ), that has the form are the Cartan weights of Y 0 and Note that here all zero-height columns of Y 0 are in the game, i.e., Y 0 is treated as a generalized Young diagram with an infinite number of columns. Elements of matrix (3.43) obey For any almost symmetric Y A we introduce the shift matrix S sh = S sh (r|Y A , Y 0 ) resulting from the intersection of the infinite almost symmetric matrix S(r|Y 0 ) (3.43) with Y A . In other words, elements of the shift matrix S sh are (3.48) The central result is

50)
and S sh i j are elements of the shift matrix S sh (r|Y A , Y 0 ) (3.47).

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The proof of Theorem 1 is given in appendix A using the homotopy trick explained in section 5.
As shown in appendix, the Kronecker YD Y δ [d 1 , d 1 , . . . , d m , d m ] can be expressed via H j (3.50) as follows For the case of one-forms described by the simplest nested hook Y A = Y[1, 1] this gives S sh In accordance with section 3.1.2 and definition (3.14), Y ′ is two-column dual to Y 0 .

Multilinear currents in M M
The construction of conserved currents in terms of closed forms proposed in [5,6] for rank-one fields admits a higher-rank generalization.
That dynamical degrees of freedom associated with the rank-one equations (1.1) live on a M -dimensional surface S ⊂ M M ⊗ R M suggests that conserved charges associated with these equations have to be built in terms of M -forms that are closed as a consequence of rank-two field equations (1.2). As shown in [35], the following is closed provided that J 2 (z,z |X) solves (1.2) with z,z (3.8). Usual conserved currents are built in terms of bilinears in solutions C j of (1.1) with parameters η that commute with 10,26].
Since modules of solutions of the rank-r equations in M M ⊗ R rM are functions of rM variables Y A i one might guess that in the rank-r case the dimension of a "local Cauchy bundle" [7] on which initial data should be given to determine a solution everywhere in M M is rM . One can see however that the following straightforward generalization of (3.53) to conserved currents in M M ⊗ R 2rM with arbitrary r is bilinear in rank-r fields C r i , does not work. The reason is that, although these forms are closed by virtue of the rank-2r unfolded equations, their pullback to M M is zero for r > 1 thus obstructing the construction of conserved charges in the form of integrals over M M .
Indeed, up to a numerical factor, the pullback of the form (3.54 (3.56) Any Young diagram associated with the combination of differentials ξ AB can only contain rows of lengths l ≤ M + 1 since otherwise it would contain symmetrization of a pair of differentials. On the other hand, because of ε-symbols, it contains r columns of the maximal height M associated with the indices A in (3.56). Anticommutativity of the differentials implies that the part of the diagram associated with the indices B should contain r total symmetrizations. However, since the first r columns are occupied by the indices A and the total number of symmetrized indices cannot exceed M + 1, this is not possible for r > 1 as at least two symmetrized indices B would belong to the same column, which implies antisymmetrization. Thus (3.56) is zero for r > 1. The reason why the naive construction does not work is that the o(r), that acts on the primary (dynamical) rank-r fields, relates different solutions to each other not affecting evolution of any given solution. This suggests that the dimension of the true local Cauchy bundle is rM − 1 2 r(r − 1). Correspondingly, the respective currents should be represented by closed r(M − 1 2 (r − 1))-forms. As shown in section 2, products of anticommuting variables ξ AB have symmetries of almost symmetric Young diagrams. The product of r totally antisymmetric tensors with 1 ≤ r < M admits a unique extension to such a tensor with (2rM −r(r − 1)) cells described is closed provided that a rank-2r field J 2r (current) obeys the rank-2r current equation , which is the first-order differential equation with respect to X derivatives, just implies that the projection to this diagram is zero. Hence, the form Ω 2r (J) (3.59) is closed on shell.
In particular, closed form (3.59) with where C j (Y j |X) solve rank-one equations (1.4), generates 2r-linear conserved currents. As in the case of bilinear currents [10], the first-order differential operators A 1 (Y |X) and A 2 (Y |X) Therefore, for any η j 1 ,...,j 2r (A), also obeys current equation (3.60), giving rise to the 2r-linear charges Q 2r η where η j 1 ,...,j 2r (A) are parameters of the symmetries generated by these multilinear charges, which are analogous to the multiparticle symmetries considered in [36].
More generally, consider X AB (4.1) with α = 1, . . . , K and α ′ = 1, . . . , K for any K. The rank-r generalization (1.8) of (1.6), is a subsystem of (1.4) with M = 2K, We set η kj = δ kj in (1.8), setting In these terms, the algebra o(r) extends to u(r) acting on the conjugated representations carried by lower and upper color indices, while sp(2M ) reduces to u(K, K) that acts on spinor indices and commutes with the u(r). Note that u(r) and u(K, K) contain the common central element.
Because of the symmetry with respect to exchange of pairs of indices, Young diagrams for color indices i(j) and spinor-like indices α(α ′ ) must have the same shape. with tensors C that obey the mutual tracelessness condition (4.5) and have definite symmetry properties in y andȳ described by a pair of Young diagrams As a consequence of (4.5) and Lemma 1 on pg. 14 at p = 0, the latter have to obey the condition In the Minkowski case, a pair of Young diagrams Y 0 and Y 0 (4.7) will be called rank-r two-column dual to the pair Y 1 (n 1 , . . . , n m , 1, . . . , 1 q ) and Y 1 (n 1 , . . . , nm, 1, . . . , 1  in the color indices and their two-column dual in the spinor ones. The following equation holds as a consequence of rank-r unfolded equations (1.8): Indeed, by virtue of (1.8), eq. (4.15) is equivalent to  ]. This yields zero by virtue of mutual tracelessness of tensor E (4.11) and Lemma 1 on pg. 14.
Since E (4.15) is a homogeneous differential operator, eq. (4.14) is obeyed by primary fields (4.6). The nontrivial part of the analysis is to show that eq. (4.14) yields the full list of dynamical equations. This follows from the analysis of the cohomology groups H 0 (σ r

4d Minkowski space
The dictionary between the tensor and two-component spinor notations is based on where σ αβ ′ a (a = 0, 1, 2, 3) are four Hermitian 2 × 2 matrices. Let us list the 4d Minkowski primary fields and their field equations for various r.
As follows from the analysis of [9], the r = 1 primary fields are C(x) , C(y|x) , C(ȳ|x) . These have symmetry properties described by the following pairs of Young diagrams The consequences of (1.8) for r = 1 impose the equations on primaries (4.18) The symmetry properties of the left-hand-sides of these equations are described by the following pairs of Young diagrams Consider the rank-two case. To obey (4.8), in the case of M = 4, r = 2 primary fields (4.6) belong to the following list C(y|x) , C(ȳ|x) , C(y,ȳ|x) Since primary fields have to satisfy (1.8), to describe C(y,ȳ|x) (4.22) it is convenient to use the language of highest-weight modules. Solutions to (1.8) form u(r)-modules, where the algebra u(r) spanned by the generators commutes with σ r − M nk (4.2) . Evidently, ρ 1 2 C(y 1 ,ȳ 2 |x) = 0 and ρ 2 1 k C(y 1 ,ȳ 2 |x) with 0 ≤ k ≤ max(n,n) form the full list of primary fields with symmetry properties (4.25). Note, that these results were used in [10] to describe free 4d conformal primary currents.
Equations of motion are projected by tensors (4.11). Since α j and α ′ j take just two values, to be nonzero at M = 4, r = 2, these should have h 1 =h 1 = 1 and h j ≤ 1,h j ≤ 1 in YDs (4.12), (4.13). Hence, the primary fields associated with diagrams (4.23) and (4.24) are off-shell obeying no field equations, while those associated with diagrams (4.25) satisfy These equations have symmetries of Y 1 = and Y 1 = . For r ≥ 3, dynamical fields (4.6) are described by Young diagrams (4.7) with at most two rows. Since H 1 (σ r − M nk ) is empty for r ≥ 3, rank-r ≥ 3 primary fields are off-shell. Due to anticommutativity of differentials, the differential forms f (ξ αβ ′ ) are described by pairs of mutually transposed Young diagrams Y and Y related by the reflection with respect to the diagonal, i.e., Such pairs of diagrams, whose role is analogous to that of almost symmetric YDs Y A (2.8) for the Sp(2M ) invariant sytems, will be denoted as Y a , Y a , describing symmetry properties of f (ξ αβ ′ ) with respect to unprimed and primed indices, respectively.

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As sketched in section 5, H p (σ r − M nk ) is characterized by a pair of mutually transposed Young diagrams Y a and Y a such that |Y a | = |Y a | = p and a pair of Young diagrams Y 0 and Y 0 obeying the condition h 1 (Y 0 ) + h 1 (Y 0 ) ≤ r.
To describe cohomology groups we introduce the following infinite shift matrix S n m (r|Y 0 , Y 0 ) S n m = △ m−n for n < m , n ≥ 1 , (4.29) S n n = △ 0 , The cohomology group H p (σ r − M nk ) consists of the following pairs of gl K ⊕gl K diagrams  Our analysis generalizes those of [5], where the rank-two case was considered, and of [34], where conformal field equations in Minkowski space of any dimension were obtained. The main tool is the standard homotopy trick. Let a linear operator Ω act in a linear space V and satisfy Ω 2 = 0. By definition, H(Ω) = ker Ω/Im Ω is the cohomology space. Let Ω * be another nilpotent operator, (Ω * ) 2 = 0. Then the operator ∆ = {Ω, Ω * } (5.1) satisfies [Ω, ∆] = [Ω * , ∆] = 0 . From (5.1) it follows that ∆(ker Ω) ⊂ Im Ω. Therefore H(Ω) ⊂ ker Ω/∆(ker Ω). Suppose now that V is a Hilbert space in which Ω * and Ω are conjugated. Then ∆ is positive semi-definite. If the operator ∆ is also quasifinitedimensional, i.e., V = ⊕V A with finite-dimensional subspaces V A such that ∆(V A ) ⊂ V A and V A is orthogonal to V B for A = B, then ∆ can be diagonalized and it is easy to see that ker Ω/∆(ker Ω) = ker ∆ ∩ ker Ω. Therefore, in this case, This formula is particularly useful for the practical analysis since, to calculate H(Ω), one can use the Homotopy equation

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An important property of ∆ r (5.11) is that it is positive semi-definite because Ω r and Ω * r are conjugated in the positive-definite Fock space generated by Y A i and ξ AB as creation operators.
By virtue of (3.19), (5.6) and (5.7), ∆ r can be represented in the form where are generators of the gl M that act on Y A i . By virtue of (3.17)-(3.21) an elementary computation yields where τ mn are o(r) generators (3.20). Hence are generators of the gl tot M that act on indices A, B, . . . carried by both Y A i and ξ AB . The first and second terms on the r.h.s. of (5.15) are the quadratic Casimir operators of the algebras gl tot M and o(r), respectively. The computation in terms of fermionic oscillator realization of generators of the algebra gl M , that makes the antisymmetrization manifest, yields (see e.g., [34]) As a result for some N . Here the pairs of symmetric indices carried by U AB (3.41) and ξ AB are symmetrized and antisymmetrized, respectively.

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Analogous computation for the orthogonal algebra yields Hence, eq. (5.12) yields By virtue of eqs. (2.4) this yields Since χ a is additive, from (5.21) it follows that where summation in χ Note that since equation (5.23) is independent of M , the assumption of section 3.2 that M is large enough to avoid the restriction on the maximal height of gl M -Young diagrams does not restrict the generality.
Analogously to [34], to find the complete set of solutions to the homotopy equation one can use the positive semi-definiteness of ∆ r . Indeed, nontrivial cohomology can appear at ∆ r = 0 which is its minimal possible value. Equation (2.3) implies that for a given set of cells of a gl(M )-diagram, the minimal value of χ 1 2 (r−1) (Y ′ \ Y 0 ) is reached when all cells are situated maximally South-West, i.e., possible values of i and j in (2.3) are maximized and minimized, respectively. This minimization will be referred to as South-West (SW) principle.
For a given traceless YD Y 0 the minimization occurs with respect to the cells associated with the differentials ξ AB forming an almost symmetric YD Y A (2.8) and the tracefull blocks U AB (3.41) forming a YD Y δ (2.7). Any solution Y ′ of (5.23) is given by for an appropriate Y δ , where SW Y 1 ⊗ Y 2 ⊗ . . . denotes maximally South-West components (∼ SW-diagrams) of the tensor product Y 1 ⊗ Y 2 ⊗ . . .. For given Y 0 and Y A , the problem is to find Y δ (2.7) such that the maximally South-West component of the tensor product Y 0 ⊗ Y δ ⊗ Y A solves (5.23).

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The details of the analysis of the Homotopy equation are given in appendix A. However, for the final results presented in section 3.2 it is easy to check that Y ′ (3.49) solves homotopy equation (5.23). Indeed, substitution of Y ′ (3.49) into (5.23) yields using (2.4) . . a n } (2.10), using the nested structure of the nested shift matrix (3.48) along with eq. (3.46), the l.h.s. of (5.25) can be rewritten in the form where One can see that A mk = 0 for all k, m by virtue of (3.47) while B mk also vanishes. The operators form gl K (C) while the operators Setting we obtain using (5.30), (4.26) where are u(r)-generators (4.26). The sum of the first two terms and the third one on the r.h.s. of (5.32) are the quadratic Casimir operators of gl K (C) and u(r), respectively. The u(r)-generators ρ n m (4.26) commute with u(K, K) (5.30). Hence u(r) acts on solutions of the higher-rank equations which form u(r)-modules. Irreducible u(r)-modules satisfy the mutual tracelessness conditions (4.4) equivalent to T α β ′ P (y,ȳ, ξ|x) = 0 . (5.33) An elementary computation in terms of fermionic oscillator realization of u(r), that makes the anti-symmetrization manifest, yields with the help of mutual tracelessness condition (5.33)  Analogously to section 5.1, in addition to indices associated with the given mutually traceless Y 0 and Y 0 , the diagrams Y ′ and Y ′ contain indices of the differentials ξ αβ ′ and of the u(r) invariant tracefull combinations Hence where, as before, summation in χ Results of this paper raise a number of interesting problems. One of the most interesting is to study conserved charges generated by the constructed currents. A single conserved current is expected to generate many different charges upon integration with different global symmetry parameters η. (For example, a traceless stress tensor generates the full conformal algebra being integrated with the parameters of translations, Lorentz rotations, dilatations and special conformal transformations.) See, e.g., [10] for the analysis of this issue for rank-two conserved currents. It is therefore important to find the full space of the corresponding symmetry parameters η leading to independent charges. An interesting peculiarity of this analysis is that, as shown in [35] for the case of rank two, to obtain nonzero charges in Minkowski subspace of M M it is necessary to consider parameters η that are singular in some of coordinates in M M transversal to Minkowski space. It is therefore necessary to find what is an appropriate singularity of η in the general case of any rank that gives rise to non-zero conserved charges.
Another peculiarity is that, being multilinear in the dynamical fields, the charges resulting from the proposed currents cannot be represented as integrals in usual Minkowski space, requiring integration over a larger space like M M or its product with the twistor space. Nevertheless, being nonlocal from the perspective of Minkowski space, the charges are well-defined and should form some algebra. An interesting question is what is this algebra and, specifically, what is its relation to the multiparticle algebra proposed recently in [36] and [10] where it was shown, in particular, that the usual bilinear (ı.e., rank-two) currents give rise to a set of charges that forms the higher-spin algebra.
A very interesting possible application of the obtained results can be related to the analysis of multi-particle amplitudes in QFT. Indeed, the higher-rank fields considered in this paper can be interpreted as being associated with the asymptotic states in the scattering processes which are on shell, i.e., obey free field equations. The idea is to associate the constructed conserved charges with the amplitudes. Of course, this will not allow us to determine amplitudes exactly since for this it is necessary to determine the parameters η in (3.65) which remain arbitrary functions of the respective twistor variables in our approach and have to be determined by the dynamics of a nonlinear model in question. However, this will determine the amplitude kinematics.
The reason why we believe that the amplitudes should be associated with the higherrank conserved charges is that, being represented by integrals of on-shell closed forms, in this case they will only be determined by certain singularities independent of local variations of the integration contour. This interpretation is not only very similar to the what happens in the Grassmanian computations (see e.g. [37,38] and references therein) but can open a unique opportunity for establishing explicit relation between the space-time computational schemes and those in the twistor space. To this end the constructed conserved charges should be extended to differential forms in the correspondence space unifying space-time with the twistor space. This program was initiated in [35] for general tensor products. We anticipate that this construction allows an extension to irreducible conserved charges considered in this paper, integrated over the full correspondence space with appropriate JHEP10(2016)067 singular parameters analogously to the construction of Minkowski currents from those in M 4 presented in [35]. We leave this exciting problem for the future.
Finally, the analysis of higher cohomology performed in this paper may have applications to the construction of equations of motion of higher gauge theories associated with higher-rank fields. The latter are related to multiparticle states in the original field theoretic model. Such higher gauge theories are likely to be related to string-like higher-spin gauge theories.

Acknowledgments
We are grateful to Nathan Berkovits for hospitality at ICTP-SAIFR Sao-Paulo where a part of this work was done. Also we thank the organizers and participants of winter 2014 Kavli Institute for Theoretical Physics in Santa Barbara program "New Methods in Nonperturbative Quantum Field Theory", where this work was continued, for creation of friendly and productive atmosphere and stimulating conversations. The authors are grateful to Leonid Bork for the stimulating discussion and Yegor Goncharov for pointing out a typo in the draft. The first version of this work was supported in part by RFBR To solve homotopy equation (5.23) we observe that at sufficiently large M a solution Y ′ (5.24) to (5.23) has the form Indeed, since there are no constraints on the heights of components of Y 1 ⊗Y 2 at sufficiently large M for given gl M -diagrams Y 1 and Y 2 , Y 1 ⊗ SW Y 2 is nonzero and maximally South-West. Note that a similar argument cannot be applied to the construction of Y ′′ since Y 0 ⊗ SW Y δ can be zero by Lemma 1 (see pg. 14). Hence, eq. (5.23) takes the form

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From (A.1) and (A.2) it follows that for any YD Y and almost symmetric YD Y A As a result, using additivity of χ a (2.3), condition (A.5) amounts to where Since ∆ r is positive semi-definite, eq. (A.6) implies that, for given Y 0 and Y ′′ , a solution to (A.7) demands Y A to minimize the last term, i.e., if Y A solves (A.7) then } that consists of two block hooks.
The black cells of the left YD exemplify an addition of a pair to form the third block hook, while the crossed cells of the right YD can be removed from the first block hook to produce another almost symmetric diagram. Another useful example is a rectangular block.

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Now let us show that the addition of cells to Y ′′ on the left from the (n + 1)-th column decreases χ 1 2 (r−1) . Let Y n 1 ,n 2 ,...n k + := Y ∪ S(h n 1 (Y) + 1, n 1 ) ∪ . . . ∪ S(h n k (Y) + 1, n k ) (A. 20) for different n i . We add here k cells into n 1 , n 2 , . . . n k -th columns of Y at the condition that the resulting diagram exists. Let Q j 1 ,...,j k + := χ 1 2 (r−1) Properties of χ a (2.3) and eq. (A.1) imply that Q j 1 ,...,j k + = k i=1 Q j i + and where h j = h j (Y A ). One can see that (j − 1 − h j ) ≤ 0 for j ≤ n + 1 because h j ≥ h n+1 while h n+1 = n (cf (2.13)). Hence, taking into account eq. (A. 19), The following statements that hold by virtue of Lemma 1 and Young properties of tensor products will be used below. Any nonzero component of the tensor product Y ∈ Y 0 ⊗ Y δ has to satisfy We prove Lemma 2 by induction. To this end we observe that if inequalities (A.27) hold for ∀k ≤ j with some j ≤ m then from the property that heights of columns of a YD do not increase from the left to the right it follows that if d l = d i for some l < i ≤ j then Y ′′ m,2j+2 + ∈ Y 0 ⊗ Y δ 2n−1,2n + in contradiction with the assumption of Lemma 2 thus completing the proof. where the starting point depends on the oddness of n ≡ N p and on whether a p − n p is zero or not. There are two reasons for this. Firstly there is a difference between diagrams with rectangular (a p = n p ) and not rectangular (a p > n p ) inner block hooks that gives different numbers of inequalities on H in these two cases. Secondly, 2m = N p + 1 for odd n and 2m = N p for even n, that leads to different relations between H j and H 2m+i . Namely for even n and a p > n p

It is convenient to introduce notations
contradicts the assumption that Y ′ solves eq. (5.23).

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