Higher-Rank Fields and Currents

$Sp(2M)$ invariant field equations in the space ${\cal M}_M$ with symmetric matrix coordinates are classified. Analogous results are obtained for Minkowski-like subspaces of ${\cal M}_M$ 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 ${\mathbf{r }}$. The infinite set of higher-spin conserved currents multilinear in rank-one fields in ${\cal M}_M$ is found. The associated conserved charges are supported by $({\mathbf{r }} M-\frac{{\mathbf{r }} ({\mathbf{r }} -1)}{2})-$dimensional differential forms in ${\cal M}_M$, that are closed by virtue of the rank-$2{\mathbf{r }}$ field equations. The cohomology groups $H^p(\sigma^{\mathbf{r }}_-)$ with all $p$ and ${\mathbf{r }}$, which determine the form of appropriate gauge fields and their field equations, are found both for ${\cal M}_M$ and for its Minkowski-like subspace.


Introduction
In [1], Sp(2M) invariant unfolded field equations corresponding to rank-r tensor products of the Fock (singleton) representation of Sp(2M) were introduced. These equations were shown to describe "branes" of different dimensions in the Sp(2M) invariant generalized space-time M M with local coordinates X AB which are symmetric M × M matrices. In [1], the case of rank-two 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 [2], which give rise to the full set of bilinear conserved charges in the rank-one theory.
Here, this analysis 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.
Also we consider a similar problem in the Minkowski-like reduction of the Sp(2M) covariant setup in which the generalized Minkowski space has local coordinates x α,α ′ with α, α ′ = 1, 2, . . . K. In particular, at M = 4, 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 [3] 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 [4,5,6] and especially, higher-spin holography (see, e.g., [7,8,9,10,11,12,13,14,15,16,17] and references therein) because, as was emphasized in [18], 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.
The rest of the paper is organized as follows. Section 2 contains summary of our results, presenting the full lists of dynamical fields of any rank-r in M M and generalized Minkowski space M M nk M (including usual four-dimensional Minkowski space), along with their field equations. The form of multilinear conserved currents in M M is also presented here. Section 3 contains details of the derivation of equations of motion and the analysis of the sigma-minus cohomology for rank-r fields and field equations in M M . In Section 4 a structure of differential forms in M M and multilinear conserved currents in M M are analyzed. Section 5 contains details of the analysis of the sigma-minus cohomology associated with the dynamical rank-r fields and field equations in generalized Minkowski space. In Section 6 some perspectives are briefly discussed. Appendix contains our Young diagram conventions.

Results
In this section we summarize our results leaving technical details for the rest of the paper.

Sp(2M) invariant space 2.1.1 Fields and equations
Rank-one Sp(2M) invariant unfolded equation is [19] dX AB ∂ ∂X AB + σ 1 − C(Y |X) = 0 , where X AB = X BA (A, B = 1, . . . , M) are matrix coordinates of M M , Y A are some auxiliary commuting variables, that will be referred to as twistor variables. To simplify formulae we will use notation Note that Eq.(2.1) admits an interesting interpretation in terms of world-like particle models of [20,21], providing also a field-theoretical realization of the observation of Fronsdal [22] that the infinite tower of all 4d massless fields enjoys sp(8) symmetry.
From (2.1) it follows that most of the component fields in the expansion are reconstructed in terms of (X-derivatives of) the primary fields that satisfy In the rank-one case, the primary fields are [19] C(X) , The symmetry properties of C and C A can be represented by the Young diagrams • and , respectively. (For more detail on Young diagrams see Appendix.) As a consequence of unfolded equations, the following equations hold and, in particular, The symmetry properties of the left-hand-sides of the respective field equations are represented by the Young diagrams and . In the language of σ − cohomology [23] (see also [24]) convenient for the analysis of the pattern of zero-form higher-rank fields, 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. (2.7),(2.8) are the only independent equations obeyed by the primary (=dynamical) fields as a consequence of (2.1). Such equations will be referred to as dynamical.
Rank-two unfolded equations are and Y A j (j = 1, 2) represents a doubled set of twistor variables. In terms of variables (2.13) The following homogeneous differential equations hold as a consequence of (2.13) where ε A 1 ,...,Am are independent rank-m totally antisymmetric tensors introduced to impose appropriate antisymmetrizations.
As shown in [1], the rank-two primary fields are Dynamical equations for the primary fields are [1] Hence, the list of the Young diagrams associated with H 0 (σ 2 − ) and 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 two 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 togeither they form a rectangular two-column block of height r + 1, i.e., where h k (Y) is the height of the k th column of Y.
We say that a pair of Young diagrams are rank-r two-column dual if they obey (2.18). From the examples of ranks one and two we observe that dynamical fields and their field equations are described by rank-r two-column dual Young diagrams. It turns out that this is true for general rank-r fields.
Rank-r unfolded equations are According to the general argument of [1], these equations are Sp(2M) symmetric. Indeed, it is well known (for more detail see e.g. [19]) 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 1-form ω(X) = dX A ω A (X) is some fixed connection of g obeying the flatness condition Note that the oscillator representation provides a standard tool of the study of representations of sp(2M) [25,26] which implies that the corresponding connection is flat.
obey the standard commutation relations [τ mp , τ qj ] = δ pq τ mj + δ mj τ pq − δ mq τ pj − δ pj τ mq . (2.26) Being mutually commuting, o(r) and sp(2M) form a Howe-dual pair [27]. As in the lower-rank cases, rank-r primary fields obey the condition In terms of the expansion 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 described by some irreducible representation of o(r) as well as of gl M with the same symmetry properties described by one or another Young diagram Y(s 1 , ..., s m ) ( s 1 + ... + s m = n).
Recall that if a tensor with respect to color indices is traceless and has symmetry properties of Y[h 1 , ..., h m ], it is nonzero only if As shown in Section 5, for any rank-r primary field C ∈ H 0 (σ r − ) associated with a Young diagram Y 0 (n 1 , . . . , n m , 1, . . . , 1 q ) (n m ≥ 2) that obeys the tracelessness condition, hence obeying 2m + q ≤ r , the left-hand-sides of the dynamical equation has the symmetry properties of the two-column dual Young diagram Y 1 (n 1 , . . . , n m , 2, . . . , 2 Rank-r two-column duality. These are gl M Young diagrams with respect to indices A, B = 1, . . . M. It should be stressed 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 presented in terms of Young diagrams with manifest antisymmetrization. For nonnegative integers m 1 ≥ m 2 , Since E Y 0 (2.35) is a homogenous differential operator, the primary fields also satisfy (2.34). The parameter (2.33) is designed in such a way that Eq. (2.34) is nontrivial only for primaries C(Y |X) with the symmetry properties of Y 0 [m 1 , . . . , m n ].
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 that the two-column dual diagram Y 1 does not respect the condition (2.30). The point is that, as an o(r)tensor, Y 1 is not traceless, containing explicitly a number of o(r) metric tensors δ ij adding additional o(r) indices to Y 1 compared to Y 0 .
The nontrivial part of the analysis is to prove that the resulting list of dynamical equations is complete. This follows from the analysis of cohomology groups H 1 (σ r − ) in Section 3.2.

Multi-linear currents in M M
The construction of conserved currents in terms of rank-one fields proposed in [2,1] admits a generalization to higher-rank fields. Recall, that the M-form Now, we are in a position to introduce differential forms in M M closed by virtue of rank-r = 2κ unfolded equations (2.19) for κ = 1, . . . , M − 1. Let where N = M − κ + 1 and ǫ A 1 ...A M is the totally antisymmetric tensor. In Section 4 it is shown that D (2.37) has the symmetry properties of the rectangular Young diagram Y(N, . . . N κ ), As shown in [28,19], the equations of motion for massless fields of all spins in 4d Minkowski space can be formulated in the unfolded form Here y α andȳ β ′ are auxiliary commuting complex conjugated two-component spinor coordinates (α, β = 1, 2; α ′ , β ′ = 1, 2) , x αβ ′ are Minkowski coordinates in twocomponent spinor notations, and ξ αα ′ = dx αα ′ .
Hence, primary fields in M M nk and have definite symmetry properties in y andȳ described by Young diagrams that have to obey the condition as a consequence of (2.48).

4d Minkowski space
The dictionary between tensor and two-component spinor notations is based on where σ αβ ′ a (a = 0, 1, 2, 3) are four Hermitian 2 × 2 matrices. In the 4d Minkowski case with various r the list of primary fields is as follows. r = 1 As follows from the analysis of [28], for r = 1 the primary fields are C(x) , C(y|x) , C(ȳ|x) .
(2.56) These have symmetry properties described by the following pairs of Young diagrams The following consequences of (2.43) impose the equations on the primaries (2.56) The symmetry properties of the left-hand-sides of these equations are described by the following pairs of Young diagrams As mentioned in the previous Section, rank-r primary fields (2.47) obey the symmetry properties of pairs of Young diagrams (2.49), that in the case of M = 4, r = 2 form the following list Since the algebra u(r) spanned by commutes to σ M nk M (2.44), then all primary fields can be described as higher weight representations of u(r). It is evident, that ς 1 2 can be chosen as a positive generator of u(2). As it is easy to see, higher weight primaries are singlets C(x) , and higher weight fields C 11 0 (y 1 ,ȳ 2 |x) , C 20 0 (y|x) , C Hence the full list of primaries consists of all descendants of C ij 0 (y,ȳ|x) (2.64), i.e., Note, that these results were used in [3] to describe 4d free conformal primary currents. Equations of motion depend on different tensors of the form (2.52). To be nonzero for M = 4 and r = 2, tensor (2.52) should have m 1 =s 1 = 1 and m j ≤ 1,s j ≤ 1, since α j and α ′ j take just two values. In particular, Eqs. (2.53) with m 1 =s 1 = 1 and m j =s j = 0 at j > 1 give where ε αβ and ε α ′ β ′ are any antisymmetric tensors. These equations have symmetry properties of Y 1 = and Y 1 = . r ≥ 3 For r ≥ 3, dynamical fields (2.47) are described by Young diagrams (2.49) with at most two rows. Since H 1 (σ r − M nk ) is empty for r ≥ 3, the rank-r ≥ 3 primary fields are off shell, not obeying any field equations.

Derivation of equations
By virtue of (2.19), Eq. (2.34) is equivalent to

Lemma 1 Let a tensor F i[m] ,j[n] be traceless and antisymmetric both in indices i and in j taking r values (it is not demanded that F i[m] ,j[n] has properties of a Young diagram). Consider the tensor
resulting from the total antisymmetrization of indices i and j. Then

Details of σ − -cohomology analysis
As shown in [23] (for more detail see [1,24]), dynamical content of (2.19) is encoded by the cohomology group H p (σ r − ) = Ker(σ r − )/Im(σ r − ) p , where p denotes the restriction to the subspace of p-forms. In particular, independent dynamical fields contained in the set of 0-forms C(Y |X) take values in H 0 (σ r − ), and there are as many independent differential equations on the dynamical 0-forms in (2.19) as the dimension of H 1 (σ r − ). Our analysis generalizes those of [1], where the case of rank-2 was considered, and of [29], 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 some other nilpotent operator, (Ω * ) 2 = 0. Then the operator satisfies [Ω, ∆] = [Ω * , ∆] = 0 . From (3.6) it follows that ∆ ker Ω ⊂ Im Ω. Therefore H ⊂ ker Ω/∆(ker Ω). Suppose now that V is a Hilbert space in which Ω * and Ω are conjugated. Then ∆ is semi-positive. If also the operator ∆ is 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, H ⊂ ker ∆ ∩ ker Ω .
This formula is particularly useful for the practical analysis.
we observe that the operators form gl(M) with the commutation relations (3.11) In this notation, with T AB (2.23). Setting An important property of so defined ∆ r is that it is semi-positive because Ω M and Ω * M are conjugated with respect to the positive-definite Fock space generated by Y A i and ξ AB treated as creation operators. By virtue of (2.24), (3.10) and (3.11 ), ∆ r can be represented in the form An elementary computation gives by virtue of (2.23)-( 2.26) are generators of gl tot M that acts on all indices A, B, . . . independently on wheither they are carried by Y A i or by ξ AB . The first and second terms on the r.h.s. of (3.17) are the quadratic Cazimir operators of the algebras gl tot M and o(r), respectively. Analogous computation in terms of fermionic oscillator realization of generators of the orthogonal group, that makes the antisymmetrization manifest, gives the well-known formula used e.g. in [29] τ mk τ mk = 2 where i enumerates columns of the gl M Young diagram Y ′ [H 1 . . . , H m ] and H i are heights of the respective columns. Note that, in addition to indices associated with the indices of Y, the diagram Y ′ can contain indices carried either by differentials ξ AB or by the o(r) invariant "tracefull" combinations where N is not smaller than the degree of the differential form.
Alternatively, the bosonic oscillator realization of the generators, that makes symmetries manifest, gives where summations are over all rows of the respective Young diagrams, while L i and l i are lengths of the rows of the respective diagrams. Hence, Eq. (3.15) gives the following two equivalent formulae The simplest way to see that this formulae are equivalent is to use the following useful observation. Any Young diagram is the unification of its elementary cells S i,j on the intersection of its i − th row and j − th column, i.e., Y = and will be called rank-r n-column dual if the condition (2.30) holds and (3.28) At n = 2 this amounts to (2.18).
Our analysis of σ − cohomology is based on the following Lemma 2 Any rank r n-column dual Young diagrams give ∆ r = 0. Indeed, Eq. (3.23) gives Now it is elementary to check that the diagrams associated with primary fields and their field equations belong to Ker∆ r . In the case of primary fields associated with H 0 , Y = Y ′ . Hence Eq. (3.23) gives zero, as anticipated. In the case of field equations associated with H 1 , ∆ r is also zero because the diagram for fields is two-column dual to that of their equations.
To show that this list of solutions is complete, analogously to [29], one can use the semi-positive definiteness of ∆ r . Indeed, the representation (3.25) for Cazimir operators tells us that for a given set of cells of a GL(M) diagram, the minimal value of the operator ∆ r is reached when all cells are situated maximally southwest, i.e., possible values of k and m in (3.25) are respectively maximized and minimized. For a given field (i.e., Y[h 1 , h 2 , . . . , h k ]) this optimization applies to the cells associated with the indices of the differentials and the traceful blocks U AB (3.21). From the semi-positivity of ∆ r , (3.25) and Lemma 2 it follows that ∆ r can only be zero for rank r n-column dual Young diagrams. However, not all of them can be nontrivial for H p (σ − ) at given p. Indeed, any rank r n-column dual Young diagram where some two columns are occupied by the indices of U AB are zero by Lemma 1. Therefore, for H 0 (σ − ) and H 1 (σ − ) the solutions are exhausted by the rank-r 0-column dual and two-column dual Young diagrams, respectively.

Differential forms and higher-rank currents in M M
That dynamical degrees of freedom associated with the rank-one equations (2.1) 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 the rank-two field equations (2.19). As shown in [30], the is closed provided that J 2 (z,z |X) solves (2.19) at r = 2. Bilinears in solutions C j of (2.19) with r = 1 and parameters η that commute to σ 2 − (2.11) give conserved currents [1,18,3].
Since, modules of solutions of the rank-κ equations in M M ⊗ R κM are functions of κM variables y α i one might guess that in the rank-r case the dimension of a "local Cauchy bundle" [31] on which initial data should be given to determine a solution everywhere in M M is κM. This is however not quite the case because the O(κ) 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 κM − 1 2 κ(κ − 1). Correspondingly, the respective currents should be represented by closed κ(M − 1 2 (κ − 1))-forms which is just the degree of the form (2.39). On the other hand, one can see that the straightforward generalization of (4.1) to conserved currents in M M ⊗ R 2κM for arbitrary κ which is, Any Young diagrams associated with the combination of differentials in the brackets can only contain rows of lengths l ≤ M + 1 since otherwise it would contain either antisymmetrization over more than M indices or symmetrization of some two anticommuting differentials. On the other hand, because of ε-symbols, it contains r columns of the maximal height M associated with the indices A in (4.3). 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, this is not possible for r > 1 as at least two symmetrized indices B will belong to the same column, which implies antisymmetrization. Thus (4.3) is zero for κ > 1.
For the proof consider first Q (4.4) obeying (4.5) in the symmetric basis of Young diagrams Y(l 1 , ..., l s ) with l 1 + . . . + l s = 2n. Let N k count a number of pairs A B that contribute to the k-th row and some k + n-th row with n > 0.
From (4.5) it follows that all pairs of the k-th class can contribute either N k or 1 + N k to the k-th row and either 0 or 1 cell different p-th rows with p ≥ k + 1. This gives the inequality l k ≤ k + N k , because for j < k each of j-th classes can contribute to the k-th row either 0 or 1 cell, while the N k -th class can contribute there either N k or N k + 1 cells. Note, that the inequality becomes the equality if and only if each of j-th classes contributes just one cell for j < k, while the N k -th class contributes N k + 1 cells.
This implies l + k ≤ N k , where l + k is the number of cells of the k th row above the diagonal. Therefore the number of cells above the diagonal is dominated by k N k . On the other hand, since the total number of cells is twice the number of pairs n, Hence, the number of cells above the diagonal is dominated by n. Now consider Q (4.4) obeying (4.5) in the antisymmetric basis of Young diagrams Y[h 1 ; ...; h m ] with h 1 + . . . + h m = 2n. Let N m count a number of pairs A B that contribute to the m-th column and some m + n-th column with n > 0 (n = 0 is impossible because on the antisymmetrization). From (4.4) it follows that all pairs of this class can only contribute no more then one cell to different p-th columns with p ≥ k + 1. This gives the inequality m is the number of cells of the m-th columns that are not above the diagonal (under and on the diagonal ). As above this inequality becomes the equality if and only if for each j < k, the j-th class contributes to the k−th column just one cell, while the N k -th class contributes N k cells.
Therefore the number of cells that are not above the diagonal is dominated by m N m . As above, 2n = 2 m N m . Hence the number of cells that are not above the diagonal is dominated by n. Therefore, the number of cells above the diagonal equals to the number of cells that are not above the diagonal, i.e., each of constructed classes contributes equals number of cells above the diagonal and not above the diagonal.
Then k l k −k = k N k and m h m −(m−1) = m N m . This gives l k = k +N k and h m = m + N m − 1 for any k and m.
Now it is not difficult to show that the number L of rows of length l k > k is equal to the number of columns of height h k ≥ k, and h k + 1 = l k for 1 ≤ k ≤ L. Indeed, the first column height h 1 is not smaller than N 1 = l 1 − 1, while the first row length l 1 is not smaller than N 1 = h 1 + 1, etc. This completes the proof. Now we are in a position to prove that Ω 2κ (J) (2.39) is closed by virtue of such that H i = r + n −h n−i+1 for i = 1, . . . , n ≤ max(l, k) , H i = h i for i = n + 1, . . . , l , (5.14) H i = r + n − h n−i+1 for i = 1, . . . , n ≤ max(l, k) , H i =h i for i = n + 1, . . . , k .
Such pair will be referred to as rank r 2n-column dual pairs of Young diagrams. For n = 1 it is equivalent to (2.51).

Lemma 3
Any rank r 2n-column dual pair of Young diagrams gives zero ∆ r M nk (5.10). Indeed, Eq. gives zero, as anticipated. In the case of H 1 , the diagrams (5.12) (5.13), that form two-column dual pairs, send ∆ r (3.23) to zero as well.
Using semi-positive definiteness of ∆ r M nk along with Lemma 3 and formula (3.25) along the lines of Section 3.2, it is not difficult to show that this gives complete list of solutions.

Conclusion
There is a number of problems raised by the analysis of this paper. 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, stress tensor generates the full conformal algebra being integrated with the parameters of translations, Lorentz rotations, dilatations and special conformal transformations.) See, e.g., [3] 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 [30] for the case of rank two, to obtain non-zero 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 tensor 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, more specifically, what is its relation with the multiparticle algebra proposed recently in [32] and [3] 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.