The Higher Spin Rectangle
The chiral algebra of the symmetric product orbifold of a single-boson CFT corresponds to a “higher spin square” algebra in the large limit. In this note, we show that a symmetrized collection of bosons defines a similar structure that we refer to as the higher spin rectangle algebra. We explore the relation of this algebra to the higher spin square algebra. The existence of such a truncated algebra hints at bulk theories interpolating between Vasiliev higher spin theory and string theory.
It has long been expected that a large symmetry underlies string theory and that this symmetry is manifest in its tensionless phase. Recently, the papers Gaberdiel:2015a (); Gaberdiel:2015b (); Gaberdiel:2014a () endeavored to unmask this symmetry via the AdS/CFT correspondence. The CFT dual to tensionless string theory on is believed to be the symmetric product orbifold theory consisting of free fermions and bosons on . In Ref. Gaberdiel:2015a () it was shown that, in the large limit, the chiral algebra of the single-trace operators for this symmetric product orbifold can be organized in terms of -algebra representations. Using the bulk/boundary dictionary Gaberdiel:2010 (); Gaberdiel:2011 (); Gaberdiel:2012 (), in this case, leads to an infinite-dimensional lie algebra as the symmetry algebra of string theory on the AdS background at the tensionless point. The associated gauge fields are towers of massless higher-spin fields, with the gauge fields of Vasiliev theory vas1 (); vas2 () appearing as a sub-sector. Because the generators of this lie algebra can be organized in a twofold way in terms of the representations of the higher spin algebras underlying Vasiliev theory, it was christened the higher spin square (). Recent work Gaberdiel:2017 () has attempted to directly identify the generators of this higher spin square algebra with massless states in tensionless string theory on the background. Regardless of its status as the symmetry algebra of string theory, however, the algebra as well as the full chiral algebra of the symmetric product orbifold theory, and its relation to conventional -algebras, are interesting objects to study in their own right.
-algebras have a long history going back to the work of Zamolodchikov:1985wn () where they appeared as the symmetry algebra of parafermionic conformal field theories. As such they are generalizations of the Virasoro algebra, with the algebra defined as an algebra with generators having spin to . In the context of string theory, historically, parafermionic models have been used to formulate world-sheet theories Argyres:1991 (); Pope:1991 () with -algebras replacing the Virasoro algebra.
In recent times, the study of -algebras has been revitalized because of their identification as the asymptotic symmetry algebras of three-dimensional Vasiliev theories Campoleoni:2010 (). In particular, the algebra is the asymptotic symmetry algebra of a Vasiliev theory with bulk gauge group , where the parameter is related to the bulk scalar mass. In fact, the -algebras do not appear in isolation but are deeply connected to lie algebras Bouwknegt:1992 (). The algebra, where denotes a lie algebra, is the Drinfel’d-Sokolov reduction of the affine algebra based on Bershadsky:1989 (). Thus, can be constructed from while can be constructed from . A reverse construction also exists. The algebra is actually a two-parameter algebra with one parameter being and the second being the central charge . One can recover the lie algebra from by the Bowcock-Watts procedure Bowcock:1991 (); Gaberdiel:2011wb (), which consists in taking the central charge to infinity while also restricting the operators such that the absolute value of their mode number is less then the spin, i.e. to the so-called wedge modes. The algebra is, therefore, often referred to as the wedge sub-algebra of . In this notation, the algebra is the wedge algebra of the chiral algebra of the large symmetric product orbifold CFT.
There are also various relations between different -algebras. For example, for changing the value of the parameter changes the algebra: is a linear algebra while generically is a non-linear algebra. However, there exists a triality symmetry Gaberdiel:2012ku () — the algebras are equivalent for generically three different values of . The algebra truncates to for and a specific value of the central charge. The algebra therefore acts as a “master algebra” in the sense that the finite -algebras can be extracted from it. Similarly, for a given , changing the value of can lead to a representation of the algebra developing null states such that it truncates to a representation of a smaller algebra.
The algebra also has an infinite-dimensional sub-algebra , which consists of fields of even spin only. This algebra, its supersymmetric extensions and holographic duals have been explored in Candu (). In the other direction, , as also algebras, can be extended by adding a spin one field: the resulting algebra is known as .
Since the higher spin square algebra is exponentially larger than the -algebras, one might expect that a truncation of the higher spin square algebra leading to a smaller algebra exists and in fact there may exist several different kind of truncations. It is the goal of this paper to construct a realization of one such truncated algebra. The operators of the full asymptotic symmetry algebra associated with the bosonic version of the can be organized in an infinite number of representations of the algebra interpreted one way or in an infinite number of representations of the algebra interpreted the second way. The operators, thus, cover a “square” with operators in the horizontal direction and operators in the vertical direction. We show that truncating the operators of the algebra such that the operators now organize in a finite number of representations still leads to a closed algebra. This set of operators can also be organized in an infinite number of representations of the algebra, thus retaining the structure of the original algebra. Because we have truncated the algebra in one direction only, we use the notation the higher spin rectangle () algebra to refer to the wedge algebra of this truncated algebra.
While the AdS/CFT correspondence has proved remarkably potent at large , it has had limited success at finite . It is, thus, important to look for scenarios where the quantum version of this duality can be verified. This is another motivation, in the larger context, to catalogue and understand finite versions of the algebra. In this paper, however, we will construct the algebra at infinite central charge with only a few observations about the truncation of the at finite central charge.
This paper is organized as follows. We first review the construction of the higher spin square algebra in Sec. (2). In Sec. (3) we construct a realization of the algebra in terms of free bosons with background charges. In Sec. (4), we show that this construction is reflected in -series identities. In Sec. (5) we discuss the possibility of a finite central charge version of the algebra. In Sec. (6) we discuss various directions for further exploration.
2 Review of the higher spin square algebra
In this paper, we will confine our analysis to bosonic theories as these capture all the essential details of the algebra. As in Ref. Gaberdiel:2015a (), we start with analyzing the symmetry algebra of the ’th symmetric orbifold of a single real boson, where we will take to . In the text, we will use the notation to refer to this bosonic chiral algebra.
In the large limit, the “single-particle” generators for the symmetric product orbifold are symmetrized products of the form
Because of the symmetrization over , this set of generators is in one-to-one correspondence with the chiral sector of a single boson. Removing the terms that are total derivatives, and in the limit, they also constitute a set of linearly independent operators.
The aim of Ref. Gaberdiel:2015a () was to the arrange these single-particle generators of the bosonic theory in terms of higher spin subalgebras. To begin this task one notes that the subset of generators of Eq. (1) of the form
define quasiprimary generators of spin , in specific linear combinations and when is even. In fact, only one independent current can be constructed at each even spin, meaning that it is not a linear combination of derivatives of lower-spin currents, and there are no independent odd-spin currents. This set of independent currents generate the even spin -algebra .
The generators in Eq. (2) are of order two, i.e., they are bilinear in the s. The currents in Eq. (1) are of arbitrary order . However, it turns out that the currents of a fixed order , suitably corrected by lower-order terms, form a representation of the wedge algebra of . This is captured in Fig. (1) where currents of a given order correspond to columns. The entire operators of the algebra are, therefore, organized into representations of .
The -algebras that we deal with in this paper are the symmetry algebras of cosets of the form . We use the coset notation to denote a representation of a -algebra where is a representation of and is a representation of . Then the statement that operators of the can be organized in representations of is captured by the following identity:
where denotes the wedge character of the representation of . Specifically it is
The LHS of Eq. (3) is the normalized partition function for a single boson. Combinatorially, the LHS is just the generating function for the number of ways one can partition an integer varying from to into an arbitrary number of parts. Each term in the sum on the RHS is number of ways one can partition an integer into exactly parts. In terms of the operators in Eq. (1), this is the spin of the operator, varying from to , being partitioned into at fixed .
One could have started sorting the operators of from a different point leading to a different organization. Operators of the form
corrected by lower-order terms (involving smaller powers of and additional derivatives) define the linear algebra. This is actually a result of bosonization of the free-fermion representation of vas2 (); Pope:1991 (); Pope:1990be (); Gaberdiel:2013 (). Similar to the previous case, the rest of the operators in Eq. (1) can be organized in terms of representations of . This is formalized by the identity
Combinatorially, each term in the sum of the RHS can be interpreted via the Durfee square construction Andrews (). The Durfee square of a partition is the largest square array that can fit in the upper left corner of its Young diagram. Each term in the RHS, therefore, is the number of partitions, of an integer, having fixed Durfee number . In terms of the fields, the Durfee number corresponds to the number of fermion bilinears needed to construct a particular operator.
3 A collection of bosons and the algebra
The algebra is a non-linear algebra which has generators ranging in spin from 1 to . It is related to as
where is a spin one scalar field. It is well-known that the algebra has a realization, at arbitrary central charge, in terms of free bosons and a background charge parameter Bouwknegt:1992 (); Prochazka:2014 (). In this section we show that a symmetrized system of free bosons can be used to realize a larger algebra that we refer to as , with its corresponding wedge algebra denoted by . As in the case of the higher spin square algebra, this comes about because a collection of bosons can be used to represent both the and the algebras. In general, to construct a -algebra with independent currents, one needs a minimum of free bosons so as to have a Hilbert space large enough to accommodate the -algebra representations. At specific values of the central charge, however, -algebra representations have a large number of null vectors so that a realization of the algebra with a smaller number of bosons can be constructed.
We first review the -boson realization of , which is commonly known as the Miura transformation, in Sec. (3.1.1). We also provide an alternate formulation of the Miura transformation in which the current operators are expressed as power sums. Then in Sec. (3.1.2) we review the -boson realization of the algebra. In Sec. (3.2), we propose that a specific set of operators that can be constructed out of free bosons, and are invariant under symmetry, are the generators of the algebra. We will show that this set of generators has closed commutation relations. This follows from the same logic as was utilized in the case in Ref. Gaberdiel:2015a (), which we will elaborate on in Sec. (3.2). This algebra can be deformed to an arbitrary (integral) value of the central charge by changing the background charge and the total number of bosons.
3.1 Free field representations
3.1.1 Free field representation of
In this section we review the representation of in terms of free bosons. This algebra has currents denoted by , where varies from to and is the conformal dimension. These currents are defined by the generating operator
The parameter is a background charge related to the central charge of the algebra as
Let us define . Then, for example, uptil the are given explicitly by
For the background charge the corresponding central charge is . Note that this central charge is equal to where is the central charge of the coset theory in the limit . At this value of the currents are of the particularly simple form
Viewed as polynomials in , the above operators correspond to the elementary symmetric polynomials in variables. The currents listed above can be thought to form an algebraically independent set of operators in the sense that the eigenvalues of the operators acting on a highest weight state are the elementary symmetric polynomials (which are algebraically independent) Bouwknegt:1992 ().
Using the Newton-Girard formulae, the elementary symmetric polynomials can be expressed as power sums. Analogously, for the bosonic fields we can make the following change of basis
where with . Under this change of basis, the generators of the algebra take the form:
where the are constants dependent on the . This form of the currents is akin to the form in which the currents are expressed in the bosonic basis. Indeed, the algebra, like the algebra, also has a formulation in terms of complex fermions. The bosonic current can be written in terms of a multi-component free fermion as
where to . This corresponds to the bosonization
In the fermonic basis, the -algebra generators can be expressed in terms of bilinears of free fermions (See, e.g., Bershtein:2017 ()). There is, therefore, a smooth way of taking the limit of the Miura construction as to give at .
3.1.2 Free field representation of
where is a positive number dependent on the spin. In this formulation the stress-energy tensor is
This is related to the spin current in Eq. (3.1.1) by a linear change of basis, as we saw in the previous section. There, thus, exists an alternate formulation of bilinear realizations of the currents in terms of bosonic fields. We can choose the current to be of the form
To find the number and form of currents at each spin we proceed as usual Pope:1991 () and write down all possible symmetric bilinear terms. At spin there exist the fields
which are not independent of each other, as they are related by a change of basis. In addition they are derivatives of the spin currents given in Eq. (19) and Eq. (18) respectively. There is, therefore, no new current at spin . At spin , the symmetric bilinear terms are
The two set of currents are again not independent of each other, so that we choose only the first set, for example. In this set, the linear combination is a derivative of . The second linear combination is an independent current. Proceeding in this manner we find a single current at each even spin. Note that the symmetrization condition has reduced the degrees of freedom in the -boson system so that the currents are in one-to-one correspondence with those that can be constructed out of a single boson. If we do not require symmetry under , the bilinears that can be constructed from two bosons, for instance, give rise to the algebra.
The above counting is somewhat naive as it does not take into account relations that may reduce the set of independent generators at finite . 111I thank Matthias Gaberdiel for this observation. However, the above analysis does capture the maximal set of independent bilinear generators that can be constructed out of bosons subject to the symmetrization condition. In actual fact, a representation of the algebra at finite central charge will truncate to a representation of a smaller algebra because of the presence of null fields Blumenhagen:1994 ().
3.2 Extending the algebra
In this section, we propose that the -boson realization of the algebra, can be extended to provide a realisation of a larger algebra. Indeed, out of bosons, one can construct not only bilinear terms, but trilinear terms, quartic terms and so forth, invariant under the symmetry, as is Sec. (2). The difference, in this case, is that we restrict the operators such that there is no operator with order .
This comes about naturally, if we start with generators, as in the top row in Figure (2). As we saw in the last section, we can choose the generators to be either of the form of the elementary symmetric polynomials or of the form of the power sums (or indeed any other symmetric polynomial basis) at . The full algebra is generated by derivatives acting on the constituent terms of each generator. The full set of operators is the realization of a algebra that we call the algebra. This construction provides a realization of the algebra at finite central charge . We show closure of this algebra in Sec. (3.2.1).
To count the total number of generators of this algebra, we point out there is a correspondence that is easily seen (in the power sum basis of Eq. (3.1.1), for instance,) between the generators and the chiral operators of a single boson. Indeed, if we restrict the set of chiral operators for a single boson field such that an operator consists of no more than , with the number of derivatives acting on the fields unrestricted, there is a one-to-one correspondence between this set and the operators in Figure (2). The total count for this set of operators is the number of ways one can partition an integer (varying from to ) into parts, where varies from to . It is therefore given by
This then is the count of generators of our proposed algebra. At finite central charge, one would expect a representation of this algebra to truncate down to a representation of a smaller, as yet unknown algebra. However, we will show in Sec. (3.2.2) that at infinite central charge, Eq. (22) represents the generating function of the algebra.
3.2.1 Closure of algebra
As in Sec. (3.1.2), using bosonic fields one can also construct a realization of the algebra. In this basis, the algebra is generated by the set of operators that form the representations of where ranges from to . This is illustrated by the columns in Figure (2). The second column corresponds to the representation and the generators of the algebra. The commutator of any operator in a given column with a generator gives rise to an operator in the same column. In the basis, the algebra is generated by operators that form representations of . Furthermore, every operator can be expressed as a sum of operators (that lie in the rectangle) and vice-versa.
Using this structure, we can postulate the existence of a closed algebra . Let us outline how to compute the commutators (or corresponding operator product expansions) of any two generators belonging to , which we denote by and . The first generator can always be expressed as a descendant of some highest weight state of ; let us denote this hw state as . Since is also a generator, the second operator has a known commutation relation with , determined by the algebra. The operators that one gets on the r.h.s of such a commutation relation will be operators that lie in the rectangle. Once we know the commutator , we can reexpress it in the basis and thus find the commutator , since is a descendant of . Note that we could have switched with in the preceding discussion.
3.2.2 Deformation to arbitrary central charge
We now show that this algebra can be deformed to arbitrary integral central charge and that in particular that we can take the central charge to infinity. The currents in Eq. (3.1.1) are a realization of the algebra at central charge given in terms of and the background charge parameter in Eq. (10). Clearly, changing the background charge changes the central charge of the theory while keeping the total number of bosons fixed to . For our purpose, however, we are interested in changing the total number of bosons as well, since we want a realization of the algebra at infinite as well. In other words, we are interested in a -boson representation of the algebra where .
Here, we take
By choosing a particular basis for the one can write down the currents explicitly. Here we make the choice in Ref. Lu (). The stress-energy tensor is then given by
corresponding to the central charge
Since the scalar field only occurs via its energy-momentum tensor in Eq. (25) and in all other currents (see Lu () for details), one can replace it with a different energy-momentum tensor with the same central charge
The total central charge remains
The stress-energy tensor is now
with the suitably chosen such that the central charge of is given by Eq. (27). This construction, thus, furnishes a representation of the algebra in terms of a total of bosons.
The goal of this section is to obtain a realization of the algebra at . This can be achieved by taking to . As a result we can take , while remains finite. Both and diverge in this case. Because , we can take the to be identically zero. As in the construction of Sec. (2), we start with an infinite number of bosons, but now we are dividing them into two parts: a collection of bosons and a collection of bosons distinguished by the differing values of the background charge. This enables us to construct a realization of the algebra using an infinite number of bosons.
The currents of the algebra can be written in a basis by adding a spin one current that commutes with all currents of the algebra. Additionally, one can check that the currents given in Eq. (3.1.1) can be written in terms of currents and an additional boson field. Applying this recursively, one will end with a realization of the algebra in terms of bosons, a spin one current and an arbitrary stress-energy tensor. Hence, the above construction for works for as well.
We can now extend the algebra by scattering additional derivatives on the constituent terms of the currents. The only change that has occurred because of the change in central charge is an increase in the total number of bosonic fields and additional lower-order terms multiplied by powers of the deformation parameter . However, for the total count of operators of the extended algebra, it is the term with the highest order in each current which matters, not the total number of bosons. The lower-order terms are irrelevant as well, since they serve as correction terms. Thus the total count of operators is still given by Eq. (22).
The construction we have outlined above no longer corresponds to the symmetric product orbifold in the large limit. As we are taking we do have an infinite number of bosons in total, however in choosing a finite subset of bosons out of this set we have broken the original symmetry. It would be interesting to find a geometrical interpretation of this construction.
The assertions in Sec. (3) can be quantitatively checked by computing the characters of the relevant -algebras. Indeed, if our assertion is true, the character in Eq. (22) should be expressible as a sum of characters of the representations of the -algebras. We show in this section that this is precisely what occurs because of the existence of non-trivial -series identities.
4.1 Vertical identity
We first show that the algebra decomposes into representations of . For this we use a finite version of the -binomial identity
This identity is a straightforward truncation of the identity in Eq. (3).
4.2 Horizontal identity
There is a second decomposition of the higher spin rectangle algebra into an infinite number of representations of the wedge algebra of a -algebra with a finite number of generators. Note that at , since the level , there are an infinite number of representations of the -algebra corresponding to the coset:
These representations can be analytically continued to by varying Gaberdiel:2012ku (). In terms of and , is given by
To take the theory to , we change from to , while the level varies from to . The theory is non-unitary in this limit.
In terms of characters, the decomposition of the higher spin rectangle algebra is given by
Here, is a character. We define it in terms of a character as
where is the quadratic Casimir for the representation . The variables are defined as , where varies from to and the variable as . Note that . The character is given in terms of Schur polynomials as Gaberdiel:2010 ()
Here is the total number of boxes in the Young diagram of the representation . The are defined in terms of the row lengths , where varies from to and , of the Young diagram as . With these definitions, Eq. (35) is identical to the following -identity:
The above identity is a special case of the Rogers-Fine identity (See, e.g., Ref. Alladi ()). The Rogers-Fine identity is in general a three-parameter identity which can be reduced to the following two-parameter identity:
where Making the substitutions and , we recover the identity in Eq. (38).
4.2.1 Interpretation of the identity
The wedge group of a -algebra typically emerges only in the limit. Indeed, for the case of the algebra for the representation where , the character is given by Perlmutter:2012 ()
in the large limit. This character reduces to
if we restrict to the wedge modes.
The term in the RHS of Eq. (38) is the character of the representation of . To illustrate for , the representation has character
Here, the expression in the outer bracket is the character for corresponding to the wedge modes:
of the algebra. One can construct the algebra using a two-boson realization, where the spin current would be order in the . In general, the standard Miura construction uses bosons to construct the algebra , with the spin of fields ranging from to and wedge group . To make contact with the bosonic construction in Sec. (3), where the highest order term in the -algebra currents is of order , we are interested in the algebra , with spins ranging from to and the wedge group . To get this algebra we need to further decompose the group as .
Let us work out the branching rule for the decomposition of the rep of into these subgroups Stanley (); goodwall (). The representation corresponds to the Young tableau , where each entry in the list is the number of boxes in the rows of the Young diagram. Henceforth, in this section, we will use the Young tableau notation to denote a representation. Specifically for the rep of , the branching rule is
where the first factor in each term on the RHS is the representation of and the second factor is the representation of . The corresponding character decomposition is
This decomposition structure is preserved at arbitrary . The rep of decomposes as
In this decomposition, the first two representations of can be combined to give the adjoint representation. This representation, corresponding to the wedge modes of the algebra, is always present in the decomposition. In terms of free bosons, the zero modes of the adjoint representation map to the bosonic realization of the algebra in Sec. (3).
For , there is an analogous branching rule for the rep of into the reps of . The identity in Eq. (38), therefore, represents the decomposition of the generating character of the algebra into characters of the representations of , which we interpret here to be the wedge group of the algebra at .
5 At finite central charge
While we have constructed the algebra at infinite central charge, it is natural to ask whether the algebra has consistent truncations at a finite value of the central charge. There are some indications for this. In particular, the algebra has a neat truncation at finite .
This assertion follows from the triality symmetry of the algebra. Due to triality symmetry the following algebras are isomorphic
at the value of central charge . Extending this triality to the algebra leads to . For our case, the central charge is for which
For the second equality we have used the fact that truncates to at . This is the symmetry algebra of the coset in the limit. In fact, much before the discovery of triality symmetry for the algebra, the relation at was already known Frenkel:1994 (); Awata:1994 (). It was shown in Ref. Frenkel:1994 () that ‘quasi-finite’ unitary representations of generically reduce to representations of . This is because at such representations develop extra null vectors. Quotienting by the submodule generated from these null vectors leads to a irreducible representation of which is isomorphic to a representation of with the same central charge.
The algebra is composed of representations at . We can analytically continue these representations to finite . As hits integer values, these representations will truncate down to reps. However, it is not clear what happens to the vertical algebra at . It is expected that this algebra too will truncate to a smaller algebra, but at present this truncation is unknown. This truncation is not as straightforward as for the case.
The algebra at does not appear to arise as a direct truncation of , this really depends on how many independent parameters the algebra has. Indeed, it could be possible that there are two independent parameters in the commutation relations of the algebra, in which case, the algebra can arise as a direct truncation.
The operators that make up the algebra at are clearly a subset of the chiral operators of the free boson theory on the symmetric product orbifold . The operators of the symmetric product orbifold theory can be enumerated using the methods of Ref. elliptic (). However, it is not clear what the significance of the algebra is to the theory of a single boson on the finite symmetric product orbifold.
In this paper we have shown that there exists an algebra , parametrized by the integer , which is similar in structure to the algebra. We provided a realization of the algebra using free bosonic fields. So far, we have not worked out the commutation relations for this algebra, but this should be straightforward for small .
By restricting to the wedge algebra of , one can postulate the existence of the lie algebra . The algebra reduces to the algebra (tensored to a field) for which is the gauge group of three-dimensional Vasiliev theory with even spin gauge fields and a scalar field with mass . For , it is synonymous with which is believed to be the symmetry algebra of string theory on the background in its bosonic incarnation. The existence of such an algebra, therefore, indicates bulk theories that interpolate between Vasiliev theory and string theory. Indeed at least in dimension , there is a bulk construction for such theories Vasiliev:2012 (). In view of this, we expect that the