Level Two String Functions and Rogers Ramanujan Type Identities

The level two string functions are calculated exactly for all simply laced Lie algebras, using a ladder coset construction. These are the characters of cosets of the type $G/U(1)^r$, where $G$ is the algebra at level two and $r$ is its rank. This coset is a theory of generalized parafermions. A conjectured Rogers Ramanujan type identity is described for these characters. Using the exact string functions, we verify the Rogers Ramanujan type expressions, that are the main focus of this work.


Introduction
One of the remarkable properties of two dimensional systems is the expression of many characters of conformal field theories (CFT) as generalized Rogers Ramanujan type sums, which we denote as GRR. Many such examples have been worked out in the literature [1,2,3,4,5,6]. Yet, the origin of these identities remains somewhat mysterious.
Some of this mystery is explained by the deep connection to two dimensional solvable lattice models, in particular, the so called, rigid solid on solid models (RSOS). This connection was first noted by Baxter, in the context of the Hard Hexagon model [7], and was considerably further developed in the works refs. [8,9]. The crux of this connection is that the local state probabilities in the RSOS models are identical to the characters of the fixed point CFT in the appropriate regime. (More precisely, the one dimensional configuration sum, which is the center piece of the local state probability, is equal to the characters.) This connection leads to the conjecture that GRR identities exist for every CFT that appears as a fixed point for some RSOS model in some regime, and vice versa.
The connection with RSOS lattice models also implies a way to prove these identities [8,9], where we consider the finite one dimensional configuration sum, which obeys some simple recursion relations. This sum is equal to a version of the characters, which is the finite GRR identities.
In this paper, we describe new GRR identities for the cosets of the type where G is any simply laced algebra, at level two, and r is its rank. The case of A r was already described in ref. [6], and we extend this work to all the other algebras. The GRR expression for the identity characters for the models H was conjectured in ref. [5], and we agree with this conjecture. We find GRR identities for all the characters, where the weight in G 2 has Dynkin index one.
We achieve these new GRR by calculating exactly the string function of G 2 , which are the characters of H (up to factors of Dedekinds eta functions). The calculation is done by utilizing a ladder coset construction, which is a product of intermediate cosets, where each of the latter can be solved exactly by identifying it as a simpler CFT.
The connection with RSOS of our GRR identities, and, in particular, their proof in this way, is an interesting question, which we are currently investigating.

The general conjecture
Our objective is to describe GRR type equalities (Generalized Rogers Ramanujan identities), for the characters of the cosets of the type where r is the rank of G and G is any of the simply laced algebras which are of the type A r , D r , E 6 , E 7 and E 8 , at level two. The case of A r was already discussed in ref. [6], where GRR identities were found for all the characters of H. Our aim here is to generalize these results to all of the simply laced algebras.
The characters of the theory H, for any G at any level, were discussed in ref. [10], and are related to generalized parafermions. Briefly, they are labeled by two weights, Φ Λ λ where Λ is a highest weight of G at level k (here k = 2) and λ is an element of the weight lattice of G. The weights obey the condition Λ − λ ∈ M , where we denote by M the root lattice of G. To each of the fields Φ Λ λ , we associate a character χ Λ λ , which is given by the corresponding string function of G k . The string functions are denoted by c Λ λ (τ ) and are defined as where L 0 is the dimension and H Λ λ is the representation of the affine G k with highest weight Λ and U (1) r charge λ. The characters of the coset H are then given by where η(τ ) denotes the Dedekind's eta function, λ is defined modulo kM and Let us present our main result which is a GRR expression for the characters of the theory H. We denote by C G the Cartan matrix of the algebra G which can be any of the algebras A r , D r , E 6 , E 7 and E 8 . Let Λ stand for any of the fundamental weights which have Dynkin index 1, Λ = Λ i or Λ = 0, where i = 1, 2, . . . , r, and θΛ i = 1, where Λ i is one of the fundamental weights and θ is the highest root. We conjecture, that the characters of H have the following expression, (1 − q m ), (2.5) and for Λ = 0 we take b i = 0. ∆ Λ λ is some dimension which we do not specify. Here q = e 2πiτ . We use the vector notation b = (b 1 , b 2 , . . . , b r ). Q is a vector of integers modulo 2 defined by where α k denotes the kth simple root ⋆ . For the algebra G = A r this reproduces the result of ref. [6]. For the case of Λ = 0 this agrees with a conjecture which was put forwards in ref. [5].
⋆ Because of the field identifications in the coset H, χ where σ is any automorphism of the Dynkin diagram of the affineĜ, at level two, it is enough to consider only Λ which are fundamental weights or zero.
The sum in eq. (2.4) consists of generalized Rogers Ramanujan identities. To establish this GRR we need to calculate the string functions of G = D r , E 6 , E 7 , E 8 at level two. We could, in principle, do this numerically using group theoretic recursion relations for the algebras in question. This is, in fact, too difficult to do by a computer program. So, instead, we will calculate the string functions exactly, finding closed analytical expressions for all the string functions. We accomplish this by generalizing the ladder coset construction of ref. [6]. Once such expressions are found we can verify the GRR, eq. (2.4).

The algebra D n
We start with D n = SO(2n) at level two. We can write the coset H = SO(2n) 2 /U (1) n as a product of cosets, where all the intermediate cosets cancel in the product. Of course, care must be taken, to make this correct in conformal field theory. It means that the same representation must be imposed when the same group appears in the numerator and the denominator, and this representation has to be summed over. In other words, for this product coset we take the special modular invariant, where all the intermediate representations are identical and are summed over. If we denote by G Λ 2 the highest weight field Λ for the group G at level two, and by G Λ 2 /H λ 2 the appropriate branching function for this coset, we can write the characters as, where µ n = Λ and α r is the rth simple root of SO(2n).
So, we proceed to calculate the characters of cosets of the type .
It is easy to see that for any r this coset has the central charge c = 1. Thus, it has to be one of the c = 1 models which were completely classified in ref. [11].
By calculating the dimensions of the fields in this coset we deduce that the coset So it remains to find the lattice M . This lattice can be constructed inductively using the ladder coset expression eq. (3.2). We use the beta method, ref. [12], to account for the summation of the intermediate weights, to find that the lattice M is the root lattice of the group SU (n) scaled by This is termed the SU (n) lattice at level two, and the characters are the classical theta functions of SU (n) level two, ref. [13]. Thus, we established: Theorem (1): The coset SO(2n) 2 /U (1) n is equivalent to the theory of n −1 free bosons, φ i , i = 1, 2, . . . , n−1, moving on the lattice Let us demonstrate how the SU (n) 2 lattice is built. We do this inductively.
First, for SO(4), we consider the theory which has the central charge 1 and is equivalent to a Z 2 orbifold of one free boson at the radius 2. This is the SU (2) torus scaled by √ 2, in accordance with the theorem.
To identify the lattice for SO(2n) 2 /U (1) n we consider the character of the identity, where we denoted by 0 the unit representation. Now, this character is equal to a theta function on some lattice M , which we wish to identify. So, the character is equal to up to factors of Dedekind's eta function.
Let us show how this lattice is obtained for SO (6). We know that the character χ 0 0 (τ ) may be written as where we denoted the representation by the superscript. For µ = 0, this is a product of two bosonic c = 1 theories, moving on the two dimensional lattice N , generated by the vectors {0, 2} and {2 √ 3, 0}, as discussed above. Now, the lattice N will be enhanced, i.e., it will be a sub lattice of M , due to the representations with µ = 0. To find M it is enough to consider µ = (1, 1), which is the second fundamental weight of SO(4). Thus we have to identify the momenta of the two fields and This we do by calculating their dimensions. The dimension of P 1 comes out to be 3/2, and that of P 2 is 1/2. Thus, they correspond to the beta vector We can continue, in this fashion, for any SO(2n) and we find that M is indeed √ 2M SU(n) , establishing theorem (1).
Theorem (1) allows to calculate easily the string functions of SO(2n) 2 , since they are given by the characters of a simple Z 2 orbifold. The fields are labeled by λ, where λ is a weight of SU (n) and their dimensions are In addition, there are the twist fields whose dimensions are ∆ = n−1 16 , n+7 16 . The characters are expressed by a level two SU (n) classical theta functions. These are defined by where λ is any element of the weight lattice of SU (n) and R denotes the root lattice of SU (n). We take the level to be m = 2 which means that this is a level two theta function.
The characters of the Z 2 orbifold are given by standard argumentation and are divided into the twisted and the untwisted sectors. In the untwisted sector we have three types of characters. The characters of the zero momenta fields are where the second term arises due to the sector twisted in the time direction. The nonzero momenta do not have such a contribution and are given simply by where λ is a weight of SU (n), n λ = 2 when λ is on the root lattice and n λ = 1, otherwise. In the twisted sector, we have two fields which have the characters, (3.16) We can determine which weight of SU (n) corresponds to each character by simply comparing the dimensions (some care is required when there is more than one field of the same dimension). Thus, we can confirm directly the GRR eq. (2.4).
We will give some examples below.
We first consider the unit character in the coset SO(2n) 2 /U (1) n . It is given by the λ = 0 character in the corresponding SU (n) 2 orbifold. Thus we have the following identity for any n, where i goes from 1 to n and χ + (τ ) was given by eq. (3.14). C n is the Cartan . This is a beautifully simple identity of the Rogers Ramanujan type that we verified directly using a Mathematica program for n = 2 to n = 8 and up to order q 20 .
Other particularly simple identities arise from the twisted sector. They correspond in the coset to the characters χ s s (τ ) and χ s s+α 1 (τ ), in the notation of eqs.
where χ t,+ (τ ) was given by eq. (3.16). For the other twist field we have, where ∆ and ∆ 1 are some dimensions. The above identities agree with our general GRR conjecture, eq. (2.4).
Let us give another example. Consider the character χ v v (τ ), for the algebra SO (8), where v is the vector representation. The field has the dimension 3/16.
The dimension determines that it corresponds to the field in the SU (4) 2 bosonic theory, which has the weight λ = ν 1 − ν 2 , where ν i is the ith fundamental weight of SU (4). Thus, the character is given by where we denoted with Θ λ,2 (τ ) the theta function on the SU (4) lattice, at level two, defined by eq. (3.13). According to our general GRR conjecture, eq. (2.4), we have the following identity, Other examples of this kind can be demonstrated. We checked all the identities for χ Λ λ (τ ), where Λθ = 1 and any λ using a Mathematica program for r = 2 to r = 8 up to order q 20 , and indeed they all hold. However, the condition Λθ = 1, where θ is the highest root (which means that Λ has a Dynkin index one), is not obeyed for some of the weights Λ and, in this case, our conjecture breaks down.
Further work is needed to find the GRR for the weights Λ such that Λθ = 1.
Let us consider now the algebra E 6 . The relevant coset here is, (4.1) We can present this coset as where the coset P is where SO(10) × U (1) is a maximal subgroup of E 6 . The characters are given by where we denoted, again, by χ Λ λ,q (τ ) the characters with, Λ, µ and q, the weights of the corresponding algebras.
Since we know the characters of SO(10) 2 , it is enough to determine the characters of P . We note that the central charge of P is It is the same central charge as that of the parafermion theory W = SU (2) 5 /U (1) [14]. Calculating the dimensions of the fields in the coset P , reveals that some of them are, indeed, as in the coset W . In fact, the theory P is equivalent to the coset W orbifolded by the symmetry A → A † , where A is any field in the theory. The dimensions of the fields in this orbifold were calculated, along with the characters, in ref. [15], and there is a complete agreement of all the dimensions, leading us to theorem (2).
It can be verified that all the fields in the coset P have dimensions which correspond either to the untwisted sector, eq. (4.7), or the twisted sector, eq.
The characters of W , orbifolded by the twist A → A † , were given in ref. [15].
Let us give some examples. We take the basis for the simple roots of E 6 which corresponds to the breaking of E 6 to SO(10) × U (1), where we denoted by α i the ith simple root. The simplest fields to consider in the P coset are where λ = s or v +s, where s (s) denotes the spinor (anti spinor) representations and v is the vector representation of SO (10). The upper index is the representation, which, here, is the singlet of E 6 . These fields have the dimensions 1/4 or 3/4, and are thus fields in the twisted sector with γ = 2, ∆ γ = 1/4, 3/4, eq. (4.11). There are two characters for these fields which are d 2 t,± , eq. (4.15), corresponding respectively to λ = s or λ = v +s. These fields are multiplied by the fields in the SO(10) 2 /U (1) 5 theory, which are the twist fields of the SU (5) 2 orbifold, whose characters are given by eq. (3.16), Thus, the two characters of the coset where ζ is either the weight ζ + = {1/2, 1/2, 1/2, 1/2, 1/2, − √ 3/2} or the weight ζ − = {3/2, 1/2, 1/2, 1/2, −1/2, − √ 3/2} are given by and We observe, now, that these two characters are given by the GRR type identity where e = 0 for r + , e = 1 for r − and C E6 stands for the Cartan matrix of E 6 .
We checked these two identities, by a Mathematica program, up to order q 20 , and indeed they hold.
Let us give another example. Consider the character, where the superescript denotes the representation and where µ 1 is the first fundamental weight of E 6 (the 27) and λ 3 is the third fundamental weight of SO(10) = D 5 . Using theorem (2), this string function can be written as where the sum is over λ which are equal to λ = λ 1 , λ 3 , λ 5 , where λ i denotes the ith fundamental weight of SO (10). We identify each of the sub characters in eq.  where [x] denotes the character of the corresponding field with the dimension x.
We used the following expressions for the dimensions of the fields. The dimension of the field ( where ρ is the sum of E 6 fundamental weights and ρ 10 that of SO(10). The This translates to the following character formula, where we denoted by ν i the ith fundamental weight of SU (5) and where Θ(ν) is the classical theta function of SU (5) at level two and the weight ν, eq. (3.13), which expresses the string functions of SO(10) 2 , in accordance with theorem (1).
Finally, we can verify the following GRR in accordance with the conjecture, eq.
where ∆ is some dimension. We verified, by Mathematica, this GRR up to order q 20 and indeed it holds.
Let us turn now to the string functions of E 7 at level two. The relevant coset here is Here we use the following ladder coset construction, where G is the coset and where there is a summation over the intermediate representations, where λ ′ andλ are the projections of λ to the U (1) and U (1) 6 , respectively.
We can calculate the central charge of G, and it is given by c = 13/10.
We note that it is the same central charge as the product of minimal models where the classical SU(2) theta functions, θ r,s (τ ) were defined by eq. (4.17). The dimension of the field in the G theory where Λ is a highest weight of E 7 at level 2, µ is a weight of SO(12) 2 , q is the weight of the U (1), is given by where ρ, or ρ 12 , are the sum of the fundamental weights of E 7 , or SO(12), respectively. We take a basis for the simple weights of E 7 , which corresponds to the breaking into SO(12) × U (1), where κ + = s and κ − =s + v and s is the weight of the spinor representation, s is the anti spinor and v is the vector, all of the algebra SO (12). µ is summed over the two values µ + = s and µ − = v +s. We can identify the fields in the G theory, by determining the dimensions of these fields, using eq. (5.9,5.6), which are 3/16 and 27/16, we infer that they correspond to the fields in the minimal models M 3 ×M 5 , which are (1, 2, 1, 2) and (1, 2, 1, 4), for g + and g − , respectively, where we denoted by (n 1 , m 1 , n 2 , m 2 ) the field in the product of minimal models M 3 × M 5 , with dimensions ∆ n 1 ,m 1 and ∆ n 2 ,m 2 , respectively. The two characters of these fields are given by eq. (5.7) and are where (n 1 , m 1 ) and (n 2 , m 2 ) take the two values indicated before. In accordance with theorem (3), these characters are multiplied by fields in the bosonic orbifold SU (6) 2 . Since the representations of SO(12) 2 are spinorial, in this example, these fields are the two twist fields of the orbifolded SU (6) 2 theory, which are given by Thus, the two characters S ± (τ ) are given by where e = 0 for r + and e = 1 for r − , and C E7 is the Cartan matrix of E 7 .
y ± are some dimensions. We checked these two identities up to order q 20 , by Mathematica, and indeed they hold.

The algebra E 8
The remaining algebra is E 8 . Our objective is to calculate the string functions of E 8 at level two. To do this, we use the following ladder coset construction, where, as usual, we denoted by the superscripts the highest weights of the representations. Since, we know from theorem (1) the string functions of SO (16) 2 it is enough to resolve the coset The central charge of this coset is calculated to be c = 1/2. Since the theory G is unitary, it has to be equivalent to the Ising model, previously denoted as M 3 .
This yields an analytic expression for the string functions of (E 8 ) 2 , and theorem (4) follows.
Theorem (4): The conformal field theory, the coset E 8 /U (1) 8 at level two, is equivalent to a product of the Ising model times the orbifold of free bosons on the SU (8) torus at level two.
In view of theorem (4), we can calculate the string functions of (E 8 ) 2 and verify our GRR conjecture, eq. (2.4). Let us give an example. Consider the field in the H theory, H 0 µ where µ = s or µ =s + v, and where we denoted by s,s and v the spinor, anti spinor and vector representations of SO (16). In the ladder coset construction, eq. (6.1), this translates to the product of characters and where we took R + for µ = s and R − for µ = v +s. Since G is a field in the Ising model, it can be identified by its dimension, which is 1/16, where we used eq. (5.6). Thus, G = χ 2,2 , which is the character of the spin field in the Ising model. Now, R ± are given by the two characters of the twist fields of the bosonic SU (8) 2 , which are (1 + q j−1/2 ) −7 . (6.6) Thus, eq. (6.6) yields these two characters, enabling the verification of our GRR conjecture, eq. (2.4). We have the expression, where Q + = (1, 0, 0, 0, 0, 0, 0, 0) for R + and Q − = (1, 0, 0, 0, 0, 1, 0, 0) for R − , where Q ± is defined modulo two and where C E 8 is the Cartan matrix of E 8 and z ± are some dimensions. Eq. (6.7) agrees with our general GRR conjecture, eq.
(2.4). We verified directly these two identities and, indeed, they hold up to order q 10 .
The reader might wonder about the result we would get if we take in the GRR, eq. (2.4), the summation on Q mod 2, other than the ones given by eq.
(2.6). The result is surprising. For any choice of Q mod 2, we find that the GRR gives one of the characters of the corresponding coset. This amounts to a great deal of interesting and nontrivial identities, and it generalizes the identity by Slater, ref. [19], for the Ising model. These identities we term generalized Slater identities. Some of these identities were already noted in ref. [6].

Discussion
In this paper, we described generalized Rogers Ramanujan (GRR) identities for the string functions of all the simply laced algebras at level two. These are the characters of the conformal field theory (CFT) of generalized parafermions [10], specialized to the appropriate algebra and level. We find identities for all the characters, such that the weight in the algebra has Dynkin index one.
These identities are surprisingly simple and pretty, and they center around the Cartan matrix of the algebra in question, in a type of ADE classification for such identities. This adds considerably to the GRR identities which were already established for characters of certain conformal field theories [1,2,3,4,5,6].
An interesting question is how to extend these identities to the weights which have Dynkin index greater than one. Another important question is how to extend these GRR identities to the levels which are greater than two and to the non simply laced algebras.
Perhaps, the most interesting issue, from the physics point of view, is the relation to RSOS lattice models. For the case of A r these models were investigated by Jimbo et al. [20]. The relation between GRR and RSOS lattice models was established before in a great deal of cases [7,8,9,4]. Importantly, this relation, once established, will furnish a way to prove the GRR identities, described here, and will also provide an expression for the finite one dimensional configuration sum in the RSOS models, which is the main piece of the local state probabilities (or, the one point functions), a central problem in this field.