Folded quantum integrable models and deformed W-algebras

Abstract

We propose a novel quantum integrable model for every non-simply laced simple Lie algebra ${{\mathfrak {g}}}$, which we call the folded integrable model. Its spectra correspond to solutions of the Bethe Ansatz equations obtained by folding the Bethe Ansatz equations of the standard integrable model associated with the quantum affine algebra $U_q(\widehat{{{\mathfrak {g}}}'})$ of the simply laced Lie algebra ${{\mathfrak {g}}}'$ corresponding to ${{\mathfrak {g}}}$. Our construction is motivated by the analysis of the second classical limit of the deformed W-algebra of ${{\mathfrak {g}}}$, which we interpret as a “folding” of the Grothendieck ring of finite-dimensional representations of $U_q(\widehat{{{\mathfrak {g}}}'})$. We conjecture, and verify in a number of cases, that the spaces of states of the folded integrable model can be identified with finite-dimensional representations of $U_q({}^L{\widehat{{{\mathfrak {g}}}}})$, where $^L{\widehat{{{\mathfrak {g}}}}}$ is the (twisted) affine Kac–Moody algebra Langlands dual to ${\widehat{{{\mathfrak {g}}}}}$. We discuss the analogous structures in the Gaudin model which appears in the limit $q \rightarrow 1$. Finally, we describe a conjectural construction of the simple ${{\mathfrak {g}}}$-crystals in terms of the folded q-characters.

Appendix: Toward constructing a folded version of the qKZ equations for non-simply laced Lie algebras

The qKZ equations [27] for $U_q({\widehat{{{\mathfrak {g}}}}})$ (with the twist parameter u, which is an element of the Cartan subgroup H as above), can be written in the form

$$\begin{aligned} \Psi (z_1,\ldots ,pz_j,\ldots ,z_N) = R^{V_jV_{j-1}}_{j,j-1}(pz_j/z_{j-1}) \cdots R^{V_jV_1}_{j,1}(pz_j/z_1) \cdot u_j \cdot \nonumber \\ R^{V_jV_N}_{j,N}(z_j/z_N) \cdots R^{V_jV_{j+1}}_{j,j+1}(z_j/z_{j+1}) \Psi (z_1,\cdots ,z_j,\cdots ,z_N). \end{aligned}$$

(10.1)

Here, each $V_i$ denotes an irreducible finite-dimensional representation of $U_q({\widehat{{{\mathfrak {g}}}}})$, $V_i(z_i)$ is its shift by the spectral parameter $z_i$, and $R^{V_jV_i}_{j,i}(z_j/z_i)$ is the R-matrix acting on $V_j(z_j) \otimes V_i(z_i)$ normalized so that it acts as the identity on $v_j \otimes v_i$, where $v_j$ and $v_i$ are highest weight vectors in $V_j(z_j)$ and $V_i(z_i)$, respectively. Also, $u_j$ denotes $u|_{V_j}$.

Let us denote the operator on the right-hand side of the jth equation by $K_j(p)$.

The critical level limit corresponds to $p \rightarrow 1$. In this limit, we have

$$\begin{aligned} K_j(1)= & {} R^{V_jV_{j-1}}_{j,j-1}(z_j/z_{j-1}) \cdots R^{V_jV_1}_{j,1}(z_j/z_1) \cdot u_j \nonumber \\&\quad \cdot R^{V_jV_N}_{j,N}(z_j/z_N) \cdots R^{V_jV_{j+1}}_{j,j+1}(z_j/z_{j+1}). \end{aligned}$$

(10.2)

Recall that for an auxiliary representation V of $U_q({\widehat{{{\mathfrak {g}}}}})$ we have the transfer matrix $t_V(z,u)$. These transfer-matrices commute with each other for a fixed u and different V and z. In what follows, we will use the same notation $t_V(z,u)$ for the action of the transfer-matrix $t_V(z,u)$ on the tensor product $V_N(z_N) \otimes \ldots \otimes V_1(z_1)$. Thus,

$$\begin{aligned} t_V(z,u) = {\text {Tr}}_a(u_a R^{VV_N}_{a,N}(z/z_N) \cdots R^{VV_1}_{a,1}(z/z_1)), \end{aligned}$$

(10.3)

where the subscript a indicates the auxiliary representation V(z).

Proposition 10.1

Suppose that $V_j$ is such that up to a scalar, $R^{V_jV_j}(1) = P$, the permutation operator on $V_j(z) \otimes V_j(z)$ sending $x \otimes y$ to $y \otimes x$. Then, $K_j(1) = t_{V_j}(z_j,u)$.

Proof

If $R^{V_jV_j}(1) = P$, then formula (10.3) with $V(z) = V_j(z_j)$ becomes

$$\begin{aligned}&t_V(z,u) \nonumber \\&\quad = {\text {Tr}}_a(u_a R^{V_jV_N}_{a,N}(z_j/z_N) \cdots R^{V_jV_{j+1}}_{a,j+1}(z_j/z_{j+1}) P_{aj} R^{V_jV_{j-1}}_{a,j-1}(z_j/z_{j-1}) \cdots R^{V_jV_1}_{a,1}(z_j/z_1)).\nonumber \\ \end{aligned}$$

(10.4)

Using the identify $A_a P_{aj} = P_{aj} A_j$, we rewrite the RHS of (10.4) as

$$\begin{aligned} {\text {Tr}}_a(P_{aj} u_j R^{V_jV_N}_{j,N}(z_j/z_N) \cdots R^{V_jV_{j+1}}_{j,j+1}(z_j/z_{j+1}) R^{V_jV_{j-1}}_{a,j-1}(z_j/z_{j-1}) \cdots R^{V_jV_1}_{a,1}(z_j/z_1)).\nonumber \\ \end{aligned}$$

(10.5)

Next, using the cyclic property of the trace, we rewrite (10.5) as:

$$\begin{aligned} {\text {Tr}}_a(R^{V_jV_{j-1}}_{a,j-1}(z_j/z_{j-1}) \cdots R^{V_jV_1}_{a,1}(z_j/z_1) P_{aj} u_j R^{V_jV_N}_{j,N}(z_j/z_N) \cdots R^{V_jV_{j+1}}_{a=j,j+1}(z_j/z_{j+1})).\nonumber \\ \end{aligned}$$

(10.6)

Using formula $A_a P_{aj} = P_{aj} A_j$ again, we rewrite (10.6) as:

$$\begin{aligned} {\text {Tr}}_a(P_{aj} R^{V_jV_{j-1}}_{j,j-1}(z_j/z_{j-1}) \ldots R^{V_jV_1}_{j,1}(z_j/z_1) u_j R^{V_jV_N}_{j,N}(z_j/z_N) \ldots R^{V_jV_{j+1}}_{a=j,j+1}(z_j/z_{j+1})).\nonumber \\ \end{aligned}$$

(10.7)

In the last formula, the only operator depending on the auxiliary space is $P_{aj}$ and its trace over the auxiliary space is the identity operator on $V_j(z_j)$. Hence, (10.7) is equal to $K_j(1)$. $\square$

Next, we discuss under what conditions $R^{VV}(1) = P$.

Lemma 10.2

Let V be an irreducible representation of $U_q({\widehat{{{\mathfrak {g}}}}})$ such that $V \otimes V$ is also irreducible. Then, $R^{VV}(1) = P$.

Proof

It follows from the definition that $P \circ R^{VV}(1)$ is an intertwining operator $V \otimes V \rightarrow V \otimes V$. If $V \otimes V$ is irreducible, then by Schur’s lemma it is a scalar operator. Under our normalization, it then has to be the identity operator, and hence $R^{VV}(1) = P$. $\square$

Note that a representation V satisfying the condition of Lemma 10.2 is called real in [34]. Not all irreducible representations of $U_q({\widehat{{{\mathfrak {g}}}}})$ are real, as shown in [44], see [34, Sect. 13.6]. However, there is a large class of real representations: Kirillov–Reshetikhin modules.

Proposition 10.3

Let V be any Kirillov–Reshetikhin module over $U_q({\widehat{{{\mathfrak {g}}}}})$. Then, $V \otimes V$ is irreducible.

Proof

There are several possible arguments. It follows from [5] that $V\otimes V$ is cyclic and cocyclic, and hence is irreducible. The statement also follows from the fact that the square of the q-character of a Kirillov–Reshetikhin module has a unique dominant monomial; namely, its highest monomial. The latter follows from the results of [35, Sect. 3.2.2]. In addition, the statement has been established by a different method in [39]. $\square$

Note however that even if $V \otimes V$ is reducible, we may still have $R^{VV}(1) = P$. It would be interesting to describe all irreducible representations V of $U_q({\widehat{{{\mathfrak {g}}}}})$ satisfying the condition $R^{VV}(1) = P$.

In any case, the above results readily imply that in the case when all $V_j$’s are Kirillov–Reshetikhin modules, the operators $K_j(1), j=1,\ldots ,N$, are commuting Hamiltonians of the XXZ-type integrable model associated with $U_q({\widehat{{{\mathfrak {g}}}}})$. It is in this sense that we say that one recovers this integrable model in the critical level limit of the qKZ equations.

Similarly, under the above condition on $V_j$, the operator $K_j(p)$ is the transfer-matrix $t_{V_j}(z_j,u)$ acting on

$$\begin{aligned} V_N(z_N) \otimes \cdots \otimes V_j(z_j) \otimes V_{j-1}(z_{j-1}p^{-1}) \otimes V_1(z_1p^{-1}). \end{aligned}$$

However, because of the shifts by $p^{-1}$ these operators do not commute with each other if $p \ne 1$.

Now suppose that ${{\mathfrak {g}}}$ is a non-simply laced simple Lie algebra. We would like to construct a “folded qKZ system” such that in the critical level limit the operators on the right hand side become the Hamiltonians of the folded quantum integrable model described in this paper. This means, in particular, that they should correspond to transfer-matrices of $U_q(\widehat{{{\mathfrak {g}}}'})$ rather than $U_q({\widehat{{{\mathfrak {g}}}}})$. Thus, each $V_i(z_i)$ should be a representation of $U_q(\widehat{{{\mathfrak {g}}}'})$.

Unfortunately, naive attempts to construct this folded qKZ system appear to fail:

• (1) For each $V_i(z_i)$, we have its subspace $(V_i(z_i))(u)$ defined as above and we can take the tensor product of these subspaces, $\otimes _{i=1}^N (V_i(z_i))(u)$. However, it is not clear why this subspace would be preserved by the operators $K_j$ or their p-deformed versions $K_j(p)$ appearing on the RHS of (10.1).

• (2) We take the subspace V(u) of the entire tensor product $V=\otimes _{i=1}^N V_i(z_i)$. According to Conjecture 5.15,(ii), it contains a subspace isomorphic to a representation M(V) of $U_q(^L\widehat{{{\mathfrak {g}}}})$. Moreover, since the algebra of transfer-matrices of $U_q(\widehat{{{\mathfrak {g}}}'})$ commutes with the Baxter operators $Q_j(z), j \in I'$, it follows that all transfer-matrices of $U_q(\widehat{{{\mathfrak {g}}}'})$ preserve this subspace V(u). In particular, the operators $K_j$ given by formula (10.2), being the transfer-matrices of $U_q(\widehat{{{\mathfrak {g}}}'})$, preserve V(u). But the problem is that on the right-hand side of these qKZ equations we have the operators $K_j(p)$ with $p\ne 1$, which are the transfer matrices acting on the tensor product of representations in some of which (namely, the ones with $i=1,\ldots ,j-1$) there is a multiplicative shift in the spectral parameter by $p^{-1}$. It is not clear why these operators should preserve V(u) (where V is the tensor product of the representations $V_i(z_i)$ without any shift by $p^{-1}$).

Hence, at the moment it is unclear to us how to fold the qKZ equations for $U_q(\widehat{{{\mathfrak {g}}}'})$ in such a way that in the critical level limit we recover the commuting Hamiltonians of the folded quantum integrable model associated with $U_q({\widehat{{{\mathfrak {g}}}}})$.