Talk on Rogers-Ramanujan identities in affine Kac-Moody algebras

abstract

The character of an irreducible representation with dominant integral highest weight of an affine Lie algebra can be written as a finite sum of theta functions with coefficients called string functions. There are still many aspects of string functions that are not well understood. In this talk I will review the basic properties of them and introduce the conjecture of Kuniba-Nakanishi-Suzuki on generalized Rogers-Ramanujan identities related to them.

review of affine Lie algebras and their integrable representations

affine Lie algebras

Affine Kac-Moody algebra
Let \(\overline{\mathfrak{g}}\) be a complex simple Lie algebra of rank \(r\)
\((a_{ij})_{i,j\in \overline{I}}\) Cartan matrix, \(\overline{I}=\{1,\cdots, r\}\) index set
untwisted affine Kac-Moody algebra

\[\tilde{\mathfrak{g}}=\overline{\mathfrak{g}}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c \oplus\mathbb{C}d\]

\((a_{ij})_{i,j\in I}\) : extended Cartan matrix \(I=\{0\}\cup \overline{I}\)
\(\theta\) : highest root
\(e_0=f_{\theta}\otimes t, f_0=e_{\theta}\otimes t^{-1}, h_0=-h_{\theta}\otimes 1+c\)
generators \(e_i,h_i,f_i , (i=0,1,2,\cdots, r)\) with relations
- \(\left[h,h'\right]=0\)
- \(\left[e_i,f_j\right]=\delta _{i,j}h_i\)
- \(\left[h,e_j\right]=\alpha_{j}(h)e_j\)
- \(\left[h,f_j\right]=-\alpha_{j}(h)f_j\)
- \(\left(\text{ad} e_i\right){}^{1-a_{i,j}}\left(e_j\right)=0\) (\(i\neq j\))
- \(\left(\text{ad} f_i\right){}^{1-a_{i,j}}\left(f_j\right)=0\) (\(i\neq j\))
basis of the Cartan subalgebra \(\mathfrak{h}\); \(h_0,h_ 1,\cdots,h_r,d\)
dual basis for \(\mathfrak{h}^{*}\); \(\Lambda_0,\Lambda_1,\cdots,\Lambda_r,\delta\)
we call \(\Lambda_0,\Lambda_1,\cdots,\Lambda_r\) the fundamental weights and \(\delta\) the imaginary root
simple roots \(\alpha_0,\alpha_1,\cdots,\alpha_r\)
distinguished elements
- central element \(c=\sum_{i=0}^{r}a_i^{\vee}h _i\)
- imaginary root \(\delta=\sum_{i=0}^{r}a_i\alpha_i\)
- Weyl vector \(\rho=\sum_{i=0}^{r}\Lambda_i\)
normalize the bilinear form \((\cdot|\cdot)\) on \(\mathfrak{h}^{*}\) so that \((\theta|\theta)=2\).

affine Weyl group

Affine Weyl group
The affine Weyl group \(W\) is generated by \(s_0,s_1,\cdots, s_r\in \operatorname{Aut}\,\mathfrak{h}^{*}\) defined by

\[s_{i}\lambda = \lambda -\lambda(h_i)\alpha_i\] for \(i=0,1, \cdots, r\).

for \(\gamma\in \mathfrak{h}^{*}\), define \(t_{\gamma} : \mathfrak{h}^{*}\to \mathfrak{h}^{*}\) by

\[t_{\gamma}(\lambda)=\lambda+\lambda(c)\gamma-\left(\frac{1}{2}\lambda(c)|\gamma|^2+(\gamma|\lambda)\right)\delta \]

let \(M\) be the lattice generated by \(\overline{W}(\theta^{\vee})\)
more explicitly, \(M=\sum_{a\in I}\Z\, \alpha_a^{\vee}\) where \(\alpha_a^{\vee}=t_a\alpha_a\) where \(t_a=\frac{2}{(\alpha_a|\alpha_a)}\)
note that \(M\subseteq Q\), where \(Q\) denotes the root lattice

thm

Let \(T=\{t_{\gamma}|\gamma\in M\}\). Then \(W=\overline{W} \ltimes T\)

integrable representations and characters

Unitary representations of affine Kac-Moody algebras
for each \(\lambda\in \mathfrak{h}^{*}\), we get an irreducible \(\tilde{\mathfrak{g}}\)-module \(L(\lambda)\) (which is a quotient of the Verma module)
character of an irreducible highest weight representation \(L(\lambda)\)

\[\operatorname{ch} L(\lambda):=\sum_{\beta\in \mathfrak{h}^{*}}\operatorname{mult}_{\lambda}(\beta) e^{\beta}\]

dominant integral weights \(\lambda(\mathfrak{h}_i)\in \mathbb{Z}_{\geq 0},\, i=0,1,\cdots,r\)
weight lattice

\[P_{+}=\{\lambda\in \mathfrak{h}^{*}|\lambda=\sum_{i=0}^{l}\lambda_{i}\Lambda_i+\xi \delta, \lambda_i \in\mathbb{Z}_{\geq 0},\xi \in \mathbb{C}\}\]

A \(\tilde{\mathfrak{g}}\)-module \(V\) is called integrable if \(V=\oplus_{\lambda\in \mathfrak{h}^{*}}V_{\lambda}\) and if \(e_i : V\to V\) and \(f_i : V\to V\) are locally nilpotent for all \(i=0,1,\cdots, r\)

thm

If \(\lambda\in P_{+}\), then \(L(\lambda)\) is an integrable representation.

thm (Kac)

Let \(\lambda\in P_{+}\). Then \[ \operatorname{ch} L(\lambda)=\frac{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\lambda+\rho})}{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho})} \]

see Weyl-Kac character formula
we call \(k=\lambda(c)\) the level of \(L(\lambda)\) and have \(k=\sum_{i=0}^{l}a_{i}^{\vee}\lambda_{i}\in \mathbb{Z}\)

theta functions

Theta functions in Kac-Moody algebras
for each \(\lambda\in P^k\), define the theta function as

\[ \Theta_{\lambda}= e^{-\frac{|\lambda|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)}=e^{k\Lambda_0}\sum_{\gamma\in M+\overline{\lambda}/k}e^{-\frac{1}{2}k|\gamma|^2 \delta + k \gamma} \]

prop

Let \(\lambda\in P_{+}^k\). We have \[ e^{-m_{\lambda}\delta}\operatorname{ch} L(\lambda)=\frac{\sum_{w\in \overline{W}} (-1)^{\ell(w)}\Theta_{w(\lambda+\rho)}}{\sum_{w\in \overline{W}} (-1)^{\ell(w)}\Theta_{w\rho}} \] where \(m_{\lambda}=\frac{(\lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}\)

note that \(m_{\lambda}=h_{\lambda}-\frac{c_{\lambda}}{24}+\xi\) where \(h_{\lambda}=\frac{(\bar{\lambda}+2\rho|\bar{\lambda})}{2(k+h^{\vee})}\) and \(c_{\lambda}=\frac{k}{k+h^{\vee}}\dim \mathfrak{\overline{g}}\)

string functions

String functions and branching functions
\(\Lambda\in P_{+}^{k}\)
A weight \(\mu\) of \(L(\Lambda)\) is called a maximal weight if \(\mu+\delta\) is not a weight
for each \(\mu\), there exists a unique integer \(n\geq 0\) such that \(\mu+n\delta\) is maximal
the set of maximal weights is stable under \(W\)

def

For each \(\lambda\in \mathfrak{h}^{*}\), the string function \(c_{\lambda }^{\Lambda}\) is defined by \[ c_{\lambda }^{\Lambda}=e^{-m_{\Lambda,\lambda}\delta}\sum_{n=-\infty}^{\infty}\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{-n \delta} \] where \[ m_{\Lambda,\lambda}=m_{\Lambda}-\frac{\lambda^2}{2k} \]

an explicit expression for the string functions is not known in general
the few that are known were guessed using the modular transformations
an exception is the case of \(A_{1}^{(1)}\)
- string functions were found to be Hecke indefinite modular forms
modular form of weight \(-r/2\)

properties

\(c_{\lambda }^{\Lambda}=c_{w\lambda }^{\Lambda}\) for \(w\in W\)

thm

We have \[ e^{-m_{\Lambda}\delta}\operatorname{ch} L(\Lambda)=\sum_{\lambda\in P^k \mod kM+\mathbb{C}\delta}c^{\Lambda}_{\lambda }\Theta_{\lambda} \]

proof

\[ \begin{aligned} \operatorname{ch} L(\Lambda)&=\sum_{\lambda\in \max{L(\Lambda)}}\sum_{n=0}^{\infty}\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{\lambda-n \delta}\\ &=\sum_{\lambda\in \max{L(\Lambda)} \mod kQ^{\vee}} \left(\sum_{\gamma\in M}e^{t_{\gamma}(\lambda)}\right)\sum_{n=0}^{\infty}\left(\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{-n\delta}\right)\\ &=\sum_{\lambda\in \max{L(\Lambda)} \mod kQ^{\vee}} e^{m_{\Lambda,\lambda}\delta}e^{-m_{\Lambda,\lambda}\delta}\left(\sum_{\gamma\in M}e^{t_{\gamma}(\lambda)}\right)\sum_{n=0}^{\infty}\left(\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{-n\delta}\right)\\ &=\sum_{\lambda\in \max{L(\Lambda)} \mod kQ^{\vee}} e^{(m_{\Lambda}-\frac{\lambda^2}{2k})\delta}e^{-m_{\Lambda,\lambda}\delta}\left(\sum_{\gamma\in M}e^{t_{\gamma}(\lambda)}\right)\sum_{n=0}^{\infty}\left(\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{-n\delta}\right)\\ &=\sum_{\lambda\in \max{L(\Lambda)} \mod kQ^{\vee}} e^{m_{\Lambda}\delta}c^{\Lambda}_{\lambda}\Theta_{\lambda} \end{aligned} \] ■

modular transformations

thm

We have \[ c_{\lambda }^{\Lambda}(-\frac{1}{\tau})=(\frac{\tau}{i})^{-r/2}\sum_{(\Lambda',\lambda')}b(\Lambda,\lambda,\Lambda',\lambda')c_{\lambda'}^{\Lambda'}(\tau) \] where \[ b(\Lambda,\lambda,\Lambda',\lambda')=(*)\exp(\frac{2\pi i(\lambda|\lambda')}{k}) \sum_{w\in \overline{W}} (-1)^{\ell(w)}\exp \left(-{\frac{2\pi i ( w(\Lambda+\rho)|\Lambda'+\rho)}{k+h^{\vee}}}\right) \] and the sum is over all \(\Lambda'\in P_{+}^k\) and \(\lambda' \in P^k \mod kM+\mathbb{C}\delta\)

asymptotic growth of coefficients

use the circle method

Rogers-Ramanujan identities for string functions

Fermionic formula for string functions and parafermion characters
now we denote the level by \(\ell\in \mathbb{Z}\) and assume \(\ell\geq 2\)
\(H_\ell=\{(a,m)|a=1,\cdots, r, 1\leq m \leq t_a \ell-1\}\), \(t_a=2/(\alpha_a|\alpha_a)\)
let

\[ K^{m n}_{a b} = \Bigl(\hbox{min}(t_bm, t_an) - {m n\over \ell}\Bigr) (\alpha_a \vert \alpha_b) \]

conjecture [KNS93]

We have \begin{equation}\label{qkns} c^{\ell\Lambda_0}_\lambda(q)\cdot \eta(\tau)^r= \sum_{\{N^{(a)}_m\}}\frac{q^{\frac{1}{2}\sum_{(a,m), (b,n) \in H_\ell} K^{mn}_{ab}N^{(a)}_mN^{(b)}_n}} {\prod_{(a,m) \in H_\ell}(1-q)(1-q^2)\cdots (1-q^{N^{(a)}_m})} \end{equation} up to a rational power of \(q\). The outer sum is over \(N^{(a)}_m \in \Z_{\ge 0}\) such that \[\sum_{(a,m) \in H_\ell}mN^{(a)}_m\alpha_a \equiv \overline{\lambda} \mod \ell M.\]

example

let \(\mathfrak{g}=A_1\)
consider the vacuum representation of level \(\ell\)

thm [Lepowski-Primc 1985]

\[ c^{\ell\Lambda_0}_{\ell\Lambda_0}(\tau)\cdot \eta(\tau)=\sum_{(N_1,\dots,N_{\ell-1})\in \mathbb{Z}_{\geq 0}^{\ell-1}}\frac{q^{\sum_{n,m=1}^{\ell-1} N_n N_m (\min (n,m) -\frac{nm}{\ell})}} {\prod_{m=1}^{\ell-1}(1-q)\cdots(1-q^{N_m})} \] where the sum is under the constraint \( \sum_{m=1}^{\ell-1} m N_m \equiv 0 \ \mathrm{mod}\ \ell\).

the associated matrix is \(2\otimes \mathcal{C}(A_{\ell-1})^{-1}\)

related items

references

Kac, Infinite dimensional Lie algebras, Chapter 12 and 13
[KNS93] Kuniba, A., T. Nakanishi, and J. Suzuki. 1993. “Characters in Conformal Field Theories from Thermodynamic Bethe Ansatz.” Mod. Phys. Lett. A8 (1993) 1649-1660 arXiv:hep-th/9301018 (January 7). doi:10.1142/S0217732393001392. http://arxiv.org/abs/hep-th/9301018.