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 $r$
The untwisted affine Kac-Moody algebra is given by the central extension of the loop algebra

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

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 $\rho=\sum_{i=0}^{r}\Lambda_i$

affine Weyl group

Affine Weyl group
The affine Weyl group $W$ is generated by $s_0,s_1,\cdots, s_l\in \operatorname{Aut}\,\mathfrak{h}^{*}$ 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 $\alpha_a^{\vee}=t_a\alpha_a$

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

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

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

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

this expression is not quite correct if the component of $\delta$ in $\lambda$ is considered

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

string functions

String functions and branching functions
let $\Lambda$ be an integral dominant weight and $k$ be the level
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 $$ c_{\lambda }^{\Lambda}=e^{-m_{\Lambda,\lambda}\delta}\sum_{n=0}^{\infty}\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{-n \delta} $$ 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 (see Indefinite theta functions)

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

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

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 $$ b(\Lambda,\lambda,\Lambda',\lambda')=\frac{i^{|\Delta_{+}|}}{|P/Q^{\vee}|\sqrt{k(k+h^{\vee})^r}}\exp(\frac{2\pi i(\lambda|\lambda')}{k}) \sum_{w\in \overline{W}} (-1)^{\ell(w)}\exp \left(-{\frac{2\pi i ( w(\Lambda+\rho)|\Lambda'+\rho)}{k+h^{\vee}}}\right) $$ 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 $\ell\geq 2$
$H_\ell=\{(a,m)|a=0,\cdots, r, 1\leq m \leq t_a \ell-1\}$
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]

Up to a powe of $q$, 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 $\sum_{(a,m) \in H_\ell}mN^{(a)}_m\alpha_a \equiv \overline{\lambda} \mod \ell M$.

note that $\eta(\tau)=\prod_{j=1}^\infty(1-q^j)$

example

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

thm [LP1985]

$$ c^{\ell\Lambda_0}_{0}(\tau)\cdot \eta(\tau)=\sum_{(N_1,\dots,N_{\ell-1})\in \mathbb{Z}_{\geq 0}^{l-1}}\frac{q^{\sum_{n,m=1}^{\ell-1} N_n N_m (\min (n,m) -\frac{nm}{\ell})}} {\prod_{m=1}^{\ell-1}(1-q)\cdots(1-q^{N_m})} $$ where the sum is under the constraint $ \sum_{m=1}^{\ell-1} m N_m \equiv 0 \ \mathrm{mod}\ 2\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.