Talk on String functions and quantum affine algebras
abstract
The character of an irreducible representation with dominant integral highest weight of an affine Lie algebra can be written as a linear combination of theta functions, with coefficients given by string functions which are modular forms. 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 explain certain connections with finite-dimensional representations of quantum affine algebras.
key message
- string functions know about Kirillov-Reshetikhin modules
- infinite vs. finite
$ \newcommand{\g}{\mathfrak{g}} \newcommand{\h}{\mathfrak{h}} \newcommand{\res}{\operatorname{res}} \newcommand{\uqg}{U_{q}(\g)} \newcommand{\ghat}{\widehat{\g}} \newcommand{\uqghat}{U_{q}(\ghat)} $
review of affine Lie algebras and their integrable representations
affine Lie algebras
- Affine Kac-Moody algebra
- $\overline{\mathfrak{g}}$ : complex simple Lie algebra of rank $r$ assoc. to Cartan matrix $(a_{ij})_{i,j\in \overline{I}}$, $\overline{I}=\{1,\cdots, r\}$
- untwisted affine Kac-Moody algebra $\mathfrak{g}$
$$\mathfrak{g}=\overline{\mathfrak{g}}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c \oplus\mathbb{C}d$$
- \((a_{ij})_{i,j\in I}\) : extended Cartan matrix $I=\{0\}\cup \overline{I}$
- can be also defined as a Lie algebra with generators \(e_i,h_i,f_i , (i=0,1,2,\cdots, r)\) and relations, for example,
- \(\left(\text{ad} e_i\right){}^{1-a_{i,j}}\left(e_j\right)=0\) (\(i\neq j\))
- 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 \(\alpha_0,\alpha_1,\cdots,\alpha_r\)
- $a_i,\, i=0,1,\dots, r$ : marks
- $a_i^{\vee},\, i=0,1,\dots, r$ : comarks
- distinguished elements
- longest root of $\overline{\mathfrak{g}}$ : $\theta = \sum_{i=1}^{r}a_i\alpha_i$
- 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 \(\rho=\sum_{i=0}^{r}\Lambda_i\)
remarks on affine weights
- call $k=\lambda(c)$ the level of $\lambda\in \mathfrak{h}^{*}$
- sometimes convenient to write $\lambda\in \mathfrak{h}^{*}$ as $\lambda=(k;\overline{\lambda};\xi)\in \mathbb{C}\times \overline{\mathfrak{h}}^{*}\times \mathbb{C}$ where $k=\lambda(c)$, $\overline{\lambda}$ is the restriction of $\lambda$ on $\overline{\mathfrak{h}}$, $\xi=\lambda(\delta)$
- $\Lambda_0 = (a_0^{\vee};0;0)$
- $\Lambda_i = (a_i^{\vee};\omega_i;0)$, for $i=1,\dots, r$ ($\omega_i$ is fundamental weight for $\overline{\mathfrak{g}}$)
- $\delta = (0;0;0)$, for $i=1,\dots, r$
- $\alpha_0 = (0;-\theta;1)$
- $\alpha_i = (0;\alpha_i;0)$, for $i=1,\dots, r$ ($\alpha_i$ simple root for $\overline{\mathfrak{g}}$)
- bilinear form $(\cdot|\cdot)$ on $\mathfrak{h}^{*}$
- $\left((k_1;\overline{\lambda}_1;\xi_1)|(k_2;\overline{\lambda}_2;\xi_2)\right) = k_1\xi_2+k_2\xi_1+(\overline{\lambda}_1|\overline{\lambda}_2)_{\overline{\mathfrak{h}}^{*}}$
- normalize $(\cdot|\cdot)$ so that $(\theta|\theta)_{\overline{\mathfrak{h}}^{*}}=2$
- sometimes write $\overline{\lambda} = (0;\overline{\lambda};0)$ by abusing notation
- let $Q=\sum_{i=1}^{r}\Z\, \alpha_i\subseteq \mathfrak{h}^{*}$ (root lattice of $\overline{\mathfrak{g}}$)
- define $M\subseteq Q$ by $M=\{\sum_{i=1}^{r}\Z\, \alpha_i^{\vee}\}$ where $\alpha_i^{\vee}=t_i\alpha_i$ where $t_i=\frac{2}{(\alpha_i|\alpha_i)}$
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
$$t_{\gamma}(\lambda)=\lambda+\lambda(c)\gamma-\left(\frac{1}{2}\lambda(c)|\gamma|^2+(\gamma|\lambda)\right)\delta $$
- 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 $\lambda\in \mathfrak{h}^{*}$, $\exists$ irreducible $\mathfrak{g}$-module $L(\lambda)$ (quotient of Verma module)
- A $\mathfrak{g}$-module $V$ is 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 $i=0,1,\cdots, r$
- $\Lambda\in \mathfrak{h}^{*}$ is dominant integral if $\Lambda(\mathfrak{h}_i)\in \mathbb{Z}_{\geq 0},\, i=0,1,\cdots,r$
- let $P_{+}$ be the set of dominant integral weights, i.e. $\{\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}\}
$
- thm
Let $V$ be an irreducible $\mathfrak{g}$-module in a certain category $\mathcal{O}$. Then $V=L(\Lambda)$ for some $\Lambda\in \mathfrak{h}^{*}$ and $L(\Lambda)$ is integrable if and only if $\Lambda\in P_{+}$
- why care irreducible and integrable representation? Weyl's character formula holds
- character of \(L(\Lambda)\)
$$\operatorname{ch} L(\Lambda):=\sum_{\lambda\in \mathfrak{h}^{*}}\operatorname{mult}_{\Lambda}(\lambda) e^{\lambda}$$
- thm (Weyl-Kac formula)
Let $\Lambda\in P_{+}$. Then $$ \operatorname{ch} L(\Lambda)=\frac{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\Lambda+\rho})}{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho})} $$
- remark
For actual computation of $m_{\lambda} = \operatorname{mult}_{\lambda}(\lambda)$, more practical to use Freudenthal multiplicity formula $$ (|\Lambda+\rho|^2-|\lambda+\rho|^2)m_{\lambda}=2\sum_{\alpha\in \Delta_{+}}\sum_{j\geq 1}(\operatorname{mult} \alpha)(\lambda+j\alpha|\alpha)m_{\lambda+j\alpha} $$
string functions
- String functions and branching functions
- Fix $\Lambda\in P_{+}^{k}$
- def
For each $\lambda\in \mathfrak{h}^{*}$, the string function $c_{\lambda }^{\Lambda}$ is $$ c_{\lambda }^{\Lambda}=e^{-m_{\Lambda,\lambda}\delta}\sum_{n=-\infty}^{\infty}\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{-n \delta} $$ where $m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}$ and $m_{\Lambda,\lambda}=m_{\Lambda}-\frac{\lambda^2}{2k}$
- note that $m_{\Lambda}=h_{\Lambda}-\frac{c(k)}{24}+\xi$ where $h_{\Lambda}=\frac{(\bar{\Lambda}+2\bar{\rho}|\bar{\Lambda})}{2(k+h^{\vee})}$ and $c(k)=\frac{k}{k+h^{\vee}}\dim \mathfrak{\overline{g}}$ (these number frequently appear in rep. theory of Virasoro algebra)
- remarks
- modular form of weight $-r/2$ after setting $q:=e^{-\delta}$
- an explicit expression for the string functions is not known in general
- the few that are known were guessed using the modular transformations
- $c_{\lambda }^{\Lambda}=c_{w\lambda }^{\Lambda}$ for $w\in W$
- Theta functions in Kac-Moody algebras
- for each $\lambda\in P^k$, define the theta function as
$$ \Theta_{\lambda}= e^{-\frac{|\lambda|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)}=e^{k\Lambda_0}\sum_{\gamma\in M+\overline{\lambda}/k}e^{-\frac{1}{2}k|\gamma|^2 \delta + k \gamma} $$
- A weight $\lambda$ of $L(\Lambda)$ is maximal if $\lambda+\delta$ is not a weight
- the set $\max(\Lambda)$ of maximal weights is stable under $W$
- thm
$$ e^{-m_{\Lambda}\delta}\operatorname{ch} L(\Lambda)=\sum_{\lambda}c^{\Lambda}_{\lambda }\Theta_{\lambda} $$
- proof
$$ \begin{aligned} \operatorname{ch} L(\Lambda)&=\sum_{\lambda\in \max{L(\Lambda)}}\sum_{n=0}^{\infty}\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{\lambda-n \delta} && \text{(any weight $\mu$ is of the form $\lambda-n \delta$ for some unique $\lambda, n$)} \\ &=\sum_{\lambda\in \max{L(\Lambda)}/T} \left(\sum_{\gamma\in M}e^{t_{\gamma}(\lambda)}\right)\left(\sum_{n=0}^{\infty}\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{-n\delta}\right)\\ &=\sum_{\lambda\in \max{L(\Lambda)}/T} e^{m_{\Lambda}\delta}c^{\Lambda}_{\lambda}\Theta_{\lambda} \end{aligned} $$ ■
asymptotic growth of coefficients
- modularity of $c_{\lambda }^{\Lambda}$ implies
- thm (Kac-Peterson)
Let $\Lambda\in P_{k}^{+},\, \lambda\in \max(\Lambda)$. As $n\to \infty$, $$ \log (\operatorname{mult}_{\Lambda}(\lambda-n\delta))\sim (\frac{2c(k)\pi^2n}{3})^{1/2} $$
conjectural formula for string functions
- Fermionic formula for string functions and parafermion characters
- denote the level by $\ell\in \mathbb{Z}$ and assume $\ell\geq 2$
- $H_\ell=\{(a,m)|a=1,\cdots, r, 1\leq m \leq t_a \ell-1\}$
- let
$$ K^{m n}_{a b} = \Bigl(\hbox{min}(t_bm, t_an) - {m n\over \ell}\Bigr) (\alpha_a \vert \alpha_b) $$
- conjecture [Kuniba-Nakanishi-Suzuki 93]
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 $q$, where $\eta$ is the Dedekind eta function .
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 $\mathfrak{g}=A_1$
- 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$.
evidence
- compare the asymptotic behavior of \ref{qkns} as $t\to 0$ with \(q=e^{-t}\)
- LHS of \ref{qkns} $\exp(\frac{\pi^2(c(\ell)-r)}{6t})$
- RHS of \ref{qkns} $\exp(\frac{\sum_{(a,m)\in H_\ell} L(x_{m}^{(a)})}{t})$
where $0<x_{m}^{(a)}<1$ is the solution of the system of equations $$ x_{m}^{(a)} = \prod_{(b,n)\in H_{\ell}}(1-x_{n}^{(b)})^{K_{ab}^{mn}},\, (a,m)\in H_{\ell} $$ and $L$ is the Rogers dilogarithm function $$ L(x) = \operatorname{Li}_ 2(x)+\frac{1}{2}\log x\log (1-x),\, 0<x<1 $$ $$ \operatorname{Li}_ 2(x)= \sum_{n=1}^\infty {x^n \over n^2},\, 0<x<1 $$
- thm (Chapoton, Nakanishi)
$$ \sum_{(a,m)\in H_\ell} L(x_{m}^{(a)}) = \frac{\pi^2}{6}(c(\ell)-r) $$
- proof uses Y-systems and cluster algebras
example
- $\overline{\mathfrak{g}} = B_2$, level $\ell = 2$, rank $r=2$
- $t_1=1,t_2=2$
- $H_{\ell} = \{(1,1),(2,1),(2,2),(2,3)\}$
- dual Coxeter number : $h^{\vee}=3$
- $\dim \overline{\mathfrak{g}}=10$
- $c(\ell)-r = 4-2=2$
- $K = \left(
\begin{array}{cccc} 2 & -1 & -2 & -1 \\ -1 & 3 & 2 & 1 \\ -2 & 2 & 4 & 2 \\ -1 & 1 & 2 & 3 \\ \end{array} \right)/2$
- equation for $x^{(a)}_m$
$$ \begin{aligned} x^{(1)}_1 & = (1-x^{(1)}_1)(1-x^{(2)}_1)^{-1/2}(1-x^{(2)}_2)^{-1}(1-x^{(2)}_3)^{-1/2} \\ x^{(2)}_1 & = (1-x^{(1)}_1)^{-1/2}(1-x^{(2)}_1)^{3/2}(1-x^{(2)}_2)^{1}(1-x^{(2)}_3)^{1/2}\\ x^{(2)}_2 & = \dots \\ x^{(2)}_3 & = \dots \\ \end{aligned} $$
- $x^{(1)}_1= 3/4,x^{(2)}_1= 2/5,x^{(2)}_2= 4/9,x^{(2)}_3= 2/5$
$$ L\left(\frac{3}{4}\right)+2 L\left(\frac{2}{5}\right)+L\left(\frac{4}{9}\right) = \frac{2\pi^2}{6} $$
quantum affine algebras and KR modules
- Q. is there more representation theoretic way to describe $x^{(a)}_m$?
- A. these numbers can be obtained from the quantum dimensions of Kirillov-Reshetikhin modules
- $\exists$ bijection between iso. classes of fin.-dim'l irr. reps of $\uqg$ and the set of $I$-tuples $\mathbf{P}=(P_i)_{i\in I}$ of polys $P_i\in \mathbb{C}[z]$ with $P_i(0)=1$, called Drinfeld poly.
- KR module $W^{(a)}_{m}(u)$ with $a\in I$, $m\in \mathbb{Z}_{\geq 0}$ and $u\in \mathbb{C}^{\times}$ is associated with Drinfeld polynomials $\mathbf{P}=(P_i)_{i\in I}$ of the form
$$ P_i(z) = \begin{cases} \prod _{s=1}^m \left(1- z u q_{a}^{2(s-1)}\right), & \text{if $i=a$}\\ 1, & \text{otherwise} \\ \end{cases} $$ where $q_{a} = q^{t/t_a}$ and $t=\max_{a\in I}t_a$.
- The quantum dimension of irr. h.w. $U_q(\overline{\mathfrak{g}})$-modules $L(\lambda)$ at level $k$ is
$$ \frac{\prod_{\alpha\in \Delta_{+}}\sin \frac{\pi(\lambda+\rho|\alpha)}{h^{\vee}+k}}{ \prod_{\in \Delta_{+}}\sin \frac{\pi (\rho|\alpha)}{h^{\vee}+k}}. $$
- recovers dimension as $k\to \infty$ (qdim is an alg. int. not necessarily positive)
- regarding $W^{(a)}_{m}(u)$ as $U_q(\overline{\mathfrak{g}})$, obtain quantum dimension of a KR module
- thm (L.)
Fix level $\ell\geq 2$. Let $Q_{m}^{(a)}$ be the qdim of $W^{(a)}_{m}(u)$ at level $\ell$. Then $Q_{m}^{(a)}$ with $(a,m)\in H_{\ell}$ is positive, $Q_{t_a\ell}^{(a)}=1$, and $x^{(a)}_m= 1-\frac{Q_{m-1}^{(a)}Q_{m+1}^{(a)}}{(Q_{m}^{(a)})^2}$.
- need fusion ring
example
- $\overline{\mathfrak{g}} = B_2$, level $\ell = 2$, rank $r=2$
- $Q_{m}^{(1)} = 1,2,1$ for $m=0,1,2$
- $Q_{m}^{(2)} = 1,\sqrt{5},3,\sqrt{5},1$ for $m=0,1,2,3,4$
memo
$$ \operatorname{mult}_{\Lambda}(\lambda-n\delta)\sim (\text{const})\times n^{-(1/4)(r+3)}e^{4\pi (a n)^{1/2}} $$