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 <math>\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} </math> ■

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 \[\sum_{(a,m) \in H_\ell}mN^{(a)}_m\overline{\alpha_a} \equiv \overline{\lambda} \mod \ell M.\]

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
• \(\exists\) bij. 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

\[ \operatorname{mult}_{\Lambda}(\lambda-n\delta)\sim (\text{const})\times n^{-(1/4)(r+3)}e^{4\pi (a n)^{1/2}} \]

related items

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