Kostant, Steinberg, and the Stokes matrices of the tt*-Toda equations

Abstract

We propose a Lie-theoretic definition of the tt*-Toda equations for any complex simple Lie algebra ${\mathfrak {g}}$, based on the concept of topological–antitopological fusion which was introduced by Cecotti and Vafa. Our main results concern the Stokes data of a certain meromorphic connection, whose isomonodromic deformations are controlled by these equations. First, by exploiting a framework introduced by Boalch, we show that this data has a remarkable structure. It can be described using Kostant’s theory of Cartan subalgebras in apposition and Steinberg’s theory of conjugacy classes of regular elements, and it can be visualized on the Coxeter Plane. Second, we compute canonical Stokes data for a certain family of solutions of the tt*-Toda equations in terms of their asymptotics. To do this, we compute the Stokes data of an auxiliary meromorphic connection, related to the original meromorphic connection by a loop group Iwasawa factorization.

Appendices

Appendix A: Formal solutions

This section is included primarily as motivation for Lemma 4.4. We sketch a proof of the existence of a formal solution to the equation

$$\begin{aligned} \frac{d\Psi }{d\xi }= \left( \sum _{k=-r-1}^\infty A_{k+1} \xi ^k \right) \Psi \end{aligned}$$

(A.1)

following [20], Proposition 1.1, but using Lie-theoretic notation as far as possible. Here $r\in {\mathbb {N}}$ and all coefficients $A_k$ belong to ${\mathfrak {g}}$, which we take to be the Lie algebra of a (simple) matrix Lie group G.

We assume that the leading coefficient $A_{-r}$ is regular, hence contained in a unique Cartan subalgebra ${\mathfrak {h}}_1$ (thus we have $\beta (A_{-r})\ne 0$ for all roots $\beta \in \Delta _1$ with respect to ${\mathfrak {h}}_1$).

Let $P\in G$. Then $\Lambda _{-r}=\mathrm {Ad}(P^{-1})A_{-r}$ is also regular, and contained in a unique Cartan subalgebra ${\mathfrak {h}}_2$ (namely $\mathrm {Ad}(P^{-1}){\mathfrak {h}}_1$). Let ${\mathfrak {g}}={\mathfrak {h}}_2\oplus (\oplus _\alpha {\mathfrak {g}}_\alpha )$ be the eigenspace decomposition for the adjoint action of ${\mathfrak {h}}_2$.

Proposition A.1

There is a unique formal fundamental solution $\Psi _f$ of equation (A.1) of the form

$$\begin{aligned} \Psi _f(\xi )=P \left( \sum _{k=0}^\infty \psi _k\xi ^k \right) \exp ^{\Lambda (\xi )}, \end{aligned}$$

(A.2)

where $\psi _0=I$ and $\displaystyle \Lambda (\xi )=\Lambda _0\log \xi +\sum _{k=-r}^{-1}\frac{\Lambda _{k}}{k}\xi ^{k}$ with all $\Lambda _k\in {\mathfrak {h}}_2$.

Proof

Consider

$$\begin{aligned} \Psi _f(\xi )=P \left( \sum _{k=0}^\infty Y_k \xi ^k \right) \exp ^{\Upsilon (\xi )}. \end{aligned}$$

(A.3)

We shall show that it is possible to find $Y_k$ and $\Upsilon (\xi )=\Lambda (\xi )+\sum _{k=1}^\infty \frac{\Lambda _k}{k}\xi ^k$ with $Y_0=I$, all $Y_k\in \oplus _\alpha {\mathfrak {g}}_\alpha$, and all $\Lambda _k\in {\mathfrak {h}}_2$, such that (A.3) formally satisfies (A.1). As (A.3) can be rewritten in the required form (A.2), this will be sufficient.

Substitution into (A.1) leads to the recurrence relations

$$\begin{aligned} \Lambda _{-r+k}+[Y_k,\Lambda _{-r}]=F_{-r+k} \end{aligned}$$

(A.4)

for $k\in {\mathbb {N}}$, where $F_{-r+1}=P^{-1}A_{-r+1}P$ and, for $k\ge 2$,

$$\begin{aligned} F_{-r+k}= & {} P^{-1}A_{-r+k}P+\sum _{n=1}^{k-1}(P^{-1}A_{-r+k-n}PY_n-Y_n\Lambda _{-r+k-n})\\&+\left\{ \begin{array}{ll}0&{}k=2,3,\ldots ,r \\ -(k-r)Y_{-r+k} &{} k=r+1,r+2,\ldots \end{array}\right. \end{aligned}$$

Taking components of (A.4) in ${\mathfrak {g}}^\prime ={\mathfrak {h}}_2$, ${\mathfrak {g}}^{\prime \prime }=\oplus _\alpha {\mathfrak {g}}_\alpha$ we obtain

$$\begin{aligned} \Lambda _{-r+k}=\text {Proj}^\prime (F_{-r+k}),\quad [Y_k,\Lambda _{-r}]=\text {Proj}^{\prime \prime }(F_{-r+k}). \end{aligned}$$

As $\mathrm {Ad}\,\Lambda _{-r}$ is invertible on $\oplus _\alpha {\mathfrak {g}}_\alpha$, all of the $\Lambda _k$, $Y_k$ can be determined recursively. $\square$

Appendix B: Singular directions and the (enhanced) Coxeter Plane

Let ${\mathfrak {g}}$ be a complex simple Lie algebra with $\mathrm {rank}\,{\mathfrak {g}}= l>1$. Let ${\mathfrak {h}}$ be a Cartan subalgebra, and $\Delta _+$ a choice of positive roots. The vectors $H_\alpha \in {\mathfrak {h}}$ are defined as in Sect. 2 by $B(h,H_\alpha )=\alpha (h)$. They span a polyhedron in the vector space ${\mathfrak {h}}$. The Coxeter Plane consists of a certain real two-dimensional subspace of ${\mathfrak {h}}$ together with the orthogonal projections of all points $H_\alpha \in {\mathfrak {h}}$ onto this plane. As a visual aid, the rays (or “spokes”) from the origin to these points, and the concentric circles (or “wheels”) passing through these points, may be drawn. For the Lie algebra ${\mathfrak {e}}_8$ this picture is very well known and appears as the frontispiece of Coxeter’s book [15]. The version in Fig. 3 is taken from [10]. There are 60 spokes and 8 wheels in this case.

Fig. 3

Coxeter Plane with root projections for ${\mathfrak {e}}_8$

We shall sketch three descriptions of the Coxeter Plane, all of them rather indirect. Then we shall explain how the Stokes data of the meromorphic o.d.e. of Sect. 4 (or that of Sect. 6) provides a more direct description of the Coxeter Plane, which also illustrates its properties effectively.

The first (and, as far as we know, original) description is aesthetic: the two-dimensional plane is chosen to give the “most symmetrical projection”.

This was made more precise by Kostant [37], and it is this second description that we shall make use of. It depends on the choice of the Lie algebra element $E_+$, hence the choice of a Cartan subalgebra ${\mathfrak {h}}^\prime ={\mathfrak {g}}^{E_+}$ in apposition to ${\mathfrak {h}}$. We have the real subspace

$$\begin{aligned} {\mathfrak {h}}^\prime _\sharp =\{ X\in {\mathfrak {h}}^\prime \ \vert \ \beta (X)\in {\mathbb {R}}\ \text {for all}\ \beta \in \Delta ^\prime \}. \end{aligned}$$

Recall that the Coxeter number of ${\mathfrak {g}}$ is $s=1+\sum _{i=1}^l q_i = \sum _{i=0}^l q_i$, where $\psi =\sum _{i=1}^l q_i \alpha _i$ and $q_0=1$, and that $\tau =\mathrm {Ad}P_0$ acts on ${\mathfrak {h}}^\prime$ as a Coxeter element, with $\tau ^s=1$. Let us choose (for simplicity) the specific coefficients $E_+=\sum _{i=0}^l \sqrt{q_i} e_{\alpha _i}$. Then $E_+\in {\mathfrak {h}}^\prime ={\mathfrak {h}}^\prime _\sharp \otimes {\mathbb {C}}$ is isotropic with respect to B, hence defines an oriented real two-dimensional subspace Y of ${\mathfrak {h}}^\prime _\sharp$. Let $Q:{\mathfrak {h}}^\prime _\sharp \rightarrow Y$ be orthogonal projection with respect to the (positive definite) inner product $B|_{{\mathfrak {h}}^\prime _\sharp }$. Then [37, section 0.2] Y is the essentially unique plane with the property that the projection Q commutes with the action of the Coxeter element $\tau$. This gives a precise meaning to “most symmetrical projection”.

The third (more abstract) description is based on the theory of Coxeter groups [10, 31, 48], and in our situation this means the Weyl group of ${\mathfrak {g}}$ with respect to a Cartan subalgebra ${\mathfrak {h}}$. We sketch this theory, referring to [10] and section 3.19 of [31] for further details. Any product

$$\begin{aligned} \tau =s_1\cdots s_l \end{aligned}$$

of reflections in simple roots is called a Coxeter element. We may assume that

$$\begin{aligned} \tau = xy,\quad x=s_1\cdots s_k, y=s_{k+1}\cdots s_l \end{aligned}$$

for some k, where $s_1,\ldots ,s_k$ commute and $s_{k+1},\ldots ,s_l$ commute. Thus $x^2=1$ and $y^2=1$ and x, y generate a dihedral group, in fact the unique dihedral subgroup of the Weyl group which contains $\tau$. The subspaces

$$\begin{aligned} \mathrm {Ker}\,\alpha _1\cap \cdots \cap \mathrm {Ker}\,\alpha _k, \mathrm {Ker}\,\alpha _{k+1}\cap \cdots \cap \mathrm {Ker}\,\alpha _l \end{aligned}$$

contain distinguished real lines $l_x,l_y$ which can be constructed explicitly from a certain “Perron–Frobenius eigenvalue” of the Cartan matrix. The Coxeter Plane is the real span of $l_x,l_y$. The lines of the Coxeter Plane are given by taking the orthogonal complements of the intersections of all root hyperplanes $\mathrm {Ker}\,\alpha _i$ with this plane.

It can be shown that $\tau$ acts on this plane as rotation through $2\pi /s$, and x, y act by reflection in adjacent lines (given by the intersections of $\mathrm {Ker}\,\alpha _1\cap \cdots \cap \mathrm {Ker}\,\alpha _k, \mathrm {Ker}\,\alpha _{k+1}\cap \cdots \cap \mathrm {Ker}\,\alpha _l$ with the plane). It follows that there are exactly s equally spaced lines (thus 2s spokes) in the Coxeter Plane, whose symmetry group is the dihedral group generated by x and y.

As explained in [10], the relation with the Cartan matrix leads to an efficient algorithm for drawing this version of the Coxeter Plane. We are grateful to Bill Casselman for the current version of [10] which contains these pictures.

It turns out that the Stokes data of the meromorphic o.d.e. of Sect. 4 provides a fourth description of the Coxeter Plane. To see this, we observe that the diagram of singular directions is related to Kostant’s plane Y, as follows. First we introduce the notation

$$\begin{aligned} E_+=\mathfrak {R}(E_+)+ {\scriptstyle \sqrt{-1}}\, \mathfrak {I}(E_+),\quad \mathfrak {R}(E_+),\mathfrak {I}(E_+)\in {\mathfrak {h}}^\prime _\sharp \end{aligned}$$

for the decomposition of $E_+$ with respect to the real subspace ${\mathfrak {h}}^\prime _\sharp$.

Proposition B.1

Let us identify Kostant’s plane Y with ${\mathbb {C}}$ by identifying the orthonormal basis $(2/s)^{\frac{1}{2}}\mathfrak {R}(E_+), (2/s)^{\frac{1}{2}}\mathfrak {I}(E_+)$ with $1,\sqrt{-1}$. Then the points $Q(H_{\beta })$ are identified with the points $(2/s)^{\frac{1}{2}}\beta (E_+)$, for all $\beta \in {\mathfrak {h}}^\prime$.

Proof

We use the key fact that the complex conjugate of $E_+$ with restect to ${\mathfrak {h}}^\prime _\sharp$ is given by ${\bar{E}}_+ = E_-$ ([37], Theorems 1.11 and 1.12). Then by direct calculation we obtain

$$\begin{aligned}&B(\mathfrak {R}(E_+),\mathfrak {R}(E_+))=\tfrac{s}{2} \\&B(\mathfrak {I}(E_+),\mathfrak {I}(E_+))=\tfrac{s}{2} \\&B(\mathfrak {R}(E_+),\mathfrak {I}(E_+))=0 \end{aligned}$$

(so $E_+$ is isotropic, as stated earlier). The projection to Y of a vector $X\in {\mathfrak {h}}^\prime _\sharp$ is

$$\begin{aligned} Q(X)=\tfrac{2}{s}\, B(X,\mathfrak {R}(E_+))\mathfrak {R}(E_+)+ \tfrac{2}{s}\, B(X,\mathfrak {I}(E_+))\mathfrak {I}(E_+). \end{aligned}$$

In particular,

$$\begin{aligned} Q(H_\beta )=\tfrac{2}{s}\, \beta (\mathfrak {R}(E_+))\mathfrak {R}(E_+)+ \tfrac{2}{s}\, \beta (\mathfrak {I}(E_+)) \mathfrak {I}(E_+) \end{aligned}$$

(B.1)

for any $\beta \in \Delta ^\prime$. Note that $\beta (\mathfrak {R}(E_+))$ and $\beta (\mathfrak {I}(E_+))$ are both real. It follows that the vector $Q(H_\beta )$ in the plane Y—under the identification of $(2/s)^{\frac{1}{2}}\mathfrak {R}(E_+),(2/s)^{\frac{1}{2}}\mathfrak {I}(E_+)$ in Y with $1,\sqrt{-1}$ in ${\mathbb {C}}$—corresponds to the complex number $(2/s)^{\frac{1}{2}}\beta (\mathfrak {R}(E_+))+ {\scriptstyle \sqrt{-1}}\, (2/s)^{\frac{1}{2}}\beta (\mathfrak {I}(E_+))$, and this is just $(2/s)^{\frac{1}{2}}\beta (E_+)$. $\square$

In the proposition and its proof we have assumed that $E_+=\sum _{i=0}^l \sqrt{q_i} e_{\alpha _i}$. In the general case $E_+=\sum _{i=0}^l c^+_i e_{\alpha _i}$, (B.1) becomes

$$\begin{aligned} Q(H_\beta )=\tfrac{2t}{s}\, \beta (\mathfrak {R}(E_+))\mathfrak {R}(E_+)+ \tfrac{2t}{s}\, \beta (\mathfrak {I}(E_+)) \mathfrak {I}(E_+) \end{aligned}$$

where $t\in {\mathbb {C}}^*$ is defined by $t( \mathfrak {R}(E_+)-\sqrt{-1}\mathfrak {I}(E_+) ) = \sum _{i=0}^l ({q_i}/{c^+_i}) e_{-\alpha _i}$ [37, Theorem 1.11]. From this we obtain $Q(H_{\beta })=((2t)/s)^{\frac{1}{2}}\beta (E_+)$. Thus the diagram of 2s singular directions, together with the points $\beta (E_+)$ (or $\beta (-E_+)$, as in Sect. 4) marked with their corresponding roots $\beta \in \Delta ^\prime$, is—up to scaling and rotation—simply the Coxeter Plane. In particular, using the fact that the symmetry group of the Coxeter Plane is a dihedral group of order 2s, we see that Theorem 4.6 holds for any such ${\mathfrak {g}}$, not just for the classical Lie algebras.

Our diagram of singular directions may in fact be regarded an “enhanced Coxeter Plane”, because of the additional choice of $E_+$, which fixes a Cartan subalgebra in apposition and a particular Coxeter element. This facilitates the assignment of a root $\beta$ to a ray in the Coxeter Plane: one simply takes the ray through the point $\beta (E_+)$. Having made this assignment, the Coxeter element acts on the roots by clockwise rotation through $2\pi /s$. There are 2s systems of positive roots corresponding to this Coxeter element (in the sense of Proposition 5.4), given by the 2s choices of “positive sectors”. The head and tail of a positive sector give the associated simple roots. The roots on any two consecutive rays generate all the roots. While this information may be implicit in the (usual) Coxeter Plane, the choice of $E_+$ makes it explicit.

In the other direction, as our diagram of singular directions arises independently of Lie theory from the Stokes data of the meromorphic connection $d+{\hat{\alpha }}$ (or $d+{\hat{\omega }}$), the possibility of reproving some classical results on root systems arises. This would be very much in the spirit of [4], where the Stokes data of a meromorphic connection (similar to $d+{\hat{\omega }}$) was used rather unexpectedly to establish a result in symplectic geometry, the Ginzburg-Weinstein isomorphism.