Musings

Unfortunately, they start off with a completely-wrong interpretation — they want the gauge group that becomes weakly-coupled in the nodal limit to be a subgroup of the centralizer of the monodromy around the pinching cycle — and then try to reverse-engineer a family that achieves that. The opposite is what’s actually true. To take the extreme case, when the gauge group associated to some cycle (note that we can always associate a gauge group to some cycle; it only becomes weakly-coupled when that cycle pinches) is the full $SU(N)$, then the monodromy around that cycle is regular semi-simple (so its centralizer is abelian).

Rather than try to explicate where they went wrong, let me instead work through an example: the “Argyres-Douglas” example of section 6.1 of their paper.

This is the 4-punctured sphere for $SL_3$, with 2 simple and 2 full punctures. On the Higgs bundle side, the Higgs field has simple poles at $p_1,p_2$ with residues in the minimal nilpotent orbit and simple poles at $p_3,p_4$ with residues in the regular nilpotent orbit.

On the character variety side, we have 4 monodromies $$\{M_1,M_2,M_3,M_4\in SL_N | M_1M_2M_3M_4=I\}$$ up to overall conjugation. Under NAH, the conjugacy classes of the $M_i$ are determined by the parabolic weights that we assign on the Higgs bundle side.

Specifically, $E|_{p}$ has a filtration

where $F_j = \ker(Res(\Phi)^j)$. We assign a set of real numbers $\alpha_j\in[-1/2,1/2)$ which sum to 0 and such that the multiplicity of $\alpha_j$ is $\dim(F_j/F_{j-1})$ and $\alpha_j\lt \alpha_{j-1}$. On the character variety side, the conjugacy class of $M$ is $[M]=exp\bigl(-2\pi i\; diag(\alpha_1,\dots,\alpha_l)\bigr)$. A nilpotent $B$ that strictly lowers the flag (1) (i.e. such that $B(F_j)\subset F_{j-1}$) is generically an element of $O$, the nilpotent orbit where the residue of $\Phi$ lies. The induced map on $Gr_{j}=F_j/F_{j-1}$ vanishes, so $M$ is indeed semisimple.

In our example, $E|_{p_{3,4}}$ have a full flag, $$E|_p= F_3\supset F_2\supset F_1\supset 0$$ so the conjugacy classes of $M_{3,4}$ have the form $[M]=exp\bigl(-2\pi i\; diag(\alpha_1,\alpha_2,-\alpha_1-\alpha_2)\bigr)$. $E|_{p_{1,2}}$ has the filtration $$E|_p \supset F \supset 0$$ So $[M] =exp\bigl(-2\pi i\; diag(\alpha,\alpha,-2\alpha))$ and $F$ is either a line or a plane, depending on the sign of $\alpha$.

The character variety in our example is 4 complex-dimensional and I will describe it in an excruciating degree of explicitness.

Warmup

Let’s warm up by considering the character variety for the 3-punctured sphere with 3 full punctures. That’s 2-dimensional and it can be described concretely as follows.

Let $A=M_3$, $B=M_4$ and let $C$ be the regular semi-simple elements of $SL_3$ satisfying $A B C =I$. Let

Of course, when the Higgs field residues are nilpotent, we have

But, with malice aforethought, let’s not make that identification just yet.

By a result of Lawton, the character variety is an affine cubic surface whose coefficients are certain polynomials in $\alpha,\tilde{\alpha},\beta,\tilde{\beta},\gamma,\tilde{\gamma}$.

Let $$X=\mathrm{tr}(A B^{-1})-\alpha\tilde{\beta},\qquad Y=\mathrm{tr}(A^{-1}B)-\tilde{\alpha}\beta,\qquad Z=\mathrm{tr}(A B A^{-1}B^{-1}).$$ These satisfy the cubic equation

where the coefficients are $$\begin{aligned} a&=\alpha\tilde\beta+\beta\tilde\gamma+\gamma\tilde\alpha\\ b&=\tilde\alpha\beta+\tilde\beta\gamma+\tilde\gamma\alpha\\ e&=\alpha\tilde\alpha+\beta\tilde\beta+\gamma\tilde\gamma-6\\ \kappa&=3+\alpha\beta\gamma+\tilde\alpha\tilde\beta\tilde\gamma-\alpha\tilde\alpha-\beta\tilde\beta-\gamma\tilde\gamma\\ f&= \alpha^2\beta+\beta^2\gamma+\gamma^2\alpha+\tilde\alpha\tilde\beta^2+\tilde\beta\tilde\gamma^2+\tilde\gamma\tilde\alpha^2 + \alpha\tilde\alpha\tilde\beta\gamma+\beta\tilde\beta\tilde\gamma\alpha+\gamma\tilde\gamma\tilde\alpha\beta - 3(\tilde\alpha\beta+\tilde\beta\gamma+\tilde\gamma\alpha)\\ g&= \tilde\alpha^2\tilde\beta+\tilde\beta^2\tilde\gamma+\tilde\gamma^2\tilde\alpha+\alpha\beta^2+\beta\gamma^2+ \gamma\alpha^2+\alpha\tilde\alpha\beta\tilde\gamma+\beta\tilde\beta\gamma\tilde\alpha+\gamma\tilde\gamma\alpha\tilde\beta -3(\alpha\tilde\beta+\beta\tilde\gamma+\gamma\tilde\alpha) \end{aligned}$$ and finally $$\begin{aligned} h ={}& \alpha^3+\tilde\alpha^3+\beta^3+\tilde\beta^3+\gamma^3+\tilde\gamma^3+9 \\ &+\alpha\tilde\beta^2\tilde\gamma^2+\tilde\alpha\beta^2\gamma^2 +\beta\tilde\gamma^2\tilde\alpha^2+\tilde\beta\gamma^2\alpha^2 +\gamma\tilde\alpha^2\tilde\beta^2+\tilde\gamma\alpha^2\beta^2 \\ &-2\bigl( \alpha^2\tilde\beta\tilde\gamma+\tilde\alpha^2\beta\gamma +\beta^2\tilde\gamma\tilde\alpha+\tilde\beta^2\gamma\alpha +\gamma^2\tilde\alpha\tilde\beta+\tilde\gamma^2\alpha\beta \bigr) \\ &+\alpha\tilde\alpha\beta\tilde\beta\gamma\tilde\gamma+\alpha\tilde\alpha\beta\tilde\beta+\beta\tilde\beta\gamma\tilde\gamma+\gamma\tilde\gamma\alpha\tilde\alpha \\ &+3(\alpha\beta\gamma+\tilde\alpha\tilde\beta\tilde\gamma)-6(\alpha\tilde\alpha+\beta\tilde\beta+\gamma\tilde\gamma). \end{aligned}$$

Since (4) is quadratic in $Z$, this cubic surface is a branched double-cover of the $X$-$Y$ plane. Away from the discriminant locus, $$D=\{0=(X Y-\kappa)^2-4(X^3+Y^3+a X^2 + b Y^2 + e X Y + f X + g Y +h)\}$$ $X,Y$ are good local coordinates.

Now let’s consider the 4-punctured sphere with 4 full punctures. We can set $A'=M_1$, $B'=M_2$ and $C'=(M_1 M_2)^{-1}=M_3M_4=C^{-1}$ is regular semi-simple. We’ll define Fenchel-Nielsen-like coordinates: $$t=\mathrm{tr}(C),\quad \tilde{t}= \mathrm{tr}(C^{-1})$$ are the “lengths” and $$\Theta\in Z(C)= T/\mathbb{Z}_3\simeq (\mathbb{C}^\ast\times\mathbb{C}^\ast)$$ are the “twists”. Here, we quotiented the maximal torus $T\subset SL_3$ by the $\mathbb{Z}_3$ center of $SL_3$, whose conjugation action is trivial.

The character variety is a $(\mathbb{C}^\ast\times\mathbb{C}^\ast)$- bundle (with fiber $\Theta$) over $\mathcal{X}$, the fiber product of two copies of the affine cubic surface: $$\mathcal{X} = S_L \times_{\mathbb{C}^2} S_R$$ with $$\begin{aligned} S_L&\coloneqq S(\alpha',\overline{\alpha}',\beta',\overline{\beta}',\tilde{t},t)\\ S_R&\coloneqq S(\alpha,\overline{\alpha},\beta,\overline{\beta},t,\tilde{t}) \end{aligned}$$

Dehn Twists

The definition of the character variety involved a choice of basis for $\pi_1$ of the punctured Riemann surface. Imagine cutting the 4-punctured sphere along a circle surrounding $p_1,p_2$, rotating by $2\pi$ and gluing back together. This acts on the monodromies (the image of $\pi_1(C_{0,4})\to SL_3$) as $$(M_1,M_2,M_3,M_4)\to (C M_1 C^{-1},C M_2 C^{-1}, M_3, M_4)$$ where, again, $C\coloneqq M_1 M_2$.

Explicitly, this does not act on $\mathcal{X}$, but it does act on the twists $\Theta$. Let $c_1,c_2,c_3$, with $c_1c_2c_3=1$ be the eigenvalues of $C$. In the basis where $C$ is diagonalized, a point in $T$ is a diagonal matrix $\mathrm{diag}(t_1,t_2,t_3)$ with $t_1t_2t_3=1$. Explicit fiber coordinates (points in $T/\mathbb{Z}_3$) are the invariant combinations $\Theta=(t_1/t_2,t_2/t_3)$. The Dehn twist acts as

Argyres-Seiberg

Finally, let’s return to the case of interest: where the Higgs-field residues at $p_1$, $p_2$ are in the minimal (rather than the regular) nilpotent orbit.

As we saw above, the parabolic weights at $p_1$, $p_2$ are $(\alpha_i,\alpha_i,-2\alpha_i)$, which give flags of the form $\mathbb{C}^3\supset F \supset 0$, with $F$ a plane or a line.

We can write the monodromies at $p_1$,$p_2$ in the following form

with $x_i,y_i\in\mathbb{C}^3$. As a consequence one of the eigenvalues of $C=M_1M_2$ is pinned to be $\nu=\lambda_1\lambda_2$. Let $s$ and $s'=1/(\nu s)$ be the remaining eigenvalues of $C$. Then the Fenchel-Nielsen length coordinates $t=\mathrm{tr}(C)$ and $\tilde{t}=\mathrm{tr}(C^{-1})$ are no longer independent. Rather $$\tilde{t} = \nu t + \nu^{-1} -\nu^2$$

Similarly, there’s just a single twist coordinate. To construct it, let $L = y_1^\perp\cap y_2^\perp$ and $W=span(x_1,x_2)$. $M_i$ acts on $L$ as $\lambda_i$ and $M_i x_j\in W$. So $C$ acts as $\nu$ on $L$ and acts with eigenvalues $s,s'$ on $W$. Generically the $2\times 2$ block is irreducible, so the stabilizer of the pair is $$T_L = \{ a I_L \oplus b I_W | a b^2=1\}$$ The twist $\theta\in T/T_L \simeq \mathbb{C}^*$. The character variety is now 4-dimensional, given by $t,\theta$ and $u\in S(\alpha,\overline{\alpha},\beta,\overline{\beta},t,\nu t + \nu^{-1} -\nu^2)$.

As before, a Dehn twist around the circle surrounding $p_1$,$p_2$ acts only on the twist parameter, taking $\theta\to (s/s')\theta = (\nu s^2)\theta$.

I want to emphasize that in both cases, discussed above, the monodromy around the circle surrounding $p_1,p_2$ was regular-semisimple (in contrast to what they wanted). What differs in the Argyres-Seiberg case is that one of the eigenvalues of that monodromy is fixed.

The other takeaway is that the action of Dehn twists (which generate the Mapping Class Group) is very transparent once we introduce the Fenchel-Nielsen-like coordinates associated to a compatible pants-decomposition of the curve: the character variety has the structure of an algebraic torus fibration over some base, and Dehn twists act by translations along the fibers.

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Note: The Argyres-Douglas example (2 minimal and 2 regular Hitchin nilpotents) generalizes straightforwardly to $SL_N$. $C=M_1M_2$ is still semi-simple, but no longer regular semi-simple. The monodromies at $p_{1,2}$ can be parametrized as $$M_i =\lambda_i(I+x_i y^t_i),\qquad 1+y^t_i x_i=\lambda_i^{-N},\qquad x_i,y_i\in \mathbb{C}^n$$ Let $\nu=\lambda_1\lambda_2$, $F=y_1^\perp\cap y_2^\perp$ and $W=\mathrm{span}(x_1,x_2)$. We have $\mathrm{dim}(F)=N-2$, $\mathrm{dim}(W)=2$ and $C$ acts as $\nu I$ on $F$ and acts with eigenvalues $s,s'$ (obeying $\nu^{N-2}s s'=1$) on $W$. The fixture on the right contributes the $2(N-2)$-dimension character variety corresponding to the 3-punctured sphere with nilpotents $[3,1^{N-3}],[N],[N]$ and there’s one pair of Fenchel-Nielsen coordinates, $t=\mathrm{tr}(C)$ and $\theta=s/s'$. In the language of our paper, pinching the sphere along the circle surround $p_1,p_2$ yields $([3,1^{N-3}],SU(2))$ as the “nilpotent at the node.”

Maybe one more example will give the flavour of the general story. Consider two punctures with residues in the nilpotent orbit $[2^n]$ of $\mathfrak{sl}_{2n}$. The conjugacy classes of the corresponding monodromies are $$[M_i] = \mathrm{exp}(-2\pi i\;\mathrm{diag}(\underset{n}{\underbrace{\alpha_i,\dots,\alpha_i}},\underset{n}{\underbrace{-\alpha_i,-\alpha_i}}))$$ We can write $$M_i = \lambda_i^{-1}(I-P_i) + \lambda_i P_i$$ where $P_i$ is a rank-$n$ idempotent. The algebra generated by a pair of idempotents has an irreducible representation of dimension at-most 2, so $\mathbb{C}^{2n}\simeq V_{1}\oplus\dots\oplus V_{n}$, where the $V_{j}$ are 2-dimensional spaces on each of which $P_1,P_2$ act as rank-1 idempotents. So $C = M_1M_2\in SL_2\times SL_2\times\dots \times SL_2\subset SL_{2n}$. $C$ has eigenvalues $x_j,x_j^{-1}$ on $V_{j}$. The stabilizer of the pair $M_{1,2}$ is $$T_L =\{\mathrm{diag}(a_1,a_1,\dots,a_n,a_n)|a_1^2a_2^2\dots a_n^2=1\}$$ The $n$ Fenchel-Nielsen length coordinates are $$t_1= \mathrm{tr}_{V_1}(C)= x_1+x_1^{-1},\quad\dots\quad,t_n= \mathrm{tr}_{V_n}(C)= x_n+x_n^{-1}$$ and the $n$ twist coordinates $$\Theta\in T/T_L\simeq (\mathbb{C}^*)^n$$ The Weyl group for $$[C]= \mathrm{diag}(x_1,x_1^{-1},x_2,x_2^{-1},\dots,x_n,x_n^{-1})$$ is $W=S_n\ltimes \mathbb{Z}_2^n$, which is the Weyl group of the $Sp(n)$ gauge group on the neck. In the language of our paper, the nodal degeneration when two $[2^n]$ punctures collide yields $4n$ free hypermultiplets transforming as the $2(\boldsymbol{2n})$ and the nilpotent-at-the-node is $([2n],Sp(n))$.