Hyperpolygon Moduli Spaces
Navigate the four-dimensional moduli space of hyperpolygons
Let \(Q\) be the four-sided star quiver with dimension vector \((2,1,1,1,1)\). A representation of \(Q\) consists of a \(2\times 4\) matrix \(x\) with columns \(x_1,\dots,x_4\), and a \(4\times 2\) matrix \(y\) with rows \(y_1,\dots,y_4\). The hyperpolygon space \(\mathcal X(\vec\beta)\) is the 4D moduli space of solutions to the moment map equations
\[\mu_{\mathrm{SL}(2,\mathbb C)}(x,y)=\sum_{i=1}^{4}x_iy_i=0,\] \[\mu_{\mathrm{GL}(1,\mathbb C),i}(x,y)=y_ix_i=0,\] \[\mu_{\mathrm{SU}(2)}(x,y)=\sum_{i=1}^{4}(x_ix_i^\dagger)_0-(y_i^\dagger y_i)_0=0,\] \[\mu_{\mathrm{U}(1),i}(x,y)=\frac12\left(|x_i|^2-|y_i|^2\right)=\beta_i.\]up to the action of the symmetry group \(G = \mathrm{SU}(2)\times\mathrm{U}(1)^4\). These equations have interpretations in terms of polygons. Set \(v_i=(x_ix_i^\dagger)_0\) and \(w_i=(y_i^\dagger y_i)_0\). Then \(\mu_{\mathrm{GL}(1,\mathbb C),i}=0\) implies \(v_i\) and \(w_i\) point in opposite directions, \(\mu_{\mathrm{U}(1),i}=\beta_i\) constrains the side lengths \(\lvert v_i\rvert-\lvert w_i\rvert=\sqrt2\,\beta_i\), and \(\mu_{\mathfrak{su}(2)}=0\) gives the closure condition \(\sum_i(v_i-w_i)=0\). Therefore the \(v_i\) and \(w_i\) vectors assemble to form a telescoping polygon in \(\mathfrak{su}(2)\cong\mathbb R^3\), considered up to \(\mathrm{SO}(3)\) rotations.
This simulation solves the moment map equations live in the browser as you vary the sliders. The right view shows your position within the 4D moduli space, and the left view displays the corresponding hyperpolygon. Click the tutorial button for a guide.