Gauge-fixing on the lattice via orbifolding

When fixing a covariant gauge, most popularly the Landau gauge, on the lattice, one encounters the Neuberger $0/0$ problem, which prevents one from formulating a Becchi–Rouet–Stora–Tyutin symmetry on the lattice. Following the interpretation of this problem in terms of Witten-type topological field theory and using the recently developed Morse theory for orbifolds, we propose a modification of the lattice Landau gauge via orbifolding of the gauge-fixing group manifold and show that this modification circumvents the orbit-dependence issue and hence can be a viable candidate for evading the Neuberger problem. Using algebraic geometry, we also show that though the previously proposed modification of the lattice Landau gauge via stereographic projection relies on delicate departure from the standard Morse theory due to the noncompactness of the underlying manifold, the corresponding gauge-fixing partition function turns out to be orbit independent for all the orbits except in a region of measure zero.

Figure 1

$Z_{\mathrm{GF}}$ corresponding to various sets of parameters $P_k$, which are defined as follows. For $P_1$, we set each parameter to a distinct angle via ${\varphi }_{\mathbf{j},\mu }=\pi /(j_2+3(j_1-1)+9(\mu -1))$. For $P_2$, we set ${\varphi }_{\mathbf{j},\mu }=\pi /2$ for all $\mathbf{j}$ and $\mu$. For $P_3$, we set ${\varphi }_{\mathbf{j},\mu }=0$ for all $\mathbf{j}$ and $\mu$. For $P_4$, we set ${\varphi }_{\mathbf{j},\mu }=\pi /3+(\pi /6)(\mu -1)$.

Figure 2

Subset of the discriminant locus projected onto two parameters for the $3\times 3$ lattice. The other $\varphi$s are fixed as listed in Table 1. While varying ${\varphi }_{(1,1),1}$ and ${\varphi }_{(1,1),2}$ and leaving all other $\varphi$s fixed for the $3\times 3$ lattice, the points in this plot are the points at which $Z_{\mathrm{GF}}$ differs from the generic value 11664.