Discretized Abelian Chern-Simons gauge theory on arbitrary graphs

In this paper, we show how to discretize the Abelian Chern-Simons gauge theory on generic planar lattices/graphs (with or without translational symmetries) embedded in arbitrary two-dimensional closed orientable manifolds. We find that, as long as a one-to-one correspondence between vertices and faces can be defined on the graph such that each face is paired up with a neighboring vertex (and vice versa), a discretized Abelian Chern-Simons theory can be constructed consistently. We further verify that all the essential properties of the Chern-Simons gauge theory are preserved in the discretized setup. In addition, we find that the existence of such a one-to-one correspondence is not only a sufficient condition for discretizing a Chern-Simons gauge theory but, for the discretized theory to be nonsingular and to preserve some key properties of the topological field theory, this correspondence is also a necessary one. A specific example will then be provided, in which we discretize the Abelian Chern-Simons gauge theory on a tetrahedron.

Figure 1

Part of a planar graph, on which a local vertex-face correspondence is defined. The disks and solid lines represent vertices and edges of the graph, respectively. Each face is marked by a cross. The local vertex-face correspondence is indicated by dotted lines, that pair up each face with one (and only one) adjacent vertex.

Figure 2

Examples of lattices and graphs that support local vertex-face correspondences. (a) A square lattice with 1 vertex and 1 face per unit cell, (b) a kagome lattice with 3 vertices and 3 faces per unit cell, (c) a dice lattice with 3 vertices and 3 faces per unit cell, and (d) a lattice that contains 9 vertices, 18 edges, and 9 faces per unit cell. The (red) parallelogram marks a unit cell with lattice vectors indicated by the two (red) arrows. It is easy to verify that for all these lattices $N_v=N_f$ and for any subgraphs the number of faces never exceeds the number of vertices, which is a sufficient and necessary condition for the existence of (at least) one local vertex-face correspondence.

Figure 3

Examples of lattices/graphs that do not support a local vertex-face correspondence. (a) A triangular lattice, which has 1 vertex and 2 faces per unit cell ($N_v<N_f$), (b) a honeycomb lattice, which has 2 vertices and 1 face per unit cell ($N_v>N_f$), and (c) a lattice with $N_v=N_f$ but some of the subgraph has more faces than vertices, e.g., the dark area, which has 18 faces and 16 vertices. The (red) parallelogram marks a unit cell with lattice vectors indicated by the two (red) arrows. Each unit cell of this lattice contains 27 vertices, 54 edges, and 27 faces ($N_v=N_f$).

Figure 4

Nonzero components of the $K$ matrix. Here, we consider two edges $e$ and $e^{\prime }$, which belong to the same face $f$ (otherwise $K_{e,e^{\prime }}=0$). Based on the local vertex-face correspondence, the face $f$ is paired up with one of its vertices, which is marked by the (red) circle. We go around the face $f$ from $e$ to $e^{\prime }$ by following the direction of the positive orientation marked by the (blue) circle at the center of the face. In (a) and (b), the path from $e$ to $e^{\prime }$ goes through the special site (marked by the red circle), and thus ${\eta }_1=+1$. For (c) and (d), the special site is not on our path, and thus ${\eta }_1=-1$. The sign of ${\eta }_2$ is determined by the orientation of $e$ and $e^{\prime }$. If their directions are both along (or opposite) to the direction of the positive orientation [(a) and (c)], ${\eta }_2=+1$. Otherwise [(b) and (d)], ${\eta }_2=-1$. Once ${\eta }_1$ and ${\eta }_2$ are determined, the value of $K_{e,e^{\prime }}$ can be obtained as $K_{e,e^{\prime }}=-{\eta }_1\times {\eta }_2/2=\pm 1/2$.

Figure 5

Gauge invariance of our theory for (a) Case II and (b) Case III. In (a), we marked two additional edges of $v, e_1$, and $e_2$, which are edges of $f_1$. In (b), we labeled two faces $f_1$ and $f_2$ and two additional edges $e_1$ and $e_2$, such that $e$ is the common edge shared by $f_1$ and $f_2$, while $e_1$ and $e_2$ are two edges of $v$, which are adjacent to $f_1$ and $f_2$, respectively. Dashed lines represent (possible) additional edges of $v$, which are irrelevant for our proof and thus are not labeled. Although we assume a specific set of orientations for edges in these two figures, none of our final conclusions rely on the choice of orientations for each edge, as proven in Appendix pp3.

Figure 6

Possible cases for computing $K_{e,e^{\prime \prime }}{K}_{e^{\prime \prime },e^{\prime }}^{*}$. Solid lines mark the edges $e, e^{\prime }$ and all edges that will contribute to $K_{e,e^{\prime \prime }}{K}_{e^{\prime \prime },e^{\prime }}^{*}$. Dashed lines are (possible) additional edges, which do not contribute to $K_{e,e^{\prime \prime }}{K}_{e^{\prime \prime },e^{\prime }}^{*}$.

Figure 7

One common vertex shared by two paths. Here, we consider two paths $P$ (thin red solid lines) and $P^{\prime }$ (thick blue solid lines). The arrows indicate the direction of each path. The disk in the middle is one common vertex shared by the two paths. Dashed lines represent other (possible) edges that are connected to the vertex, and they do not contribte to the commutator that we want to compute. (a) shows a right-handed intersection between $P$ and $P^{\prime }$ and (b) is a left-handed one. In (c), the two paths do not intersect. (d) Shows a special case, where one path terminates at this vertex. Here, the number of intersections can be $\pm 1$ or 0 depending on microscopic details. In Sec. 7e, a method will be introduced to obtain the value of $\nu$ for (d) by defining a dual path in the dual graph.

Figure 8

A cycle in a graph and the dual cycle in the dual graph. Here, we consider a planar graph with a local face-vertex correspondence. The vertices in the original graph are marked by disks, while the crosses label the faces, i.e., vertices of the dual graph. The local face-vertex correspondence is marked using the dotted lines, which pair up each face with one of its neighboring vertex. The thick solid (blue) lines marks a contractible cycle (a closed path) and the orientation of the cycle is marked by the arrows. For a contractible cycle, its interior is formed by a set of faces (dark region). For each face inside the dark region, we find the corresponding vertex using the local face-vertex correspondence. These vertices are marked by circles. Then, we draw a loop in the dual graph, which encloses these vertices (red dashed lines connecting neighboring crosses). This loop in the dual graph is the dual of the original loop in the original graph. And, we require the two loops to have the same orientation.

Figure 9

Noncontractible cycles on a surface with nonzero genus. For a genus $g$ surface, we can choose $g$-independent noncontractible cycles, which do not intersect with one another. These $g$ cycles will be used as vector ${\xi }_i$ with $i=N_f,N_f+1,...,N_f+g-1$, in our complete basis for the edge space. Here, we show an example with $g=3$. The three red loops mark three independent noncontractible cycles without any intersections.

Figure 10

A tetrahedron viewing from the top. The circles represent the vertices, and the arrows are the edges (direction assigned). Here, we label each vertices, edges, and faces using integers. The last face (i.e., the face 4) is on the other side of the tetrahedron invisible from the current view point.

Figure 11

Examples of graphs that are not simple. The figures demonstrate situations that are not allowed for a simple graph. (a) Shows a pair of sites connected by three different edges. In (b), one of the edges connects a site with itself (i.e., the two ends of an edge coincide).