Emergent Kalb-Ramond fields from a dimer model

We discuss the emergence of a Kalb-Ramond field and string charge in the lattice and consider the local bosonic model with rotor variables placed on the faces of a cubic lattice. We give the coupling model consisting of the Maxwell fields and the Kalb-Ramond field. This construction naturally incorporates the emerging coupling between both gauge and string fields. In the process, an object that resembles a D-brane on the lattice is introduced.

Figure 1

Continuous and lattice D-branes. Diagram (a) shows a 0-brane on the lattice, (b) shows a 1-brane on the lattice, and (c) shows a 2-brane.

Figure 2

Continuous and lattice strings attached to 2-branes.

Figure 3

Continuous and lattice string charge attached to 2-branes. The string charge is shown in blue, while the electric field is shown in green.

Figure 4

The kinetic term flips the dimers as in Eq. (19).

Figure 5

The argument of the cosine function in Eq. (22) is visible, and we show how it gives rise to a dual magnetic field $-B_{az}$. The positive field potentials are shown in blue, while the negative ones are shown in red. The magnetic field that is generated is in the opposite direction and negative, as shown in green.

Figure 6

A defect at site $i$ when Gauss’s constraint is violated in the $\widehat{x}$ direction. Again, blue is positive, red negative, and green is the charge. One can see that the 2 in the current on Eq. (28) comes from the difference of the links.

Figure 7

The dimer $(i,x)$ (in green) is created by the interaction of the positive boson numbers (in blue) and the negative ones (in red) as the constraint $H_0$ dictates.

Figure 8

The terms inside of the cosine of Eq. (39) are plaquettes instead of dimers.