# A Covariant Lattice and its Application to String Bits

We explore a possibility of constructing lattice theories from a basic principle where lattice is viewed as a restriction of the continuum. This allows us to derive a discrete derivative, nonlocal on the lattice, that satisfies Liebniz rule. As a result a theorem related to locality follows which states that any local equation in the classical continuum theory holds true also locally on the lattice. Using this one is able to preserve a remnant of all the global and gauge symmetries, which, in two dimensions, can be lifted to include local diffeomorphism and Weyl symmetries as well. Consequently, one is able, for example, to carry out classical BRST construction of covariant sting bits (for the first time).

This lattice derivative is, in some sense, similar to SLAC derivative discussed earlier in the literature. However, we point out that its action suffers from branch cut ambiguities while acting on Fourier basis. This may indicate that quantization is not straightforward and may have to be done entirely in position space.

https://uky.zoom.us/j/98298147329