# Lattice gauge theory + Chern-Simons Theory

Let $M$ be a 4-dimensional manifold with boundary $\partial M$. Then the general gauge invariant action of a Yang-Mills theory with field strength 2-form $F$ and constants $\theta, g_{YM}$ has the form

$$S = \int_M \left(\frac{1}{2g_{YM}^2}\mathrm{tr}(F \wedge *F) + \theta\ \mathrm{tr}(F \wedge F)\right)$$

The second term of this action is a topological term and is equivalent to

$$\theta \int_{\partial M} \left(\mathrm{tr}(A \wedge F) - \frac{1}{3}\mathrm{tr}(A \wedge A \wedge A)\right)$$

by the use of the Gaussian law and is a Chern-Simons theory on the manifold boundary.

Now I want to discretize this action and replace connections $A$ with Wilson lines $U_{x,\mu}$ that is located at lattice with number $x$ and goes over the corresponding edge $\mu$. It is clear that a plausible discretization of the first Yang-Mills theory term of the action is proportional to the trace over a Wilson loop (in low orders of the lattice length $a$ it is indeed the action of non-topological Yang-Mills theory).

My question is: Does the topological term $F \wedge F$ really need a boundary of the manifold $M$ to be different from zero? From continuum mechanics I know that also singular surfaces (surfaces in which fields have non-continuous behavior) will lead to an extra contribution over the singular surface. Can such a singular surface be introduced in a Lattice Gauge theory e.g. if two lattices $L_A,L_B$ are connected by a (singular) surface $S_{AB} = L_A \cap L_B$ where the values of the fermion fields on the one side of $S_{AB}$ are not equal to values of fermion fields on the other side of $S_{AB}$ (is called discontinuity)?

In lattice gauge theories Fermion doubling problem may occur, can this problem be fixed when assuming discontinuities in fermion fields? I think that fermion fields will then be defined almost everywhere, but are not defined on discont. surfaces like $S_{AB}$ that changes the structure of the inverse fermion propagator (see https://en.wikipedia.org/wiki/Fermion_doubling)

(The generalized Gaussian law is $\int_M df = \int_{\partial M}f - \int_{Discont.Surface}f$ if discontinuous surface lies within $\partial M$)

• The question about fermion doubling seems unrelated to the issue of the $F\wedge F$ term; I'd advise you to ask it separately. – ACuriousMind Jan 17 '17 at 17:56

No, you do not need a boundary to have a non-zero $F\wedge F$ term. Unless your gauge group is $\mathrm{U}(1)$, in which case you need non-zero second cohomology of spacetime to have non-trivial bundles (see this answer of mine), any instanton does what you want; the BPST instanton gives an explicit construction of such field configurations, see also this answer of mine.