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 $\partial M$)

  • 1
    $\begingroup$ The question about fermion doubling seems unrelated to the issue of the $F\wedge F$ term; I'd advise you to ask it separately. $\endgroup$
    – ACuriousMind
    Jan 17, 2017 at 17:56

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.