I have a question about Constructuion of Chern-Simon Action. In its paper "Non-commutative geometry and string field theory", Witten construct the Action of the String Field Theory inspiring on the Chern-Simon theory one.

In the paper, there are few objects, with analogy with gauge field objects (I will not define precisely, I guess what I write here is enough to answer my question): $F$ - a field strength, similar to Curvature and its corresponding connection field $A$; $\star$ - star operator, similar to wedge product, actin on fields; $Q$ BRST operator acting on fields, similar to differential operator $d$ in such a way that $Q^2=0$.

In that case, he argues that if the action would had the form $P=\int F \star F$ then the action $P$ would be a topological invariant - "a generalisation of the first Pontryangin class in YM theory".

Another possibility is to imitate the YM theory and think something like $P = \int \langle F,F \rangle =\int F_{\mu\nu}F^{\mu\nu}$. But in the way Witten constructs this theory, such a "product" of fields doesn't exists.

The next possibility is to consider Chern-Simon action $P = \int(A\star QA + \frac{3}{2}A\star A\star A) $ and now everything is all right: $P$ is NOT a topological invariant and has good properties (doesn't matter here).

My Question is: What I conclude is that Chern-Simon theory is constructed in such a way that its action is NOT a topological invariant, as in the construction of string field theory. Am I right?

If I am right, why do we want the action not to be a topological invariant? (In the case of String Field Theory Witten says explicitly that he wants to construct an action that is not topological invariant, so this question yet apply even if this is not true for Chern-Simon)


1 Answer 1


String theory is a quantum theory of gravity. Quantum consistency forces the appearance of closed string states in any quantum theory of open strings as a consequence of the fact that certain one-loop open string diagrams can be interpreted as tree level closed string exchange (see section 4.3 in Introduction to M Theory for details).

Topological invariants are not good observables in quantum theories of gravity (or string theories) because one generically expects background topology change due to phenomena such as black hole condensation, non-perturbative contributions comming from baby universes and other gravitational instantons in such quantum theory of gravity.

The same story is true even for topological string theories. Any formulation of topological string theory (GW/DT/GV) compute very subtle and powerful invariants that depend more on the "birrational class" of the target space rather that in its topology. By the very construction of the A and B models, all possible backgrounds "contribute equally" to the partition function independently of the specific topological details (values of the moduli) of the manifold; more precisely, the A(/B) path integrals integrate over the entire moduli space of all Kahler(/complex structure) moduli.

Extreme examples include the ones at which the very notion of Zariski topology stops to make sense or the formulation of the topological string via target quantum spacetime foam.

Now let's return to your actual question. Why is the Chern-Simons form the most natural choice for a theory of interacting open strings?

The answer is that the basic interaction open string vertex is the cubic one (2->1).enter image description here

Recall that topological gauge theories does not have physical excitations(otherwise depend on quantum corrections) and are such that the variation of its actions under gauge transformation are closed forms (otherwise a physical charge must be coupled to the theory to compensate this last term). Now the fact is that the Chern-Simons form is the most general free ($QA=0$) polynomial action in the connection $A$ such that $F=0$ (no physical excitations) and $\delta A=QA$ that respects the open string BRST $Q^{2}=0$ symmetry. Also you must easily verify that its variation under gauge transformations is a closed form.


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