# Topology-dependent groud state degeneracy of $B \wedge F + B \wedge B$ and $B \wedge F + B \wedge B \wedge B$

There are some examples of topological BF theory with extra terms allow it still being topological. See this Ref. paper

In 4d (3+1D), we have the trace of: $$\int\frac{k}{2\pi}\text{Tr}[B \wedge F + \frac{\Lambda}{12}B \wedge B]$$

question 1: What is the ground state degeneracy on $\mathbb{T}^3$ spatial 3-torus?

In 3d (2+1D), we have the trace of: $$\int \frac{k}{2\pi}\text{Tr}[B \wedge F + \frac{\Lambda}{3}B \wedge B \wedge B]$$

question 2: What is the ground state degeneracy on $\mathbb{T}^2$ spatial 2-torus?

Topology-dependent ground state degeneracy($GSD$) means the number of ground states of this topological field theory.

If we set the $\Lambda=0$, and suppose F=dA are U(1) gauge-symmetry 2-form, and $A$ is a 1-form. The B is 2-form in 4d and 1-form in 3d.

In 4d (3+1D), we have this term: $$\int \frac{k}{2\pi} B \wedge F$$ with its topology-dependent ground state degeneracy($GSD$) of this action on $\mathbb{T}^2$ torus as $$GSD=k^2$$

In 4d (3+1D), we again have this term: $$\int \frac{k}{2\pi} B \wedge F$$ with its topology-dependent ground state degeneracy($GSD$) of this action on $\mathbb{T}^3$ torus as

## $$GSD=k^3$$

question 3: How the $\Lambda \neq 0$ modifies the topology-dependent ground state degeneracy on $\mathbb{T}^2$, $\mathbb{T}^3$ spatial 2-torus, 3-torus? Please provide any example possible to show the truncation(?) of ground state degeneracy.

Simply listing down useful Ref is still welcome. Please do so.

Thanks. :-)

-