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

+ 7 like - 0 dislike
395 views

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.

Thanks. :-)

This post imported from StackExchange Physics at 2014-05-23 10:42 (UCT), posted by SE-user mysteriousness
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsO$\varnothing$erflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.