# Large gauge transformations for higher p-form gauge fields

+ 3 like - 0 dislike
95 views

Question: What is the large gauge transformations for higher p-form gauge field on a spatial d-dimensional torus $T^d$ or a generic (compact) manifold $M$? for p=1,2,3, etc or any other integers. Is there a homotopy group to label distinct classes of large gauge transformations for p-form gauge field on $d$-dimensional torus $T^d$ or any $M$ manifold ? (shall we assume the theory is a topological field theory, or not necessary?) References are welcome.

Background: Large gauge transformation has been of certain interests. The Wiki introduces it as

Given a topological space M, a topological group G and a principal G-bundle over M, a global section of that principal bundle is a gauge fixing and the process of replacing one section by another is a gauge transformation. If a gauge transformation isn't homotopic to the identity, it is called a large gauge transformation. In theoretical physics, M often is a manifold and G is a Lie group.

1-form: The well-known example is a connection $A$ as Lie algebra value 1-form. We have the finite gauge transformation. $$A \to g(A+d)g^{-1}$$ An example of a large gauge transformation of a Schwarz-type Chern-Simons theory, $\int A \wedge dA$, on 2-dimensional $T^2$ torus of the size $L_1 \times L_2$ with spatial coordinates $(x_1,x_2)$ can be $g=\exp[i 2\pi(\frac{n_1 x_1}{L_1}+\frac{n_2 x_2}{L_2})]$. This way, for the constant gauge profile $(a_1(t),a_2(t))$ (constant respect to the space, satisfying EOM $dA=0$), the large gauge transformation identifies: $$(a_1,a_2)\to (a_1,a_2)+2\pi (\frac{n_1}{L_1},\frac{n_2 }{L_2})$$

This seems the two $\mathbb{Z}^2$ integer indices $(n_1,n_2)$ remind me the homotopy group: $\pi_1(T^2)=\pi_1(S^1\times S^1)=\mathbb{Z}^2$.

2-form: If we consider a 2-form $B$ field for a Schwarz-type TQFT, do we have the identification by $\pi_2(M)$ on the $M$ as the based manifold? (Note that $\pi_2(T^d)=0$ - ps. from the math fact that $\pi_2(G)=0$ for any compact connected Lie group $G$.) Is this the correct homotopy group description? How does large gauge transformation work on $T^d$ or $M$?

3-form: is there a homotopy group description on large gauge transformation? How does its large gauge transform on $T^d$ or $M$?

This post imported from StackExchange Physics at 2014-06-04 11:38 (UCT), posted by SE-user Idear
 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:$\varnothing\hbar$ysicsOverflowThen 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.