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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 $M$?

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