# Large gauge transformations for higher p-form gauge fields

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 $M$?

-

The mathematical framework that I am familiar with for abelian p-form gauge theory (the one promoted by Freed, Moore and others) is that of Cheeger-Simons differential forms. In this framework, the space of topologically trivial p-form gauge fields over a manifold $X$ (the analogue of 1-form gauge fields on the trivial $U(1)$ bundle) are identified with the quotient of $\Omega^p(X)$ by the group of gauge transformation $\Omega^p_{\mathbb{Z}}(X)$. $\Omega^p_{\mathbb{Z}}(X)$ denotes the set of $p$-forms with integer values when integrated along any closed $p$-dimensional sub-manifold $\Sigma \subset M$. In the $p = 1$ case, $\Omega^1_{\mathbb{Z}}(X)$ is precisely the space of all closed $1$-forms of the form $dg g^{-1}$ for some gauge transformation $g:X \to U(1)$. For more details, I recommend this paper by Belov and Moore, particularly p. 17-19.
In this framework, there is a notion of large gauge transformations for p-form gauge fields. In the 1-form case, the gauge transformations $g(x)$ that are not connected to the identity are precisely those $g(x)$ such that $dg g^{-1}$ has non-zero value when integrated around some closed, non-contractible loop $\gamma$. In the above language, we can say that the connected components of the space $\Omega^p_{\mathbb{Z}}(X)$ correspond to $H^p(X;\mathbb{Z})/\text{torsion}$ (via the map sending the $p$-forms to their de-Rham class) and that the large gauge transformations lie in the components which are not sent to $0$ by this map.