A Lie based 4-dimensional higher Chern-Simons theory (Q2814215)

Based on the theory of 2-groups and Lie 2-algebras, the author has developed higher gauge theory and 4-dimensional Chern-Simons theory [``AKSZ models of semistrict higher gauge theory'', Preprint, \url{arXiv:1112.2819}; \textit{E. Soncini} and the author, ``4-d semistrict higher Chern-Simons theory. I'', Preprint, \url{arXiv:1406.2197}; \textit{M. Kulaxizi} et al., J. High Energy Phys. 2014, No. 9, Article ID 010, 24 p. (2014; Zbl 1333.81372)].NEWLINENEWLINEIn this paper, after exposing physical origin of higher gauge theory and 4-d Chern-Simos theory (2-Chern-Simons theory) in \S 1 B, the author's construction of higher gauge theory and higher Chern-Simons theory are reviewed in \S 2. Definitions and properties of 2-groups and Lie 2-algebras are reviewed in the appendix. NEWLINENEWLINEIn \S 3, special higher gauge theory is studied in detail as follows: Let \(G\) be a Lie group with the center \(Z=Z(G)\) and Lie algebra \(\mathfrak g\), and let \(N\) a smooth Riemannian manifold. Then the set \(\mathrm{Gau}_1(N,G)\) of special \(G\)-1-gauge transformation consists of all quadruples \((\gamma,\zeta_\gamma,\alpha_\gamma,\chi_\gamma)\) with \(\gamma\in\mathrm{Map}(N, G)\), \(\zeta_\gamma\in\Omega^2(N,R)\), \(\alpha_\gamma\in\Omega 0(N,\mathrm{End}(\mathfrak g))\), \(\chi_\gamma\in\Omega^1(N,\mathfrak g)\) satisfying \((\gamma^{-1}d\gamma,k)=0\), \(k\in Z\) and NEWLINE\[NEWLINE(\gamma^{-1}d\gamma, [\gamma^{-1}d\gamma,\gamma^{-1}d\gamma])=6d\zeta_\gamma,\quad (x,\alpha_\gamma (y))+(y,\alpha_\gamma (x))=0,NEWLINE\]NEWLINE where \(x, y\in\mathfrak g\). \(\beta,\gamma\in\mathrm{Gau}_1(N, G)\) are said to be 2-gauge compatible whenever \(\gamma\beta^{-1}=K\), \(K\in\mathrm{Map}(N, G)\) is a constant element taking the value in \(Z\), and \(\zeta_\beta=\zeta_\gamma\). The set \(\mathrm{Gau}_2(N,G)\) of special \(G\)-2-gauge transformations \(\gamma\Rightarrow\beta\), where \(\beta\), \(\gamma\) are 2-gauge compatible, consists of all triples of \((K,\Phi_K,P_K)\), \(K\in\mathrm{Map}(N, Z)\), \(\Phi_K\in\Omega^0(N,\mathrm{End}(\mathfrak g)\), \(PK\in\Omega^1(N,G)\) satisfying \(K^{-1}dK=0\), \((x,\Phi_K (y))+(y,\Phi_K(x))=0\) and NEWLINE\[NEWLINE\gamma\beta-1=K,\quad \alpha_\gamma(\pi)=\alpha_\beta(\pi) =\Phi_K(\pi)=0,\quad x_\gamma-x_\beta=P_K=0.NEWLINE\]NEWLINE Starting these definitions, special gauge transformation 2-group and algebra are defined in \S 3.A.. Definition of 2-algebra depends on the choice of \(k\in z(G)\), so is is denoted by \(\mathfrak v_k\)). Their infinitesimal versions are also defined in \S 3,A. The component \(\gamma\in\mathrm{Map}(N,G)\) of a special 1-gauge transformation can be viewed as an ordinary gauge transformation. Its topology is governed by NEWLINE\[NEWLINE w(\gamma) =\frac{1}{48\pi^2}(\gamma^{-1}d\gamma,[\gamma^{-1}d\gamma,[\gamma^{-1}d\gamma,\gamma^{-1}d\gamma]).NEWLINE\]NEWLINE By definitions, we have \(w(\gamma)=\frac{1}{8\pi^2}d\zeta_\gamma\). Its role is investigated in \S 3.A.7. Then after dealing with special 2-connections and curvatures 1in \S 3.B, there geometric realizations (special principal 2-bundles) are discussed in \S 3.C, and show principal \(\mathfrak v_k(g)\)-2-bundles with 2-connections and 1-gauge transformation are characterized by cohomology classes with values in the lattice NEWLINE\[NEWLINE\mathfrak l(\mathfrak g)=\mathrm{ker}(\exp|_{\mathfrak z(\mathfrak g)} )\subset \mathfrak z(\mathfrak g)NEWLINE\]NEWLINE and the sheaves \(\underline{Z(G)}\), \(\underline{G/Z(G)}\) of sheaves of smooth \(Z(G)\), \(G/Z(G)\)-valued functions (cf. [\textit{B. Jurco}, \textit{C. Saemann}, and \textit{M. Wolf}, ``Semistrict higher gauge theory'', Preprint, \url{arXiv:1403.7185}]). These constructions seems similar to those in non abelian de Rham theory. But it lacks the theory of symmetry (higher group and Lie algebra theory) (cf. [\textit{A. Asada}, in: Prospects of mathematical science, Proc. Symp., Tokyo/Jap. 1986. 13--40 (1988; Zbl 0682.58044)]).NEWLINENEWLINEHigher Chern-Simons action already presented in \S 2.B. In the framework of special higher gauge theory, it takes the form NEWLINE\[NEWLINE\mathrm{CS}_2(\omega)=\kappa_2\int_N[(d\omega+\frac{1}{2}[\omega,\omega],\Omega_\omega)-\frac{1}{6} (\omega,k)(\omega, [\omega, \omega])]NEWLINE\]NEWLINE (\S 4.(296)). Then to take advantage from the richer structure of the Lie 2-algebra \(\mathfrak v_k(\mathfrak g)\), a coupling term to a background closed 3-form \(H\); \(\Delta\mathrm{CS}_2(\omega,H)=8\pi\kappa_2\int_N (\omega, k)H\) is added to this action. The resulting action is denoted by \(\overline{\mathrm{CS}}_2(\omega, H)\). The 2-connection space \(\mathrm{Conn}_2(M, G)\) carries a symplectic structure \(\sum_{M,G}=\int_M(\delta\omega,\delta\Omega_\omega)\). Hence we use several tools of symplectic geometry to the study of higher connection space. These are exposed in \S 3.B. Then adopting these tools the partition function of special \(G\)-2-Chern-Simons theory NEWLINE\[NEWLINEZ_{\mathrm{sCS}}(H)=\frac{1}{V}\int D\omega D\Omega_\omega\exp(i\overline{\mathrm{CS}}_2(\omega,\Omega_\omega H))NEWLINE\]NEWLINE where \(V\) is the functional volume of the gauge modulo gauge for gauge symmetry, is computed. (\S 4 (366)). It express the partition function as an integral over the \(\mathrm{Map}_k(N,G)\)-orbit space \(\mathcal M_k(N,G)\) by factoring the volume of the residual unfixed \(\mathrm{Map}_k(N,G)/\mathrm{Map}_{kc}(N,G)\) symmetry. Then via the analysis of the cohomology class of the ordinary Chern-Simons 3-form, it is concluded that the non vanishing of the partition function detects the path connected component structure of \(\mathcal M_k(N,G)\). Since the functional volume \(V\) depends on the metric \(g\) of \(N\), this paper is concluded to give brief discussion on influence of \(g\) to the partition function.