\(L_\infty\)-algebra models and higher Chern-Simons theories (Q2829631)

According to the authors, this paper is intended as a starting point for future work, studying the dynamics of quantized multisymplectic manifolds, along the lines of [\textit{H. S. Yang}, Mod. Phys. Lett. A 22, No. 16, 1119--1132 (2007; Zbl 1117.83082); \textit{H. Steinacker}, Classical Quantum Gravity 27, No. 13, Article ID 133001, 46 p. (2010; Zbl 1255.83007)], using \(L_\infty\) models. NEWLINENEWLINEThe authors state topology and geomtry of spacetime in a physical theory should emerge from a more fundamental descripiton. One of the closest model to solve this problem is the IKKT matrix model [\textit{N. Ishibashi} et al., ``A large \(N\) reduced models superstring'', Nucl.Phys. B 498, No. 1--2, 467--491 (1997; \url{doi:10.1016/S0550-3213(97)00290-3})]. This model is equivalent to the 10-dimensional super Yang-Mills theory dimensionally reduced to a point. In this paper, how to generalize the IKKT model in terms of higher Lie algebras, or equivalently, \(L_\infty\)-algebras, is discussed.NEWLINENEWLINE\(L_\infty\)-algebras appear in BV-quantization of classical field theory. Authors discussed models employing 2-term \(L_\infty\)-algebras [the authors, J. High Energy Phys. 2014, No. 4, Paper No. 066, 44 p. (2014; Zbl 1333.81358)]. They are generalized to arbitrary (truncated) \(L_\infty\)-algebras in this paper, and refer to the resulting model as \(L_\infty\)-algebra models. The authors state beyond the dimensional reduction of the bosonic part of the six-dimensional (2,0)-theory (discussed in \S6.2), physically interesting action of higher Yang-Mills theory is not known. While higher versions of Chern-Simons theories are constructed using AKSZ formalism [\textit{M. Alexandrov} et al., Int. J. Mod. Phys. A 12, No. 7, 1405--1429 (1997; Zbl 1073.81655)]. In this construction, both spacetime and higher gauge-algebroid are regarded as N\(Q\)-maniflods; graded manifold having \(\mathbb{N}\) as grade, equipped with a nilpotent vector field \(Q\).NEWLINENEWLINEDefinitions and properties of N\(Q\) manifolds and \(L_\infty\)-algebras are reviewed in \S2 [\textit{D. Roytenberg}, Contemp. Math. 315, 169--185 (2002; Zbl 1036.53057)], [the authors, ``Automorphisms of strong homotopy Lie algebras of local observables'', Preprint, \url{arXiv:1507.00972}]. In \S3, after explaining higher gauge theory with symplectic N\(Q\)-manifolds, higher Chenr-Simons action is derived by using AKSZ formlism(\S3.4). It takes the following form NEWLINE\[NEWLINES_{\mathrm{CS}}=\int_\Sigma\bigl(\langle\phi,\mathrm{d}_\Sigma\phi\rangle+\sum_{k=1}^{p+1}\frac{(-1)^{\sigma_k}}{(k+1)!}\langle\mu_k(\phi,\ldots,\phi),\phi\rangle\bigr),NEWLINE\]NEWLINE where \(\mu_k\)'s are higher bracket and \(\sigma_k=k(k+1)/2\). Since the IKKT model is obtained by dimensionally reducing 10-dimensional maximally supersymmetric Yang-Mills theory, \S4 deals with dimensional reduction and supersymetric gauge theory using N\(Q\)-manifolds.NEWLINENEWLINEThe noncommutative spaces arising in the IKKT model are Kähler manifolds, which are restrictive for gravitational physics. In [\textit{J. Arnlind} and \textit{G. Huisken}, Lett. Math. Phys. 104, No. 12, 1507--1521 (2014; Zbl 1308.53123)], it is suggested one should turn to Nambu-Poissson manifolds, which overlap with multisymplectic manifolds. On the multisymplectic manifolds, the Poisson algebra is replaced by a higher Lie algebra of observables [\textit{J. C. Baez} et al., Commun. Math. Phys. 293, No. 3, 701--725 (2010; Zbl 1192.81208)]. These are reviewed in \S5 with special attention to the Heisenberg Lie \(p\)-algebra, which is sufficient for the description of quantized symplectic manifolds as solutions of the IKKT model. After these preparations, \(L_\infty\)-models are discussed in \S6, the last section. It begins to investigate Yang-Mills type models. Then they show how dimensionally reduced higher Chern-Simons theories provide a unifying picture.