Chirality, a new key for the definition of the connection and curvature of a Lie-Kac super-algebra
Jean Thierry-Mieg
备注：
10 pages, 24 references. This is the revised version published in JHEP. Our new definition of the superconnection is unchanged relative to version 1 of this preprint, but thanks to the careful help of the referee, the discussion of the literature is streamlined and much improved
A natural generalization of a Lie algebra connection, or Yang-Mills field, to the case of a Lie-Kac superalgebra, for example SU(m/n), just in terms of ordinary complex functions and differentials, is proposed. Using the chirality $χ$ which defines the supertrace of the superalgebra: $STr(...) = Tr (χ...)$, we construct a covariant differential: $D = χ(d + A) + Φ$, where A is the standard even Lie-subalgebra connection 1-form and $Φ$ a scalar field valued in the odd module. Despite the fact that $Φ$ is a scalar, $Φ$ anticommutes with $(χA)$ because $χ$ anticommutes with the odd generators hidden in $Φ$. Hence the curvature $F = DD$ is a superalgebra-valued linear map which respects the Bianchi identity and correctly defines a chiral parallel transport compatible with a generic Lie superalgebra structure.
We demonstrate that the near horizon symmetries of black holes in Einstein--Yang-Mills (EYM) theory are generated by an infinite-dimensional algebra that contains, in addition to supertranslations and superrotations, a non-Abelian loop algebra. This means that the Virasoro--Kac-Moody structure of EYM in asymptotically flat spacetimes has an exact analog in the near horizon region.
All finite dimensional irreducible representations of the simple Lie-Kac super algebra SU(2/1) are explicitly constructed in the Chevalley basis as complex matrices. For typical representations, the distinguished Dynkin label is not quantized. We then construct the generic atypical indecomposable quivers classified by Marcu, Su and Germoni and typical indecomposable N-generations block triangular extensions for any irreducible module and any integer N. In addition to the quadratic and cubic super-Casimir operators $C_2$ and $C_3$, the supercenter of the enveloping algebra contains a chiral gho…
I examine the Generalized K$\ddot{a}$hler geometry of classical $N=(2,2)$ superconformal WZW model on a compact group and relate the right-moving and left-moving Kac-Moody superalgebra currents to the Generalized K$\ddot{a}$hler geometry data using Hamiltonian formalism. It is shown that canonical Poisson homogeneous space structure induced by the Generalized K$\ddot{a}$hler geometry of the group manifold is crucial to provide $N=(2,2)$ superconformal sigma-model with the Kac-Moody superalgebra symmetries. Biholomorphic gerbe geometry is used to prove that Kac-Moody superalgebra currents are g…
Tensor hierarchy algebras are infinite-dimensional generalisations of Cartan-type Lie superalgebras. They are not contragredient, exhibiting an asymmetry between positive and negative levels. These superalgebras have been a focus of attention due to the fundamental role they play for extended geometry. In the present paper, we examine tensor hierarchy algebras which are super-extensions of over-extended (often, hyperbolic) Kac--Moody algebras. They contain novel algebraic structures. Of particular interest is the extension of a over-extended algebra by its fundamental module, an extension that…
The representation theory of involutory (or 'maximal compact') subalgebras of infinite-dimensional Kac-Moody algebras is largely terra incognita, especially with regard to fermionic (double-valued) representations. Nevertheless, certain distinguished such representations feature prominently in proposals of possible symmetries underlying M theory, both at the classical and the quantum level. Here we summarise recent efforts to study spinorial representations systematically, most notably for the case of the hyperbolic Kac-Moody algebra $E_{10}$ where spinors of the involutory subalgebra $K(E_{10…
It has been constructed the quantum super fuzzy Dirac and chirality operators on q-deformed super fuzzy sphere. Using the quantum super fuzzy Ginsparg-Wilson algebra, it has been studied the q-deformed super gauged fuzzy Dirac and chirality operators in instanton sector. It has been showed that they have correct commutative limit in the limit case when noncommutative parameter $ l $ tends to infinity and q tends to unit.
We construct a Borcherds Kac-Moody (BKM) superalgebra on which the Conway group Co$_0$ acts faithfully. We show that the BKM algebra is generated by the BRST-closed states in a chiral superstring theory. We use this construction to produce denominator identities for the chiral partition functions of the Conway module $V^{s\natural}$, a supersymmetric $c=12$ chiral conformal field theory whose (twisted) partition functions enjoy moonshine properties and which has automorphism group isomorphic to Co$_0$. In particular, these functions satisfy a genus zero property analogous to that of monstrous …
We consider the subalgebras of split real, non-twisted affine Kac-Moody Lie algebras that are fixed by the Chevalley involution. These infinite-dimensional Lie algebras are not of Kac-Moody type and admit finite-dimensional unfaithful representations. We exhibit a formulation of these algebras in terms of $\mathbb{N}$-graded Lie algebras that allows the construction of a large class of representations using the techniques of induced representations. We study how these representations relate to previously established spinor representations as they arise in the theory of supergravity.
Construction of matryoshka nested indecomposable N-replications of Kac-modules of quasi-reductive Lie superalgebras, including the sl(m/n) and osp(2/2n) series
本文构造了一类新的有限维不可分解表示，通过递归嵌入 N 个不可约表示副本并用广义 Cabibbo 角耦合，以解释标准模型中三代费米子的存在，并将构造推广到拟约化李超代数。
We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a $\widehat{sl(2)}$ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of $\widehat{sl(2)}$ characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients …
In this paper we shall consider an axionic Chern-Simons corrected $f(R)$ gravity theoretical framework, and we shall study the chirality of the generated primordial gravitational waves. Particularly, we shall consider two main axion models, the canonical misalignment axion model and the kinetic axion model, both of which provide an interesting particle phenomenology, in the presence of $R^2$ terms in the inflationary Lagrangian. The axion does not affect significantly the background evolution during the inflationary era, which is solely controlled by $R^2$ gravity. However, the due to the pres…
We study particle and chirality production from the vacuum in the presence of a slow strong parallel electromagnetic field superimposed by a fast weak perturbative electromagnetic field. We derive an analytical formula for the particle and chirality production number based on the perturbation theory in the Furry picture. With the formula, we analytically discuss the interplay and dynamical assistance between the Schwinger mechanism and one-photon pair production and clarify effects of a parallel slow strong magnetic field. We also show that the dynamical assistance can significantly enhance ch…
