Scattering amplitudes of any four-dimensional theory with nonabelian gauge group $\mathcal G$ may be recast as two-dimensional correlation functions on the asymptotic two-sphere at null infinity. The soft gluon theorem is shown, for massless theories at the semiclassical level, to be the Ward identity of a holomorphic two-dimensional $\mathcal G$-Kac-Moody symmetry acting on these correlation functions. Holomorphic Kac-Moody current insertions are positive helicity soft gluon insertions. The Kac-Moody transformations are a $CPT$ invariant subgroup of gauge transformations which act nontrivially at null infinity and comprise the four-dimensional asymptotic symmetry group.
Starting from noncommutative Fermi theory in two-dimensions, we construct a deformed Kac-Moody algebra between its vector and Chiral currents . The higher-order corrections to the deformed Kac-Moody algebra are explicitly calculated. We observe that the ordinary Schwinger terms are not affected by noncommutativity. Finally we conclude that the deformed Kac-Moody algebra can be given in term of ordinary Kac-Moody algebra plus infinitely many Lie algebraic structures at each non-zero power of the antisymmetric coefficient $θ$.
We show that the theory of gravity constructed from the non-linear realisation of the semi-direct product of the Kac-Moody algebra A1+++ with its vector representation does not allow a cosmological constant. The existence of a cosmological constant in this theory is related to the breaking of the gravitational duality symmetry.
Kac-Moody groups $G$ over $\mathbb{R}$ have been conjectured to occur as symmetry groups of supergravities in dimensions less than 3, and their integer forms $G(\mathbb{Z})$ are conjecturally U-duality groups. Mathematical descriptions of $G(\mathbb{Z})$, due to Tits, are functorial and not amenable to computation or applications. We construct Kac-Moody groups over $\mathbb{R}$ and $\mathbb{Z}$ using an analog of Chevalley's constructions in finite dimensions and Garland's constructions in the affine case. We extend a construction of Eisenstein series on finite dimensional semisimple algebraic…
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.
Given a critical quantum spin chain with a microscopic Lie-group symmetry, corresponding e.g. to $U(1)$ or $SU(2)$ spin isotropy, we numerically investigate the emergence of Kac-Moody symmetry at low energies and long distances. In that regime, one such critical quantum spin chain is described by a conformal field theory where the usual Virasoro algebra associated to conformal invariance is augmented with a Kac-Moody algebra associated to conserved currents. Specifically, we first propose a method to construct lattice operators corresponding to the Kac-Moody generators. We then numerically sho…
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.
We investigate a class of Kac-Moody algebras previously not considered. We refer to them as n-extended Lorentzian Kac-Moody algebras defined by their Dynkin diagrams through the connection of an $A_n$ Dynkin diagram to the node corresponding to the affine root. The cases $n=1$ and $n=2$ correspond to the well studied over and very extended Kac-Moody algebras, respectively, of which the particular examples of $E_{10}$ and $E_{11}$ play a prominent role in string and M-theory. We construct closed generic expressions for their associated roots, fundamental weights and Weyl vectors. We use these q…
We define Eisenstein series on rank 2 hyperbolic Kac--Moody groups over R, induced from quasi--characters. We prove convergence of the constant term and hence the almost everywhere convergence of the Eisenstein series. We define and calculate the degenerate Fourier coefficients. We also consider Eisenstein series induced from cusp forms and show that these are entire functions.
Kac--Moody groups $G$ over $\mathbb{R}$ have been conjectured to occur as symmetry groups of supergravity theories dimensionally reduced to dimensions less than 3, and their integral forms $G(\mathbb{Z})$ conjecturally encode quantized symmetries. In this review paper, we briefly introduce the conjectural symmetries of Kac--Moody groups in supergravity as well as the known evidence for these conjectures. We describe constructions of Kac--Moody groups over $\R$ and $\Z$ using certain choices of fundamental modules that are considered to have physical relevance. Eisenstein series on certain fini…
We introduce a quantum model for the Universe at its early stages, formulating a mechanism for the expansion of space and matter from a quantum initial condition, with particle interactions and creation driven by algebraic extensions of the Kac-Moody Lie algebra $\mathbf{e_9}$. We investigate Kac-Moody and Borcherds algebras, and we propose a generalization that meets further requirements that we regard as fundamental in quantum gravity.
We introduce a higher dimensional generalization of the affine Kac-Moody algebra using the language of factorization algebras. In particular, on any complex manifold there is a factorization algebra of "currents" associated to any Lie algebra. We classify local cocycles of these current algebras, and compare them to central extensions of higher affine algebras recently proposed by Faonte-Hennion-Kapranov. A central goal of this paper is to witness higher Kac-Moody algebras as symmetries of a class of holomorphic quantum field theories. In particular, we prove a generalization of the free field…
