Infinite Charge Algebra of Gravitational Instantons

Using a formalism of minitwistors, we derive infinitely many conserved charges for the sl(∞)-Toda equation which accounts for gravitational instantons with a rotational Killing symmetry. These charges are shown to form an infinite dimensional algebra through the Poisson bracket which is isomorphic to two dimensional area preserving diffeomorphism with central extentions. E-mail address; [email protected] 2 E-mail address; [email protected] It has been known for some time that certain large N limits of two dimensional field theories yield higher dimensional field theories (see e.g.[1]). In particular, a large N limit of the two dimensional sl(N)-Toda equation becomes a three dimensional equation for a scalar field u(w, w̄, t), ∂∂̄u = −∂ t e u ; ∂ = ∂ ∂w , ∂̄ = ∂ ∂w̄ (1) which is also the self-dual Einstein equation with a rotational Killing symmetry and the metric: ds = 1 u,t (4edwdw̄ + dt) + u,t(dθ + iu,wdw + iu,w̄dw̄) 2 . (2) Eq.(1) as a large N limit of the sl(N)-Toda equation is expected to possess infinite symmetries. Indeed, the infinitesimal action of such symmetries has been obtained previously and these were shown to form an algebra of area preserving diffeomorphisms. However, generators of such symmetries, i.e. conserved charges of Eq.(1), are not known explicitly except for few cases such as spin 2 charge. Moreover the associated charge algebra which is essential in understanding the quantum aspect of Eq.(1) is presently unknown. Even though these charges are expected to arise from large N limits of conserved charges of the sl(N)-Toda equation, unlike the Toda equation case, the large N limit procedure for conserved charges is more involved. As explained in this letter, they are correctly described by the language of twistor theory. In this letter, we first derive conserved charges explicitly through the first order differential equations which determine the infinitesimal symmetries of Eq.(1). Then, by defining a minitwistor space for Eq.(1), we provide a general closed form of spin-s conserved charges. These charges are also shown to form a symmetry algebra via a Poisson bracket which is isomorphic to the centrally extended area preserving diffeomorphisms. We first recall that the infinitesimal spin-s symmetry of Eq.(1), ∂∂̄δu = −∂ t (e δu) , (3) is given by the following recursive equations; δu = ∂tA (s) 0 ∂tA (s) r−1 = (∂̄ + r∂̄u)A (s) r ; r = 1, 2, · · ·s− 1 (4) with A (s) s−1 = f(w̄) and f an arbitrary anti-holomorphic function. Defining ∂tq ≡ u ; p ≡ ∂̄q and Kr ≡ 1 ∂t (∂̄ + rpt) ; Mr ≡ (∂̄ − rpt) 1 ∂t , (5)