Noah Miller

A bstract: Self-dual spacetimes can be thought of as spacetimes containing only positive helicity gravitons. In this work we give a perturbiner expansion for self-dual spacetimes based on Plebański’s second heavenly equation. The expansion is naturally organized as a sum over “marked tree graphs” where each node corresponds to a positive helicity graviton and can have an arbitrary number of edges. Negative helicity gravitons must be added in by hand. We then use this perturbiner expansion to give a first principles derivation of the NSVW tree formula for the MHV amplitude in Einstein gravity. A unique feature of this proof is that it does not use BCFW recursion or twistor theory. It works by plugging the spacetime with arbitrarily many + gravitons and two – gravitons into the on-shell gravitational action and evaluating it. The action we use is the self-dual Plebański action plus an additional boundary term, and the amplitude itself comes entirely from the boundary term. Along the way, we also find an interesting new generalization of the NSVW formula which has not previously appeared in the literature. In the appendix we give another way to express the perturbiner expansion using binary tree graphs instead of marked tree graphs, and prove the equivalence of these two expansions diagrammatically. We also provide an introduction to self-dual gravity aimed at non-experts, as well as a proof of the Parke-Taylor formula in Yang Mills theory analogous to our proof of the NSVW formula in gravity.

BFSS proposed that asymptotically flat M-theory is dual to a large N limit of the matrix quantum mechanics describing N nonrelativistic D0-branes. Recent insights on the soft symmetries of any quantum theory of gravity in asymptotically flat space are applied to the BFSS matrix model. It is shown that soft gravitons are realized by submatrices whose rank is held fixed in the large N M-theory limit, rather than the usual linear scaling with N for hard gravitons. The soft expansion is identified with the large N expansion and the soft theorem becomes a universal formula for the quantum mechanical scattering of such submatrix excitations. This formula is shown to be the Ward identity of large type IIA U(1)_RR asymptotic gauge symmetry in the matrix model, whose asymptotic boundaries are at future and past timelike infinity.

We present a discrete basis of solutions of the massless Klein-Gordon equation in 3 + 1 Minkowski space which transform as 𝔰𝔩(2, ℂ) Lorentz/conformal primaries and descendants, and whose elements all have integer conformal dimension. We show that the basis is complete in the sense that the Wightman function can be expressed as a quadratic sum over the basis elements.

We present a discrete basis of solutions of the massless Klein-Gordon equation in 3+1 Minkowski space which transform as sl(2,C) Lorentz/conformal primaries and descendants, and whose elements all have integer conformal dimension. We show that the basis is complete in the sense that the Wightman function can be expressed as a quadratic sum over the basis elements.

The analytic continuation of the general signature (1, 3) Lorentzian Kerr-Taub-NUT black holes to signature (2, 2) Kleinian black holes is studied. Their global structure is characterized by a toric Penrose diagram resembling their Lorentzian counterparts. Kleinian black holes are found to be self-dual when their mass and NUT charge are equal for any value of the Kerr rotation parametera. Remarkably, it is shown that the rotationacan be eliminated by a large diffeomorphism; this result also holds in Euclidean signature. The continuation from Lorentzian to Kleinian signature is naturally induced by the analytic continuation of the S-matrix. Indeed, we show that the geometry of linearized black holes, including Kerr-Taub-NUT, is captured by (2, 2) three-point scattering amplitudes of a graviton and a massive spinning particle. This stands in sharp contrast to their Lorentzian counterparts for which the latter vanishes kinematically and enables a direct link to the S-matrix.

The bulk-to-boundary dictionary for 4D celestial holography is given a new entry defining 2D boundary states living on oriented circles on the celestial sphere. The states are constructed using the 2D CFT state-operator correspondence from operator insertions corresponding to either incoming or outgoing particles which cross the celestial sphere inside the circle. The BPZ construction is applied to give an inner product on such states whose associated bulk adjoints are shown to involve a shadow transform. Scattering amplitudes are then given by BPZ inner products between states living on the same circle but with opposite orientations. 2D boundary states are found to encode the same information as their 4D bulk counterparts, but organized in a radically different manner.

We derive a thermodynamic first law for the electrically charged C-metric with vanishing cosmological constant. This spacetime describes a pair of identical accelerating black holes each pulled by a cosmic string. Treating the "boost time" of this spacetime as the canonical time, we find a thermodynamic first law in which every term has an unambiguous physical meaning. We then show how this first law can be derived using Noetherian methods in the covariant phase space formalism. We argue that the area of the acceleration horizon contributes to the entropy and that the appropriate notion of energy of this spacetime is a "boost mass", which vanishes identically. The recovery of the Reissner-Nordstrom first law in the limit of small string tension is also demonstrated. Finally, we compute the action of the Euclidean section of the C-metric and show it agrees with the thermodynamic grand potential, providing an independent confirmation of the validity of our first law. We also briefly speculate on the significance of firewalls in this spacetime.