Stabilizing the Richardson Algorithm by Controlling Chaos

By viewing the operations of the Richardson purification algorithm as a discrete time dynamical process, we propose a method to overcome the instability of the algorithm by controlling chaos. We present theoretical analysis and numerical results on the behavior and performance of the stabilized algorithm.Comment: Send email to [email protected] or [email protected] for uuencoded tarred gzipped postscript files for the five figure

Conformal bootstrap to R\'enyi entropy in 2D Liouville and super-Liouville CFTs

The R\'enyi entanglement entropy (REE) of the states excited by local operators in two-dimensional irrational conformal field theories (CFTs), especially in Liouville field theory (LFT) and $\mathcal{N}=1$ super-Liouville field theory (SLFT), has been investigated. In particular, the excited states obtained by acting on the vacuum with primary operators were considered. {We start from evaluating the second REE in a compact $c=1$ free boson field theory at generic radius, which is an irrational CFT. Then we focus on the two special irrational CFTs, e.g., LFT and SLFT. In these theories, the second REE of such local excited states becomes divergent in early and late time limits. For simplicity, we study the memory effect of REE for the two classes of the local excited states in LFT and SLFT. In order to restore the quasiparticles picture, we define the difference of REE between target and reference states, which belong to the same class. The variation of the difference of REE between early and late time limits always coincides with the log of the ratio of the fusion matrix elements between target and reference states. Furthermore, the locally excited states by acting generic descendent operators on the vacuum have been also investigated. The variation of the difference of REE is the summation of the log of the ratio of the fusion matrix elements between the target and reference states, and an additional normalization factor. Since the identity operator (or vacuum state) does not live in the Hilbert space of LFT and SLFT and no discrete terms contribute to REE in the intermediate channel, the variation of the difference of REE between target and reference states is no longer the log of the quantum dimension which is shown in the 1+1-dimensional rational CFTs (RCFTs).Comment: 53 pages,add a new section (Section 2.4

T-duality to Scattering Amplitude and Wilson Loop in Non-commutative Super Yang-Mills Theory

We first perform bosonic T-duality transformation on one of the marginal TsT (T-duality, shift, T-duality)-deformed $AdS_5\times S_5$ spacetime, which corresponds to 4D $\mathcal{N}=4$ non-commutative super Yang-Mills theory (NCSYM). We then construct the solution to killing spinor equations of the resulting background, and perform the fermionic T-duality transformation. The final dual geometry becomes the usual $AdS_5\times S_5$ but with the constant NS-NS B-field depending on the non-commutative parameter. As applications, we study the gluon scattering amplitude and open string (Wilson loop) solution in the TsT-deformed $AdS_5\times S_5$ spacetime, which are dual to the null polygon Wilson loop and the folded string solution respectively in the final dual geometry.Comment: 24 pages, latex, references added, published versio

A note on connected formula for form factors

In this note we study the connected prescription, originally derived from Witten's twistor string theory, for tree-level form factors in ${\cal N}=4$ super-Yang-Mills theory. The construction is based on the recently proposed four-dimensional scattering equations with $n$ massless on-shell states and one off-shell state, which we expect to work for form factors of general operators. To illustrate the universality of the prescription, we propose compact formulas for super form factors with chiral stress-tensor multiplet operator, and bosonic ones with scalar operators ${\rm Tr}(\phi^m)$ for arbitrary $m$.Comment: 13 page

An Etude on Recursion Relations and Triangulations

Following~\cite{Arkani-Hamed:2017thz}, we derive a recursion relation by applying a one-parameter deformation of kinematic variables for tree-level scattering amplitudes in bi-adjoint $\phi^3$ theory. The recursion relies on properties of the amplitude that can be made manifest in the underlying kinematic associahedron, and it provides triangulations for the latter. Furthermore, we solve the recursion relation and present all-multiplicity results for the amplitude: by reformulating the associahedron in terms of its vertices, it is given explicitly as a sum of "volume" of simplicies for any triangulation, which is an analogy of BCFW representation/triangulation of amplituhedron for ${\cal N}=4$ SYM.Comment: 26 pages, 3 figure

A Causal Set Black Hole

We explicitly compute the causal structure of the Schwarzschild black hole spacetime, by providing an algorithm to decide if any pair of events is causally related. The primary motivation for this study comes from discrete quantum gravity, in particular the causal set approach, in which the fundamental variables can be thought of as the causal ordering of randomly selected events in spacetime. This work opens the way to simulating non-conformally flat spacetimes within the causal set approach, which may allow one to study important questions such as black hole entropy and Hawking radiation on a full four dimensional causal set black hole.Comment: 22 pages, 9 figures, LaTeX; response to referee comment

New Relations for Gauge-Theory and Gravity Amplitudes at Loop Level

In this letter, we extend the tree-level Kawai--Lewellen--Tye (KLT) and Bern--Carrasco--Johansson (BCJ) amplitude relations to loop integrands of gauge theory and gravity. By rearranging the propagators of gauge and gravity loop integrands, we propose the first manifestly gauge- and diffeomorphism invariant formulation of their double-copy relations. The one-loop KLT formula expresses gravity integrands in terms of more basic gauge invariant building blocks for gauge-theory amplitudes, dubbed partial integrands. The latter obey a one-loop analogue of the BCJ relations, and both KLT and BCJ relations are universal to bosons and fermions in any number of spacetime dimensions and independent on the amount of supersymmetry. Also, one-loop integrands of Einstein--Yang--Mills (EYM) theory are related to partial integrands of pure gauge theories.Comment: 6 pages; v2: references added, minor corrections, published version with updated reference on work in progres