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Register a new userKinematic numerators from the worldsheet: cubic trees from labelled trees
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In this note we revisit the problem of explicitly computing tree-level scattering amplitudes in various theories in any dimension from worldsheet formulas. The latter are known to produce cubic-tree expansion of tree amplitudes with kinematic numerators automatically satisfying Jacobi-identities, once any half-integrand on the worldsheet is reduced to logarithmic functions. We review a natural class of worldsheet functions called Cayley functions, which are in one-to-one correspondence with labelled trees, and natural expansions of known half-integrands onto them with coefficients that are particularly compact building blocks of kinematic numerators. We present a general formula expressing kinematic numerators of all cubic trees as linear combinations of coefficients of labelled trees, which satisfy Jacobi identities by construction and include the usual combinations in terms of master numerators as a special case. Our results provide an efficient algorithm, which is implemented in a Mathematica package, for computing all tree amplitudes in theories including non-linear sigma model, special Galileon, Yang-Mills-scalar, Einstein-Yang-Mills and Dirac-Born-Infeld.
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We show that open strings living on a D-brane which lies outside an AdS black hole can tunnel into the black hole through worldsheet instantons. These instantons have a simple interpretation in terms of thermal quarks in the dual Yang-Mills (YM) theory. As an application we calculate the width of a meson in a strongly coupled quark-gluon plasma which is described holographically as a massless mode on a D7 brane in AdS_5 times S_5. While the width of the meson is zero to all orders in the 1/sqrt{lambda} expansion with lambda the t Hooft coupling, it receives non-perturbative contributions in 1/sqrt{lambda} from worldsheet instantons. We find that the width increases quadratically with momentum at large momentum and comment on potential phenomenological implications of this enhancement for heavy ion collisions. We also comment on how this non-perturbative effect has important consequences for the phase structure of the YM theory obtained in the classical gravity limit.
In this paper we study the minimum number of reversals needed to transform a unicellular fatgraph into a tree. We consider reversals acting on boundary components, having the natural interpretation as gluing, slicing or half-flipping of vertices. Our main result is an expression for the minimum number of reversals needed to transform a unicellular fatgraph to a plane tree. The expression involves the Euler genus of the fatgraph and an additional parameter, which counts the number of certain orientable blocks in the decomposition of the fatgraph. In the process we derive a constructive proof of how to decompose non-orientable, irreducible, unicellular fatgraphs into smaller fatgraphs of the same type or trivial fatgraphs, consisting of a single ribbon. We furthermore provide a detailed analysis how reversals affect the component-structure of the underlying fatgraphs. Our results generalize the Hannenhalli-Pevzner formula for the reversal distance of signed permutations.
The $ell$-deck of a graph $G$ is the multiset of all induced subgraphs of $G$ on $ell$ vertices. In 1976, Giles proved that any tree on $ngeq 6$ vertices can be reconstructed from its $ell$-deck for $ell geq n-2$. Our main theorem states that it is enough to have $ellgeq (8/9+o(1))n$, making substantial progress towards a conjecture of Nydl from 1990. In addition, we can recognise connectedness from the $ell$-deck if $ellgeq 9n/10$, and reconstruct the degree sequence from the $ell$-deck if $ellge sqrt{2nlog(2n)}$. All of these results are significant improvements on previous bounds.
We present new formulas for one-loop ambitwistor-string correlators for gauge theories in any even dimension with arbitrary combinations of gauge bosons, fermions and scalars running in the loop. Our results are driven by new all-multiplicity expressions for tree-level two-fermion correlators in the RNS formalism that closely resemble the purely bosonic ones. After taking forward limits of tree-level correlators with an additional pair of fermions/bosons, one-loop correlators become combinations of Lorentz traces in vector and spinor representations. Identities between these two types of traces manifest all supersymmetry cancellations and the power counting of loop momentum. We also obtain parity-odd contributions from forward limits with chiral fermions. One-loop numerators satisfying the Bern-Carrasco-Johansson (BCJ) duality for diagrams with linearized propagators can be extracted from such correlators using the well-established tree-level techniques in Yang-Mills theory coupled to biadjoint scalars. Finally, we obtain streamlined expressions for BCJ numerators up to seven points using multiparticle fields.
The modular decomposition of a symmetric map $deltacolon Xtimes X to Upsilon$ (or, equivalently, a set of symmetric binary relations, a 2-structure, or an edge-colored undirected graph) is a natural construction to capture key features of $delta$ in labeled trees. A map $delta$ is explained by a vertex-labeled rooted tree $(T,t)$ if the label $delta(x,y)$ coincides with the label of the last common ancestor of $x$ and $y$ in $T$, i.e., if $delta(x,y)=t(mathrm{lca}(x,y))$. Only maps whose modular decomposition does not contain prime nodes, i.e., the symbolic ultrametrics, can be exaplained in this manner. Here we consider rooted median graphs as a generalization to (modular decomposition) trees to explain symmetric maps. We first show that every symmetric map can be explained by extended hypercubes and half-grids. We then derive a a linear-time algorithm that stepwisely resolves prime vertices in the modular decomposition tree to obtain a rooted and labeled median graph that explains a given symmetric map $delta$. We argue that the resulting tree-like median graphs may be of use in phylogenetics as a model of evolutionary relationships.
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