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Stampedes I: Fishnet OPE and Octagon Bootstrap with Nonzero Bridges
TL;DR: In this article, the authors introduce a notion of ''stampede'' which is a simple time-evolution of a bunch of particles which start their life in a corner and hop their way to the opposite corner through the repeated action of a quantum Hamiltonian.
Abstract: Some quantities in quantum field theory are dominated by so-called $\mathit{leading\,logs}$ and can be re-summed to all loop orders. In this work we introduce a notion of $\mathit{stampede}$ which is a simple time-evolution of a bunch of particles which start their life in a corner - on the very right say - and $\mathit{hop}$ their way to the opposite corner - on the left - through the repeated action of a quantum Hamiltonian. Such stampedes govern leading logs quantities in certain quantum field theories. The leading euclidean OPE limit of correlation functions in the fishnet theory and null double-scaling limits of correlators in $\mathcal{N}=4$ SYM are notable examples. As an application, we use these results to extend the beautiful bootstrap program of Coronado [1] to all octagons functions with arbitrary diagonal bridge length.
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TL;DR: In this paper , it was shown that all singular limits of the fishnet can be attained within the double scaling limit, including the null limit with the four points approaching the cusps of a null square.
Abstract: A bstract Basso-Dixon integrals evaluate rectangular fishnets — Feynman graphs with massless scalar propagators which form a m × n rectangular grid — which arise in certain one-trace four-point correlators in the ‘fishnet’ limit of $$ \mathcal{N} $$ N = 4 SYM. Recently, Basso et al. explored the thermodynamical limit m → ∞ with fixed aspect ratio n/m of a rectangular fishnet and showed that in general the dependence on the coordinates of the four operators is erased, but it reappears in a scaling limit with two of the operators getting close in a controlled way. In this note I investigate the most general double scaling limit which describes the thermodynamics when one of two pairs of operators become nearly light-like. In this double scaling limit, the rectangular fishnet depends on both coordinate cross ratios. I show that all singular limits of the fishnet can be attained within the double scaling limit, including the null limit with the four points approaching the cusps of a null square. A direct evaluation of the fishnet in the null limit is presented any m and n .
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TL;DR: In this article , a broad class of d-dimensional conformal field theories of SU(N ) adjoint scalar fields generalising the 4 d Fishnet CFT (FCFT) discovered by Ö. Gürdogan and one of the authors as a special limit of γ -deformed $$ \mathcal{N} $$ was proposed.
Abstract: A bstract We propose a broad class of d -dimensional conformal field theories of SU( N ) adjoint scalar fields generalising the 4 d Fishnet CFT (FCFT) discovered by Ö. Gürdogan and one of the authors as a special limit of γ -deformed $$ \mathcal{N} $$ N = 4 SYM theory. In the planar N → ∞ limit the FCFTs are dominated by the “fishnet” planar Feynman graphs. These graphs are explicitly integrable, as was shown long ago by A. Zamolodchikov. The Zamolodchikov’s construction, based on the dual Baxter lattice (straight lines on the plane intersecting at arbitrary slopes) and the star-triangle identities, can serve as a “loom” for “weaving” the Feynman graphs of these FCFTs, with certain types of propagators, at any d . The Baxter lattice with M different slopes and any number of lines parallel to those, generates an FCFT consisting of M ( M – 1) fields and a certain number of chiral vertices of different valences with distinguished couplings. These non-unitary, logarithmic CFTs enjoy certain reality properties for their spectrum due to a symmetry similar to the PT-invariance of non-hermitian hamiltonians proposed by C. Bender and S. Boettcher. We discuss in more detail the theories generated by a loom with M = 2, 3, 4, and the generalisation of the loom FCFTs for spinning fields in 4d.
TL;DR: In this article , the authors considered correlation functions of single trace operators approaching the cusps of null polygons in a double-scaling limit where so-called cusp times were held fixed and the Hooft coupling was small.
Abstract: We consider correlation functions of single trace operators approaching the cusps of null polygons in a double-scaling limit where so-called cusp times ${t}_{i}^{2}={g}^{2}\mathrm{log}{x}_{i\ensuremath{-}1,i}^{2}\mathrm{log}{x}_{i,i+1}^{2}$ are held fixed and the `t Hooft coupling is small. With the help of stampedes, symbols, and educated guesses, we find that any such correlator can be uniquely fixed through a set of coupled lattice PDEs of Toda type with several intriguing novel features. These results hold for most conformal gauge theories with a large number of colors, including planar $\mathcal{N}=4$ SYM.
TL;DR: In this paper , it was shown that for any single-trace operator in 4 SYM theory there is a large twist double-scaling limit in which the Feynman graphs have an iterative structure.
Abstract: A bstract We argue that for any single-trace operator in $$ \mathcal{N} $$ N = 4 SYM theory there is a large twist double-scaling limit in which the Feynman graphs have an iterative structure. Such structure can be recast using a graph-building operator. Generically, this operator mixes between single trace operators with different scaling limits. The mixing captures both the finite coupling spectrum and corrections away from the large twist limit. We first consider a class of short operators with gluons and fermions for which such mixing problems do not arise, and derive their finite coupling spectra. We then focus on a class of long operators with gluons that do mix. We invert their graph-building operator and prove its integrability. The picture that emerges from this work opens the door to a systematic expansion of $$ \mathcal{N} $$ N = 4 SYM theory around the large twist limit.
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04 May 1999
TL;DR: The harmonic polylogarithms (hpl's) as mentioned in this paper are a generalization of Nielsen's poly logarithm, satisfying a product algebra (the product of two hpl's is in turn a combination of hpls) and forming a set closed under the transformation of the arguments x=1/z and x=(1-t)/(1+t)
Abstract: The harmonic polylogarithms (hpl's) are introduced They are a generalization of Nielsen's polylogarithms, satisfying a product algebra (the product of two hpl's is in turn a combination of hpl's) and forming a set closed under the transformation of the arguments x=1/z and x=(1-t)/(1+t) The coefficients of their expansions and their Mellin transforms are harmonic sums
1,100 citations
TL;DR: In this article, the psu(2,2|4) dilatation operator of N = 4 Super YangMills theory is presented, which generates the matrix of one-loop anomalous dimensions for all local operators in the theory.
634 citations
TL;DR: In this article, each irreducible representation of the symmetric group S n may be identified by a partition [λ] of n into non-negative integral parts λ 1 ≥ λ 2 ≥ … λ n ≥ 0, of which the first λ'j parts are ≥j.
Abstract: Each irreducible representation [λ] of the symmetric group S n may be identified by a partition [λ] of n into non-negative integral parts λ1 ≥ λ2 ≥ … λ n ≥ 0, of which the first λ'j parts are ≥j, or by a right (Young) diagram also called [λ], that contains λi nodes in its ith row and λ'j
nodes in its jth column.
488 citations
TL;DR: The first result is that the correlation functions of the Schur process are determinants with a kernel that has a nice contour integral representation in terms of the parameters of the process.
Abstract: Schur process is a time-dependent analog of the Schur measure on partitions studied in math.RT/9907127. Our first result is that the correlation functions of the Schur process are determinants with a kernel that has a nice contour integral representation in terms of the parameters of the process. This general result is then applied to a particular specialization of the Schur process, namely to random 3-dimensional Young diagrams. The local geometry of a large random 3-dimensional diagram is described in terms of a determinantal point process on a 2-dimensional lattice with the incomplete beta function kernel (which generalizes the discrete sine kernel). A brief discussion of the universality of this answer concludes the paper.
452 citations
TL;DR: In this paper, the authors investigated some measures induced by families of non-intersecting paths in domino tilings of the Aztec diamond, rhombus tilings, a dimer model on a cylindrical brick lattice and a growth model.
Abstract: We investigate certain measures induced by families of non-intersecting paths in domino tilings of the Aztec diamond, rhombus tilings of an abc-hexagon, a dimer model on a cylindrical brick lattice and a growth model. The measures obtained, e.g. the Krawtchouk and Hahn ensembles, have the same structure as the eigenvalue measures in random matrix theory like GUE, which can in fact can be obtained from non-intersecting Brownian motions. The derivations of the measures are based on the Karlin-McGregor or Lindstrom-Gessel-Viennot method. We use the measures to show some asymptotic results for the models.
270 citations