14 research outputs found
O(N) and O(N) and O(N)
Three related analyses of theory with symmetry are presented.
In the first, we review the model over the -adic numbers and the
discrete renormalization group transformations which can be understood as spin
blocking in an ultrametric context. We demonstrate the existence of a
Wilson-Fisher fixed point using an expansion, and we show how to
obtain leading order results for the anomalous dimensions of low dimension
operators near the fixed point. Along the way, we note an important aspect of
ultrametric field theories, which is a non-renormalization theorem for kinetic
terms. In the second analysis, we employ large methods to establish
formulas for anomalous dimensions which are valid equally for field theories
over the -adic numbers and field theories on . Results for
anomalous dimensions agree between the first and second analyses when they can
be meaningfully compared. In the third analysis, we consider higher derivative
versions of the model on , the simplest of which has been
studied in connection with spatially modulated phases. Our general formula for
anomalous dimensions can still be applied. Analogies with two-derivative
theories hint at the existence of some interesting unconventional field
theories in four real Euclidean dimensions.Comment: 44 pages, 8 figure
Higher melonic theories
We classify a large set of melonic theories with arbitrary -fold
interactions, demonstrating that the interaction vertices exhibit a range of
symmetries, always of the form for some , which may be .
The number of different theories proliferates quickly as increases above
and is related to the problem of counting one-factorizations of complete
graphs. The symmetries of the interaction vertex lead to an effective
interaction strength that enters into the Schwinger-Dyson equation for the
two-point function as well as the kernel used for constructing higher-point
functions.Comment: 43 pages, 12 figure
Edge length dynamics on graphs with applications to -adic AdS/CFT
We formulate a Euclidean theory of edge length dynamics based on a notion of
Ricci curvature on graphs with variable edge lengths. In order to write an
explicit form for the discrete analog of the Einstein-Hilbert action, we
require that the graph should either be a tree or that all its cycles should be
sufficiently long. The infinite regular tree with all edge lengths equal is an
example of a graph with constant negative curvature, providing a connection
with -adic AdS/CFT, where such a tree takes the place of anti-de Sitter
space. We compute simple correlators of the operator holographically dual to
edge length fluctuations. This operator has dimension equal to the dimension of
the boundary, and it has some features in common with the stress tensor.Comment: 42 pages, 6 figure
Melonic theories over diverse number systems
Melonic field theories are defined over the p-adic numbers with the help of a sign character. Our construction works over the reals as well as the p-adics, and it includes the fermionic and bosonic Klebanov-Tarnopolsky models as special cases; depending on the sign character, the symmetry group of the field theory can be either orthogonal or symplectic. Analysis of the Schwinger-Dyson equation for the two-point function in the leading melonic limit shows that power law scaling behavior in the infrared arises for fermionic theories when the sign character is non-trivial, and for bosonic theories when the sign character is trivial. In certain cases, the Schwinger-Dyson equation can be solved exactly using a quartic polynomial equation, and the solution interpolates between the ultraviolet scaling controlled by the spectral parameter and the universal infrared scaling. As a by-product of our analysis, we see that melonic field theories defined over the real numbers can be modified by replacing the time derivative by a bilocal kinetic term with a continuously variable spectral parameter. The infrared scaling of the resulting two-point function is universal, independent of the spectral parameter of the ultraviolet theory