1,882 research outputs found
Uncolored Random Tensors, Melon Diagrams, and the SYK Models
Certain models with rank- tensor degrees of freedom have been shown by
Gurau and collaborators to possess a novel large limit, where is
held fixed. In this limit the perturbative expansion in the quartic coupling
constant, , is dominated by a special class of "melon" diagrams. We study
"uncolored" models of this type, which contain a single copy of real rank-
tensor. Its three indexes are distinguishable; therefore, the models possess
symmetry with the tensor field transforming in the tri-fundamental
representation. Such uncolored models also possess the large limit
dominated by the melon diagrams. The quantum mechanics of a real anti-commuting
tensor therefore has a similar large limit to the model recently introduced
by Witten as an implementation of the Sachdev-Ye-Kitaev (SYK) model which does
not require disorder. Gauging the symmetry in our quantum mechanical
model removes the non-singlet states; therefore, one can search for its
well-defined gravity dual. We point out, however, that the model possesses a
vast number of gauge-invariant operators involving higher powers of the tensor
field, suggesting that the complete gravity dual will be intricate. We also
discuss the quantum mechanics of a complex 3-index anti-commuting tensor, which
has symmetry and argue that it is equivalent in the large
limit to a version of SYK model with complex fermions. Finally, we discuss
similar models of a commuting tensor in dimension . While the quartic
interaction is not positive definite, we construct the large
Schwinger-Dyson equation for the two-point function and show that its solution
is consistent with conformal invariance. We carry out a perturbative check of
this result using the expansion.Comment: 26 pages, 16 figures, v2: sections 3 and 5 expanded, minor
corrections, references added, v3: minor corrections, a reference added, v4:
minor corrections, v5: spectrum of the complex model corrected; a note added
about "uncolored" higher rank tensor
On Large Limit of Symmetric Traceless Tensor Models
For some theories where the degrees of freedom are tensors of rank or
higher, there exist solvable large limits dominated by the melonic
diagrams. Simple examples are provided by models containing one rank- tensor
in the tri-fundamental representation of the symmetry group. When the
quartic interaction is assumed to have a special tetrahedral index structure,
the coupling constant must be scaled as in the melonic large
limit. In this paper we consider the combinatorics of a large theory of one
fully symmetric and traceless rank- tensor with the tetrahedral quartic
interaction; this model has a single symmetry group. We explicitly
calculate all the vacuum diagrams up to order , as well as some diagrams
of higher order, and find that in the large limit where is held
fixed only the melonic diagrams survive. While some non-melonic diagrams are
enhanced in the symmetric theory compared to the one, we have
not found any diagrams where this enhancement is strong enough to make them
comparable with the melonic ones. Motivated by these results, we conjecture
that the model of a real rank- symmetric traceless tensor possesses a smooth
large limit where is held fixed and all the contributing diagrams
are melonic. A feature of the symmetric traceless tensor models is that some
vacuum diagrams containing odd numbers of vertices are suppressed only by
relative to the melonic graphs.Comment: 18 pages, 12 figures; v2: minor improvements, references adde
Bosonic Tensor Models at Large and Small
We study the spectrum of the large quantum field theory of bosonic
rank- tensors, whose quartic interactions are such that the perturbative
expansion is dominated by the melonic diagrams. We use the Schwinger-Dyson
equations to determine the scaling dimensions of the bilinear operators of
arbitrary spin. Using the fact that the theory is renormalizable in , we
compare some of these results with the expansion, finding perfect
agreement. This helps elucidate why the dimension of operator
is complex for : the large fixed point in
has complex values of the couplings for some of the
invariant operators. We show that a similar phenomenon holds in the
symmetric theory of a matrix field , where the double-trace operator
has a complex coupling in dimensions. We also study the spectra of
bosonic theories of rank tensors with interactions. In
dimensions there is a critical value of , above which we have not
found any complex scaling dimensions. The critical value is a decreasing
function of , and it becomes in . This raises a
possibility that the large theory of rank- tensors with sextic potential
has an IR fixed point which is free of perturbative instabilities for
. This theory may be studied using renormalized perturbation theory
in .Comment: 20 pages, 3 figures, v2: minor corrections, references adde
On and in Conformal QED
QED with a large number of massless fermionic degrees of freedom has a
conformal phase in a range of space-time dimensions. We use a large
diagrammatic approach to calculate the leading corrections to , the
coefficient of the two-point function of the stress-energy tensor, and ,
the coefficient of the two-point function of the global symmetry current. We
present explicit formulae as a function of and check them versus the
expectations in 2 and dimensions. Using our results in higher even
dimensions we find a concise formula for of the conformal Maxwell theory
with higher derivative action . In , QED has a topological symmetry current, and we calculate the
correction to its two-point function coefficient, . We
also show that some RG flows involving QED in obey and discuss possible implications of this inequality for the
symmetry breaking at small values of .Comment: 29 pages, 9 figures. v3: minor improvements, references adde
- …