6,581 research outputs found
Strings from Feynman Graph counting : without large N
A well-known connection between n strings winding around a circle and
permutations of n objects plays a fundamental role in the string theory of
large N two dimensional Yang Mills theory and elsewhere in topological and
physical string theories. Basic questions in the enumeration of Feynman graphs
can be expressed elegantly in terms of permutation groups. We show that these
permutation techniques for Feynman graph enumeration, along with the Burnside
counting lemma, lead to equalities between counting problems of Feynman graphs
in scalar field theories and Quantum Electrodynamics with the counting of
amplitudes in a string theory with torus or cylinder target space. This string
theory arises in the large N expansion of two dimensional Yang Mills and is
closely related to lattice gauge theory with S_n gauge group. We collect and
extend results on generating functions for Feynman graph counting, which
connect directly with the string picture. We propose that the connection
between string combinatorics and permutations has implications for QFT-string
dualities, beyond the framework of large N gauge theory.Comment: 55 pages + 10 pages Appendices, 23 figures ; version 2 - typos
correcte
Feynman Integrals and Intersection Theory
We introduce the tools of intersection theory to the study of Feynman
integrals, which allows for a new way of projecting integrals onto a basis. In
order to illustrate this technique, we consider the Baikov representation of
maximal cuts in arbitrary space-time dimension. We introduce a minimal basis of
differential forms with logarithmic singularities on the boundaries of the
corresponding integration cycles. We give an algorithm for computing a basis
decomposition of an arbitrary maximal cut using so-called intersection numbers
and describe two alternative ways of computing them. Furthermore, we show how
to obtain Pfaffian systems of differential equations for the basis integrals
using the same technique. All the steps are illustrated on the example of a
two-loop non-planar triangle diagram with a massive loop.Comment: 13 pages, published versio
P versus NP and geometry
I describe three geometric approaches to resolving variants of P v. NP,
present several results that illustrate the role of group actions in complexity
theory, and make a first step towards completely geometric definitions of
complexity classes.Comment: 20 pages, to appear in special issue of J. Symbolic. Comp. dedicated
to MEGA 200
Deformation of string topology into homotopy skein modules
Relations between the string topology of Chas and Sullivan and the homotopy
skein modules of Hoste and Przytycki are studied. This provides new insight
into the structure of homotopy skein modules and their meaning in the framework
of quantum topology. Our results can be considered as weak extensions to all
orientable 3-manifolds of classical results by Turaev and Goldman concerning
intersection and skein theory on oriented surfaces.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-34.abs.htm
Still Wrong Use of Pairings in Cryptography
Several pairing-based cryptographic protocols are recently proposed with a
wide variety of new novel applications including the ones in emerging
technologies like cloud computing, internet of things (IoT), e-health systems
and wearable technologies. There have been however a wide range of incorrect
use of these primitives. The paper of Galbraith, Paterson, and Smart (2006)
pointed out most of the issues related to the incorrect use of pairing-based
cryptography. However, we noticed that some recently proposed applications
still do not use these primitives correctly. This leads to unrealizable,
insecure or too inefficient designs of pairing-based protocols. We observed
that one reason is not being aware of the recent advancements on solving the
discrete logarithm problems in some groups. The main purpose of this article is
to give an understandable, informative, and the most up-to-date criteria for
the correct use of pairing-based cryptography. We thereby deliberately avoid
most of the technical details and rather give special emphasis on the
importance of the correct use of bilinear maps by realizing secure
cryptographic protocols. We list a collection of some recent papers having
wrong security assumptions or realizability/efficiency issues. Finally, we give
a compact and an up-to-date recipe of the correct use of pairings.Comment: 25 page
Stochastic aspects of easy quantum groups
We consider several orthogonal quantum groups satisfying the easiness
assumption axiomatized in our previous paper. For each of them we discuss the
computation of the asymptotic law of Tr(u^k) with respect to the Haar measure,
u being the fundamental representation. For the classical groups O_n, S_n we
recover in this way some well-known results of Diaconis and Shahshahani.Comment: 28 page
A & B model approaches to surface operators and Toda theories
It has recently been argued by Alday et al that the inclusion of surface
operators in 4d N=2 SU(2) quiver gauge theories should correspond to insertions
of certain degenerate operators in the dual Liouville theory. So far only the
insertion of a single surface operator has been treated (in a semi-classical
limit). In this paper we study and generalise this proposal. Our approach
relies on the use of topological string theory techniques. On the B-model side
we show that the effects of multiple surface operator insertions in 4d N=2
gauge theories can be calculated using the B-model topological recursion
method, valid beyond the semi-classical limit. On the mirror A-model side we
find by explicit computations that the 5d lift of the SU(N) gauge theory
partition function in the presence of (one or many) surface operators is equal
to an A-model topological string partition function with the insertion of (one
or many) toric branes. This is in agreement with an earlier proposal by Gukov.
Our A-model results were motivated by and agree with what one obtains by
combining the AGT conjecture with the dual interpretation in terms of
degenerate operators. The topological string theory approach also opens up new
possibilities in the study of 2d Toda field theories.Comment: 43 pages. v2: Added references, including a reference to unpublished
work by S.Gukov; minor changes and clarifications
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