76 research outputs found
Tessellating cushions: four-point functions in N=4 SYM
We consider a class of planar tree-level four-point functions in N=4 SYM in a
special kinematic regime: one BMN operator with two scalar excitations and
three half-BPS operators are put onto a line in configuration space;
additionally, for the half-BPS operators a co-moving frame is chosen in flavour
space. In configuration space, the four-punctured sphere is naturally
triangulated by tree-level planar diagrams. We demonstrate on a number of
examples that each tile can be associated with a modified hexagon form-factor
in such a way as to efficiently reproduce the tree-level four-point function.
Our tessellation is not of the OPE type, fostering the hope of finding an
independent, integrability-based approach to the computation of planar
four-point functions.Comment: 10 pages, 2 figure
Integrable S matrix, mirror TBA and spectrum for the stringy WZW model
We compute the tree-level bosonic S matrix in light-cone gauge for
superstrings on pure-NSNS
. We show that
it is proportional to the identity and that it takes the same form as for
and for flat space. Based on
this, we make a conjecture for the exact worldsheet S matrix and derive the
mirror thermodynamic Bethe ansatz (TBA) equations describing the spectrum.
Despite a non-trivial vacuum energy, they can be solved in closed form and
coincide with a simple set of Bethe ansatz equations - again much like
and flat space. This suggests
that the model may have an integrable spin-chain interpretation. Finally, as a
check of our proposal, we compute the spectrum from the worldsheet CFT in the
case of highest-weight representations of the underlying Ka\v{c}-Moody
algebras, and show that the mirror-TBA prediction matches it on the nose.Comment: 38 pages, Version accepted for publication in JHE
Three-point functions in SYM: the hexagon proposal at three loops
Basso, Komatsu and Vieira recently proposed an all-loop framework for the
computation of three-point functions of single-trace operators of
super-Yang-Mills, the "hexagon program". This proposal results in several
remarkable predictions, including the three-point function of two protected
operators with an unprotected one in the and sectors. Such
predictions consist of an "asymptotic" part---similar in spirit to the
asymptotic Bethe Ansatz of Beisert and Staudacher for two-point functions---as
well as additional finite-size "wrapping" L\"uscher-like corrections. The focus
of this paper is on such wrapping corrections, which we compute at three-loops
in the sector. The resulting structure constants perfectly match the
ones obtained in the literature from four-point correlators of protected
operators.Comment: 18 pages, 3 tables; v2: note added, ref. added, (some) misprints
corrected; v3: more ref. added, more misprints correcte
Long Strings and Symmetric Product Orbifold from the AdS Bethe Equations
A particularly rich class of integrable systems arises from the AdS/CFT
duality. There, the two-dimensional quantum field theory living on the string
worldsheet may be understood in terms of a non-relativistic factorized S
matrix, and the energy spectrum may be derived by techniques such as the mirror
thermodynamic Bethe ansatz or the quantum spectral curve. In the case of
AdS/CFT without Ramond-Ramond fluxes, the worldhseet theory is a
Wess-Zumino-Witten model with continous and discrete representations which, for
the lowest allowed level, is dual to the symmetric product orbifold of a free
theory. I will show how continuous representations may arise from
integrability, and that at lowest level the Bethe equations yield the symmetric
product orbifold partition function on the nose.Comment: 6 page
Towards integrability for AdS3/CFT2
We review the recent progress towards applying worldsheet integrability
techniques to the correspondence to find its all-loop S matrix
and Bethe-Yang equations. We study in full detail the massive sector of
superstrings supported by pure Ramond-Ramond (RR)
fluxes. The extension of this machinery to accommodate massless modes, to the
pure-RR background and to backgrounds
supported by mixed background fluxes is also reviewed. While the results
discussed here were found elsewhere, our presentation sometimes deviates from
the one found in the original literature in an effort to be pedagogical and
self-contained.Comment: Review, 152 pages, 29 figures; v2: minor changes, references added;
v3: more minor changes, more references; v4: misprints corrected, references
updated, as publishe
Renormalization: an advanced overview
We present several approaches to renormalization in QFT: the multi-scale
analysis in perturbative renormalization, the functional methods \`a la
Wetterich equation, and the loop-vertex expansion in non-perturbative
renormalization. While each of these is quite well-established, they go beyond
standard QFT textbook material, and may be little-known to specialists of each
other approach. This review is aimed at bridging this gap.Comment: Review, 130 pages, 33 figures; v2: misprints corrected, refs. added,
minor improvements; v3: some changes to sect. 5, refs. adde
An introduction to universality and renormalization group techniques
These lecture notes have been written for a short introductory course on
universality and renormalization group techniques given at the VIII Modave
School in Mathematical Physics by the author, intended for PhD students and
researchers new to these topics.
First the basic ideas of dynamical systems (fixed points, stability, etc.)
are recalled, and an example of universality is discussed in this context: this
is Feigenbaum's universality of the period doubling cascade for iterated maps
on the interval. It is shown how renormalization ideas can be applied to
explain universality and compute Feigenbaum's constants.
Then, universality is presented in the scenario of quantum field theories,
and studied by means of functional renormalization group equations, which allow
for a close comparison with the case of dynamical systems. In particular,
Wetterich equation for a scalar field is derived and discussed, and then
applied to the computation of the Wilson-Fisher fixed point and critical
exponent for the Ising universality class.
References to more advanced topics and applications are provided.Comment: 64 pages, 18 figures, to be published in the proceedings of the VIII
Modave School in Mathematical Physics; v2: minor changes; v3: Proceedings of
Science versio
A dynamic su(1|1)^2 S-matrix for AdS3/CFT2
We derive the S-matrix for the d(2,1;alpha)^2 symmetric spin-chain of
AdS3/CFT2 by considering the centrally extended su(1|1)^2 algebra acting on the
spin-chain excitations. The S-matrix is determined uniquely up to four scalar
factors, which are further constrained by a set of crossing relations. The
resulting scattering includes non-trivial processes between magnons of
different masses that were previously overlooked.Comment: 41 pages, 4 figures. v2: corrected a misprint in appendix E, updated
references, corrected some typos. v3: added a new appendix F with comparison
to the literature, changed notation for the crossing equations, added
references. Published versio
All-loop Bethe ansatz equations for AdS3/CFT2
Using the S-matrix for the d(2,1;alpha)^2 symmetric spin-chain of AdS3/CFT2,
we propose a new set of all-loop Bethe equations for the system. These
equations differ from the ones previously found in the literature by the choice
of relative grading between the two copies of the d(2,1;alpha) superalgebra,
and involve four undetermined scalar factors that play the role of dressing
phases. Imposing crossing symmetry and comparing with the near-BMN form of the
S-matrix found in the literature, we find several novel features. In
particular, the scalar factors must differ from the Beisert-Eden-Staudacher
phase, and should couple nodes of different masses to each other. In the
semiclassical limit the phases are given by a suitable generalization of
Arutyunov-Frolov-Staudacher phase.Comment: 26 pages, 2 figures. v2: references added. v3: changed notation for
the crossing equations, added references. Published versio
deformations with supersymmetry
We investigate the behaviour of two-dimensional quantum field theories with
supersymmetry under a deformation induced by the
`' composite operator. We show that the deforming operator can be
defined by a point-splitting regularisation in such a way as to preserve
supersymmetry. As an example of this construction, we work
out the deformation of a free theory and compare to that
induced by the Noether stress-energy tensor. Finally, we show that the
supersymmetric deformed action actually possesses
symmetry, half of which is non-linearly realised
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