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The Higher Spin AdS/CFT correspondence and Mathieu Moonshine

English title The Higher Spin AdS/CFT correspondence and Mathieu Moonshine
Applicant Gaberdiel Matthias
Number 146186
Funding scheme Project funding (Div. I-III)
Research institution Institut für Theoretische Physik ETH Zürich
Institution of higher education ETH Zurich - ETHZ
Main discipline Theoretical Physics
Start/End 01.04.2013 - 31.03.2016
Approved amount 525'563.00
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All Disciplines (2)

Discipline
Theoretical Physics
Mathematics

Keywords (3)

Higher Spin Theory; String Theory; Conformal Field Theory

Lay Summary (German)

Lead
Stringtheorie ist gegenwaertige der erfolgversprechendste Kandidatfuer eine vereinheitlichende Theorie aller bekanntenWechselwirkungen. Sie hat ueber die Jahre ausserdem zu vielenAnregungen und Ideen in theoretischer Physik und Mathematikgefuehrt. In der juengeren Vergangenheit sind viele dieserEntwicklungen durch die AdS/CFT Korrespondenz inspiriert worden.
Lay summary

Die AdS/CFT Korrespondenz ist eine der bemerkenswertesten Entwicklungen der theoretischen Physik der letzten Jahre. Sie stellt eine nicht-triviale Beziehung zwischen einer Gravitationstheorie (einer Verallgemeinerung der Einstein'schen Allgemeinen Relativitaetstheorie) und einer Quantenfeldtheorie (so wie sie zum Beispiel im Standardmodell der Teilchenphysik auftritt) her. Obgleich in den letzten Jahren viel ueberzeugende Evidenz fuer diese Beziehung zwischen den beiden Theorien gefunden worden ist, fehlt bislang ein tiefes Verstaendnis der zu Grunde liegenden Mechanismen. Durch die Analyse eines vereinfachten Modells, bei der auf der Gravitationsseite eine sogenannte hoehere Spintheorie betrachtet wird, soll dieser Frage auf den Grund gegangen werden.

Vor ein paar Jahren wurde eine bemerkenswerte Relation zwischen einer Funktion, die natuerlicherweise in der Stringtheorie auftritt (dem sogenannten elliptischen Geschlecht auf K3), und der Mathieu Gruppe M24, einer der sporadischen endlichen Gruppen, gefunden. Diese Relation ist inzwischen mathematisch bewiesen worden; allerdings fehlt weiterhin ein konzeptionelles Verstaendnis dafuer, warum genau M24 in dem elliptischen Geschlecht von K3 auftauchen sollte. Das zweite Ziel dieses Projektes ist es, den mathematischen oder physikalischen Grund fuer diese Beobachtung zu verstehen.

Direct link to Lay Summary Last update: 27.03.2013

Lay Summary (English)

Lead
String theory is currently the most promising candidate for a unifying framework to describe all known interactions. It has also, over the years, given rise to many new ideas and concepts in theoretical physics and mathematics. In the recent past, many of these inspirations have come from the so-called AdS/CFT correspondence. Another intriguing new development concerns Mathieu Moonshine, i.e. the emergence of the largest Mathieu group in string theory.
Lay summary

The AdS/CFT correspondence represents one of the most intriguing recent developments in theoretical physics. It establishes a highly non-trivial relation between gravitational theories such as Einstein's General Relativity on the one hand, and quantum field theories similar to those that appear in the standard model of particle physics on the other. While there is very strong evidence for various aspects of the AdS/CFT correspondence, and certain incarnations of it can be derived from string theory, a deep understanding of the inner workings of the
duality is still missing. By analysing a simpler (lower-dimensional) toy model involving a higher spin theory (rather than a string theory) on AdS3 the project aims to uncover some of these underlying mechanisms. 

A few years ago, an intriguing relation between a certain function that appears naturally in string theory (the elliptic genus on K3), and the Mathieu group M24, one of the sporadic finite groups, was noted. While there is by now overwhelming evidence for this connection, the underlying reason for it remains mysterious. The other aim of the project is to understand the physical or mathematical reason for this phenomenon. 


Direct link to Lay Summary Last update: 27.03.2013

Responsible applicant and co-applicants

Employees

Publications

Publication
String Theory as a Higher Spin Theory
Gaberdiel Matthias, Gopakumar Rajesh (2016), String Theory as a Higher Spin Theory, in JHEP, 1609, 085.
The symmetric orbifold of N=2 minimal models
Gaberdiel Matthias, Kelm Maximilian (2016), The symmetric orbifold of N=2 minimal models, in Journal of High Energy Physics, 1607, 113.
Higgsing the stringy higher spin symmetry
Gaberdiel Matthias, Peng Cheng, Zadeh Ida (2015), Higgsing the stringy higher spin symmetry, in JHEP, 1510, 101.
Higher spins in the symmetric orbifold of K3
Baggio Marco, Gaberdiel Matthias, Peng Cheng (2015), Higher spins in the symmetric orbifold of K3, in Phys. Rev. D, 92, 026007.
Stringy Symmetries and the Higher Spin Square
Gaberdiel Matthias, Gopakumar Rajesh (2015), Stringy Symmetries and the Higher Spin Square, in J.Phys.A: Math. Theor., 48, 185402.
The so-Kazama-Suzuki Models at Large Level
Ferreira Kevin, Gaberdiel Matthias (2015), The so-Kazama-Suzuki Models at Large Level, in JHEP, 1504, 017.
A K3 sigma model with Z_2^8:M_{20} symmetry
Gaberdiel Matthias, Taormina Anne, Volpato Roberto, Wendland Katrin (2014), A K3 sigma model with Z_2^8:M_{20} symmetry, in JHEP, 1402, 022.
Extended supersymmetry in AdS3 higher spin theories
Candu Constantin, Peng Cheng, Vollenweider Carl (2014), Extended supersymmetry in AdS3 higher spin theories, in JHEP, 1412, 113.
Higher Spins & Strings
Gaberdiel Matthias, Gopakumar Rajesh (2014), Higher Spins & Strings, in JHEP, 1411, 044.
On the coset duals of extended higher spin theories
Candu Constantin, Vollenweider Carl (2014), On the coset duals of extended higher spin theories, in JHEP, 1404, 145.
The Continuous Orbifold of N=2 Minimal Model Holography
Gaberdiel Matthias, Kelm Maximilian (2014), The Continuous Orbifold of N=2 Minimal Model Holography, in JHEP, 1408, 084.
The large N=4 superconformal W∞ algebra
Beccaria Matteo, Candu Constantin, Gaberdiel Matthias (2014), The large N=4 superconformal W∞ algebra, in JHEP, 1406, 117.
The Spectrum of Light States in Large N Minimal Models
Gaberdiel Matthias, Gopakumar Rajesh, Rangamani Mukund (2014), The Spectrum of Light States in Large N Minimal Models, in JHEP, 1401, 116.
The symmetry of large N=4 holography
Gaberdiel Matthias, Peng Cheng (2014), The symmetry of large N=4 holography, in JHEP, 1405, 152.
Large N = 4 holography
Gaberdiel Matthias, Gopakumar Rajesh (2013), Large N = 4 holography, in JHEP, 1309, 036.
N=1 extension of minimal model holography
Beccaria Matteo, Cand Constantin, Gaberdiel Matthias, Groher Michael (2013), N=1 extension of minimal model holography, in JHEP, 1307, 174.
Perturbations of W∞ CFTs
Gaberdiel Matthias, Li Wei, Jin Kewang (2013), Perturbations of W∞ CFTs, in JHEP, 1310, 162.
Probing higher spin black holes from CFT
Gaberdiel Matthias, Jin Kewang, Perlmutter Eric (2013), Probing higher spin black holes from CFT, in JHEP, 1310, 045.
The N=1 algebra W∞[μ] and its truncations
Candu Constantin, Vollenweider Carl (2013), The N=1 algebra W∞[μ] and its truncations, in JHEP, 1311, 032.

Collaboration

Group / person Country
Types of collaboration
Prof. Mukund Rangamani, Durham University Great Britain and Northern Ireland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Prof. Wei Li, ITP Beijing, Chinese Academy of Science China (Asia)
- in-depth/constructive exchanges on approaches, methods or results
Prof. Terry Gannon, Mathematics Department, University of Alberta Canada (North America)
- in-depth/constructive exchanges on approaches, methods or results
Prof. Rajesh Gopakumar, Harish-Chandra Research Institute India (Asia)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Prof. Matthias Blau, Universität Bern Switzerland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
Prof. Marcos Marino, University of Geneva Switzerland (Europe)
- in-depth/constructive exchanges on approaches, methods or results

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
LMS Symposium on ‘New Moonshines, Mock Modular Forms, and String Theory’ Talk given at a conference Symmetries of K3 sigma models 05.08.2015 Durham, Great Britain and Northern Ireland Gaberdiel Matthias;
The Mathematics of Conformal Field Theory Talk given at a conference Higher Spins & Strings 13.07.2015 Canberra, Australia Gaberdiel Matthias;
Strings '15 Talk given at a conference Strings from a Higher Spin Perspective 22.06.2015 Bangalore, India Gaberdiel Matthias;
Irish Quantum Foundations 2015 Talk given at a conference Higher Spins & Strings 28.05.2015 Dublin, Ireland Gaberdiel Matthias;
Eurostrings ’15 Talk given at a conference Higher Spins & Strings 24.03.2015 Cambridge, Great Britain and Northern Ireland Gaberdiel Matthias;
Higher Spin Symmetries and Conformal Bootstrap Talk given at a conference Higher Spins & Strings 06.11.2014 Princeton, United States of America Gaberdiel Matthias;
Quantum Symmetries and Strings Talk given at a conference Minimal Model Holography 16.09.2014 Corfu, Greece Gaberdiel Matthias;
String Math '14 Talk given at a conference Higher Spins & Strings 10.06.2014 Edmonton, Canada Gaberdiel Matthias;
Holography for Black Holes and Cosmology Talk given at a conference Higher Spins & Strings 16.04.2014 Brussels, Belgium Gaberdiel Matthias;
3rd Bangkok workshop on high-energy theory Talk given at a conference Minimal Model Holography 20.01.2014 Bangkok, Thailand Gaberdiel Matthias;
Annual Theory Meeting, IPPP Durham Talk given at a conference Higher Spin — Minimal Model Duality’ 16.12.2013 Durham, Great Britain and Northern Ireland Gaberdiel Matthias;
Higher-Spin and Higher-Curvature Gravity Talk given at a conference Minimal Model Holography 06.11.2013 Sao Paulo, Brazil Gaberdiel Matthias;
Integrability in Gauge and String Theory Talk given at a conference Large N=4 Holography 21.08.2013 Utrecht, Netherlands Gaberdiel Matthias;
Strings '13 Talk given at a conference Large N=4 Holography 25.06.2013 Seoul, Korean Republic (South Korea) Gaberdiel Matthias;


Self-organised

Title Date Place
All about AdS3 16.11.2015 ETH Zurich, Switzerland
(Mock) Modularity, Moonshine and String Theory 13.04.2015 Perimeter Institute, United States of America
Recent Developments in String Theory 21.07.2014 Monte Verita, Ascona, Switzerland
Mock Modular Forms, Moonshine & String Theory, 26.08.2013 Simons Center, Stony Brook, United States of America

Associated projects

Number Title Start Funding scheme
129542 The AdS/CFT correspondence, pure spinors and logarithmic conformal field theories 01.04.2010 Project funding (Div. I-III)
165667 Higher Spins & Strings and Mathieu Moonshine 01.04.2016 Project funding (Div. I-III)

Abstract

One of the most important topics in string theory in recent years has been the AdS/CFT correspondence. In its original form it relates string theory on a 5-dimensional Anti-de Sitter background to N=4 superconformal Young Mills theory in 4 dimensions. This duality is of great importance since it allows one to gain insight into quantum gravity on the one hand, and into strongly coupled (gauge) theories on the other. Unfortunately, however, a real understanding of why this correspondence works and which ingredients are crucial for it is still missing. This is an important question given that for many interesting applications of the AdS/CFT correspondence, say to QCD or condensed matter physics, other versions of the duality (that cannot be directly justified from string theory) are relevant. In order to gain insight into this problem it is instructive to consider the limit in which the conformal field theory is weakly coupled. In that case, the dual string theory is believed to possess a consistent higher spin subsector that should be dual to the nearly free field theory. This special case is of quite some interest since both sides of the duality are then weakly coupled, and hence one may hope to find a perturbative proof of the AdS/CFT correspondence in this regime. About two years ago I made (together with Rajesh Gopakumar) a concrete proposal for such a duality. It relates a specific higher spin theory on AdS3 (that is characterised by an infinite-dimensional Lie algebra that depends on a continuous parameter) to a 't Hooft like large N limit of a family of 2-dimensional minimal model conformal field theories, whose 't Hooft coupling constant is identified with the parameter appearing in the higher spin algebra. By now substantial support for this proposal has been obtained, but there are also many interesting open questions. In particular, it should be possible to prove the duality, and to understand how precisely it can be embedded into string theory. Furthermore, since the duality is also understood in detail at finite N, the 2d CFT makes concrete predictions for quantum gravity effects in AdS3, which it would be very interesting to understand from a perturbative quantum gravity point of view. Both of these directions (as well as others in this area) will be pursued.In 2010, Eguchi, Ooguri and Tachikawa observed that the first few Fourier coefficients of the elliptic genus of K3 can be written in terms of dimensions of Mathieu M24 representations. (The Mathieu group M24 is one of the sporadic finite simple groups.) In previous work I have given very convincing evidence that this observation is true for all Fourier coefficients, and that a similar structure also exists for the various twisted sectors. These results suggest that all of these observations can be explained in terms of an underlying VOA, as in the analogous case of Monstrous Moonshine. The other project is to construct this vertex operator algebra.
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