Pulling Yourself up by Your Bootstraps in Quantum Field Theory
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Conformal Bootstrap Signatures of the Tricritical Ising Universality Class
PHYSICAL REVIEW D 101, 116020 (2020) Conformal bootstrap signatures of the tricritical Ising universality class Chethan N. Gowdigere , Jagannath Santara, and Sumedha School of Physical Sciences, National Institute of Science Education and Research, Jatni 752050, India and Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India (Received 28 November 2018; revised manuscript received 17 April 2020; accepted 12 June 2020; published 30 June 2020) We study the tricritical Ising universality class using conformal bootstrap techniques. By studying bootstrap constraints originating from multiple correlators on the conformal field theory (CFT) data of multiple operator product expansions (OPEs), we are able to determine the scaling dimension of the spin field Δσ in various noninteger dimensions 2 ≤ d ≤ 3. Here, Δσ is connected to the critical exponent η that governs the (tri)critical behavior of the two-point function via the relation η ¼ 2 − d þ 2Δσ. Our results for Δσ match with the exactly known values in two and three dimensions and are a conjecture for noninteger 3 dimensions. We also compare our CFT results for Δσ with ϵ-expansion results, available up to ϵ order. Our techniques can be naturally extended to study higher-order multicritical points. DOI: 10.1103/PhysRevD.101.116020 I. INTRODUCTION the restrictions imposed by conformal invariance and found that the conformal field theory corresponding to The consequences of the conformal hypothesis which the Ising universality class sits on the boundary of the posits conformal invariance to the behavior of physical allowed region at a kinklike point in the space of scaling systems at criticality, in addition to scale invariance, are dimensions of the only two relevant operators. -
Modular Invariance of Characters of Vertex Operator Algebras
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Number 1, January 1996 MODULAR INVARIANCE OF CHARACTERS OF VERTEX OPERATOR ALGEBRAS YONGCHANG ZHU Introduction In contrast with the finite dimensional case, one of the distinguished features in the theory of infinite dimensional Lie algebras is the modular invariance of the characters of certain representations. It is known [Fr], [KP] that for a given affine Lie algebra, the linear space spanned by the characters of the integrable highest weight modules with a fixed level is invariant under the usual action of the modular group SL2(Z). The similar result for the minimal series of the Virasoro algebra is observed in [Ca] and [IZ]. In both cases one uses the explicit character formulas to prove the modular invariance. The character formula for the affine Lie algebra is computed in [K], and the character formula for the Virasoro algebra is essentially contained in [FF]; see [R] for an explicit computation. This mysterious connection between the infinite dimensional Lie algebras and the modular group can be explained by the two dimensional conformal field theory. The highest weight modules of affine Lie algebras and the Virasoro algebra give rise to conformal field theories. In particular, the conformal field theories associated to the integrable highest modules and minimal series are rational. The characters of these modules are understood to be the holomorphic parts of the partition functions on the torus for the corresponding conformal field theories. From this point of view, the role of the modular group SL2(Z)ismanifest. In the study of conformal field theory, physicists arrived at the notion of chi- ral algebras (see e.g. -
Solving the 3D Ising Model with the Conformal Bootstrap Sheer El-Showk, Miguel Paulos, David Poland, Slava Rychkov, David Simmons-Duffin, Alessandro Vichi
Solving the 3D Ising model with the conformal bootstrap Sheer El-Showk, Miguel Paulos, David Poland, Slava Rychkov, David Simmons-Duffin, Alessandro Vichi To cite this version: Sheer El-Showk, Miguel Paulos, David Poland, Slava Rychkov, David Simmons-Duffin, et al.. Solving the 3D Ising model with the conformal bootstrap. Physical Review D, American Physical Society, 2012, 86 (2), pp.25022. 10.1103/PhysRevD.86.025022. cea-00825766 HAL Id: cea-00825766 https://hal-cea.archives-ouvertes.fr/cea-00825766 Submitted on 17 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. This is the accepted manuscript made available via CHORUS. The article has been published as: Solving the 3D Ising model with the conformal bootstrap Sheer El-Showk, Miguel F. Paulos, David Poland, Slava Rychkov, David Simmons-Duffin, and Alessandro Vichi Phys. Rev. D 86, 025022 — Published 20 July 2012 DOI: 10.1103/PhysRevD.86.025022 LPTENS{12/07 Solving the 3D Ising Model with the Conformal Bootstrap Sheer El-Showka, Miguel F. Paulosb, David Polandc, Slava Rychkovd, David Simmons-Duffine, -
3D Ising Model and Conformal Bootstrap
3D Ising Model and Conformal Bootstrap Slava Rychkov (ENS Paris & CERN) CERN Winter School on Supergravity, Strings, and Gauge Theory 2013 see also “Lectures on CFT in D≥3” @ sites.google.com/site/slavarychkov Part 1 Conformal symmetry (Physical foundations & Basics, Ising model as an example) 2 /60 The subject of these lectures is: The simplest - experimentally relevant - unsolved Conformal Field Theory is 3D Ising Model @ T=Tc It’s also an ideal playground to explain the technique of conformal bootstrap... 3 /60 Basics on the Ising Model cubic lattice → Paradigmatic model of ferromagnetism M- spont. magnetization Tc T Critical temperature (Curie point) 4 /60 Correlation length Critical point can also be detected by looking at the spin-spin correlations For T>Tc : correlation length At T=Tc: Critical theory is scale invariant: It is also conformally invariant [conjectured by Polyakov’71] 5 /60 2D Ising Model • free energy solved by Onsager’44 on the lattice and for any T • Polyakov noticed that is conf. inv. at T=Tc • In 1983 Belavin-Polyakov-Zamolodchikov identified the critical 2D Ising model with the first unitary minimal model 6 /60 3D Ising Model • Lattice model at generic T is probably not solvable [many people tried] • Critical theory (T=Tc) in the continuum limit might be solvable [few people tried, conformal invariance poorly used] 7 /60 Existing approaches to 3D Ising • Lattice Monte-Carlo • High-T expansion on the lattice [~strong coupling expansion] Expand exponential in Converges for T>> Tc , extrapolate for T→ Tc by Pade -
From String Theory and Moonshine to Vertex Algebras
Preample From string theory and Moonshine to vertex algebras Bong H. Lian Department of Mathematics Brandeis University [email protected] Harvard University, May 22, 2020 Dedicated to the memory of John Horton Conway December 26, 1937 – April 11, 2020. Preample Acknowledgements: Speaker’s collaborators on the theory of vertex algebras: Andy Linshaw (Denver University) Bailin Song (University of Science and Technology of China) Gregg Zuckerman (Yale University) For their helpful input to this lecture, special thanks to An Huang (Brandeis University) Tsung-Ju Lee (Harvard CMSA) Andy Linshaw (Denver University) Preample Disclaimers: This lecture includes a brief survey of the period prior to and soon after the creation of the theory of vertex algebras, and makes no claim of completeness – the survey is intended to highlight developments that reflect the speaker’s own views (and biases) about the subject. As a short survey of early history, it will inevitably miss many of the more recent important or even towering results. Egs. geometric Langlands, braided tensor categories, conformal nets, applications to mirror symmetry, deformations of VAs, .... Emphases are placed on the mutually beneficial cross-influences between physics and vertex algebras in their concurrent early developments, and the lecture is aimed for a general audience. Preample Outline 1 Early History 1970s – 90s: two parallel universes 2 A fruitful perspective: vertex algebras as higher commutative algebras 3 Classification: cousins of the Moonshine VOA 4 Speculations The String Theory Universe 1968: Veneziano proposed a model (using the Euler beta function) to explain the ‘st-channel crossing’ symmetry in 4-meson scattering, and the Regge trajectory (an angular momentum vs binding energy plot for the Coulumb potential). -
Feynman Diagrams and the Large Charge Expansion in 3−Ε Dimensions
Physics Letters B 802 (2020) 135202 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Feynman diagrams and the large charge expansion in 3 − ε dimensions ∗ ∗ ∗ ∗ Gil Badel a, , Gabriel Cuomo a, , Alexander Monin a,b, , Riccardo Rattazzi a, a Theoretical Particle Physics Laboratory (LPTP), Institute of Physics, EPFL, Lausanne, Switzerland b Department of Theoretical Physics, University of Geneva, 24 quai Ernest-Ansermet, 1211 Geneva, Switzerland a r t i c l e i n f o a b s t r a c t Article history: In arXiv:1909 .01269 it was shown that the scaling dimension of the lightest charge n operator in the Received 12 December 2019 U (1) model at the Wilson-Fisher fixed point in D = 4 − ε can be computed semiclassically for arbitrary Accepted 6 January 2020 values of λn, where λ is the perturbatively small fixed point coupling. Here we generalize this result to Available online 10 January 2020 the fixed point of the U (1) model in 3 − ε dimensions. The result interpolates continuously between Editor: N. Lambert diagrammatic calculations and the universal conformal superfluid regime for CFTs at large charge. In 0 particular it reproduces the expectedly universal O(n ) contribution to the scaling dimension of large charge operators in 3D CFTs. © 2020 Published by Elsevier B.V. This is an open access article under the CC BY license 3 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . 1. Introduction EFT construction, or, in fact, deriving it. This was recently illus- trated in [13], by focusing on the two-point function of the charge n It is known that, even in a weakly coupled Quantum Field n operator φ in the U (1) invariant Wilson-Fisher fixed point in Theory (QFT), there exist situations where the ordinary Feynman 4 − ε dimensions. -
COSETS of AFFINE VERTEX ALGEBRAS INSIDE LARGER STRUCTURES Vertex Algebra Are a Fundamental Class of Algebraic Structures That Ar
COSETS OF AFFINE VERTEX ALGEBRAS INSIDE LARGER STRUCTURES THOMAS CREUTZIG AND ANDREW R. LINSHAW ABSTRACT. Given a finite-dimensional reductive Lie algebra g equipped with a nondegen- erate, invariant, symmetric bilinear form B, let Vk(g;B) denote the universal affine vertex algebra associated to g and B at level k. Let Ak be a vertex (super)algebra admitting a homomorphism Vk(g;B) !Ak. Under some technical conditions on Ak, we characterize the commutant Ck = Com(Vk(g;B); Ak) for generic values of k. We establish the strong 0 0 0 0 finite generation of Ck in the following cases: Ak = Vk(g ;B ), Ak = Vk−l(g ;B ) ⊗ F, and 0 0 00 00 0 00 Ak = Vk−l(g ;B ) ⊗ Vl(g ;B ). Here g and g are finite-dimensional Lie (super)algebras containing g, equipped with nondegenerate, invariant, (super)symmetric bilinear forms B0 and B00 which extend B, l 2 C is fixed, and F is a free field algebra admitting a homomor- phism Vl(g;B) !F. Our approach is essentially constructive and leads to minimal strong finite generating sets for many interesting examples. 1. INTRODUCTION Vertex algebra are a fundamental class of algebraic structures that arose out of confor- mal field theory and have applications in a diverse range of subjects. The coset or commu- tant construction is a standard way to construct new vertex algebras from old ones. Given a vertex algebra V and a subalgebra A ⊂ V, Com(A; V) is the subalgebra of V which com- mutes with A. This was introduced by Frenkel and Zhu in [FZ], generalizing earlier constructions in representation theory [KP] and physics [GKO], where it was used to con- struct the unitary discrete series representations of the Virasoro algebra. -
Chiral Algebras and Partition Functions
Provided by the author(s) and NUI Galway in accordance with publisher policies. Please cite the published version when available. Title Partition functions and chiral algebras. Author(s) Tuite, Michael P. Publication Date 2007-05 Publication Mason, G., Tuite, M.P. (2007) Partition functions and chiral Information algebras Lie algebras, vertex operator algebras and their applications, CONTEMPORARY 442 Publisher American Mathematical Society Link to publisher's http://www.ams.org/bookstore-getitem/item=conm-442 version Item record http://hdl.handle.net/10379/4848 Downloaded 2021-09-29T23:55:39Z Some rights reserved. For more information, please see the item record link above. Chiral Algebras and Partition Functions Geoffrey Mason∗ Department of Mathematics, University of California Santa Cruz, CA 95064, U.S.A. MichaelP.Tuite† Department of Mathematical Physics, National University of Ireland, Galway, Ireland. In Honor of Jim Lepowsky and Robert Wilson Abstract We discuss recent work of the authors concerning correlation functions and partition functions for free bosons/fermions and the b-c or ghost system. We compare and contrast the nature of the 1-point functions at genus 1, and explain how one may understand the free boson partition function at genus 2 via vertex operators and sewing complex tori. 1 Introduction This paper is based on the talk given by one of the authors at the North Carolina State Conference honoring Jim Lepowsky and Robert Wilson. The paper concerns the idea of partition functions in the theory of chiral algebras. The genus 1 partition function of a vertex operator algebra - a.k.a. the graded dimension - has been studied extensively, but the case when either the genus is greater than 1 or else the chiral algebra is not a vertex operator algebra ∗Partial support provided by NSF DMS-0245225 and the Committee on Research, University of California, Santa Cruz †Supported by the Millenium Fund, National University of Ireland, Galway 1 has received little attention from mathematicians thus far. -
Introduction to Vertex Operator Algebras I 1 Introduction
数理解析研究所講究録 904 巻 1995 年 1-25 1 Introduction to vertex operator algebras I Chongying Dong1 Department of Mathematics, University of California, Santa Cruz, CA 95064 1 Introduction The theory of vertex (operator) algebras has developed rapidly in the last few years. These rich algebraic structures provide the proper formulation for the moonshine module construction for the Monster group ([BI-B2], [FLMI], [FLM3]) and also give a lot of new insight into the representation theory of the Virasoro algebra and affine Kac-Moody algebras (see for instance [DL3], [DMZ], [FZ], [W]). The modern notion of chiral algebra in conformal field theory [BPZ] in physics essentially corresponds to the mathematical notion of vertex operator algebra; see e.g. [MS]. This is the first part of three consecutive lectures by Huang, Li and myself. In this part we are mainly concerned with the definitions of vertex operator algebras, twisted modules and examples. The second part by Li is about the duality and local systems and the third part by Huang is devoted to the contragradient modules and geometric interpretations of vertex operator algebras. (We refer the reader to Li and Huang’s lecture notes for the related topics.) So many exciting topics are not covered in these three lectures. The book [FHL] is an excellent introduction to the subject. There are also existing papers [H1], [Ge] and [P] which review the axiomatic definition of vertex operator algebras, geometric interpretation of vertex operator algebras, the connection with conformal field theory, Borcherds algebras and the monster Lie algebra. Most work on vertex operator algebras has been concentrated on the concrete exam- ples of vertex operator algebras and the representation theory. -
Maximally Supersymmetric String Theories in D< 10
FERMILAB-PUB-95/099-T NSF-ITP-94-37 hep-th/9505054 Maximally Supersymmetric String Theories in D<10 Shyamoli Chaudhuri∗ Institute for Theoretical Physics University of California Santa Barbara, CA 93106-4030 George Hockney and Joseph D. Lykken† Fermi National Accelerator Laboratory P.O. Box 500 Batavia, IL 60510 Abstract The existence of maximally supersymmetric solutions to heterotic string the- ory that are not toroidal compactifications of the ten-dimensional superstring is established. We construct an exact fermionic realization of an N=1 super- symmetric string theory in D=8 with non-simply-laced gauge group Sp(20). Toroidal compactification to six and four dimensions gives maximally extended supersymmetric theories with reduced rank (4, 12) and (6, 14) respectively. arXiv:hep-th/9505054v1 10 May 1995 ∗[email protected] †[email protected]; [email protected] Finiteness is a robust property of the perturbative amplitudes of the known su- perstring theories. N=4 supersymmetric Yang-Mills theory is known to be finite in four dimensions [1], and there is growing evidence that the theory exhibits an ex- tension of Olive-Montonen strong-weak coupling duality known as S-duality[2] [3]. A generalization of the Olive-Montonen duality of N=4 theories has also been identi- fied in N=1 supersymmetric Yang-Mills theory [4]. In string theory, conjectures for S-duality have mostly been explored in the context of toroidal compactifications of the ten-dimensional heterotic string to spacetime dimensions D<10 [5]. It would be helpful to have insight into the generic moduli space, and the generic duality group, of such maximally supersymmetric string theories. -
Conformal Bootstrap in Two Dimensions
Conformal Bootstrap in Two Dimensions The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Lin, Ying-Hsuan. 2016. Conformal Bootstrap in Two Dimensions. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:33493283 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Conformal Bootstrap in Two Dimensions A dissertation presented by Ying-Hsuan Lin to The Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts April 2016 ©2016 - Ying-Hsuan Lin All rights reserved. Thesis advisor Author Xi Yin Ying-Hsuan Lin Conformal Bootstrap in Two Dimensions Abstract In this dissertation, we study bootstrap constraints on conformal field theories in two dimensions. The first half concerns two-dimensional (4; 4) superconformal field theories of cen- tral charge c = 6, corresponding to nonlinear sigma models on K3 surfaces. The superconformal bootstrap is made possible through a surprising relation between the BPS = 4 superconformal blocks with c = 6 and bosonic Virasoro conformal blocks N with c = 28, and an exact moduli dependence of a certain integrated BPS four-point function. Nontrivial bounds on the non-BPS spectrum in the K3 CFT are obtained as functions of the CFT moduli, that interpolate between the free orbifold points and singular CFT points. -
Asymptotic Symmetry Groups and Operator Algebras
Asymptotic symmetry groups and operator algebras Waldemar Schulgin Universit´eLibre de Bruxelles and International Solvay Institutes ULB-Campus Plaine CP231 B-1050 Brussels, Belgium waldemar.schulgin AT ulb.ac.be Jan Troost Laboratoire de Physique Th´eorique1 Ecole Normale Sup´erieure 24 rue Lhomond, 75005 Paris, France Abstract: We associate vertex operators to space-time diffeomorphisms in flat space string theory, and compute their algebra, which is a diffeomorphism algebra with higher derivative corrections. As an application, we realize the asymptotic symmetry group BMS3 of three- dimensional flat space in terms of vertex operators on the string worldsheet. This provides an embedding of the BMS3 algebra in a consistent theory of quantum gravity. Higher deriva- ′ arXiv:1307.3423v2 [hep-th] 5 Sep 2013 tive corrections vanish asymptotically. An appendix is dedicated to α corrected algebras in conformal field theory and string theory. 1Unit´eMixte du CNRS et de l’Ecole Normale Sup´erieure associ´ee `al’universit´ePierre et Marie Curie 6, UMR 8549. Contents 1 Introduction 3 2 The embedding in AdS3 4 2.1 The embedding in AdS3 gravity............................ 4 2.2 The embedding in AdS3 stringtheory ........................ 4 3 The algebra of diffeomorphism vertex operators 5 3.1 Diffeomorphism vertex operators . ...... 5 3.2 The commutator of diffeomorphism vertex operators . .......... 6 3.3 Other worldsheet quantum corrections . ......... 8 3.4 Quantum corrections and marginality . ........ 8 3.5 Asymptotically marginal diffeomorphisms . ......... 9 4 Three-dimensional flat space 9 4.1 The BMS3 diffeomorphismvertexoperators. 10 4.2 The BMS3 algebra................................... 12 4.3 Thecentralcharge ................................ 13 5 Conclusion 14 A Diffeomorphism operators are BRST exact 15 B Higher derivative corrections to operator algebras 16 B.1 Chiralalgebras .................................