DOCSLIB.ORG

Valdo Tatitscheff –

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Valdo Tatitscheff – 59, rue de Zurich 67 000 Strasbourg France H +33 6 51 35 18 12 Valdo Tatitscheff B valdo.tatitcheff@normalesup.org Education and Research Experiences sept.2018-sept.2021 PhD in Mathematics, IRMA, Strasbourg, under the supervision of Prof. V. Fock. Subject: Dimer integrable systems jan.2018-jun.2018 Research internship, City, UoL, London, under the supervision of Prof. Y.H He. Dessins d’enfants, Monstrous Moonshine and cusps of Hecke congruence groups nov.2017-dec.2017 Solvay doctoral school in theoretical physics, Paris (nov.) and Amsterdam (dec.). Topics: Supersymmetry (Paris session) and AdS/CFT correspondence (Amsterdam session) oct.2017 Master’s degree in Mathematics, UPMC, Paris, with honours. Subject of the thesis: Super-Teichmüller spaces, V. Fock (IRMA) 2015-2017 Theoretical Physics, ICFP, Paris. During the academic year 2015-2016, I followed and validated the classes Quantum Field Theories, Quantum Chromodynamics and Introduction to the bosonic string of the M2 ICFP. The next year I discovered large-N expansions in QFTs and two-dimensional supersymmetric QFTs under the kind supervision of Prof. A.K. Kashani-Poor. sept.2015 Bachelor’s degrees in Mathematics and Physics, Paris XIII, Paris, with honours. Subject: In which sense do Maxwell equations already contain hints of QFTs ? T. Lévy (UPMC) BSM Physics through effective approaches of the di-Higgs coupling at LHC. R. Salerno (LLR) jul. 2014 Admission at the Ecole Normale Supérieure de la rue d’Ulm. Research Interests Cluster varieties, with V. Fock. One of the topics I’m working on at the time of writing (and which forms the main subject of my PhD) consists of the theory of cluster integrable systems. These are finite-dimensional integrable system, with a phase space which enjoys nice properties commonly shared by cluster varieties. In particular: canonical deformation quantization for the X -version, and a motivic avatar in the A-case. The link with algebraic K-theory let us hope for some relationship between cluster integrable systems, and the higher-Teichmüller theory - as well as exact WKB, BPS states in string theory and the Spectral Networks defined by Gaiotto, Moore and Neitzke. Brane tilings and dynamical supersymmetry breaking, with R. Argurio, M. Bertolini, E. García-Valdecasas, A. Pasternak and S. Meynet. One way to extend the original setting of the AdS/CFT correspondence has been to consider IIB string backgrounds in which regular D3-branes are placed at the tip of an affine toric Calabi-Yau 3-fold X. In that case, one gets an equivalence between supergravity on some Sasaki-Einstein 5-manifold Y5 over which X is the real cone, and a special type of 4d quiver gauge theory with N = 1 supersymmetry (and in some special cases, N = 2), which can be completely encoded in an embedding of a bipartite graph on a torus, in such a way that the faces are topological disks, plus an additional minimality condition. These theory are believed to flow to superconformal fixed points in the IR. If one places fractional branes on top of the regular ones in the geometric background, the superconfomality is broken and now the theory flows. The question is to understand whether in the extreme IR one can obtain some of the "canonical" N = 1 dynamical supersymmetry breaking models, such as the SU(5) model, or the 3 − 2 model, using such tools. Duality cascades and dimer integrable systems. Brane tilings, which are a natural setting for generalisations of Strassler and Klebanov duality cascades, seem to be very related to a special class of cluster integrable systems defined by Goncharov and Kenyon using bipartite surface graphs and dimer techniques. I would like to understand more deeply these links. Exceptional objects. I am also interested in Monstrous Moonshine type of correspondences, on which I had the chance to work for a while, when in London, with Prof. Yang-Hui He. Teaching Experiences jan.-jun. 2020 Maths pour la Science, 2., Université de Strasbourg, Strasbourg. This is a class for first-year students from the physics department, which could be called a first course in Linear Algebra. mar. 2018 Teaching replacements, City, UoL, London. Replacements of Prof. Yang-Hui He { Introduction to Mathematical Physics, 3rd year of Bachelor (2 hours) { Applied Mathematics, 2nd year of Bachelor (6 hours) oct.2014-jun.2017 Khôlles, Lycée Stanislas, Paris. The goal of khôlles in Classes Préparatoires is to provide the students a regular and specific training for the competition that they will pass at the end of these two years of Classes Préparatoires, called Concours pour les grandes écoles. As during the real oral exams, each student has to solve exercises on the blackboard, that are chosen by the examiner. A generic khôlle lasts one hour, during which three students are given different exercices at the same time. { 2014-2015 Chemistry in second-year PSI∗ (one hour a week) { 2015-2016 - Physics and Chemistry in second-year PSI∗ (one hour a week) - Mathematics in first-year MPSI (one hour a week) - Mathematics in first-year PCSI (one hour a week) { 2016-2017 Mathematics in first-year PCSI (one hour a week) mar. 2015 Exam marking, Lycée Stanislas, Paris. Marking of the mathematics mock exam of second-year students (approx. 180 examination scripts) Publications 7th Feb. 2019 A short introduction to Monstrous Moonshine, arXiv:1902.03118. 31st Dec. 2018 Cusps, Congruence Groups and Monstrous Dessins, arXiv:1812.11752. Co-authored with Prof. Y.H He and Prof. J. McKay. Languages French Motherthongue English Bilingal German Basic working knowledge Miscellanous 2013-2018 Youth monitoring, in the french Scout Association EEUdF. I have been scout leader of 8 to 12 years-old children between sept. 2013 and aug. 2015, then scout leader of 12 to 16 years-old children between sept. 2015 and aug. 2018. This activity consisted of approx. one week-end leadership each month (during the academic year), and of three weeks of summer camp in July. I got the national youth monitoring qualification BAFA in 2016 and the first half of the camp-director qualification BAFD in 2017. 2005-2016 Trumpet studies, at the conservatoire municipal Charles Munch, Paris. I still play the trumpet nowadays, and enjoy music a lot. 2013 45th International Chemistry Olympiads, for the french national team, Moscow, Russia. My final rank was 84th (silver medal) ICFP Theoretical Physics - 2015-2016 Quantum field theories, A. Bilal. Introduction to QCD. String theory (bosonic strings), J. Troost, I chose the topic of my final project to be on a homological picture of the Fadeev-Popov procedure and BRST quantization.. UPMC Fundamental Mathematics - 2016-2017 Géométrie différentielle et Riemannienne, J. Marché. Introduction à la géometrie algébrique, I. Itenberg. Introduction aux surfaces de Riemann, M. Maculan. Algèbre homologique et cohomologie des faisceaux, F. Loeser. Géométrie complexe et théorie de Hodge, J. Cao. Topologie algébrique, A. Oancea. Variations de structures de Hodge, B. Klinger. Théorie de Hurwitz, P. Georgevia. Master thesis Description of super-Teichmüller spaces as moduli spaces of graph connections, under the kind supervision of V. V. Fock. Solvay doctoral school - Paris, nov. 2017 4-dimensional supersymmetry and MSSM, U. Ellwanger. Introduction to Supergravity, A. van Proeyen. Introduction to Superstrings, C. Bachas. Lie algebras in Physics, A. Kleinschmidt. Solvay doctoral school - Amsterdam, dec. 2017 QFT in curved spaces, G. Pimentel. Topics in large N, D. Anninos. Introduction to AdS/CFT, K. Papadodimas. Applied AdS/CFT, J. Sonner. Introduction to Resurgence, M. Vonk..
Load more

Recommended publications
  • Flavor Moonshine
    Prog. Theor. Exp. Phys. 2020, 013B03 (20 pages) DOI: 10.1093/ptep/ptz137 Flavor moonshine Shotaro Shiba Funai1,∗ Hirotaka Sugawara2,∗ 1Physics and Biology Unit, Okinawa Institute of Science and Technology (OIST), 1919-1 Tancha, Onna-son, Kunigami-gun, Okinawa 904-0495, Japan 2High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan ∗E-mail: [email protected], [email protected] Received August 30, 2019; Revised October 17, 2019; Accepted October 23, 2019; Published January 13, 2020 ................................................................................................................... The flavor moonshine hypothesis is formulated to suppose that all particle masses (leptons, quarks, Higgs, and gauge particles—more precisely, their mass ratios) are expressed as coef- ficients in the Fourier expansion of some modular forms just as, in mathematics, dimensions of representations of a certain group are expressed as coefficients in the Fourier expansion of some modular forms. The mysterious hierarchical structure of the quark and lepton masses is thus attributed to that of the Fourier coefficient matrices of certain modular forms. Our inten- tion here is not to prove this hypothesis starting from some physical assumptions but rather to demonstrate that this hypothesis is experimentally verified and, assuming that the string theory correctly describes the natural law, to calculate the geometry (Kähler potential and the metric) of the moduli space of the Calabi–Yau manifold, thus providing a way to calculate the metric of the Calabi–Yau manifold itself directly from the experimental data. ................................................................................................................... Subject Index B41, B55 1. Introduction Some researchers, including one of the authors of this work (H.S.), have been working on flavor physics, assuming that some discrete symmetry plays an important role in its understanding [1–9]; S3, S4, A4, etc.
    [Show full text]
  • Umbral Moonshine and K3 Surfaces Arxiv:1406.0619V3 [Hep-Th]
    SLAC-PUB-16469 Umbral Moonshine and K3 Surfaces Miranda C. N. Cheng∗1 and Sarah Harrisony2 1Institute of Physics and Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, the Netherlandsz 2Stanford Institute for Theoretical Physics, Department of Physics, and Theory Group, SLAC, Stanford University, Stanford, CA 94305, USA Abstract Recently, 23 cases of umbral moonshine, relating mock modular forms and finite groups, have been discovered in the context of the 23 even unimodular Niemeier lattices. One of the 23 cases in fact coincides with the so-called Mathieu moonshine, discovered in the context of K3 non-linear sigma models. In this paper we establish a uniform relation between all 23 cases of umbral moonshine and K3 sigma models, and thereby take a first step in placing umbral moonshine into a geometric and physical context. This is achieved by relating the ADE root systems of the Niemeier lattices to the ADE du Val singularities that a K3 surface can develop, and the configuration of smooth rational curves in their resolutions. A geometric interpretation of our results is given in terms of the marking of K3 surfaces by Niemeier lattices. arXiv:1406.0619v3 [hep-th] 18 Mar 2015 ∗[email protected] [email protected] zOn leave from CNRS, Paris. 1 This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-76SF00515 and HEP. Umbral Moonshine and K3 Surfaces 2 Contents 1 Introduction and Summary 3 2 The Elliptic Genus of Du Val Singularities 8 3 Umbral Moonshine and Niemeier Lattices 14 4 Umbral Moonshine and the (Twined) K3 Elliptic Genus 20 5 Geometric Interpretation 27 6 Discussion 30 A Special Functions 32 B Calculations and Proofs 34 C The Twining Functions 41 References 48 Umbral Moonshine and K3 Surfaces 3 1 Introduction and Summary Mock modular forms are interesting functions playing an increasingly important role in various areas of mathematics and theoretical physics.
    [Show full text]
  • The Romance Between Maths and Physics
    The Romance Between Maths and Physics Miranda C. N. Cheng University of Amsterdam Very happy to be back in NTU indeed! Question 1: Why is Nature predictable at all (to some extent)? Question 2: Why are the predictions in the form of mathematics? the unreasonable effectiveness of mathematics in natural sciences. Eugene Wigner (1960) First we resorted to gods and spirits to explain the world , and then there were ….. mathematicians?! Physicists or Mathematicians? Until the 19th century, the relation between physical sciences and mathematics is so close that there was hardly any distinction made between “physicists” and “mathematicians”. Even after the specialisation starts to be made, the two maintain an extremely close relation and cannot live without one another. Some of the love declarations … Dirac (1938) If you want to be a physicist, you must do three things— first, study mathematics, second, study more mathematics, and third, do the same. Sommerfeld (1934) Our experience up to date justifies us in feeling sure that in Nature is actualized the ideal of mathematical simplicity. It is my conviction that pure mathematical construction enables us to discover the concepts and the laws connecting them, which gives us the key to understanding nature… In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed. Einstein (1934) Indeed, the most irresistible reductionistic charm of physics, could not have been possible without mathematics … Love or Hate? It’s Complicated… In the era when Physics seemed invincible (think about the standard model), they thought they didn’t need each other anymore.
    [Show full text]
  • Monstrous Entanglement JHEP10(2017)147 ] (Or Its Chiral 8 [ 3 20 6 22 11 15 4 22 28 9 16 29 26 , As Its Automorphism Group
    Published for SISSA by Springer Received: September 1, 2017 Accepted: October 12, 2017 Published: October 20, 2017 Monstrous entanglement JHEP10(2017)147 Diptarka Das,a Shouvik Dattab and Sridip Pala aDepartment of Physics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, U.S.A. bInstitut f¨urTheoretische Physik, Eidgen¨ossischeTechnische Hochschule Z¨urich, Wolfgang-Pauli-Strasse 27, 8049 Z¨urich,Switzerland E-mail: [email protected], [email protected], [email protected] Abstract: The Monster CFT plays an important role in moonshine and is also conjectured to be the holographic dual to pure gravity in AdS3. We investigate the entanglement and R´enyi entropies of this theory along with other extremal CFTs. The R´enyi entropies of a single interval on the torus are evaluated using the short interval expansion. Each order in the expansion contains closed form expressions in the modular parameter. The leading terms in the q-series are shown to precisely agree with the universal corrections to R´enyi entropies at low temperatures. Furthermore, these results are shown to match with bulk computations of R´enyi entropy using the one-loop partition function on handlebodies. We also explore some features of R´enyi entropies of two intervals on the plane. Keywords: Conformal Field Theory, AdS-CFT Correspondence, Field Theories in Lower Dimensions, Discrete Symmetries ArXiv ePrint: 1708.04242 Open Access, c The Authors. https://doi.org/10.1007/JHEP10(2017)147 Article funded by SCOAP3. Contents 1 Introduction1 2 Short interval expansion
    [Show full text]
  • Final Report (PDF)
    Novel Approaches to the Finite Simple Groups John McKay (Concordia University), Roland Friedrich (Humboldt University Berlin) 22. April–29. April 2012 1 Motivation It is now accepted that the classification of finite simple groups (CFSG) is complete. Our purpose is in con- structions which we re-examine in the light of observations that suggest that a substantial amount of insight is to be gained if we pursue novel connections to other, apparently disparate, areas of mathematics. This may lead to a natural home for the 26 sporadic simple groups which today are considered the complement of the Lie-Chevalley groups. 2 Overview of the Field In order to understand the importance of the new ideas and conjectures linked with Monstrous Moon- shine (MM) that we set out to develop, we have to understand its history, see [15, 19, 20]. Although the Monster is the most spectacular of the sporadic groups, it is just one of 26 groups under our wing. We discern three major evolutionary periods linked with the Monster group, the early and foundational one [5, 6, 7, 9, 10, 11, 17, 18, 22, 23], the time when the fruits of previous labour could be picked, cul- minating in Borcherds’ Fields medal rewarded proof of the conjecture [2], and finally the post Moonshine one, where it is becoming evident that we are not dealing with one particular isolated phenomenon but rather with a (possible) new class of objects of which MM is the most prominent representative so far, and its new mathematical connections [16]. An electronic collection of relevant articles for the subject has been posted at: http://www.fields.utoronto.ca/˜jplazas/MoonshineRoadmap.html 3 Recent Developments and Open Problems During the focused research group various parts of this picture became clearer.
    [Show full text]
  • Modular Forms in String Theory and Moonshine
    (Mock) Modular Forms in String Theory and Moonshine Miranda C. N. Cheng Korteweg-de-Vries Institute of Mathematics and Institute of Physics, University of Amsterdam, Amsterdam, the Netherlands Abstract Lecture notes for the Asian Winter School at OIST, Jan 2016. Contents 1 Lecture 1: 2d CFT and Modular Objects 1 1.1 Partition Function of a 2d CFT . .1 1.2 Modular Forms . .4 1.3 Example: The Free Boson . .8 1.4 Example: Ising Model . 10 1.5 N = 2 SCA, Elliptic Genus, and Jacobi Forms . 11 1.6 Symmetries and Twined Functions . 16 1.7 Orbifolding . 18 2 Lecture 2: Moonshine and Physics 22 2.1 Monstrous Moonshine . 22 2.2 M24 Moonshine . 24 2.3 Mock Modular Forms . 27 2.4 Umbral Moonshine . 31 2.5 Moonshine and String Theory . 33 2.6 Other Moonshine . 35 References 35 1 1 Lecture 1: 2d CFT and Modular Objects We assume basic knowledge of 2d CFTs. 1.1 Partition Function of a 2d CFT For the convenience of discussion we focus on theories with a Lagrangian description and in particular have a description as sigma models. This in- cludes, for instance, non-linear sigma models on Calabi{Yau manifolds and WZW models. The general lessons we draw are however applicable to generic 2d CFTs. An important restriction though is that the CFT has a discrete spectrum. What are we quantising? Hence, the Hilbert space V is obtained by quantising LM = the free loop space of maps S1 ! M. Recall that in the usual radial quantisation of 2d CFTs we consider a plane with 2 special points: the point of origin and that of infinity.
    [Show full text]
  • Characterizing Moonshine Functions by Vertex-Operator-Algebraic Conditions
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 114, 8 pages Characterizing Moonshine Functions by Vertex-Operator-Algebraic Conditions Scott CARNAHAN, Takahiro KOMURO and Satoru URANO Division of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571 Japan E-mail: [email protected], [email protected], [email protected] URL: http://www.math.tsukuba.ac.jp/~carnahan/ Received May 07, 2018, in final form October 15, 2018; Published online October 25, 2018 https://doi.org/10.3842/SIGMA.2018.114 Abstract. Given a holomorphic C2-cofinite vertex operator algebra V with graded dimen- sion j − 744, Borcherds's proof of the monstrous moonshine conjecture implies any finite order automorphism of V has graded trace given by a \completely replicable function", and by work of Cummins and Gannon, these functions are principal moduli of genus zero modu- lar groups. The action of the monster simple group on the monster vertex operator algebra produces 171 such functions, known as the monstrous moonshine functions. We show that 154 of the 157 non-monstrous completely replicable functions cannot possibly occur as trace functions on V . Key words: moonshine; vertex operator algebra; modular function; orbifold 2010 Mathematics Subject Classification: 11F22; 17B69 1 Introduction In this paper, we show that the monstrous moonshine functions are nearly completely charac- terized by vertex-operator-algebra-theoretic properties of trace functions. In particular, we strongly constrain the possible exotic automorphisms of any counterexample to the conjectured uniqueness of the monster vertex operator algebra V \ (also known as the moonshine module) constructed in [15].
    [Show full text]
  • MONSTROUS MOONSHINE OVER $\Mathbb{Z}$?
    91 MONSTROUS MOONSHINE OVER \mathbb{Z} ? SCOTT CARNAHAN ABSTRACT. Monstrous Moonshine was extended in two complementary directions during the 1980s and 1990s , giving rise to Norton’s Generalized Moonshine conjecture and Ryba’s Modular Moon‐ shine conjecture. Both conjectures have been unconditionally established in the ıast few years, so we describe some speculative conjectures that may extend and unify them. ı. MONSTROUS MOONSHINE AND ITS EXTENSIONS Monstrous moonshine began in the 1970s with an examination of the modular J‐invariant, which we normalize as: J(\tau)=q^{-1}+196884q+21493760q^{2}+ Here, \tau ranges over the complex upper half‐plane \mathfrak{H} , and the expansion in powers of q=e^{2\pi i\tau} is the Fourier expansion. The J‐function is a Hauptmodul for SL_{2}(\mathbb{Z}) , meaning J is invariant under the standard action of SL_{2}(\mathbb{Z}) on \mathfrak{H} by Möbius transformations, and induces a complex‐analytic isomorphism from the quotient SL_{2}(\mathbb{Z})\backslash S to the complex projective line \mathbb{P}_{\mathbb{C}}^{1} minus finitely many points. McKay first observed a possible relationship between J and the Monster sporadic simple group \mathbb{M} , pointing out that the irreducible representations of \mathbb{M} have lowest dimension ı and ı96883, and that ı96884 =196883+1 . Thompson then showed that the next few coefficients of J also decompose into a small number of irreducible representations of \mathbb{M} [Thompson 1979]. This suggested that the coefficients of J , which are all non‐negative integers, describe a special class of representations of M. The observations by McKay and Thompson were explained by the construction of the Monster vertex operator algebra V^{\mathfrak{h}}=\oplus_{n=0}^{\infty}V_{n}^{\#} in [Fhenkel‐Lepowsky‐Meurman 1988].
    [Show full text]
  • Characterizing Moonshine Functions by Vertex-Operator-Algebraic Conditions
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 114, 8 pages Characterizing Moonshine Functions by Vertex-Operator-Algebraic Conditions Scott CARNAHAN, Takahiro KOMURO and Satoru URANO Division of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571 Japan E-mail: [email protected], [email protected], [email protected] URL: http://www.math.tsukuba.ac.jp/~carnahan/ Received May 07, 2018, in final form October 15, 2018; Published online October 25, 2018 https://doi.org/10.3842/SIGMA.2018.114 Abstract. Given a holomorphic C2-cofinite vertex operator algebra V with graded dimen- sion j − 744, Borcherds's proof of the monstrous moonshine conjecture implies any finite order automorphism of V has graded trace given by a \completely replicable function", and by work of Cummins and Gannon, these functions are principal moduli of genus zero modu- lar groups. The action of the monster simple group on the monster vertex operator algebra produces 171 such functions, known as the monstrous moonshine functions. We show that 154 of the 157 non-monstrous completely replicable functions cannot possibly occur as trace functions on V . Key words: moonshine; vertex operator algebra; modular function; orbifold 2010 Mathematics Subject Classification: 11F22; 17B69 1 Introduction In this paper, we show that the monstrous moonshine functions are nearly completely charac- terized by vertex-operator-algebra-theoretic properties of trace functions. In particular, we strongly constrain the possible exotic automorphisms of any counterexample to the conjectured uniqueness of the monster vertex operator algebra V \ (also known as the moonshine module) constructed in [15].
    [Show full text]
  • Monstrous Moonshine and Monstrous Lie Superalgebras
    Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109, 405-444 (1992). Richard E. Borcherds, Department of pure mathematics and mathematical statistics, 16 Mill Lane, Cam- bridge CB2 1SB, England. We prove Conway and Norton’s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by Frenkel, Lepowsky and Meur- man. To do this we use the no-ghost theorem from string theory to construct a family of generalized Kac-Moody superalgebras of rank 2, which are closely related to the monster and several of the other sporadic simple groups. The denominator formulas of these su- peralgebras imply relations between the Thompson functions of elements of the monster (i.e. the traces of elements of the monster on Frenkel, Lepowsky, and Meurman’s repre- sentation), which are the replication formulas conjectured by Conway and Norton. These replication formulas are strong enough to verify that the Thompson functions have most of the “moonshine” properties conjectured by Conway and Norton, and in particular they are modular functions of genus 0. We also construct a second family of Kac-Moody super- algebras related to elements of Conway’s sporadic simple group Co1. These superalgebras have even rank between 2 and 26; for example two of the Lie algebras we get have ranks 26 and 18, and one of the superalgebras has rank 10. The denominator formulas of these algebras give some new infinite product identities, in the same way that the denominator formulas of the affine Kac-Moody algebras give the Macdonald identities. 1 Introduction. 2 Introduction (continued). 3 Vertex algebras.
    [Show full text]
  • Introduction to Stringy Moonshine ” June, 2018 Tohru Eguchi, University of Tokyo
    ”Introduction to Stringy Moonshine ” June, 2018 Tohru Eguchi, University of Tokyo Monstrous Moonshine Several years ago we found some curious phenomenon in string theory [1], i.e. appearance of exotic discrete symmetries in the theory. This is now called as moon- shine phenomenon and is now under intensive study. Today I would like to give you a brief introduction to moonshine phenomena which may possibly play an in- teresting role in string theory in the future. Before going to the moonshine phenomenon in string theory let me briefly recall the story of monstrous moon- shine which is well-known. Modular J function has a q-series expansion 1 J(q) = + 744 + 196884q + 21493760q2 + 864299970q3 q +20245856256q4 + 333202640600q5 + ··· : ( ) aτ + b a b q = e2πiτ ; Im(τ ) > 0;J(τ ) = J( ); 2 SL(2;Z) cτ + d c d It turns out q-expansion coefficients of J-function are decomposed into a sum of dimensions of irreducible representations of the monster group M as 196884 = 1 + 196883; 21493760 = 1 + 196883 + 21296876; 864299970 = 2 × 1 + 2 × 196883 + 21296876 + 842609326; 20245856256 = 1 × 1 + 3 × 196883 + 2 × 21296876 +842609326 + 19360062527; ··· : 1 Dimensions of irreducible representations of monster are in fact given by f1; 196883; 21296876; 842609326; 18538750076; 19360062527 · · · g Monster group is the largest sporadic discrete group, of order ≈ 1053 and the strange relationship between modular form and the largest discrete group was first noted by McKay. To be precise we may write as X n J1(τ ) = J(q) − 744 = c(n)q ; c(0) = 0 X n=−1 n = T rV (n) 1 × q ; dimV (n) = c(n) n=−1 McKay-Thompson series is given by X n Jg(τ ) = T rV (n) g × q ; g 2 M n=−1 where T rV (n) g denotes the character of a group ele- ment g in the representation V (n).
    [Show full text]
  • The Hidden Symmetry of the Heterotic String
    i \5-Kachru" | 2018/3/1 | 17:59 | page 1729 | #1 i i i ADV. THEOR. MATH. PHYS. Volume 21, Number 7, 1729{1745, 2017 The hidden symmetry of the heterotic string Shamit Kachru and Arnav Tripathy We propose that Borcherds' Fake Monster Lie algebra is a broken symmetry of heterotic string theory compactified on T 7 × T 2. As evidence, we study the fully flavored counting function for BPS instantons contributing to a certain loop amplitude. The result is controlled by Φ12, an automorphic form for O(2; 26; Z). The degen- eracies it encodes in its Fourier coefficients are graded dimensions of a second-quantized Fock space for this large symmetry algebra. This construction provides a concrete realization of Harvey and Moore's proposed relationship between Generalized Kac-Moody symmetries and supersymmetric string vacua. 1. Introduction It has long been a hope that string theory exhibits infinite-dimensional sym- metries which constrain the form of string effective actions and S-matrices [1{4]. Natural candidates are symmetries which generalize the normal struc- ture of Lie groups and even affine Kac-Moody symmetries | the so-called Generalized Kac-Moody (GKM) algebras. The Cartan matrices of these pro- posed larger symmetry algebras allow for negative diagonal entries, thereby greatly enlarging the root systems familiar from Lie theory to include whole lattices of \imaginary roots". That these extremely large symmetries might indeed exist as symmetries of sufficiently supersymmetric theories is of ob- vious interest for constraining the theory and possibly even formulating a classification of such theories. As such, the question has generated a consid- erable literature (including, in addition to the references above, papers such as [5{8] and references therein).
    [Show full text]