## Liouville п¬Ѓeld Modular forms and Elliptic Genera

### Mathieu Moonshine and the Geometry of K3 Surfaces

Lecture Notes on Automorphic Forms and Physics. The Virasoro conjecture proposed by Eguchi-Hori-Xiong and S. Katz predicts that the generating function of Gromov-Witten invariants is annihilated by infinitely many differential operators which form a half branch of the Virasoro algebra. In this paper, we study the genus-1 case of the conjecture, Virasoro group representations fall into the Virasoro coadjoint orbits where each orbit is labelled by at most two quantum numbers (an integer and a continuous parameter) and states in a given orbit are specified by their charges under Virasoro generators L_n's. We show that local (classical) gravity probes can only measure orbit invariant quantities. Details of the Virasoro charges associated.

### CWRU PAT Coffee Agenda glamdring.case.edu

Double-Weyl Phonons in Transition-Metal Monosilicides. An elliptic Virasoro symmetry in 6d Fabrizio Nieri Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden. E-mail: fb.nieri@gmail.com Abstract: We de ne an elliptic deformation of the Virasoro algebra. We conjecture that the R4 Г—T2 Nekrasov partition function reproduces the chiral blocks of this algebra. We support this proposal by showing that at special, The Virasoro conjecture proposed by Eguchi-Hori-Xiong and S. Katz predicts that the generating function of Gromov-Witten invariants is annihilated by infinitely many differential operators which form a half branch of the Virasoro algebra..

An elliptic Virasoro symmetry in 6d 2149 this work. The R4 Nekrasov partition function and the Virasoro algebra have both a natural trigonometric deformation. pdf. Quantum field theory in curved graphene spacetimes, Lobachevsky geometry, Weyl symmetry, Hawking effect, and all that. 76 Pages. Quantum field theory in curved graphene spacetimes, Lobachevsky geometry, Weyl symmetry, Hawking effect, and all that. Author. Gaetano Lambiase. Download with Google Download with Facebook or download with email. Quantum field theory in вЂ¦

In this paper, we obtain a class of irreducible Virasoro modules by taking tensor products of the irreducible Virasoro modules О© (О», b) with irreducible highest weight modules V (Оё, h) or with irreducible Virasoro modules Ind Оё (N) defined in Mazorchuk and Zhao (Selecta Math. Elliptic deformations of quantum Virasoro and Wn algebras Work in collaboration with L. Frappat and E. Ragoucy (LAPTH Annecy) Extension of work by J.A, L.F., M. Rossi, P. Sorba, 1997-99

We define an elliptic deformation of the Virasoro algebra. We argue that the $\mathbb{R}^4\times \mathbb{T}^2$ Nekrasov partition function reproduces the chiral blocks of this algebra. algebras of virasoro type, riemann surfaces and structures of the theory of solitons i. m. krichever and s. p. novikov udc 517.9

The elliptic genus of K3 and CFT International Workshop \Mock Modular Forms and Physics" IMSc, Chennai, India, April 14-18, 2014 Katrin Wendland 2.3. qв†’ в€’1 limit of q-Virasoro generators and N= 1 superconformal algebra In this subsection, we consider directly the qв†’ в€’ 1 limit of the generating function T ( z ) of the q -Virasoro generators (2.4).

Physics Research Publications from the Research School of Physics and Engineering CLASSIFYING SPACES, VIRASORO EQUIVARIANT BUNDLES, ELLIPTIC COHOMOLOGY AND MOONSHINE ANDREW BAKER & CHARLES THOMAS Introduction This work explores some connections between the elliptic cohomology of classifying spaces

An elliptic Virasoro symmetry in 6d 2149 this work. The R4 Nekrasov partition function and the Virasoro algebra have both a natural trigonometric deformation. Octob er 1999 DPSU-99-5 Bey ond CFT: Deformed Virasoro and Elliptic Algebras 1 Sa tor u Od ake Dep artment of Physics, F aculty Scienc e Shinshu University, Matsumoto 390-8621,

Virasoro conjecture says that, for each k в‰Ґ 0, the derivative of the generating function of genus-1 primitive GromovвЂ“Witten invariants along a vector п¬Ѓeld E k (the k th quantum power of the Euler vector п¬Ѓeld) is equal to a function, denoted by П† k , which is deп¬Ѓned Mathieu Moonshine and the Geometry of K3 Surfaces Thomas Creutzig and Gerald H ohny Abstract We compare the moonshine observation of Eguchi, Ooguri and Tachikawa relating the Mathieu group M 24 and the complex elliptic genus of a K3 surface with the symmetries of geometric structures on K3 surfaces. Two main results are that the complex elliptic genus of a K3 surface is a virtual module for

S.V. Kryukov, Deformatsiya algebry Virasoro i integraly dvizheniya kvantovogo uravneniya sinus-Gordon, PisвЂ™ma v ZhETF, 63 (5), 375-380 (1996) [S.V. Kryukov, Deformation of a Virasoro algebra and the integrals of motion of the quantum sine-Gordon equation, JETP Lett., 63 (5), 390-397 (1996)]. Mathieu Moonshine and the Geometry of K3 Surfaces Thomas Creutzig and Gerald H ohny Abstract We compare the moonshine observation of Eguchi, Ooguri and Tachikawa relating the Mathieu group M 24 and the complex elliptic genus of a K3 surface with the symmetries of geometric structures on K3 surfaces. Two main results are that the complex elliptic genus of a K3 surface is a virtual module for

simplest nontrivial Riemann surfaces, the hyper-elliptic ones, and п¬Ѓnd that the full theory features a consistent Virasoro algebra with the correct action on its space of states. i 334 determination of the polarization vectors of lattice waves of an aiom under the influence of a lattice wave is {- W,)li_. dB!:) may be considered as the scatt~ing elliptic.

S.V. Kryukov, Deformatsiya algebry Virasoro i integraly dvizheniya kvantovogo uravneniya sinus-Gordon, PisвЂ™ma v ZhETF, 63 (5), 375-380 (1996) [S.V. Kryukov, Deformation of a Virasoro algebra and the integrals of motion of the quantum sine-Gordon equation, JETP Lett., 63 (5), 390-397 (1996)]. Keywords: elliptic algebra, deformed Virasoro symmetry, root of unity limit 1. Introduction Since the proposal of AGT(W) relation [2], 2d/4d correspondence and its generalizations 1, have been intensively studied. Originally, it was the correspondence between the Nekrasov partition function of 4d N = 2 supersymmetric SU(N) gauge theory and the conformal block of 2d CFT with the W N symmetry

Modular invariance and Virasoro algebra in geometry Kefeng Liu liu@math.ucla.edu Department of Mathematics UCLA Box 951555 Los Angeles, CA 90095-1555 USA Abstract I will discuss certain applications of modular invariance and Vi-rosoro algebra in geometry and topology, through elliptic genera and moduli spaces of Riemann surfaces. Created Date: 3/13/2007 2:04:04 PM Lecture Notes on Automorphic Forms and Physics Miranda C. N. Cheng [;\ [Department of Mathematics, Harvard University, Cambridge, MA 02138, USA \Department of Physics, Harvard University, Cambridge, MA 02138, USA Abstract This is a type-up of the lecture notes for a series of lectures on the topic \Automorphic Forms and Physics", given in the prepara-tory school for the вЂ¦

### CLASSIFYING SPACES VIRASORO EQUIVARIANT BUNDLES ELLIPTIC

q-alg/9705012 PDF arXiv - MAFIADOC.COM. Modular Invariant Virasoro Modules and Elliptic Curves MURRAY R. BREMNER* Department of Mathematics, Yale University, Box 2155 Yale Station, New Haven, CT 06520, U.S.A. (Received: 1 October 1989) Abstract. Given a rational number ~, when does the tensor product of two modular invariant Virasoro modules have central charge (? I show that the resulting equations define elliptic вЂ¦, Physics Research Publications from the Research School of Physics and Engineering.

### Elliptic deformations of quantum Virasoro and Wn algebras

Quantum field theory in curved graphene spacetimes. Adimurthi, ; Yadava, S. L. (1992) Nonexistence of nodal solutions of elliptic equations with critical growth in R 2 Transactions of the American Mathematical Societ, 332 (1). pp. 449-458. ISSN 0002-9947 the Weierstrass elliptic and related functions are used. All the necessary formulae can All the necessary formulae can be found in standard mathematical literature, e.g. [38], or вЂ¦.

Symmetries in Conformal Field Theory Chris Elliott These are elementary notes on Virasoro and a ne Lie algebra symmetries in 2d conformal eld theory, prepared for Elliptic deformation of the Virasoro algebra and the W N-algebra In this section we review the elliptic deformation of the Virasoro algebra and the W N -algebra. We fix three parameters x , r , s such that 0 < x < 1 , Re ( r ) > 0 and Re ( s ) > 0 .

q-Virasoro algebra and by Feigin et al[13,21] for q-deformed W algebra, which is some elliptic deforma- tion of aп¬ѓne algebra. The elliptic algebra A q,p;Л†ПЂ (glc The temperature dependences of phonon linewidth and lifetime of E 2 (TO) modes are analyzed in terms of an anharmonic damping effect induced by thermal and growth conditions. The results show that the lifetime of E 2 (TO) mode increases when the quality of the sample improves.

Using properties of the latter, it is possible to find appropriate solutions of the equations of motion that additionally satisfy the geometric and Virasoro constraints, effectively inverting the Pohlmeyer reduction for the class of elliptic solutions of the reduced system. The corresponding string solutions read The Virasoro conjecture proposed by Eguchi-Hori-Xiong and S. Katz predicts that the generating function of Gromov-Witten invariants is annihilated by infinitely many differential operators which form a half branch of the Virasoro algebra. In this paper, we study the genus-1 case of the conjecture

In this lecture we discuss вЂbeyond CFT вЂ™ from symmetry point of view. After reviewing the Virasoro algebra, we introduce deformed Virasoro algebras and elliptic algebras. These algebras appear in solvable lattice models and we study them by free field approach. hep-th/9910226 CLASSIFYING SPACES, VIRASORO EQUIVARIANT BUNDLES, ELLIPTIC COHOMOLOGY AND MOONSHINE ANDREW BAKER & CHARLES THOMAS Introduction This work explores some connections between the elliptic cohomology of classifying spaces

Phonon calculations.вЂ” Phonons are quantized excited vibrational states of interacting atoms. Solids with more than one atom in the primitive cell have both acoustic and Read the latest articles of Journal of Geometry and Physics at ScienceDirect.com, ElsevierвЂ™s leading platform of peer-reviewed scholarly literature

CWRU Coffee Voting System Toggle navigation CWRU Coffee Home; Current Votes; My Votes pdf. Quantum field theory in curved graphene spacetimes, Lobachevsky geometry, Weyl symmetry, Hawking effect, and all that. 76 Pages. Quantum field theory in curved graphene spacetimes, Lobachevsky geometry, Weyl symmetry, Hawking effect, and all that. Author. Gaetano Lambiase. Download with Google Download with Facebook or download with email. Quantum field theory in вЂ¦

## [1511.00574] An elliptic Virasoro symmetry in 6d

Free п¬Ѓeld constructions for the elliptic algebra Aqp sl. Analog condensed matter systems present an exciting opportunity to simulate early Universe models in table-top experiments. We consider a recent proposal for an analog condensed matter experiment to simulate the relativistic quantum decay of the false vacuum., q-Virasoro algebra and by Feigin et al[13,21] for q-deformed W algebra, which is some elliptic deforma- tion of aп¬ѓne algebra. The elliptic algebra A q,p;Л†ПЂ (glc.

### Elliptic deformations of quantum Virasoro and Wn algebras

Symmetries in Conformal Field Theory Accueil - IHES. elliptic genus from supergravity. In carrying out the computation we consider п¬Ѓrst, in In carrying out the computation we consider п¬Ѓrst, in section 6, the supergravity contributions, and next, in section 7, the contributions from, The elliptic genus of K3 and CFT International Workshop \Mock Modular Forms and Physics" IMSc, Chennai, India, April 14-18, 2014 Katrin Wendland.

Octob er 1999 DPSU-99-5 Bey ond CFT: Deformed Virasoro and Elliptic Algebras 1 Sa tor u Od ake Dep artment of Physics, F aculty Scienc e Shinshu University, Matsumoto 390-8621, MIRZAKHANIвЂ™S RECURSION RELATIONS, VIRASORO CONSTRAINTS AND THE KDV HIERARCHY MOTOHICO MULASE1 AND BRAD SAFNUK2 Abstract. We вЂ¦

The Virasoro conjecture predicts that the generating function of Gromov-Witten invariants is annihilated by infinitely many differential operators which form a half branch of the Virasoro algebra. This conjecture was proposed by Eguchi, Hori and Xiong [EHX2] and вЂ¦ Read the latest articles of Journal of Geometry and Physics at ScienceDirect.com, ElsevierвЂ™s leading platform of peer-reviewed scholarly literature

Liouville п¬Ѓeld, Modular forms and Elliptic Genera Anne Taormina Dubna, June 2007 Based on hep-th/0611338 (with T. Eguchi and Y. Sugawara) 1 How does one describe strings propagating on вЂ¦ An elliptic Virasoro symmetry in 6d Fabrizio Nieri 0 0 Department of Physics and Astronomy, Uppsala University , Box 516, 75120 Uppsala , Sweden We define an elliptic deformation of the Virasoro вЂ¦

CLASSIFYING SPACES, VIRASORO EQUIVARIANT BUNDLES, ELLIPTIC COHOMOLOGY AND MOONSHINE ANDREW BAKER & CHARLES THOMAS Introduction This work explores some connections between the elliptic cohomology of classifying spaces Examples of Lie algebras Kac-Moody algebras Elliptic Afп¬Ѓne algebras DJKM algebra Outline - Virasoro algebra - Heisenberg/Weyl algebras - Kac-Moody algebras

Abstract: We define an elliptic deformation of the Virasoro algebra. We argue that the $\mathbb{R}^4\times \mathbb{T}^2$ Nekrasov partition function reproduces the вЂ¦ Mirror symmetry for elliptic curves Matthias Ihl1 and Alexander Kahle2 1Dept. of Physics, University of Texas, Austin, Texas 78712, USA 2Dept. of Mathematics, University of Texas, Austin, Texas 78712, USA This is a termpaper for the M390C Abelian Varieties class taught by Gavril Farkas in Spring 2006 at the University of Texas. We try to provide a survey of basic ideas relating to mirror

Mathieu Moonshine and the Geometry of K3 Surfaces Thomas Creutzig and Gerald H ohny Abstract We compare the moonshine observation of Eguchi, Ooguri and Tachikawa relating the Mathieu group M 24 and the complex elliptic genus of a K3 surface with the symmetries of geometric structures on K3 surfaces. Two main results are that the complex elliptic genus of a K3 surface is a virtual module for CWRU Coffee Voting System Toggle navigation CWRU Coffee Home; Current Votes; My Votes

Amer Iqbal is a Pakistani American theoretical physicist. He is primarily known for his work in string theory and mathematical physics Amer Iqbal is a Pakistani American theoretical physicist. He is primarily known for his work in string theory and mathematical physics

Abstract: Contour dynamics is a classical subject both in physics and in complex analysis. We show that the dynamics provided by the L\"owner-Kufarev ODE and PDE possesses a rigid algebraic structure given by the Virasoro algebra. 1/31/1981 2/21/2014 458 1. 1/1/1982 3/5/2013 322. 1/1/1982 3/20/2013 410. 1/1/1982 8/14/2012 484. 1/1/1982 5/19/2011 224. 1/1/1982 3/12/2013 1104. 1/31/1983 10/29/2014

The Virasoro conjecture predicts that the generating function of Gromov-Witten invariants is annihilated by infinitely many differential operators which form a half branch of the Virasoro algebra. This conjecture was proposed by Eguchi, Hori and Xiong [EHX2] and вЂ¦ S.V. Kryukov, Deformatsiya algebry Virasoro i integraly dvizheniya kvantovogo uravneniya sinus-Gordon, PisвЂ™ma v ZhETF, 63 (5), 375-380 (1996) [S.V. Kryukov, Deformation of a Virasoro algebra and the integrals of motion of the quantum sine-Gordon equation, JETP Lett., 63 (5), 390-397 (1996)].

### Modular invariant Virasoro modules and elliptic curves

Tan Zhao Irreducible Virasoro modules from tensor products. the Weierstrass elliptic and related functions are used. All the necessary formulae can All the necessary formulae can be found in standard mathematical literature, e.g. [38], or вЂ¦, Keywords: elliptic algebra, deformed Virasoro symmetry, root of unity limit 1. Introduction Since the proposal of AGT(W) relation [2], 2d/4d correspondence and its generalizations 1, have been intensively studied. Originally, it was the correspondence between the Nekrasov partition function of 4d N = 2 supersymmetric SU(N) gauge theory and the conformal block of 2d CFT with the W N symmetry.

### Research publications RSPE - ANU

www.slac.stanford.edu. Examples of Lie algebras Kac-Moody algebras Elliptic Afп¬Ѓne algebras DJKM algebra Outline - Virasoro algebra - Heisenberg/Weyl algebras - Kac-Moody algebras This monograph deals with two aspects of the theory of elliptic genus: its topological aspect involving elliptic functions, and its representation theoretic aspect involving vertex operator super-algebras. For the second aspect, elliptic genera are shown to have the structure of modules over.

Phonon calculations.вЂ” Phonons are quantized excited vibrational states of interacting atoms. Solids with more than one atom in the primitive cell have both acoustic and CWRU Coffee Voting System Toggle navigation CWRU Coffee Home; Current Votes; My Votes

i 334 determination of the polarization vectors of lattice waves of an aiom under the influence of a lattice wave is {- W,)li_. dB!:) may be considered as the scatt~ing elliptic. Modular Invariant Virasoro Modules and Elliptic Curves MURRAY R. BREMNER* Department of Mathematics, Yale University, Box 2155 Yale Station, New Haven, CT 06520, U.S.A. (Received: 1 October 1989) Abstract. Given a rational number ~, when does the tensor product of two modular invariant Virasoro modules have central charge (? I show that the resulting equations define elliptic вЂ¦

Mathieu Moonshine and the Geometry of K3 Surfaces Thomas Creutzig and Gerald H ohny Abstract We compare the moonshine observation of Eguchi, Ooguri and Tachikawa relating the Mathieu group M 24 and the complex elliptic genus of a K3 surface with the symmetries of geometric structures on K3 surfaces. Two main results are that the complex elliptic genus of a K3 surface is a virtual module for We show that phonon-induced dephasing of exciton states confined in a quantum dot may be interpreted in terms of which way information transferred from the carrier system to the lattice environment. Using distinguishability of quantum states as a measure for this information transfer we cast this interpretation in the form of a quantitative complementarity relation.

Phonon calculations.вЂ” Phonons are quantized excited vibrational states of interacting atoms. Solids with more than one atom in the primitive cell have both acoustic and Physics Research Publications from the Research School of Physics and Engineering

MIRZAKHANIвЂ™S RECURSION RELATIONS, VIRASORO CONSTRAINTS AND THE KDV HIERARCHY MOTOHICO MULASE1 AND BRAD SAFNUK2 Abstract. We вЂ¦ MODULAR FORMS AND TOPOLOGY KEFENG LIU We want to discuss various applications of modular forms in topology. The starting point is elliptic genus and its generalizations.