Hypergeometric Modular Transformations

2017 ◽  
pp. 595-612
Author(s):  
Shaun Cooper
Keyword(s):  
Modular Transformations

Defining Modular Transformations

1999 ◽  
Vol 21 (2) ◽  
pp. 200-229 ◽  
Author(s):  
Matthew Santa
Keyword(s):  
Modular Transformations

scholarly journals Modular invariance in finite temperature Casimir effect

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Francesco Alessio ◽  
Glenn Barnich
Keyword(s):  
Partition Function ◽  
Chemical Potential ◽  
Casimir Effect ◽  
Temperature Inversion ◽  
Massless Scalar ◽  
Partition Functions ◽  
Parallel Plates ◽  
Modular Transformations ◽  
Real Analytic ◽  
Set Up

Abstract The temperature inversion symmetry of the partition function of the electromagnetic field in the set-up of the Casimir effect is extended to full modular transformations by turning on a purely imaginary chemical potential for adapted spin angular momentum. The extended partition function is expressed in terms of a real analytic Eisenstein series. These results become transparent after explicitly showing equivalence of the partition functions for Maxwell’s theory between perfectly conducting parallel plates and for a massless scalar with periodic boundary conditions.

scholarly journals Metaplectic flavor symmetries from magnetized tori

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Yahya Almumin ◽  
Mu-Chun Chen ◽  
Víctor Knapp-Pérez ◽  
Saúl Ramos-Sánchez ◽  
Michael Ratz ◽  
...  
Keyword(s):  
Zero Mode ◽  
Metaplectic Groups ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Yukawa Couplings ◽  
Flavor Symmetries ◽  
Euler’S Theorem ◽  
Modular Transformations ◽  
Analytic Expressions ◽  
Geometric Picture

Abstract We revisit the flavor symmetries arising from compactifications on tori with magnetic background fluxes. Using Euler’s Theorem, we derive closed form analytic expressions for the Yukawa couplings that are valid for arbitrary flux parameters. We discuss the modular transformations for even and odd units of magnetic flux, M, and show that they give rise to finite metaplectic groups the order of which is determined by the least common multiple of the number of zero-mode flavors involved. Unlike in models in which modular flavor symmetries are postulated, in this approach they derive from an underlying torus. This allows us to retain control over parameters, such as those governing the kinetic terms, that are free in the bottom-up approach, thus leading to an increased predictivity. In addition, the geometric picture allows us to understand the relative suppression of Yukawa couplings from their localization properties in the compact space. We also comment on the role supersymmetry plays in these constructions, and outline a path towards non-supersymmetric models with modular flavor symmetries.

Conformal Field Theory of Free Bosonic and Fermionic Fields

2020 ◽  
pp. 443-475
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Field Theories ◽  
Majorana Fermion ◽  
Partition Functions ◽  
Conformal Field Theories ◽  
Bosonic Field ◽  
Conformal Field ◽  
Mathematical Identities ◽  
Modular Transformations ◽  
Fermionic Fields ◽  
The Subject

Free theories are usually regarded as trivial examples of quantum systems. This chapter proves that this is not the case of the conformal field theories associated to the free bosonic and fermionic fields. The subject is not only full of beautiful mathematical identities but is also the source of deep physical concepts with far reaching applications. Chapter 12 also covers quantization of the bosonic field, vertex operators, the free bosonic field on a torus, modular transformations, the quantization of the free Majorana fermion, the Neveu–Schwarz and Ramond sectors, fermions on a torus, calculus for anti-commuting quantities and partition functions.

scholarly journals Grove Arctic Curves from Periodic Cluster Modular Transformations

2020 ◽  
Author(s):  
Terrence George
Keyword(s):  
Generating Functions ◽  
Gibbs Measures ◽  
The Arctic ◽  
Probability Measures ◽  
Finite Region ◽  
Resistor Network ◽  
Modular Transformations ◽  
Spanning Forests ◽  
Limit Shapes

Abstract Groves are spanning forests of a finite region of the triangular lattice that are in bijection with Laurent monomials that arise in solutions of the cube recurrence. We introduce a large class of probability measures on groves for which we can compute exact generating functions for edge probabilities. Using the machinery of asymptotics of multivariate generating functions, this lets us explicitly compute arctic curves, generalizing the arctic circle theorem of Petersen and Speyer. Our class of probability measures is sufficiently general that the limit shapes exhibit all solid and gaseous phases expected from the classification of ergodic Gibbs measures in the resistor network model.

FINITE EUCLIDEAN MAGNETIC GROUP AND THETA FUNCTIONS

1992 ◽  
Vol 07 (19) ◽  
pp. 4671-4691 ◽  
Author(s):  
S. FUBINI
Keyword(s):  
Magnetic Field ◽  
Phase Transitions ◽  
Constant Magnetic Field ◽  
Theta Functions ◽  
Type Ii ◽  
Finite Rotations ◽  
Type Ii Superconductors ◽  
Modular Transformations ◽  
Jacobi Theta Functions

The Euclidean magnetic group of translations and rotations in a constant magnetic field is discussed in detail. The eigenfunctions of finite magnetic translations are shown to be related to the quasi periodic Jacobi theta functions, whose group theoretical properties under modular transformations are simply discussed. Invariance under finite rotations is very important; it leads to the two fundamental lattices of 60° and 90° already appearing in the theory of the phase transitions of Type II superconductors.

QUANTIZATION OF REDUCED CHERN–SIMONS GRAVITY WITH A SOURCE

2004 ◽  
Vol 19 (28) ◽  
pp. 4883-4897 ◽  
Author(s):  
A. V. NAZARENKO
Keyword(s):  
Continuous Spectrum ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Quantum States ◽  
Theory Of Constraints ◽  
Volume Operator ◽  
Modular Transformations ◽  
Probability Of Transition ◽  
Chern Simons

Using the Dirac's theory of constraints, procedure of reduction of field degrees of freedom, whose number is restricted by equations of motion and topological conditions, is proposed. Such a procedure is applied in the case of space with the topology of a torus to the Chern–Simons gravity generalized by inclusion of a source. It is shown that in this system some modular transformations preserving the volume do not lead to physically equivalent states. Such a breaking of modular symmetry reduces the degeneration of quantum states with preservation of continuous spectrum of the volume operator. Probability of transition between spaces of different volumes is computed.

Modular transformations and one-polygon tessellation

1998 ◽  
Vol 418 (1-2) ◽  
pp. 42-45 ◽  
Author(s):  
R. Franzosi ◽  
M. Ghilardi ◽  
E. Guadagnini
Keyword(s):  
Modular Transformations
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