scholarly journals Lessons from the Ramond sector

2020 ◽  
Vol 9 (5) ◽  
Author(s):  
Nathan Benjamin ◽  
Ying-Hsuan Lin
Keyword(s):  
Partition Function ◽  
Field Theories ◽  
Partition Functions ◽  
Modular Invariance ◽  
Conformal Field Theories ◽  
Modular Functions ◽  
Conformal Field ◽  
Underlying Theory ◽  
Rule Out

We revisit the consistency of torus partition functions in (1+1)d fermionic conformal field theories, combining old ingredients of modular invariance/covariance with a modernized understanding of bosonization/fermionization dualities. Various lessons can be learned by simply examining the oft-ignored Ramond sector. For several extremal/kinky modular functions in the bootstrap literature, we can either rule out or identify the underlying theory. We also revisit the N = 1 Maloney-Witten partition function by calculating the spectrum in the Ramond sector, and further extending it to include the modular sum of seed Ramond characters. Finally, we perform the full N = 1 RNS modular bootstrap and obtain new universal results on the existence of relevant deformations preserving different amounts of supersymmetry.

scholarly journals Harmonic analysis of 2d CFT partition functions

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Nathan Benjamin ◽  
Scott Collier ◽  
A. Liam Fitzpatrick ◽  
Alexander Maloney ◽  
Eric Perlmutter
Keyword(s):  
Partition Function ◽  
Harmonic Analysis ◽  
Moduli Space ◽  
Target Space ◽  
Field Theories ◽  
Partition Functions ◽  
Two Dimensional ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Operator Spectrum

Abstract We apply the theory of harmonic analysis on the fundamental domain of SL(2, ℤ) to partition functions of two-dimensional conformal field theories. We decompose the partition function of c free bosons on a Narain lattice into eigenfunctions of the Laplacian of worldsheet moduli space ℍ/SL(2, ℤ), and of target space moduli space O(c, c; ℤ)\O(c, c; ℝ)/O(c)×O(c). This decomposition manifests certain properties of Narain theories and ensemble averages thereof. We extend the application of spectral theory to partition functions of general two-dimensional conformal field theories, and explore its meaning in connection to AdS3 gravity. An implication of harmonic analysis is that the local operator spectrum is fully determined by a certain subset of degeneracies.

scholarly journals Quantum criticality in many-body parafermion chains

2021 ◽  
Vol 4 (2) ◽  
Author(s):  
Ville Lahtinen ◽  
Teresia Mansson ◽  
Eddy Ardonne
Keyword(s):  
Partition Function ◽  
Boundary Conditions ◽  
Quantum Criticality ◽  
Field Theories ◽  
Partition Functions ◽  
Conformal Field Theories ◽  
Many Body ◽  
Modular Invariant ◽  
Potts Models ◽  
Conformal Field

We construct local generalizations of 3-state Potts models with exotic critical points. We analytically show that these are described by non-diagonal modular invariant partition functions of products of Z_3Z3 parafermion or u(1)_6u(1)6 conformal field theories (CFTs). These correspond either to non-trivial permutation invariants or block diagonal invariants, that one can understand in terms of anyon condensation. In terms of lattice parafermion operators, the constructed models correspond to parafermion chains with many-body terms. Our construction is based on how the partition function of a CFT depends on symmetry sectors and boundary conditions. This enables to write the partition function corresponding to one modular invariant as a linear combination of another over different sectors and boundary conditions, which translates to a general recipe how to write down a microscopic model, tuned to criticality. We show that the scheme can also be extended to construct critical generalizations of kk-state clock type models.

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 BITS AND PIECES IN LOGARITHMIC CONFORMAL FIELD THEORY

2003 ◽  
Vol 18 (25) ◽  
pp. 4497-4591 ◽  
Author(s):  
MICHAEL A. I. FLOHR
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Quantum Hall ◽  
Field Theories ◽  
Partition Functions ◽  
Conformal Field Theories ◽  
Fractional Quantum Hall ◽  
Conformal Field ◽  
Logarithmic Conformal Field Theory ◽  
Verlinde Formula

These are notes of my lectures held at the first School & Workshop on Logarithmic Conformal Field Theory and its Applications, September 2001 in Tehran, Iran. These notes cover only selected parts of the by now quite extensive knowledge on logarithmic conformal field theories. In particular, I discuss the proper generalization of null vectors towards the logarithmic case, and how these can be used to compute correlation functions. My other main topic is modular invariance, where I discuss the problem of the generalization of characters in the case of indecomposable representations, a proposal for a Verlinde formula for fusion rules and identities relating the partition functions of logarithmic conformal field theories to such of well known ordinary conformal field theories. The two main topics are complemented by some remarks on ghost systems, the Haldane-Rezayi fractional quantum Hall state, and the relation of these two to the logarithmic c=-2 theory.

scholarly journals TOWERS OF ALGEBRAS IN RATIONAL CONFORMAL FIELD THEORIES

1991 ◽  
Vol 06 (12) ◽  
pp. 2045-2074 ◽  
Author(s):  
CÉSAR GOMEZ ◽  
GERMAN SIERRA
Keyword(s):  
Sufficient Conditions ◽  
Field Theories ◽  
Partition Functions ◽  
Polynomial Equations ◽  
Conformal Field Theories ◽  
Modular Invariant ◽  
Conformal Field ◽  
Ade Classification

Jones fundamental construction is applied to rational conformal field theories. The Jones algebra which emerges in this application is realized in terms of duality operations. The generators of the algebra are an open version of Verlinde’s operators. The polynomial equations appear in this context as sufficient conditions for the existence of Jones algebra. The ADE classification of modular invariant partition functions is put in correspondence with Jones classification of subfactors.

scholarly journals CLONING SO(N) LEVEL 2

1999 ◽  
Vol 14 (08) ◽  
pp. 1283-1291 ◽  
Author(s):  
A. N. SCHELLEKENS
Keyword(s):  
Infinite Number ◽  
Essential Role ◽  
Field Theories ◽  
Partition Functions ◽  
Conformal Field Theories ◽  
Modular Invariant ◽  
Chiral Algebra ◽  
Conformal Field ◽  
N Level ◽  
Level 2

For each N an infinite number of conformal field theories is presented that has the same fusion rules as SO (N) level 2. These new theories are obtained as extensions of the chiral algebra of SO (NM2) level 2, and correspond to new modular invariant partition functions of these theories. A one-to-one map between the c=1 orbifolds of radius R2=2r and Dr level 2 plays an essential role.

CONFORMAL FIELD THEORIES ON SURFACES WITH BOUNDARIES AND CROSSCAPS

1989 ◽  
Vol 04 (02) ◽  
pp. 161-168 ◽  
Author(s):  
TETSUYA ONOGI ◽  
NOBUYUKI ISHIBASHI
Keyword(s):  
Crucial Role ◽  
Model Building ◽  
Open String ◽  
Theory Model ◽  
Field Theories ◽  
Partition Functions ◽  
Conformal Field Theories ◽  
Modular Invariant ◽  
Conformal Field ◽  
Loop Channel

We classify the possible operator contents of the minimal conformal field theories when boundaries and crosscaps are present by imposing loop channel-tree channel duality conditions. These are the open string analogues of modular invariant partition functions, which play a crucial role in string theory model building.

scholarly journals THE CAPPELLI–ITZYKSON–ZUBER A-D-E CLASSIFICATION

2000 ◽  
Vol 12 (05) ◽  
pp. 739-748 ◽  
Author(s):  
TERRY GANNON
Keyword(s):  
Mathematical Structure ◽  
The Other ◽  
Field Theories ◽  
Partition Functions ◽  
Conformal Field Theories ◽  
Modular Invariant ◽  
Conformal Field ◽  
Affine Algebras ◽  
Order Of Magnitude ◽  
The Rich

In 1986 Cappelli, Itzykson and Zuber classified all modular invariant partition functions for the conformal field theories associated to the affine A1 algebra; they found they fall into an A-D-E pattern. Their proof was difficult and attempts to generalise it to the other affine algebras failed — in hindsight the reason is that their argument ignored most of the rich mathematical structure present. We give here the "modern" proof of their result; it is an order of magnitude simpler and shorter, and much of it has already been extended to all other affine algebras. We conclude with some remarks on the A-D-E pattern appearing in this and other RCFT classifications.

LOCALIZED MODES IN SINGULAR CALABI-YAU CONFORMAL FIELD THEORIES

2008 ◽  
Vol 23 (14n15) ◽  
pp. 2184-2186
Author(s):  
SHUN'YA MIZOGUCHI
Keyword(s):  
Discrete Series ◽  
Field Theories ◽  
Partition Functions ◽  
Type Ii ◽  
Conformal Field Theories ◽  
Modular Invariant ◽  
Heterotic Strings ◽  
Conformal Field ◽  
Series Representations ◽  
Discrete Series Representations

We construct spacetime supersymmetric, modular invariant partition functions for type II and heterotic strings on the conifold-type singularities such that they include contributions coming from the discrete-series representations of SL(2, R). In particular for the E8 × E8 heterotic case, they are in the 27 representation of E6 and localized on a four-dimensional "brane" at the tip of the cigar geometry.

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