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 On rational points in CFT moduli spaces

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Nathan Benjamin ◽  
Christoph A. Keller ◽  
Hirosi Ooguri ◽  
Ida G. Zadeh
Keyword(s):  
Perturbation Theory ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Sigma Model ◽  
Rational Points ◽  
Target Space ◽  
Field Theories ◽  
Two Dimensional ◽  
Conformal Field Theories ◽  
Conformal Field

Abstract Motivated by the search for rational points in moduli spaces of two-dimensional conformal field theories, we investigate how points with enhanced symmetry algebras are distributed there. We first study the bosonic sigma-model with S1 target space in detail and uncover hitherto unknown features. We find for instance that the vanishing of the twist gap, though true for the S1 example, does not automatically follow from enhanced symmetry points being dense in the moduli space. We then explore the supersymmetric sigma-model on K3 by perturbing away from the torus orbifold locus. Though we do not reach a definite conclusion on the distribution of enhanced symmetry points in the K3 moduli space, we make several observations on how chiral currents can emerge and disappear under conformal perturbation theory.

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 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.

scholarly journals Defects and perturbation

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Enrico M. Brehm
Keyword(s):  
Entanglement Entropy ◽  
Topological Defects ◽  
Field Theories ◽  
Two Dimensional ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Conformal Field

Abstract We investigate perturbatively tractable deformations of topological defects in two-dimensional conformal field theories. We perturbatively compute the change in the g-factor, the reflectivity, and the entanglement entropy of the conformal defect at the end of these short RG flows. We also give instances of such flows in the diagonal Virasoro and Super-Virasoro Minimal Models.

scholarly journals On 2d CFTs that interpolate between minimal models

2019 ◽  
Vol 6 (6) ◽  
Author(s):  
Sylvain Ribault
Keyword(s):  
Correlation Functions ◽  
Central Charge ◽  
Field Theories ◽  
Two Dimensional ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Exactly Solvable ◽  
Structure Constants

We investigate exactly solvable two-dimensional conformal field theories that exist at generic values of the central charge, and that interpolate between A-series or D-series minimal models. When the central charge becomes rational, correlation functions of these CFTs may tend to correlation functions of minimal models, or diverge, or have finite limits which can be logarithmic. These results are based on analytic relations between four-point structure constants and residues of conformal blocks.

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.

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