scholarly journals Rademacher expansions and the spectrum of 2d CFT

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Luis F. Alday ◽  
Jin-Beom Bae
Keyword(s):  
Partition Function ◽  
Modular Forms ◽  
Analytic Number Theory ◽  
Exact Formula ◽  
Conformal Field Theories ◽  
Light Spectrum ◽  
Chiral Algebra ◽  
Positive Weight ◽  
Conformal Field ◽  
Pure Gravity

Abstract A classical result from analytic number theory by Rademacher gives an exact formula for the Fourier coefficients of modular forms of non-positive weight. We apply similar techniques to study the spectrum of two-dimensional unitary conformal field theories, with no extended chiral algebra and c > 1. By exploiting the full modular constraints of the partition function we propose an expression for the spectral density in terms of the light spectrum of the theory. The expression is given in terms of a Rademacher expansion, which converges for spin j ≠ 0. For a finite number of light operators the expression agrees with a variant of the Poincare construction developed by Maloney, Witten and Keller. With this framework we study the presence of negative density of states in the partition function dual to pure gravity, and propose a scenario to cure this negativity.

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.

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 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 RATIONAL CONFORMAL FIELD THEORY AND MULTIWORMHOLE PARTITION FUNCTION IN THREE-DIMENSIONAL GRAVITY

1993 ◽  
Vol 08 (22) ◽  
pp. 3909-3932 ◽  
Author(s):  
SHUN’YA MIZOGUCHI
Keyword(s):  
Partition Function ◽  
Initial Data ◽  
Three Dimensional ◽  
Lens Space ◽  
Conformal Field Theories ◽  
Positive Cosmological Constant ◽  
Modular Invariant ◽  
Conformal Field ◽  
Dimensional Gravity ◽  
Chern Simons

We study the Turaev-Viro (TV) invariant as the Euclidean Chern-Simons-Witten gravity partition function with positive cosmological constant. After explaining why it can be identified as the partition function of three-dimensional gravity, we show that the initial data of the TV invariant can be constructed from the duality data of a certain class of rational conformal field theories, and that, in particular, the original TV initial data are associated with the Ak+1 modular invariant SU(2) WZW model. As a corollary we then show that the partition function Z(M) is bounded from above by [Formula: see text], where g is the smallest genus of handlebodies with which M can be presented by Hegaard splitting. Z(M) is generically very large near Λ~+0 if M is neither S3 nor a lens space, and many-wormhole configurations dominate near Λ~+0 in the sense that Z(M) generically tends to diverge faster as the “number of wormholes” g becomes larger.

scholarly journals KRAMERS–WANNIER DUALITIES FOR WZW THEORIES AND MINIMAL MODELS

2008 ◽  
Vol 10 (05) ◽  
pp. 773-789 ◽  
Author(s):  
CHRISTOPH SCHWEIGERT ◽  
EFROSSINI TSOUCHNIKA
Keyword(s):  
Partition Function ◽  
Charge Conjugation ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Chiral Symmetries ◽  
Conformal Field

We study Kramers–Wannier dualities for Wess–Zumino–Witten theories and (super-) minimal models in the Cardy case, i.e. the case with bulk partition function given by charge conjugation. Using the TFT approach to full rational conformal field theories, we classify those dualities that preserve all chiral symmetries. Dualities turn out to exist for small levels only.

scholarly journals Chaos and pole skipping in CFT2

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
David M. Ramirez
Keyword(s):  
Energy Density ◽  
Quantum Chaos ◽  
Central Charge ◽  
Finite Size ◽  
Complex Frequency ◽  
Conformal Field Theories ◽  
Point Function ◽  
Light Spectrum ◽  
Conformal Field ◽  
Stress Energy

Abstract Recent work has suggested an intriguing relation between quantum chaos and energy density correlations, known as pole skipping. We investigate this relationship in two dimensional conformal field theories on a finite size spatial circle by studying the thermal energy density retarded two-point function on a torus. We find that the location ω* = iλ of pole skipping in the complex frequency plane is determined by the central charge and the stress energy one-point function 〈T〉 on the torus. In addition, we find a bound on λ in c > 1 compact, unitary CFT2s identical to the chaos bound, λ ≤ 2πT. This bound is saturated in large c CFT2s with a sparse light spectrum, as quantified by [1], for all temperatures above the dual Hawking-Page transition temperature.

CONFORMAL FIELD THEORIES WITH ADDITIONAL ZN SYMMETRY

1987 ◽  
Vol 02 (01) ◽  
pp. 165-178 ◽  
Author(s):  
M. BERSHADSKY ◽  
A. RADUL
Keyword(s):  
String Theory ◽  
Partition Function ◽  
Correlation Functions ◽  
Bosonic String ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Loop Amplitude ◽  
Bosonic String Theory

In this paper we construct the correlation functions for ZN twists. These correlation functions are connected with partition function of string propagating on orbifold. Using the correlation functions of Z2 twists we reproduce the two-loop amplitude in the bosonic string theory.

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.

ARITHMETIC TRACES OF NON-HOLOMORPHIC MODULAR INVARIANTS

2010 ◽  
Vol 06 (01) ◽  
pp. 69-87 ◽  
Author(s):  
ALISON MILLER ◽  
AARON PIXTON
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Algebraic Numbers ◽  
Heegner Points ◽  
Positive Weight ◽  
Poincare Series ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Modular Invariants

We extend results of Bringmann and Ono that relate certain generalized traces of Maass–Poincaré series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences [Formula: see text] and [Formula: see text]. We show that if f is a modular form of non-positive weight 2 - 2 λ and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms [Formula: see text].

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.

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