scholarly journals Logarithmic CFT at generic central charge: from Liouville theory to the $Q$-state Potts model

2021 ◽  
Vol 10 (1) ◽  
Author(s):  
Rongvoram Nivesvivat ◽  
Sylvain Ribault
Keyword(s):  
Potts Model ◽  
Free Parameter ◽  
Central Charge ◽  
Liouville Theory ◽  
Conformal Dimension ◽  
Conformal Blocks ◽  
Potts Models ◽  
Dimension 2 ◽  
Arbitrary Precision ◽  
Derivatives Of

Using derivatives of primary fields (null or not) with respect to the conformal dimension, we build infinite families of non-trivial logarithmic representations of the conformal algebra at generic central charge, with Jordan blocks of dimension 2 or 3. Each representation comes with one free parameter, which takes fixed values under assumptions on the existence of degenerate fields. This parameter can be viewed as a simpler, normalization-independent redefinition of the logarithmic coupling. We compute the corresponding non-chiral conformal blocks, and show that they appear in limits of Liouville theory four-point functions.As an application, we describe the logarithmic structures of the critical two-dimensional O(n) and Q-state Potts models at generic central charge. The validity of our description is demonstrated by semi-analytically bootstrapping four-point connectivities in the Q-state Potts model to arbitrary precision. Moreover, we provide numerical evidence for the Delfino-Viti conjecture for the three-point connectivity. Our results hold for generic values of Q in the complex plane and beyond.

scholarly journals Geometrical four-point functions in the two-dimensional critical Q-state Potts model: the interchiral conformal bootstrap

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Yifei He ◽  
Jesper Lykke Jacobsen ◽  
Hubert Saleur
Keyword(s):  
Potts Model ◽  
Correlation Functions ◽  
Liouville Theory ◽  
Analytical Determination ◽  
Conformal Weight ◽  
Two Dimensional ◽  
Conformal Blocks ◽  
Conformal Bootstrap ◽  
Point Correlation

Abstract Based on the spectrum identified in our earlier work [1], we numerically solve the bootstrap to determine four-point correlation functions of the geometrical connectivities in the Q-state Potts model. Crucial in our approach is the existence of “interchiral conformal blocks”, which arise from the degeneracy of fields with conformal weight hr,1, with r ∈ ℕ*, and are related to the underlying presence of the “interchiral algebra” introduced in [2]. We also find evidence for the existence of “renormalized” recursions, replacing those that follow from the degeneracy of the field $$ {\Phi}_{12}^D $$ Φ 12 D in Liouville theory, and obtain the first few such recursions in closed form. This hints at the possibility of the full analytical determination of correlation functions in this model.

scholarly journals A NUMERICAL TEST OF KPZ SCALING: POTTS MODELS COUPLED TO TWO-DIMENSIONAL QUANTUM GRAVITY

1992 ◽  
Vol 07 (17) ◽  
pp. 1519-1533 ◽  
Author(s):  
C.F. BAILLIE ◽  
D.A. JOHNSTON
Keyword(s):  
Exact Solution ◽  
Potts Model ◽  
Cluster Algorithm ◽  
Central Charge ◽  
Numerical Test ◽  
Potts Models ◽  
First Order ◽  
First Order Phase Transition ◽  
Spherical Topology ◽  
First Order Phase

We perform Monte-Carlo simulations using the Wolff cluster algorithm of the q=2 (Ising), 3, 4 and q=10 Potts models on dynamical phi-cubed graphs of spherical topology with up to 5000 nodes. We find that the measured critical exponents are in reasonable agreement with those from the exact solution of the Ising model and with those calculated from KPZ scaling for q=3, 4 where no exact solution is available. Using Binder’s cumulant we find that the q=10 Potts model displays a first order phase transition on a dynamical graph, as it does on a fixed lattice. We also examine the internal geometry of the graphs generated in the simulation, finding a linear relationship between ring length probabilities and the central charge of the Potts model.

scholarly journals Analytic conformal bootstrap and Virasoro primary fields in the Ashkin-Teller model

2021 ◽  
Vol 11 (5) ◽  
Author(s):  
Nikita Nemkov ◽  
Sylvain Ribault
Keyword(s):  
Correlation Functions ◽  
Central Charge ◽  
Theta Functions ◽  
Conformal Dimension ◽  
Conformal Blocks ◽  
Conformal Bootstrap ◽  
Crossing Symmetry ◽  
Point Correlation ◽  
Structure Constants ◽  
Twist Fields

We revisit the critical two-dimensional Ashkin–Teller model, i.e. the \mathbb{Z}_2ℤ2 orbifold of the compactified free boson CFT at c=1c=1. We solve the model on the plane by computing its three-point structure constants and proving crossing symmetry of four-point correlation functions. We do this not only for affine primary fields, but also for Virasoro primary fields, i.e. higher twist fields and degenerate fields. This leads us to clarify the analytic properties of Virasoro conformal blocks and fusion kernels at c=1c=1. We show that blocks with a degenerate channel field should be computed by taking limits in the central charge, rather than in the conformal dimension. In particular, Al. Zamolodchikov’s simple explicit expression for the blocks that appear in four-twist correlation functions is only valid in the non-degenerate case: degenerate blocks, starting with the identity block, are more complicated generalized theta functions.

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.

scholarly journals LIOUVILLE THEORY: QUANTUM GEOMETRY OF RIEMANN SURFACES

1993 ◽  
Vol 08 (37) ◽  
pp. 3529-3535 ◽  
Author(s):  
LEON A. TAKHTAJAN
Keyword(s):  
Riemann Surfaces ◽  
Quantum Correction ◽  
Central Charge ◽  
Perturbation Expansion ◽  
Functional Integral ◽  
Liouville Theory ◽  
Conformal Anomaly ◽  
Quantum Geometry ◽  
Loop Contribution ◽  
Classical Expression

Inspired by Polyakov’s original formulation1,2 of quantum Liouville theory through functional integral, we analyze perturbation expansion around a classical solution. We show the validity of conformal Ward identities for puncture operators and prove that their conformal dimension is given by the classical expression. We also prove that the total quantum correction to the central charge of Liouville theory is given by one-loop contribution, which is equal to 1. Applied to the bosonic string, this result ensures the vanishing of total conformal anomaly along the lines different from those presented by KPZ3 and Distler-Kawai.4

scholarly journals The non-rational limit of D-series minimal models

2020 ◽  
Vol 3 (1) ◽  
Author(s):  
Sylvain Ribault
Keyword(s):  
Central Charge ◽  
Structure Constant ◽  
Liouville Theory ◽  
Two Dimensional ◽  
Minimal Models ◽  
Distribution Factor ◽  
Smooth Functions ◽  
Limit Theory ◽  
Conformal Field ◽  
Verlinde Formula

We study the limit of D-series minimal models when the central charge tends to a generic irrational value c\in (-\infty, 1)c∈(−∞,1). We find that the limit theory’s diagonal three-point structure constant differs from that of Liouville theory by a distribution factor, which is given by a divergent Verlinde formula. Nevertheless, correlation functions that involve both non-diagonal and diagonal fields are smooth functions of the diagonal fields’ conformal dimensions. The limit theory is a non-trivial example of a non-diagonal, non-rational, solved two-dimensional conformal field theory.

scholarly journals On four-point connectivities in the critical 2d Potts model

2019 ◽  
Vol 7 (4) ◽  
Author(s):  
Marco Picco ◽  
Sylvain Ribault ◽  
Raoul Santachiara
Keyword(s):  
Monte Carlo ◽  
Half Plane ◽  
Potts Model ◽  
Central Charge ◽  
Symmetry Algebra ◽  
Double Cover ◽  
Minimal Models ◽  
Virasoro Symmetry ◽  
Special Cases

We perform Monte-Carlo computations of four-point cluster connectivities in the critical 2d Potts model, for numbers of states Q\in (0,4)Q∈(0,4) that are not necessarily integer. We compare these connectivities to four-point functions in a CFT that interpolates between D-series minimal models. We find that 3 combinations of the 4 independent connectivities agree with CFT four-point functions, down to the 22 to 44 significant digits of our Monte-Carlo computations. However, we argue that the agreement is exact only in the special cases Q=0, 3, 4Q=0,3,4. We conjecture that the Potts model can be analytically continued to a double cover of the half-plane \{\Re c <13\}{ℜc<13}, where cc is the central charge of the Virasoro symmetry algebra.

scholarly journals LIOUVILLE THEORY: WARD IDENTITIES FOR GENERATING FUNCTIONAL AND MODULAR GEOMETRY

1994 ◽  
Vol 09 (25) ◽  
pp. 2293-2299 ◽  
Author(s):  
LEON A. TAKHTAJAN
Keyword(s):  
Moduli Spaces ◽  
Correlation Functions ◽  
Riemann Surfaces ◽  
Central Charge ◽  
Perturbation Expansion ◽  
Liouville Theory ◽  
Energy Tensor ◽  
Ward Identities ◽  
Generating Functional ◽  
Local Index

We continue the study of quantum Liouville theory through Polyakov’s functional integral,1,2 started in Ref. 3. We derive the perturbation expansion for Schwinger’s generating functional for connected multi-point correlation functions involving stress-energy tensor, give the “dynamical” proof of the Virasoro symmetry of the theory and compute the value of the central charge, confirming previous calculation in Ref. 3. We show that conformal Ward identities for these correlation functions contain such basic facts from Kähler geometry of moduli spaces of Riemann surfaces, as relation between accessory parameters for the Fuchsian uniformization, Liouville action and Eichler integrals, Kähler potential for the Weil-Petersson metric, and local index theorem. These results affirm the fundamental role that universal Ward identities for the generating functional play in Friedan-Shenker modular geometry.4

scholarly journals The large-c Virasoro identity block is a semi-classical Liouville correlator

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Gideon Vos
Keyword(s):  
Correlation Function ◽  
Liouville Equation ◽  
Classical Limit ◽  
Central Charge ◽  
Equation Of Motion ◽  
Einstein Gravity ◽  
Liouville Theory ◽  
Vertex Operators

Abstract It will be shown analytically that the light sector of the identity block of a mixed heavy-light correlator in the large central charge limit is given by a correlation function of light operators on an effective background geometry. This geometry is generated by the presence of the heavy operators. It is shown that this background geometry is a solution to the Liouville equation of motion sourced by corresponding heavy vertex operators and subsequently that the light sector of the identity block matches the Liouville correlation function in the semi-classical limit. This method effectively captures the spirit of Einstein gravity as a theory of dynamical geometry in AdS/CFT. The reason being that Liouville theory is closely related to semi-classical asymptotically AdS3 gravity.

A Continuum Approach to 2D-Quantum Gravity for c>1

1997 ◽  
Vol 11 (26n27) ◽  
pp. 3247-3279
Author(s):  
M. Martellini ◽  
M. Spreafico ◽  
K. Yoshida
Keyword(s):  
Quantum Gravity ◽  
Scalar Curvature ◽  
Free Parameter ◽  
Central Charge ◽  
Flat Space ◽  
String Tension ◽  
Physical Significance ◽  
Two Dimensional ◽  
Continuum Approach ◽  
Gauge Fixing

Two dimensional induced quantum gravity with matter central charge c>1 is studied by carefully treating both diffeomorphism and Weyl symmetries. It is shown that, for the gauge fixing condition R(g) (scalar curvature) = const, one obtains a modification of the David–Distler–Kawai version of KPZ scaling. We obtain a class of models with real string tension for all values c>1. They contain a free parameter which is, however, strongly constrained by the requirement of the non triviality of the model. The possible physical significance of the new model is discussed. In particular we note that it describes smooth surfaces imbedded in d-dimensional flat space time for arbitrary d, which is consistent with recent numerical results for d=3.

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