Exploring field theories via the Conformal Bootstrap

2021 ◽  
Author(s):  
Στέφανος-Ρόμπερτ Κούσβος
Keyword(s):  
Correlation Functions ◽  
Field Theories ◽  
Reflection Positivity ◽  
Conformal Bootstrap

Στην εργασία μελετώνται σύμμορφες θεωρίες πεδίου που ενδιαφέρουν λόγω φαινομενολογικών όπως και θεωρητικών ερωτημάτων. Η μελέτη των εν λόγω θεωριών γίνεται με χρήση συνθηκών αυτοσυνέπειας. Η μελέτη (σύμμορφων) θεωριών με χρήση συνθηκών αυτοσυνέπειας είναι γνωστή ως “the bootstrap”. Στην παρούσα εργασία θα κάνουμε χρήση του αριθμητικού bootstrap όπως αυτό εμφανίστηκε στην πρόσφατη αναβίωση του. Δηλαδή, θα επιβάλουμε την συμμετρία crossing και τις συνθήκες unitarity/reflection positivity σε συγκεκριμένες συναρτήσεις συσχέτισης (correlation functions), και ως αποτέλεσμα θα βρούμε περιορισμούς στον παραμετρικό χώρο. Ο παραμετρικός χώρος παραμετροποιείται με βάση της ποσότητες γνωστές ως scaling dimensions και OPE coefficients. Αυτές οι ποσότητες καθορίζουν διάφορα μετρήσιμα μεγέθη, όπως οι κρίσιμοι εκθέτες που παρατηρούνται σε κρίσιμα σημεία. Με την χρήση του αριθμητικού bootstrap θα παρέχουμε ισχυρούς περιορισμούς των επιτρεπόμενων τιμών που μπορούν να πάρουν συγκεκριμένοι κρίσιμοι εκθέτες. Σε κάποιες περιπτώσεις οι περιορισμοί θα είναι τόσο ισχυροί που στην ουσία αποτελούν υπολογισμό των εν λόγω ποσοτήτων. Με άλλα λόγια, μόνο μικρές απομονωμένες περιορισμένες περιοχές του παραμετρικού χώρου θα είναι συμβατές με τις συνθήκες αυτοσυνέπειας. Μερικά παραδείγματα θεωριών που θα μελετηθούν είναι οι θεωρίες με υπερ-κυβική, Ο(m) x O(n)/Z2 και “ΜΝ” συμμετρία. Θα βρούμε μικρές επιτρεπόμενες απομονωμένες περιοχές στον παραμετρικό χώρο αυτών των θεωριών. Επίσης, κατά την διάρκεια της μελέτης μας θα προσδιορίσουμε διάφορες τανυστικές δομές που είναι χρήσιμες πέραν του αριθμητικού bootstrap.

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 Generalized hydrodynamics regime from the thermodynamic bootstrap program

2020 ◽  
Vol 8 (1) ◽  
Author(s):  
Axel Cortes Cubero ◽  
Milosz Panfil
Keyword(s):  
Energy Density ◽  
Correlation Functions ◽  
Form Factors ◽  
Finite Energy ◽  
Axiomatic Approach ◽  
Field Theories ◽  
Relativistic Field ◽  
Generalized Hydrodynamics ◽  
Quantum Integrable Models ◽  
Integrable Field

Within the generalized hydrodynamics (GHD) formalism for quantum integrable models, it is possible to compute simple expressions for a number of correlation functions at the Eulerian scale. Specializing to integrable relativistic field theories, we show the same correlators can be computed as a sum over form factors, the GHD regime corresponding to the leading contribution with one particle-hole pair on a finite energy-density background. The thermodynamic bootstrap program (TBP) formalism was recently introduced as an axiomatic approach to computing such finite-energy-density form factors for integrable field theories. We derive a new axiom within the TBP formalism from which we easily recover the predicted GHD Eulerian correlators. We also compute higher form factor contributions, with more particle-hole pairs, within the TBP, allowing for the computation of correlation functions in the diffusive, and beyond, GHD regimes. The two particle-hole form factors agree with expressions recently conjectured within the GHD.

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 Gauge Dressing of 2D Field Theories

1997 ◽  
Vol 12 (13) ◽  
pp. 2425-2436 ◽  
Author(s):  
Ian I. Kogan ◽  
Alex Lewis ◽  
Oleg A. Soloviev
Keyword(s):  
Correlation Functions ◽  
Field Theories ◽  
Ward Identities ◽  
Zamolodchikov Equation

By using the gauge Ward identities, we study correlation functions of gauged WZNW models. We show that the gauge dressing of the correlation functions can be taken into account as a solution of the Knizhnik–Zamolodchikov equation. Our method is analogous to the analysis of the gravitational dressing of 2D field theories.

CORRELATION FUNCTIONS OF INTEGRABLE 2D MODELS OF THE RELATIVISTIC FIELD THEORY: ISING MODEL

1991 ◽  
Vol 06 (19) ◽  
pp. 3419-3440 ◽  
Author(s):  
V.P. YUROV ◽  
AL. B. ZAMOLODCHIKOV
Keyword(s):  
Magnetic Field ◽  
Correlation Functions ◽  
Form Factors ◽  
Field Theories ◽  
Scattering Data ◽  
Exact Form ◽  
Relativistic Field ◽  
Spin Correlations ◽  
Ising Spin ◽  
Infinite Sums

A program is proposed to study numerically the correlation functions in massive integrable 2D relativistic field theories. It relies crucially on the exact form factors of fields which can be reconstructed from the factorized scattering data. The correlation functions are expressible as infinite sums over intermediate asymptotic states. We suggest using computer power to perform the summation numerically. The convergence of the sum is tested for the simplest example of the scaling Ising spin-spin correlations (without magnetic field).

scholarly journals Bootstrap experiments on higher dimensional CFTs

2018 ◽  
Vol 33 (07) ◽  
pp. 1850036 ◽  
Author(s):  
Yu Nakayama
Keyword(s):  
Empirical Relationship ◽  
Adjoint Representation ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Time Dimension ◽  
Unitarity Bound ◽  
Conformal Field ◽  
Dimension Formula ◽  
Conformal Bootstrap ◽  
Higher Dimensional

Recent programs on conformal bootstrap suggest an empirical relationship between the existence of nontrivial conformal field theories and nontrivial features such as a kink in the unitarity bound of conformal dimensions in the conformal bootstrap equations. We report the existence of nontrivial kink-like behaviors in the unitarity bound of scalar operators in the adjoint representation of the [Formula: see text] symmetric conformal field theories. They have interesting properties: (1) the kink-like behaviors exist in [Formula: see text] dimensions; (2) the location of kink-like behaviors are when the unitarity bound hits the space–time dimension [Formula: see text]; (3) there exists a “conformal window” of [Formula: see text], where [Formula: see text] in [Formula: see text] and [Formula: see text] in [Formula: see text].

scholarly journals A roadmap for bootstrapping critical gauge theories: decoupling operators of conformal field theories in $d>2$ dimensions

2021 ◽  
Vol 11 (6) ◽  
Author(s):  
Yin-Chen He ◽  
Junchen Rong ◽  
Ning Su
Keyword(s):  
Gauge Group ◽  
Gauge Theories ◽  
Higher Dimensions ◽  
Equation Of Motion ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Gauge Groups ◽  
Conformal Field ◽  
Conformal Bootstrap ◽  
Null Operator

We propose a roadmap for bootstrapping conformal field theories (CFTs) described by gauge theories in dimensions d>2d>2. In particular, we provide a simple and workable answer to the question of how to detect the gauge group in the bootstrap calculation. Our recipe is based on the notion of decoupling operator, which has a simple (gauge) group theoretical origin, and is reminiscent of the null operator of 2d2d Wess-Zumino-Witten CFTs in higher dimensions. Using the decoupling operator we can efficiently detect the rank (i.e. color number) of gauge groups, e.g., by imposing gap conditions in the CFT spectrum. We also discuss the physics of the equation of motion, which has interesting consequences in the CFT spectrum as well. As an application of our recipes, we study a prototypical critical gauge theory, namely the scalar QED which has a U(1)U(1) gauge field interacting with critical bosons. We show that the scalar QED can be solved by conformal bootstrap, namely we have obtained its kinks and islands in both d=3d=3 and d=2+\epsilond=2+ϵ dimensions.

scholarly journals Logarithmic Operators in Conformal Field Theory and the ${\mathcal W}_\infty$-Algebra

1997 ◽  
Vol 12 (21) ◽  
pp. 3723-3738 ◽  
Author(s):  
A. Shafiekhani ◽  
M. R. Rahimi Tabar
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Correlation Functions ◽  
Operator Product Expansion ◽  
Field Theories ◽  
Operator Product ◽  
Tensor Operator ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Point Correlation

It is shown explicitly that the correlation functions of conformal field theories (CFT) with the logarithmic operators are invariant under the differential realization of Borel subalgebra of [Formula: see text]-algebra. This algebra is constructed by tensor-operator algebra of differential representation of ordinary [Formula: see text]. This method allows us to write differential equations which can be used to find general expression for three- and four-point correlation functions possessing logarithmic operators. The operator product expansion (OPE) coefficients of general logarithmic CFT are given up to third level.

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