Smoothness of Correlation Functions in Liouville Conformal Field Theory

Oikarinen, Joona

doi:10.1007/s00023-019-00789-0

Cited by 7 publications

(14 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Instead of deriving the conformal Ward identities by varying the background metric, the authors of [ 13 ] defined (the zz -component of) the stress-energy tensor directly as the field and computed the correlation functions ( 1.5 ) for for a specific metric by Gaussian integration by parts. Generalizing this approach to arbitrary n was obstructed by a lack of proof of smoothness of the correlation functions ( 1.1 ) (which was later proven [ 17 ]) and the difficulty of simplifying the expressions coming from the integration by parts. It is however necessary to have ( 1.8 ) for arbitrary n in order to construct the representation of the Virasoro algebra for LCFT.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…This is the motivation and the objective of the present paper. Its main technical input is the recent proof of smoothness of the LCFT correlation functions by the second author [17].…”

Section: Path Integrals and Liouville Conformal Field Theorymentioning

confidence: 99%

“…We will now argue that smoothness of Fĝ in will follow from smoothness of ϕ and u in (recall that ψ(z) = z + u(z)). First, to prove smoothness of the expectation on the righthand side of (3.20) we use the result of the second author [17] that the correlation function…”

Section: N Have Disjoint Compact Supports and Setmentioning

confidence: 99%

“…By Proposition 3.1 for small enough we have Writing the Beltrami coefficient ( 3.5 ) is and the function is given by Now by the diffeomorphism and Weyl transformation laws in Proposition 2.4 We will now argue that smoothness of in will follow from smoothness of and u in (recall that ). First, to prove smoothness of the expectation on the right-hand side of ( 3.20 ) we use the result of the second author [ 17 ] that the correlation function is smooth in the region of non-coinciding points. Since is a diffeomorphism the points are non-coinciding as well and smoothness of the expectation in follows.…”

Section: Conformal Ward Identitiesmentioning

confidence: 99%

See 3 more Smart Citations

Stress-Energy in Liouville Conformal Field Theory

Kupiainen

Oikarinen

2020

J Stat Phys

Self Cite

View full text Add to dashboard Cite

We construct the stress-energy tensor correlation functions in probabilistic Liouville conformal field theory (LCFT) on the two-dimensional sphere S 2 by studying the variation of the LCFT correlation functions with respect to a smooth Riemannian metric on S 2. In particular we derive conformal Ward identities for these correlation functions. This forms the basis for the construction of a representation of the Virasoro algebra on the canonical Hilbert space of the LCFT. In Kupiainen et al. (Commun Math Phys 371:1005-1069, 2019) the conformal Ward identities were derived for one and two stress-energy tensor insertions using a different definition of the stress-energy tensor and Gaussian integration by parts. By defining the stress-energy correlation functions as functional derivatives of the LCFT correlation functions and using the smoothness of the LCFT correlation functions proven in Oikarinen (Ann Henri Poincaré 20(7):2377-2406, 2019) allows us to control an arbitrary number of stress-energy tensor insertions needed for representation theory.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…This is the motivation and the objective of the present paper. Its main technical input is the recent proof of smoothness of the LCFT correlation functions by the second author [17].…”

Section: Path Integrals and Liouville Conformal Field Theorymentioning

confidence: 99%

Section: N Have Disjoint Compact Supports and Setmentioning

confidence: 99%

Section: Conformal Ward Identitiesmentioning

confidence: 99%

See 2 more Smart Citations

Stress-Energy in Liouville Conformal Field Theory

Kupiainen

Oikarinen

2020

J Stat Phys

Self Cite

View full text Add to dashboard Cite

show abstract

“…This is the motivation and the objective of the present paper. Its main technical input is the recent proof of smoothness of the LCFT correlation functions by the second author [14].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Stress-Energy in Liouville Conformal Field Theory

Kupiainen,

Oikarinen

2019

Preprint

Self Cite

View full text Add to dashboard Cite

We construct the Stress-Energy tensor correlation functions in probabilistic Liouville Conformal Field Theory (LCFT) on the two dimensional sphere S 2 by studying the variation of the LCFT correlation functions with respect to a smooth Riemannian metric on S 2 . In particular we derive Conformal Ward identities for these correlations. This forms the basis for the construction of a representation of the Virasoro algebra on the canonical Hilbert space of the LCFT.

show abstract

A Second-Quantized Kolmogorov–Chentsov Theorem via the Operator Product Expansion

Abdesselam

2020

Commun. Math. Phys.

View full text Add to dashboard Cite

We establish a direct connection between two fundamental topics: one in probability theory and one in quantum field theory. The first topic is the problem of pointwise multiplication of random Schwartz distributions which has been the object of recent progress thanks to Hairer's theory of regularity structures and the theory of paracontrolled distributions introduced by Gubinelli, Imkeller and Perkowski. The second topic is Wilson's operator product expansion which is a general property of models of quantum field theory and a cornerstone of the bootstrap approach to conformal field theory. Our main result is a general theorem for the almost sure construction of products of random distributions by mollification and suitable additive as well as multiplicative renormalizations. The hypothesis for this theorem is the operator product expansion with precise bounds for pointwise correlations. We conjecture these bounds to be universal features of quantum field theories with gapped dimension spectrum. Our theorem can accommodate logarithmic corrections, anomalous scaling dimensions and even lack of translation invariance. However, it only applies to fields with short distance singularities that are milder than white noise. As an application, we provide a detailed treatment of a scalar conformal field theory of mean field type, i.e., the fractional massless free field also known as the fractional Gaussian field. Contents 4.3. Graph construction 33 4.4. Two-scale pin and sum argument 34 4.5. Power-counting 36 5. Proof of the main theorem 37 6. A detailed example: the fractional massless free field 38 6.1. The explicit abstract system of pointwise correlations and the BNNFB 38 6.2. The OPE structure 40 6.3. A proof of the ENNFB 44 6.4. A proof of the soft and hard hypotheses on the OPE coefficients 47 6.5. Application of the main theorem to the construction of local Wick monomials as random distributions 48 6.6. Conformal invariance 49 References 50 1.2. Abstract systems of pointwise correlations. Let A be a finite set which we will call an alphabet and will serve to label fields. Define the big diagonal

show abstract

Smoothness of Correlation Functions in Liouville Conformal Field Theory

Cited by 7 publications

References 34 publications

Stress-Energy in Liouville Conformal Field Theory

Stress-Energy in Liouville Conformal Field Theory

Stress-Energy in Liouville Conformal Field Theory

A Second-Quantized Kolmogorov–Chentsov Theorem via the Operator Product Expansion

Contact Info

Product

Resources

About