**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Concept# Conformal bootstrap

Summary

The conformal bootstrap is a non-perturbative mathematical method to constrain and solve conformal field theories, i.e. models of particle physics or statistical physics that exhibit similar properties at different levels of resolution.
Unlike more traditional techniques of quantum field theory, conformal bootstrap does not use the Lagrangian of the theory. Instead, it operates with the general axiomatic parameters, such as the scaling dimensions of the local operators and their operator product expansion coefficients. A key axiom is that the product of local operators must be expressible as a sum over local operators (thus turning the product into an algebra); the sum must have a non-zero radius of convergence. This leads to decompositions of correlation functions into structure constants and conformal blocks.
The main ideas of the conformal bootstrap were formulated in the 1970s by the Soviet physicist Alexander Polyakov and the Italian physicists Sergio Ferrara, Raoul Gatto and Aurelio Grillo. Other early pioneers of this idea were Gerhard Mack and Ivan Todorov.
In two dimensions, the conformal bootstrap was demonstrated to work in 1983 by Alexander Belavin, Alexander Polyakov and Alexander Zamolodchikov. Many two-dimensional conformal field theories were solved using this method, notably the minimal models and the Liouville field theory.
In higher dimensions, the conformal bootstrap started to develop following the 2008 paper by Riccardo Rattazzi, Slava Rychkov, Erik Tonni and Alessandro Vichi. The method was since used to obtain many general results about conformal and superconformal field theories in three, four, five and six dimensions. Applied to the conformal field theory describing the critical point of the three-dimensional Ising model, it produced the most precise predictions for its critical exponents.
The international Simons Collaboration on the Nonperturbative Bootstrap unites researchers devoted to developing and applying the conformal bootstrap and other related techniques in quantum field theory.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related courses (7)

Related publications (60)

Related people (17)

Related concepts (7)

Related lectures (32)

Related units (3)

Ontological neighbourhood

This course is an introduction to the non-perturbative bootstrap approach to Conformal Field Theory and to the Gauge/Gravity duality, emphasizing the fruitful interplay between these two ideas.

The course builds on the course QFT1 and QFT2 and develops in parallel to the course on Gauge Theories and the SM.

A fundamental subject that students workiing with scattering amplitudes and boot-straps and effective theories must know, and in order to constrain and improve theories.

Operator product expansion

In quantum field theory, the operator product expansion (OPE) is used as an axiom to define the product of fields as a sum over the same fields. As an axiom, it offers a non-perturbative approach to quantum field theory. One example is the vertex operator algebra, which has been used to construct two-dimensional conformal field theories. Whether this result can be extended to QFT in general, thus resolving many of the difficulties of a perturbative approach, remains an open research question.

Two-dimensional conformal field theory

A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations. In contrast to other types of conformal field theories, two-dimensional conformal field theories have infinite-dimensional symmetry algebras. In some cases, this allows them to be solved exactly, using the conformal bootstrap method. Notable two-dimensional conformal field theories include minimal models, Liouville theory, massless free bosonic theories, Wess–Zumino–Witten models, and certain sigma models.

Critical exponent

Critical exponents describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on some of its general features. For instance, for ferromagnetic systems, the critical exponents depend only on: the dimension of the system the range of the interaction the spin dimension These properties of critical exponents are supported by experimental data.

Conformal Blocks in Quantum Physics

Explains conformal blocks in quantum physics and the superposition theorem for total electric field.

Conformal Black Body Radiation

Explores conformal black body radiation, ANEC, and quantum physics applications.

Operator Product Expansion

Explores Operator Product Expansion (OPE) and its role in Conformal Bootstrap.

Alessandro Vichi, Maria Refinetti, Marten Jan Reehorst

We use numerical bootstrap techniques to study correlation functions of traceless sym-metric tensors of O(N) with two indices ti j. We obtain upper bounds on operator dimen-sions for all the relevant representations and several values of N. We discover sev ...

Riccardo Rattazzi, Alexander Monin, Gil Badel

The Large Charge sector of Conformal Field Theory (CFT) can generically be described through a semiclassical expansion around a superfluid background. In this work, focussing on U(1) invariant Wilson-Fisher fixed points, we study the spectrum of spinning l ...

The expectation value of a smooth conformal line defect in a CFT is a conformal invariant functional of its path in space-time. For example, in large N holographic theories, these fundamental observables are dual to the open-string partition function in Ad ...