Periodic Reporting for period 1 - TcCFT (Tools to Carve out Conformal Field Theories)
Reporting period: 2017-09-25 to 2019-09-24
The Bootstrap philosophy employs mathematical consistency (e.g. symmetries and quantum mechanics) to map out the space of quantum theories - including strongly coupled models - and make
sharp predictions for their physical observables.
While the Bootstrap idea dates back to the early 1940s with its roots in the S-matrix approach to the strong nuclear force, in recent years it has experienced a wave of renewed interest and successes owing to the
development of new computational tools and physical insights. A notable development has been the application of the Bootstrap to Conformal Field Theories (CFTs) in d>2 dimensions, known as the Conformal Bootstrap. The constraint of Conformal Symmetry makes CFTs easier to study than generic QFTs since, given a spectrum and operator product expansion (OPE) coefficients, it specifies all observables. The OPE coeffients can be determined by solving the crossing equation (fig 1), which enforces associativity of the OPE - known as crossing symmetry.
The ubiquity and universality of Conformal Field Theories in physical systems means that successes of the Conformal Bootstrap have potentially far-reaching impact in various branches of theoretical physics.
Indeed, CFTs describe a wealth of physical phenomena such as: Critical phenomena, the String Theory world sheet and, via gauge-gravity duality, UV complete theories of gravity, and more, for which there are few quantitative methods available to study their properties – in particular non-perturbatively.
The most celebrated results of the Conformal Bootstrap programme to date have been borne out of the development of powerful numerical machinery based on convex optimisation, which provide bounds on OPE data consistent with crossing symmetry. In contrast, the fruits of efforts to tackle the Conformal Bootstrap problem analytically are taking longer to mature. The last couple of years however have seen pivotal progress, to which this project has contributed directly. Most significantly, powerful inversion techniques have been developed which give a recipe to extract OPE data from a given CFT correlator. This provides a long-awaited CFT analogue of the celebrated S-matrix inversion formula of Froissart and Gribov. Inversion techniques also give rise to the decomposition of individual operator exchanges under crossing, known as crossing kernels, which provide basic building blocks needed to solve the crossing equation analytically.
Ultimately the development of analytic tools to solve non-perturbatively a QFT is of utmost importance to push forward our understanding of quantum phenomena in general, phase transitions in condensed matter and via gauge-gravity duality also the emergent nature of quantum gravity which are among the most fundamental problems in nature and for which even a small step forward would constitute an important advance for the whole society.
sharp predictions for their physical observables.
While the Bootstrap idea dates back to the early 1940s with its roots in the S-matrix approach to the strong nuclear force, in recent years it has experienced a wave of renewed interest and successes owing to the
development of new computational tools and physical insights. A notable development has been the application of the Bootstrap to Conformal Field Theories (CFTs) in d>2 dimensions, known as the Conformal Bootstrap. The constraint of Conformal Symmetry makes CFTs easier to study than generic QFTs since, given a spectrum and operator product expansion (OPE) coefficients, it specifies all observables. The OPE coeffients can be determined by solving the crossing equation (fig 1), which enforces associativity of the OPE - known as crossing symmetry.
The ubiquity and universality of Conformal Field Theories in physical systems means that successes of the Conformal Bootstrap have potentially far-reaching impact in various branches of theoretical physics.
Indeed, CFTs describe a wealth of physical phenomena such as: Critical phenomena, the String Theory world sheet and, via gauge-gravity duality, UV complete theories of gravity, and more, for which there are few quantitative methods available to study their properties – in particular non-perturbatively.
The most celebrated results of the Conformal Bootstrap programme to date have been borne out of the development of powerful numerical machinery based on convex optimisation, which provide bounds on OPE data consistent with crossing symmetry. In contrast, the fruits of efforts to tackle the Conformal Bootstrap problem analytically are taking longer to mature. The last couple of years however have seen pivotal progress, to which this project has contributed directly. Most significantly, powerful inversion techniques have been developed which give a recipe to extract OPE data from a given CFT correlator. This provides a long-awaited CFT analogue of the celebrated S-matrix inversion formula of Froissart and Gribov. Inversion techniques also give rise to the decomposition of individual operator exchanges under crossing, known as crossing kernels, which provide basic building blocks needed to solve the crossing equation analytically.
Ultimately the development of analytic tools to solve non-perturbatively a QFT is of utmost importance to push forward our understanding of quantum phenomena in general, phase transitions in condensed matter and via gauge-gravity duality also the emergent nature of quantum gravity which are among the most fundamental problems in nature and for which even a small step forward would constitute an important advance for the whole society.
"The work performed during the first 16 months of this project has been focused on two main workpackages. The first workpackage was devoted to the development of new analytic methods for the conformal bootstrap. The effort has been focused on building up new tools based on the Mellin space formalism for CFT correlators with focus on arbitrary conformal correlators with spinning external and internal legs. Among the main result achieved so far we have been able to identify a convenient basis of conformal structures at cubic order, which was motivated by the gauge-gravity duality. This basis of conformal structures was demonstrated to have remarkable properties. It allowed to write a relatively simple closed form expression for the bulk-to-boundary map for 3pt and 4pt amplitudes. It also allowed to obtain closed form expressions for the shadow transform of infinite sets of spinning 3pt conformal structures. Furthermore, when using this basis of structures to evaluate conformal blocks and conformal partial waves with spinning external legs, this basis provided remarkable simplifications allowing to obtain closed form expressions which we have used to write down inversion formulas in Mellin space for the decomposition of spinning conformal 4pt functions into conformal blocks.
The inversion formulas obtained in Mellin space were then used to study the so called crossing kernels which describe the expansion of a conformal partial wave in, say, the t-channel as an infinite sum of conformal blocks in the crossed channel. The study of these crossing kernels was linked successfully with the previously known large-spin expansions known for correlators involving scalar operators and was shown to provide a resummation of these expansions including the exponentially small terms, which are crucial for analiticity in spin of the final expression.
A second direction was devoted to a numerical implementation of the analytic tools developed in the first workpackage. The main programming language used has been ""mathematica"" chosen for its versatility in this phase of the project. The main task on which we have focused was at this stage to select which functions to compile. The main result so far is a set of functions which evaluate Witten diagrams, compute conformal integrals of various complexity and combine these intermediate tools to obtain the conformal block expansion of a given conformal four point functions.
As for regards the dissemination of the above results I have been invited to give seminars in institutions in US, Canada, Europe and Asia, and I have participated to the Simons collaboration workshop on the bootstrap held in Caltech (California) in August 2018. I have also participated to the Simons Bootstrap Collaborations annual meetings held in New York both in 2017 and 2018. These meetings have provided a stimulating environment to disseminate my research and to discuss our findings with experts on the subject.
"
The inversion formulas obtained in Mellin space were then used to study the so called crossing kernels which describe the expansion of a conformal partial wave in, say, the t-channel as an infinite sum of conformal blocks in the crossed channel. The study of these crossing kernels was linked successfully with the previously known large-spin expansions known for correlators involving scalar operators and was shown to provide a resummation of these expansions including the exponentially small terms, which are crucial for analiticity in spin of the final expression.
A second direction was devoted to a numerical implementation of the analytic tools developed in the first workpackage. The main programming language used has been ""mathematica"" chosen for its versatility in this phase of the project. The main task on which we have focused was at this stage to select which functions to compile. The main result so far is a set of functions which evaluate Witten diagrams, compute conformal integrals of various complexity and combine these intermediate tools to obtain the conformal block expansion of a given conformal four point functions.
As for regards the dissemination of the above results I have been invited to give seminars in institutions in US, Canada, Europe and Asia, and I have participated to the Simons collaboration workshop on the bootstrap held in Caltech (California) in August 2018. I have also participated to the Simons Bootstrap Collaborations annual meetings held in New York both in 2017 and 2018. These meetings have provided a stimulating environment to disseminate my research and to discuss our findings with experts on the subject.
"
The results obtained so far have already achieved substantial progress beyond the state of the art providing new tools to analyse correlators with spinning external leg. We are currently working towards extending the techniques and the result developed in this project in various directions which include:
1) Generalising the Mellin space techniques to include the case of correlators with four spinning operators. Within this class of correlators we plan to investigate in particular the case of correlators involving four stress tensors which would provide an important mile-stone for the project.
2) Developing epsilon expansion for spinning correlators using our methods with potential new results on various physically relevant CFTs, including vector and tensor models.
3) Building a mathematica package providing an explicit implementation of the tools developed during this project. These include the explicit form of the Mack polynomials and inversion formulas.
1) Generalising the Mellin space techniques to include the case of correlators with four spinning operators. Within this class of correlators we plan to investigate in particular the case of correlators involving four stress tensors which would provide an important mile-stone for the project.
2) Developing epsilon expansion for spinning correlators using our methods with potential new results on various physically relevant CFTs, including vector and tensor models.
3) Building a mathematica package providing an explicit implementation of the tools developed during this project. These include the explicit form of the Mack polynomials and inversion formulas.