Conformal Bootstrap: Non-perturbative QFT's under siege Alessandro Vichi Iberian Strings 2017 Lecture 1 Physical Foundations of Conformal Ubiquitous CFT's Symmetry 1.1 Fixed points I CFT's are the building blocks of quantum field theories: they are signposts in the Quantum Field Theory (QFT) is, in most general terms, the study of Renormalization Group (RG) flows, i.e. how the theory evolves from the Ultraviolet (UV) to the Infrared space of quantum(IR) theories regimes: One can ask which IR phases are possible. A priori, there are three possibilities: I Promising way to lean more about quantum gravity via AdS/CFT correspondence A. atheorywithamassgap, I Many real physicalB. systemsatheorywithmasslessparticlesintheIR, where scale invariance shows up C. aScaleInvariant(SI)theorywithacontinuousspectrum. It is the last class that we will call CFT and will mostly study in these lectures. But first let’s look at some examples corresponding to these phases. Nonabelian Yang-Mills (YM) theory in D = 4 dimensions belongs to type A. 1 L = F a F aµ⌫ . (1.1.1) −4g2 µ⌫ 4 2 Lecture 1 Physical Foundations of Conformal Ubiquitous CFT's Symmetry 1.1 Fixed points CFT's are the building blocks of quantum field theories: they are signposts in the I Quantum Field Theory (QFT) is, in most general terms, the study of Renormalization Group (RG) flows, i.e. how the theory evolves from the Ultraviolet (UV) to the Infrared space of quantum(IR) theories regimes: One can ask which IR phases are possible. A priori, there are three possibilities: I Promising way to lean more about quantum gravity via AdS/CFT correspondence A. atheorywithamassgap, I Many real physicalB. systemsatheorywithmasslessparticlesintheIR, where scale invariance shows up C. aScaleInvariant(SI)theorywithacontinuousspectrum. It is the last class that we will call CFT and will mostly study in these lectures. But first Condensed matter: let’s look at some examples corresponding to these phases. • 4 Nonabelian Yang-Mills (YM) theory in D Superfluid= 4 dimensions belongsHe to type A. O(2)-model critical point of many fluids, 1 ! L = F a F aµ⌫ . (1.1.1) −4g2 µ⌫ 4 2 Lecture 1 Physical Foundations of Conformal Ubiquitous CFT's Symmetry 1.1 Fixed points CFT's are the building blocks of quantum field theories: they are signposts in the I Quantum Field Theory (QFT) is, in most general terms, the study of Renormalization Group (RG) flows, i.e. how the theory evolves from the Ultraviolet (UV) to the Infrared space of quantum(IR) theories regimes: One can ask which IR phases are possible. A priori, there are three possibilities: I Promising way to lean more about quantum gravity via AdS/CFT correspondence A. atheorywithamassgap, I Many real physicalB. systemsatheorywithmasslessparticlesintheIR, where scale invariance shows up C. aScaleInvariant(SI)theorywithacontinuousspectrum. Percolation Q = 1It Potts is the last classmodel that we will call CFT and will mostly study in these lectures. But first • ! let’s look at some examples corresponding to these phases. Nonabelian Yang-Mills (YM) theory in D = 4 dimensions belongs to type A. 1 L = F a F aµ⌫ . (1.1.1) −4g2 µ⌫ 4 2 Lecture 1 Physical Foundations of Conformal Ubiquitous CFT's Symmetry 1.1 Fixed points CFT's are the building blocks of quantum field theories: they are signposts in the I Quantum Field Theory (QFT) is, in most general terms, the study of Renormalization Group (RG) flows, i.e. how the theory evolves from the Ultraviolet (UV) to the Infrared space of quantum(IR) theories regimes: One can ask which IR phases are possible. A priori, there are three possibilities: I Promising way to lean more about quantum gravity via AdS/CFT correspondence A. atheorywithamassgap, I Many real physicalB. systemsatheorywithmasslessparticlesintheIR, where scale invariance shows up C. aScaleInvariant(SI)theorywithacontinuousspectrum. Percolation Q = 1It Potts is the last classmodel that we will call CFT and will mostly study in these lectures. But first • ! let’s look at some examples corresponding to these phases.Polymers Self-avoiding random Nonabelian Yang-Mills (YM) theory in D = 4 dimensions• belongs to type! A. 1 walk (O(N = 0) vector model) L = F a F aµ⌫ . (1.1.1) −4g2 µ⌫ 4 2 Many real physical systems do not fall in the regime of validity of any of the above assumptions. Ex: O(N) vector model: I N = 1 (Ising model): Ferromagnetism, Vapor to liquid transition,... 4 I N = 2 (XY model): superfluid He, I N = 3 (Heisenberg model): magnetism Genuine, non-supersymmetric, small-N, non perturbative systems. Nature is perverse... I In 2D, great results have been achieved thanks to a richer structure (infinite dimensional symmetry) I In D>2, general believe that no quantitative result could be obtained... I perturbative expansion (in couplings, dimensions, number of fields,...) I supersymmetry I strong/weak dualities (AdS/CFT, ...) 3 Nature is perverse... I In 2D, great results have been achieved thanks to a richer structure (infinite dimensional symmetry) I In D>2, general believe that no quantitative result could be obtained... I perturbative expansion (in couplings, dimensions, number of fields,...) I supersymmetry I strong/weak dualities (AdS/CFT, ...) Many real physical systems do not fall in the regime of validity of any of the above assumptions. Ex: O(N) vector model: I N = 1 (Ising model): Ferromagnetism, Vapor to liquid transition,... 4 I N = 2 (XY model): superfluid He, I N = 3 (Heisenberg model): magnetism Genuine, non-supersymmetric, small-N, non perturbative systems. 3 Tools at disposal Alternative approaches to non-perturbative theories: how to compute critical exponents (η,ν,...)? I More exotic field theory approaches I MonteCarlo simulations (spin systems) 0.042 0.0370 0.040 HT exp 0.0368 0.038 ✏ exp 0.0366 η η 0.036 − MC (2003) 0.034 0.0364 3D exp 0.032 − 0.0362 MC (2010) 0.030 0.627 0.628 0.629 0.630 0.631 0.632 0.633 0.634 0.6299 0.6300 0.6301 0.6302 0.6303 ν ν I QFT on the lattice I Truncated Conformal Space Approach I ..... 4 Conformal bootstrap The conformal bootstrap aims to develop a systematic and rigorous method to study the properties of conformal invariant fixed points in any dimension, complementary to existing techniques. No are 1, 2,..., n Conformal {O O O } bootstrap part of a CFT? Maybe 5 Table of contents 1. Conformal bootstrap main ideas 2. A few applications 3. Going Beyond 4. Conclusions 6 Conformal bootstrap main ideas What is a CFT? Theories invariant under the conformal algebra SO(D 2) which includes: j I translations I Lorentz transformations I dilatations I "inversion" They are described by three ingredients: 1) Operator content: representations totally characterized by scaling dimension ∆ and spin ` of the primary (lowest dimension operator). All other are called descendants. 2) Interactions between operators: encoded in the Operator Product Expansion (OPE) X ∆ ∆ C12O ∆ + descendants O 1 × O 2 ∼ O ;` O | {z } fixed by conformal symmetry C12O are called OPE coefficients 3) Crossing symmetry constraints: see next slides... 8 The power of conformal invariance In CFT we are interested in computing correlations functions i (x1)::: j (xn) : hO O i Two point functions of primaries: completely fixed δij i (x1) j (x2) = x12 x1 x2 ∆i = [ i ] hO O i 2∆i ≡ j − j O x12 Three point functions of primaries: fixed modulo a constant. Use OPE to reduce higher point functions to smaller ones 8 C ( + descendants ) if > 123 3 3 3 1 2 > |hO O i {z } O 2 O × O < fixed by conformal symmetry 1 2 3 hO O O i / > > : 0 otherwise 9 1 Introduction and formulation of the problem Our knowledge about non-supersymmetric Conformal Field Theories (CFTs) in four dimensions (4D) is still quite incomplete. Suffices it to say that not a single nontrivial example is known which would be solvable to the same extent as, say, the 2D Ising model. However, we do not doubt that CFTs must be ubiquitous. For example, non-supersymmetric gauge theories with Nc colors and Nf flavors are widely believed to have “conformal windows” in which the theory has a conformal 1 Introductionfixedand point in formulation the IR, with evidence of from the large problemNc analysis [1], supersymmetric analogues [2], and lattice simulations [3]. Since these fixed points are typically strongly coupled, we do not have much control over them. In this situation particularly important are general, model-independent Our knowledge about non-supersymmetric Conformal Field Theories (CFTs) in four dimensions properties. (4D) is still quite incomplete. Suffices it to say that not a single nontrivial example is known which would be solvable to the sameOne extent example as, of say, such the a 2D property Ising model. is the However,famous unitarity we do not bound doubt [4] that on the dimension ∆ of a 1 CFTs must be ubiquitous.spin l Forconformal example, primary non-supersymmetric operator O∆,l : gauge theories with Nc colors and Nf flavors are widely believed to have “conformal windows” in which the theory has a conformal ∆ 1(l =0), (1.1) fixed point in the IR, with evidence from large Nc analysis [1], supersymmetric≥ analogues [2], and Four point functions ∆ l +2 (l 1) . lattice simulations [3]. Since these fixed points are typically strongly≥ coupled,≥ we do not have much control over them. In this situation particularly important are general, model-independent properties. These bounds are derived by imposing that the two point function OO have a positive spectral density. h i One example of such a property is the famous unitarity bound [4] on the dimension ∆of a As is well known,1 3-point functions in CFT are fixed by conformal symmetry up to a few arbi- spin l conformal primary operator O∆,l : trary constants (Operator Product Expansion (OPE) coefficients).