# Haut PDF Minimal lectures on two-dimensional conformal field theory

### Minimal lectures on two-dimensional conformal field theory

In the first three sections, most axioms are common to all two-dimensional conformal field theories. These axioms specify in particular how the Virasoro symmetry algebra acts on fields, and the existence and properties of the operator product expansion. Next, we introduce additional axioms that single out either Liouville theory, or minimal models. In particular, these axioms determine the three-point functions. Finally we check that these uniquely defined theories do exist, by studying their four-point functions.

### Conformal field theory of critical Casimir interactions in 2D

Massachusetts Institute of Technology, Department of Physics, Cambridge, Massachusetts 02139, USA (Dated: July 16, 2013) Thermal fluctuations of a critical system induce long-ranged Casimir forces between objects that couple to the underlying field. For two dimensional (2D) conformal field theories (CFT) we derive an exact result for the Casimir interaction between two objects of arbitrary shape, in terms of (1) the free energy of a circular ring whose radii are determined by the mutual capacitance of two conductors with the objects’ shape; and (2) a purely geometric energy that is proportional to conformal charge of the CFT, but otherwise super-universal in that it depends only on the shapes and is independent of boundary conditions and other details.
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### CONFORMAL BOOTSTRAP IN LIOUVILLE THEORY

so called structure constants of the CFT (see Section 1.3 ). Hence the correlation functions of a CFTs are determined in the bootstrap approach if one knows its spectrum and the structure constants. In the case of two dimensional conformal field theories (d = 1 or n = 2 above) the conformal symmetry constrains the possible CFTs particularly strongly and Belavin, Polyakov and Zamolodchikov [ BPZ84 ] (BPZ from now on) showed the power of the bootstrap hypothesis by producing explicit expressions for the correlation functions of a large family of CFTs of interest to statistical physics among which the CFT that is believed to coincide with the scaling limit of critical Ising model. In a nutshell, BPZ argued that one could parametrize CFTs by a unique parameter c called the central charge and they found the correlation functions for certain rational values of c where the number of primary fields is finite (the minimal models). During the last decade the bootstrap approach has also led to spectacular predictions of critical exponents in the three dimensional case [ PoRyVi ]. The reader may consult [ Ga ] for some background on 2d CFTs.
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### Conformal bootstrap in Liouville theory

In the case of two dimensional conformal field theories (d = 1 or n = 2 above) the conformal symmetry constrains possible CFT’s particularly strongly and Belavin, Polyakov and Zamolodchikov [BPZ84] (BPZ from now on) showed the power of the bootstrap approach by producing explicit expressions for the correla- tion functions of a large family of CFT’s of interest to statistical physics among them the Ising model. In a nutshell, BPZ argued that one could parametrize CFT’s by a unique parameter c called the central charge and they found the correlation functions for certain rational values of c where the number of primary fields is finite (the minimal models). During the last decade the bootstrap approach has also led to spectacular predictions of critical exponents in the physically interesting three dimensional case [PoRyVi].
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### Conformal two-boundary loop model on the annulus

1 Introduction The study of conformal boundary conditions (CBC) and boundary operators is one of the most fruitful aspects of the vast problem of solving two dimensional field theories and string theories. There are many reasons for this. In the equivalent 1+1 dimensional systems, CBC describe possible fixed points in quantum impurity problems, such as the multichannel Kondo problem [1], while boundary operators decide the stability of these fixed points as well as RG flows. In string theory, CBC describe possible branes, while RG flows in this language decide issues of (open string) tachyon decay [2]. In statistical mechanics, boundaries are roughly where couplings to the outside take place— for instance couplings to electrodes in quantum Hall effect type problems and their Chalker-Coddington type lattice formulations [3, 4].
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### Non compact conformal field theories in statistical mechanics

L + . . . . (1.31) Let us now look back on the way we have taken so far. From a given classical or quan- tum critical problem, namely that of polymer collapse in two dimensions or the plateau transition in the Integer Quantum Hall Effect, we have claimed that one may derive classical two-dimensional geometrical models on the square lattice, hoping that a fine- tuning of the corresponding Boltzmann weights should recover the universal behaviour of interest in each case. Once the corresponding transfer matrix or quantum Hamilto- nian clearly defined in either its loop, vertex, or any of its representations, criticality can be investigated and the associated set of critical exponents in principle extracted from the finite-size scaling of the corresponding eigenvalues. Additionally, peculiar algebraic features of the continuum limit such as indecomposability in the case of a logarithmic conformal field theory can be dug out from algebraic features of the finite size transfer matrix itself, or its associated quantum Hamiltonian [8]. What remains hidden behind our use of the term ‘in principle’ is that the finite-size scaling analysis can turn out to be very difficult in practice. The dimension of the transfer matrix usually grows up expo- nentially with the system size L, limiting the range of sizes tractable by usual numerical methods to, at best, L ∼ 10 or 20 for usual systems. Adding to this the fact that in some cases (namely, whenever marginal operators are present, or, more fundamentally, in the case of the non compact CFTs which are of central interest for us) the corrections to the conformal charges in (1.30) and (1.31) may show a really slow convergence of the order of log L, we cannot escape as a conclusion that the critical exponents can sometimes simply turn out to be untractable from straight numerics. What is left for us to rely on is the possibility to turn the corresponding models into exactly-solvable ones, a procedure whose main lines we shall depict now.
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### Geometry of integrable systems : from topological Lax systems to conformal field theories

Chapter 3 W-symmetric conformal ﬁeld theories 1 Lightning review of conformal ﬁeld theory Conformal ﬁeld theories in two dimensions have appeared in the physics litterature as powerful tools to study numerous systems, from critical (pos- sibly quantum) statistical models in two dimensions (in some thermody- namic limit) to the worldsheet conformal symmetry of string theories [ 7 ]. They have the particularity to exhibit inﬁnite dimensional conformal al- gebras of symmetries, namely extensions of the Virasoro algebra (that are not necessarily Lie algebras as we shall see). It was in turn argued in some cases [ 4 ] that there exists an underlying structure of quantum integrable system with commuting transfer matrices and such.
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### Strongly $\gamma$-Deformed $\mathcal{N}=4$ Supersymmetric Yang-Mills Theory as an Integrable Conformal Field Theory

Conclusions.—We demonstrated by explicit multiloop calculation that the strongly γ-deformed planar N ¼ 4 SYM has two nontrivial fixed points whose position depends on the properly rescaled ’t Hooft coupling. We also provided evidence that, at the fixed points, it is described by an integrable nonunitary four-dimensional conformal field theory. Namely, we found a closed expres- sion for the four-point correlation function of the simplest protected operators and used it to compute the exact conformal data (scaling dimensions and OPE coefficients) of twist-2 and twist-4 operators with arbitrary Lorentz spin. In general, correlation functions in this theory are domi- nated by fishnet graphs [15,25] which admit a description in terms of integrable noncompact Heisenberg spin chains
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### Conformal field theories in a periodic potential: Results from holography and field theory

Cambridge, Massachusetts 02139, USA 2 Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (Received 22 August 2013; published 21 January 2014) We study (2 þ 1)-dimensional conformal field theories (CFTs) with a globally conserved U(1) charge, placed in a chemical potential which is periodically modulated along the spatial direction x with zero average: μðxÞ ¼ V cosðkxÞ. The dynamics of such theories depends only on the dimensionless ratio V=k, and we expect that they flow in the infrared to new CFTs whose universality class changes as a function of V=k. We compute the frequency-dependent conductivity of strongly coupled CFTs using holography of the Einstein-Maxwell theory in four-dimensional anti –de Sitter space. We compare the results with the corresponding computation of weakly coupled CFTs, perturbed away from the CFT of free, massless Dirac fermions (which describes graphene at low energies). We find that the results of the two computations have significant qualitative similarities. However, differences do appear in the vicinities of an infinite discrete set of values of V=k: the universality class of the infrared CFT changes at these values in the weakly coupled theory, by the emergence of new zero modes of Dirac fermions which are remnants of local Fermi surfaces. The infrared theory changes continuously in holography, and the classical gravitational theory does not capture the physics of the discrete transition points between the infrared CFTs. We briefly note implications for a nonzero average chemical potential.
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### Null-vectors in integrable field theory

F-91191, Gif-sur-Yvette, France. Abstract. The form factor bootstrap approach allows to construct the space of local fields in the massive restricted sine-Gordon model. This space has to be isomorphic to that of the corresponding minimal model of conformal field theory. We describe the subspaces which correspond to the Verma modules of primary fields in terms of the commutative algebra of local integrals of motion and of a fermion (Neveu-Schwarz or Ramond depending on the particular primary field). The description of null-vectors relies on the relation between form factors and deformed hyper-elliptic integrals. The null-vectors correspond to the deformed exact forms and to the deformed Riemann bilinear identity. In the operator language, the null-vectors are created by the action of two operators Q (linear in the fermion) and C (quadratic in the fermion). We show that by factorizing out the null-vectors one gets the space of operators with the correct character. In the classical limit, using the operators Q and C we obtain a new, very compact, description of the KdV hierarchy. We also discuss a beautiful relation with the method of Whitham.
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### Conformal bootstrap in two-dimensional conformal field theories with with non-diagonal spectrums

In this work we focus on conformal symmetry, and its consequences on two dimensional quantum field theories. Conformal transformations are symmetry transformations that leave angles invariant, and they form a larger class of trans- formations than those of the Poincaré group. Then, conformal field theories have an enhanced symmetry that makes them more tractable than a generic QFT. Sometimes conformal field theories can become completely solvable, in the sense that all their correlation functions can in principle be computed. Among confor- mal transformations there are dilatations, or scale changes. That these transfor- mations are a symmetry of a system might seem unphysical, because it is known that physical laws are usually strongly dependent on scale. The exceptions come when the characteristic distances of a system become either 0 of ∞, and there are many examples where a conformal field theory description is possible. In the next section we list some of them.
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### Lax matrix solution of c = 1 conformal field theory

model to each correlation function of Liouville theory, so that the variables of the corre- lation function, such as field positions, correspond to parameters of the model, such as coupling constants. We will follow this second approach. This approach was already shown to provide a perturbative expansion of Liouville correlation functions around the value c = ∞ of the central charge [ 4 ]. This perturbative expansion is encoded in a non-commutative spectral curve, as was demonstrated by the explicit computation of the first two terms of the three-point function. Here we will use this approach for exactly solving Liouville theory at c = 1. Liouville theory is supposed to exist as a consistent, unitary conformal field theory for c > 1 [ 5 ], although its quantum gravity interpretation is clear only for c > 25. The c = 1 limit of Liouville theory is another consistent, unitary conformal field theory [ 6 ], which was originally found by Runkel and Watts as a limit of Minimal Models [ 7 ], as we illustrate in the following diagram:
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### Field theory on the light-front

intermediate state without the need to include them explicitly, provided the effective Lagrangian has the right dynamics to generate these resonances. 5. Perspectives The general framework we presented above to study the structure of relativistic compound systems in light-front dynamics in a nonperturbative way relies on three main advances: i) the con- struction of a covariant formulation of light-front dynamics [ 2 , 3 ] in order to control any violation of rotational invariance; ii) the development of an appropriate renormalization scheme — the so- called Fock sector dependent renormalization scheme — to deal with the truncation of the Fock expansion [ 4 ]; iii) the use of an appropriate regularization scheme — the so-called Taylor-Lagrange regularization scheme — very well adapted to systematic calculations in light-front dynamics [ 11 ]. These advances should enable us to have a predictive framework order by order in the Fock expansion. We shall complete in the future this description by considering physical systems involv- ing spontaneous symmetry breaking. It is known that these systems can be described in light-front dynamics by the consideration of zero-mode contributions, in the λ φ 4 theory in 1 + 1 dimension for instance [ 15 ]. Their full calculation in 3 + 1 dimensions within the general framework presented above remains to be done.
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### Field theory for recurrent mobility

~ W angle analysis performed in Supplementary Note 11 (Supplementary Figs. 45–50) [49]. Circular infrastructures are not so uncommon in cities, besides circular metro lines many highways are organized as concentric rings when there is no major geographical impediment as in Paris or London. One may, thus, won- der why typically we do not observe rotational compo- nents in the cities vector field. To have such components, it would be necessary to have an unbalanced flow of peo- ple living in an area and working in another over the ring following one of the rotation senses. At the scale that we are using, this is not seen anywhere in the cities under study. The main factor that could favor the emergence of rotational components is thus the segregation of land use. However, land use mixing is strong enough in large cities [50] to prevent this sort of loops in the mobility flows at mesoscopic scales, leading to hierarchical configurations of the mobility with a few clear attraction centers.
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### Three-point functions in c <= 1 Liouville theory and conformal loop ensembles

tors with identical values of ˆ α is normalized to have unit residue. The consistency of the c ≤ 1 theory with ˆ α = Q 2 ˆ +p and p ∈ R—viz. ∆ = p 2 − Q ˆ 4 2 by (2)—was recently demon- strated in [11], including by extensive numerical checks of crossing symmetry for the corresponding four-point functions. Meanwhile, the relevance of Liouville theory to conformal models of fluctuating loops was pointed out in [12, 13]. These works, based on the so-called geomet- rical Coulomb gas (CG) construction [14], were however limited to two-point functions. But following the sug- gestion in [5], an interesting proposal was made [6] that the probability for three points to belong to the same Fortuin-Kasteleyn (FK) cluster in the critical Q-state Potts model was simply related to the three-point cou- pling (3) for a particular value of the charges ˆ α i = ˆ α(Q).
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### Constructing two-qubit gates with minimal couplings

J 12 共t兲 = g 0 + g + 共t兲cos共 ␻ + t 兲 + g − 共t兲cos共 ␻ − t 兲. 共2兲 To perform single-qubit operations we use Rabi oscillations driven by a resonant microwave control field, u j共t兲 = 2⍀j共t兲cos关⌬ j t + ␾ j共t兲兴. 共3兲 In this setup all the temporal dependence of the Hamiltonian is assumed to arise from the time-dependent flux of the ap- plied fields. The rotating wave approximation, which is also valid if cross couplings are taken into account, results in a rotating frame Hamiltonian of the form

### A Multi-Step Procedure for Enriching Limited Two-Dimensional Acoustic Far-Field Pattern Measurements.

5 Summary and Conclusion In response to the pressure need for FFP data in large apertures ( 90 o -apertures at least) for solving efficiently inverse obstacle problems with iterative solution methodologies, we have designed a procedure that extends few noisy FFP measurements to larger aper- tures. The proposed method is a three-stage procedure where the total variation of the FFP field is used for regularization. Using synthetic FFP data, we have perofrmed numerical experiments to assess the performance of the proposed method. The nu- merical results indicate that the procedure extends successfully noisy backscattering measurements to full aperure ( 360 o -aperture). Such an efficiency is achieved even in the presence of high level of noise, but for measured data corresponding to relatively small frequency values. Note that in the particular case of backscattering measure- ments and low noise level (less than 1%), the field is reconstructed accurately over
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### Conformal invariance of the subleading soft theorem in gauge theory

(Received 5 September 2014; published 2 October 2014) In this note, I show that the recently proposed subleading soft factor in massless gauge theory uniquely follows from conformal symmetry of tree-level gauge theory amplitudes in four dimensions. DOI: 10.1103/PhysRevD.90.087701 PACS numbers: 12.38.Bx, 11.15.-q

### Two-Dimensional Colloidal Nanocrystals

smaller than the {111} facets, thus the growth of atoms on these latest is favored. 145 On the other hand, the calculated values of the adsorption energy of oleic acid on both plans show that oleic acid would stabilize {100} plans less than {111} plans. 145 Thus, depending on the temperature of reaction and the ligands, it is in some points possible to tune the shape of the nanocrystals. Here, during the synthesis of nanoplatelets, a short carboxylate aliphatic chain (less than three carbons) with a low boiling point is introduced. The acetate or propionate salt will lower the influence of the facet stabilization energy by the ligands, due to a fast kinetic of adsorption and desorption with short chain ligands. This will favor the growth of nanocrystals with low-index crystallographic planes such as the {001} facets. The nanoplatelets are indeed exhibiting the top and bottom facets as cadmium rich {001} facets stabilized by oleate ligands, not acetate ligands which are too small to stabilize NPLs in non-polar solvent. Nevertheless, it does not explain the breaking in the isotropic growth which, instead of giving cubic nanocrystals, leads to nanoplatelets. For the four existing populations of nanoplatelets (2 to 5MLs), the seeds diameter would be inferior to 1.7 nm with a number of atoms inferior to 200 with more than 30% of them on the surface (see Figure 23). For the two thickest populations of nanoplatelets, the synthesis requires a first step of growth without the acetate (resp. propionate) salt, where spherical zinc-blende nanocrystals are obtained with mostly cadmium rich surfaces, since cadmium is in excess and the only ligands are oleate which can mostly bind cations. Once the acetate salt is introduced at high temperature (superior to 190°C), equilibrium of exchange may happen on the different facets between acetate and oleate. Presumably, regarding the oleate steric
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### Connections over spaces of conformal field theories : testing for integrablity

states are smooth sections in theory space. We then conclude that the divergences in the second covariant derivative of the basis states must be canceled by other [r]