Clément Hongler Spring 2016 Lecture Series EPFL

... structures that allow for extremely precise investigations, yielding in particular nonperturbative descriptions with exact formulae. This series will explain how 2D CFT works, in particular how planar lattice models can be understood using the Minimal Models of CFT. While this connection remains lar ...

... structures that allow for extremely precise investigations, yielding in particular nonperturbative descriptions with exact formulae. This series will explain how 2D CFT works, in particular how planar lattice models can be understood using the Minimal Models of CFT. While this connection remains lar ...

Very brief introduction to Conformal Field Theory

... Similar chiral correlators have been considered in the Fractional Quantum Hall effect at filling fraction 5/2. This is the so called Pfaffian state due to Moore and Read. FQHE/CFT correspondence ...

... Similar chiral correlators have been considered in the Fractional Quantum Hall effect at filling fraction 5/2. This is the so called Pfaffian state due to Moore and Read. FQHE/CFT correspondence ...

1 The free boson on the sphere, normal ordering, and all that

... employed above. Give the general relation between normal ordered and radially ordered operators. b) Give the correlator hX(z)X(w)i for the field X(z). Compare this with the correlator of two primary fields in a general CFT. Deduce the correlator h∂X(z)∂X(w)i from hX(z)X(w)i. c) Review the derivation ...

... employed above. Give the general relation between normal ordered and radially ordered operators. b) Give the correlator hX(z)X(w)i for the field X(z). Compare this with the correlator of two primary fields in a general CFT. Deduce the correlator h∂X(z)∂X(w)i from hX(z)X(w)i. c) Review the derivation ...

Title: Some Combinatorial Problems Inherent in and Related

... Speaker: G.H.E.Duchamp (LIPN, Université Paris XIII) We consider two aspects of the product formula for formal power series applied to combinatorial field theories. Firstly, we remark that the case when the functions involved in the product formula are free exponentials (like in the derivation of Be ...

... Speaker: G.H.E.Duchamp (LIPN, Université Paris XIII) We consider two aspects of the product formula for formal power series applied to combinatorial field theories. Firstly, we remark that the case when the functions involved in the product formula are free exponentials (like in the derivation of Be ...

Symmetries in Conformal Field Theory

... Where does this all come from? One interpretation is that to get these natural equations of motion we’d like a 2-form β whose derivative looks like χ, then we could just add to the density a term like φ∗ β. However χ is not globally exact, so we need to choose an extension of φ to make the term we w ...

... Where does this all come from? One interpretation is that to get these natural equations of motion we’d like a 2-form β whose derivative looks like χ, then we could just add to the density a term like φ∗ β. However χ is not globally exact, so we need to choose an extension of φ to make the term we w ...

PASCOS - CERN Indico

... Starting with a generic SFT one knows (Wess 1960) that the trace of the energy momentum obeys a local equation : where is local , the “virial current” . If the virial current is a gradient i.e. ...

... Starting with a generic SFT one knows (Wess 1960) that the trace of the energy momentum obeys a local equation : where is local , the “virial current” . If the virial current is a gradient i.e. ...

PDF

... Hamiltonian algebroids are generalizations of the Lie algebras of canonical transformations, but cannot be considered just a special case of Lie algebroids. They are instead a special case of a quantum algebroid. Definition 0.1. Let X and Y be two vector fields on a smooth manifold M , represented h ...

... Hamiltonian algebroids are generalizations of the Lie algebras of canonical transformations, but cannot be considered just a special case of Lie algebroids. They are instead a special case of a quantum algebroid. Definition 0.1. Let X and Y be two vector fields on a smooth manifold M , represented h ...

1 Towards functional calculus

... • Complex-valued functions f : C → C form an algebra under point-wise multiplication, and by ‘an algebra of functions’ we mean some subalgebra of this one. • The most important example of an algebra of functions is the polynomials. • The linear transformations of H form an algebra under composition. ...

... • Complex-valued functions f : C → C form an algebra under point-wise multiplication, and by ‘an algebra of functions’ we mean some subalgebra of this one. • The most important example of an algebra of functions is the polynomials. • The linear transformations of H form an algebra under composition. ...

Schweigert.pdf

... theory they play the role of precorrelators; they also form the state spaces of the associated three-dimensional topological ﬁeld theory. The monodromy properties of these conformal blocks provide us with a collection of data – fusing matrices, braiding matrices, conformal weights, representations o ...

... theory they play the role of precorrelators; they also form the state spaces of the associated three-dimensional topological ﬁeld theory. The monodromy properties of these conformal blocks provide us with a collection of data – fusing matrices, braiding matrices, conformal weights, representations o ...

Noncommutative space-time and Dirac constraints - Indico

... the relationship between energy and entropy in our system is given by ...

... the relationship between energy and entropy in our system is given by ...

Symmetry and Integrability of Nonsinglet Sectors in MQM

... Operator algebra which does not change the representation = spectrum generating algebra for specific representation U(N) invariant operator ...

... Operator algebra which does not change the representation = spectrum generating algebra for specific representation U(N) invariant operator ...

Abstracts of the talks

... Given a surface S with a collection of special points on the boundary modulo isotopy, and a split reductive group G, we define a moduli space M (G, S) of G-local systems on S with some special data at the special points. We introduce a function W on M (G, S), the potential. It determines a set of W ...

... Given a surface S with a collection of special points on the boundary modulo isotopy, and a split reductive group G, we define a moduli space M (G, S) of G-local systems on S with some special data at the special points. We introduce a function W on M (G, S), the potential. It determines a set of W ...

Lecture 1: conformal field theory

... sphere S 2 = R with the standard coordinate z and two holomorphic holes of radius one around 0 and 1. 5. Smoothness: Sometimes one requires that the operator ji depends smoothly (or continuously) on the Riemann surface . To make this assumption, one needs to assume that V is at least a topological ...

... sphere S 2 = R with the standard coordinate z and two holomorphic holes of radius one around 0 and 1. 5. Smoothness: Sometimes one requires that the operator ji depends smoothly (or continuously) on the Riemann surface . To make this assumption, one needs to assume that V is at least a topological ...

Titles and Abstracts

... Title: Character and dimension formulas for queer Lie superalgebras Abstract: Using Brundan's algorithm, we obtain closed character and dimension formulas for queer Lie superalgebras. This is a recent joint work with R.B.Zhang. Anne Taormina (University of Durham, UK) Title: K3 elliptic genus: glimp ...

... Title: Character and dimension formulas for queer Lie superalgebras Abstract: Using Brundan's algorithm, we obtain closed character and dimension formulas for queer Lie superalgebras. This is a recent joint work with R.B.Zhang. Anne Taormina (University of Durham, UK) Title: K3 elliptic genus: glimp ...

Superintegrability as an organizing principle for special function theory

... possible, but of course not all commuting). If the independent symmetries can all be chosen of order k or less as differential operators the system is kth order superintegrable. Superintegrability is much more restrictive than integrability. Washington DC talk – p. 2/26 ...

... possible, but of course not all commuting). If the independent symmetries can all be chosen of order k or less as differential operators the system is kth order superintegrable. Superintegrability is much more restrictive than integrability. Washington DC talk – p. 2/26 ...

A Conformal Field Theory Primer

... as we move them around each another. This in turn can be expressed very elegantly in terms of properties of a three-dimensional topological field theory, with remarkable consequences in knot theory. In understanding how all this works, a whole new algebraic structure of the operators, the fusion alg ...

... as we move them around each another. This in turn can be expressed very elegantly in terms of properties of a three-dimensional topological field theory, with remarkable consequences in knot theory. In understanding how all this works, a whole new algebraic structure of the operators, the fusion alg ...

Determinant formulas for the reflection equation algebra

... where Fσ : C → C op denotes the identity functor, equipped with a tensor structure σ, induces a functor from C C op -algebras to C-algebras, under which the REA is the image of the FRT algebra. 4. The center of REA The center of the FRT algebra has been thoroughly studied and described: it is isom ...

... where Fσ : C → C op denotes the identity functor, equipped with a tensor structure σ, induces a functor from C C op -algebras to C-algebras, under which the REA is the image of the FRT algebra. 4. The center of REA The center of the FRT algebra has been thoroughly studied and described: it is isom ...

Hawking Radiation by Kerr Black Holes and Conformal Symmetry Ivan Agullo,

... modes defining the jini j0i and jouti vacuum states are related by a conformal transformation. We can use expression (1) to evaluate this expectation value. Integrating by parts in (1) and taking into account that the field modes fiout vanish at spacelike infinity, one finds that the two-point fun ...

... modes defining the jini j0i and jouti vacuum states are related by a conformal transformation. We can use expression (1) to evaluate this expectation value. Integrating by parts in (1) and taking into account that the field modes fiout vanish at spacelike infinity, one finds that the two-point fun ...

Gravity Duals for Nonrelativistic Conformal Field Theories Please share

... Discussion.—In discussions of AdS/CFT, one often hears that the CFT ‘‘lives at the boundary’’ of the bulk spacetime. The spacetime (2) is conformal to a pp-wave spacetime and hence has a boundary which is onedimensional —for z > 1, gtt grows faster than the other components at small r. While one mig ...

... Discussion.—In discussions of AdS/CFT, one often hears that the CFT ‘‘lives at the boundary’’ of the bulk spacetime. The spacetime (2) is conformal to a pp-wave spacetime and hence has a boundary which is onedimensional —for z > 1, gtt grows faster than the other components at small r. While one mig ...

on line

... groups are likewise defined by polynomial equations and have corresponding algebras C[G] , as well as versions k[G] defined over general fields with the same relations. Meanwhile, working over C , a “real form” means the additional structure of a compatible complex-linear involution making the coord ...

... groups are likewise defined by polynomial equations and have corresponding algebras C[G] , as well as versions k[G] defined over general fields with the same relations. Meanwhile, working over C , a “real form” means the additional structure of a compatible complex-linear involution making the coord ...

Gravity Duals for Nonrelativistic Conformal Field

... Discussion.—In discussions of AdS/CFT, one often hears that the CFT ‘‘lives at the boundary’’ of the bulk spacetime. The spacetime (2) is conformal to a pp-wave spacetime and hence has a boundary which is onedimensional —for z > 1, gtt grows faster than the other components at small r. While one mig ...

... Discussion.—In discussions of AdS/CFT, one often hears that the CFT ‘‘lives at the boundary’’ of the bulk spacetime. The spacetime (2) is conformal to a pp-wave spacetime and hence has a boundary which is onedimensional —for z > 1, gtt grows faster than the other components at small r. While one mig ...

Studying Quantum Field Theory

... decomposition into a positive and a negative frequency part. Adding the requirement of Hilbert space (or Wightman) positivity, i.e. demanding that the space generated by the action of a GCI (smeared quantum) field on the vacuum admits a (non-trivial) subspace realizing a unitary positive energy irre ...

... decomposition into a positive and a negative frequency part. Adding the requirement of Hilbert space (or Wightman) positivity, i.e. demanding that the space generated by the action of a GCI (smeared quantum) field on the vacuum admits a (non-trivial) subspace realizing a unitary positive energy irre ...

LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS Setting. W

... an isomorphism. Then we prove that the natural morphism RepΓ (H0,c , CΓ)// GL(CΓ)Γ → Spec(eH0,c e) is an isomorphism. Let y1 , . . . , yn be the tautological basis in Cn = h and x1 , . . . , xn be the dual basis in h∗ . The elements xn , yn still act on N Sn−1 ∼ = Cn . Show that [xn , yn ] ∈ O = {A| ...

... an isomorphism. Then we prove that the natural morphism RepΓ (H0,c , CΓ)// GL(CΓ)Γ → Spec(eH0,c e) is an isomorphism. Let y1 , . . . , yn be the tautological basis in Cn = h and x1 , . . . , xn be the dual basis in h∗ . The elements xn , yn still act on N Sn−1 ∼ = Cn . Show that [xn , yn ] ∈ O = {A| ...

Whittaker Functions and Quantum Groups

... We will soon enter territory where each Cartan type must be handled individually, and although results are available for other Cartan types we will restrict ourselves to Type A, that is, GL(n). Here is the Casselman-Shalika formula for GL(n). (It was proved earlier by Shintani for this case.) If G = ...

... We will soon enter territory where each Cartan type must be handled individually, and although results are available for other Cartan types we will restrict ourselves to Type A, that is, GL(n). Here is the Casselman-Shalika formula for GL(n). (It was proved earlier by Shintani for this case.) If G = ...