Conformal Field Theories

28 Nov 2022 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

Conformal Field Theories - Overview of the Renormalisation Group Flow and fixed points. Explicit examples of RG transformations: - decimation in the one dimensional Ising model: exact solution and proof of no non-trivial fixed point at finite temperature. - decimation in the two dimensional Ising Model: approximate solutions, fixed points, relevant and irrelevant couplings

30 Nov 2022 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

Wilson approach to Renormalisation in the continuum limit: integrating out shell of momenta. Evolution of interactions: relevant, irrelevant and marginal couplings. Ward identities in QFT: Path integral derivation. Conserved currents and charges as topological operators. Conserved currents, charges and generator of symmetry transformation.

07 Dic 2022 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

Consequences of having a traceless stress tensor: conformal killing equation and their solution. Infinitesimal and finite form of conformal transformations in general dimension. Conformal algebra: action of conformal charges on the stress tensor and extraction of commutation relation. Algebra of conformal group in general dimension.

19 Dic 2022 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

Transformation properties of operators under global conformal group. Irreducible representations of the conformal algebra on fields. Primary operators and descendants. General form of correlation functions of local operators consistent with conformal symmetry: 2 and 3 point functions of scalars. Conformal field theories in 2 dimensions. Global and local conformal transformations. Transformations properties of fields. Primaries and quasi-primaries.

21 Dic 2022 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

Recap of Ward identities for translations, rotation and dilations in a general QFT. Ward identities of the stress tensor in two dimensions and holomorphic properties. General form of correlation functions of primary operators in two dimensions.

09 Gen 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

Recap of Path Integral in QFT: time evolution and space-time foliation, definition of states, and vacuum. Radial quantisation and cylinder interpretation in two dimensions: conjugation in radial quantization as inversion; incoming and out coming states; radial ordering and commutators.

11 Gen 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

- State operator correspondence: insertion of local operators inside a sphere defines a state on the sphere and vice-versa. Operator Product Expansion. Operator Product Expansion. Validity of the OPE

13 Gen 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

Free Boson in two dimensions. Correlation functions, stress tensor and its OPE, central charge. Free boson in radial quantization: canonical quantization on the cylinder. Exercise: Free boson with twisted boundary conditions.

16 Gen 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

Expansion in normal modes of primary operators, in particular the stress tensor. Generators of the local conformal algebra. Virasoro algebra and central charge. Irreducible representations of the Virasoro Algebra: Vema modules.

18 Gen 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

Action of Virasoro generator on primary operators: Virasoro descendants. Reducible Verma modules and singular states. Unitary representation of Virasoro Algebra: negative norm states, Kac determinant.

20 Gen 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

Roots of Kac determinant. Examples of null descendants at level 1 and two. Differential equation for correlation functions from existence of null states. Fusion rules for Reducible Verma modules. Truncation of Fusion rule to prevent violation of unitarity.

23 Gen 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

- Truncated fusion rules and minimal models. Kac table. Unitary Minimal Models. Examples: Ising model, Potts Model. Vertex operators.

25 Gen 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

- Embedding Formalism in general dimension. Projective null hyperboloid and Poincaré section. Implementation of conformal transformation. Embedding of scalar and tensor operators. Most general form of correlation function in embedding space and projection to original space. Examples of 2,3,4 point functions. Encoding tensors in polynomials.

27 Gen 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

Encoding global symmetry tensors in polynomials. Recap of radial quantiazation in general dimension. Relation between correlation functions on the d-dimensional cylinder and in flat space. Conjugation in Euclidean space: scalars and tensors. EXERCISE: compute the correlation function of two scalar fields on the cylinder.

30 Gen 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

Unitarity bounds in general dimension. Explicit proof for traceless symmetric tensors. Examples of other representations. Recap of Operator product expansion in general dimension, consistency with conformal symmetry and relation with three point functions.

01 Feb 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

Recap of correlation functions. Comments about OPE: presence of identity and conserved currents, selection rules for SO(d) and for global symmetries. Constraints from permutations symmetries.

03 Feb 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

- Iterative use of OPE to reduce n-point functions to combination of smaller-n correlators. Four point functions and crossing equations. Crossing equations as constraints on the CFT-data. Definition of a CFT. Four point function of four identical scalar operators: OPE decomposition and crossing equation. Conformal blocks: definition in terms of OPE and in terms of projectors. Quadratic Casimir and differential equation for conformal blocks. Asymptotic behaving of the solution from OPE expansion.

06 Feb 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

Explicit solution of Casimir equation for conformal blocks in 2 and 4 dimensions. Radial expansion for conformal blocks: existence and convergence

08 Feb 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

Application of numerical bootstrap to two dimensions. Generalization to three dimensions: the Ising model. Generalized free theories: definitions and basic properties

10 Feb 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

Large N gauge theories: diagrammatic expansion and planar diagrams dominance. Factoriaztion of connected correlation functions. Scaling behaviour of stress tensor correlators and central charge. Anti de Sitter (AdS) spacetime: defintion, topology, choice of coordinates, boundary, isometries. Klein-Gordon equation in AdS, relation with Casimir operators of conformal group. Solutions.

13 Feb 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

QFT in Anti de Sitter: propagator and perturbation theory in terms of Witten diagrams. Boundary correlators as correlation functions of a CFT and their properties: conformal covariance, crossing symmetry, unitarity, conformal block decomposition. Alternative formulation of boundary theory: connection between boundary generating functional and AdS path integral with given boundary conditions. Inclusion of gravity: properties of gravity Path Integral with AdS boundary conditions: locality, Weyl invariance. Limit of semi-classical gravity defines a large N CFT dual.

15 Feb 2023 (2h 00m)

Alessandro Vichi - Corso (attività didattica) - In presenza

: Concrete example of AdS/CFT duality: N=4 SYM in four dimensiona and Type IIB string theory on AdS5xS5 background. Symmetries and properties of the two sides. Intuitive derivation of the conjecture staring from D-branes picture. Relation between string loops expansion and large N expansion. Relation between string tension and t'Hooft coupling.