# Program

### Wednesday, December 17 - Sala Stemmi:

8.45 - 9.00 *Opening*

9.00 - 11.00

**Giuseppe Policastro** (ENS Paris)

**Erik Tonni **(SISSA and INFN)

*Entanglement and holography*

### Abstract and references

**Abstract**

We will review the basic ideas underlying the computations of the entanglement entropy and the Renyi entropies in conformal field theories and of their holographic counterparts. The topics that will be covered are

-) path integral representation of the reduced density matrix

-) replica trick for the entanglement entropy

-) twist fields and Riemann surfaces for the Renyi entropies

-) reduced density matrix for a sphere in higher dimensional CFT

-) Ryu-Takayanagi formula for the holographic entanglement entropy

-) covariant generalizations of the RT prescription to time-dependent situations

-) connection to black hole entropy via the "generalized gravitational entropy"

Additional topics:

-) strong subadditivity

-) holographic Renyi entropies

**References:**

reviews:

http://arxiv.org/abs/1204.2450

http://arxiv.org/abs/0905.4013

CFT:

http://arxiv.org/abs/hep-th/9403108

http://arxiv.org/abs/hep-th/0405152

http://arxiv.org/abs/1102.0440

Holography:

http://arxiv.org/abs/hep-th/0605073

http://arxiv.org/abs/0704.3719

http://arxiv.org/abs/arXiv:0705.0016

http://arxiv.org/abs/1110.1084

http://arxiv.org/abs/1304.4926

11.00 - 11.30 Coffee break

11.30 - 12.30

12:30 - 14:30 Lunch break

14.30 - 16.30

** Andrea Campoleoni **(Université Libre de Bruxelles)

**Massimo Taronna** (AEI Potsdam)

*Higher-spins and strings*

### Abstract and references

**Abstract**

We give an introduction to Vasiliev equations aimed at deciphering some recent results on the relations between higher-spin gauge theories and string theory. After reviewing the main ingredients of the unfolded formulation we take advantage of it in a holographic context, emphasizing the bulk-to-bulk duality between Vasiliev theories and the ABJM model. We conclude by providing a brief overview of the current status of higher spin holography.

**References**

S. Giombi and X. Yin, ``The Higher Spin/Vector Model Duality,'' J. Phys. A46 (2013) 214003

http://arxiv.org/abs/arXiv:1208.4036

M.A. Vasiliev, ``Holography, Unfolding and Higher-Spin Theory,'' J. Phys. A46 (2013) 214013.

http://arxiv.org/abs/arXiv:1203.5554

C.M. Chang, S. Minwalla, T. Sharma and X. Yin, ``ABJ Triality: from Higher Spin Fields to Strings,'' J. Phys. A46 (2013) 214009

http://arxiv.org/abs/arXiv:1207.4485

M.R. Gaberdiel and R. Gopakumar, ``Higher Spins & Strings,'' JHEP 1411 (2014) 044

http://arxiv.org/abs/arXiv:1406.6103

16.30 - 17.00 Coffee break

17.00 - 18.00

### Thursday, December 18 - Sala Stemmi:

9.00 - 11.00

**Agnese Bissi** (Oxford University)

**Alessandro Vichi **(CERN)

*Conformal bootstrap*

### Abstract and references

**Abstract**

The conformal bootstrap is a program that aims to exploit general properties of a Conformal Field Theories (CFT), such as unitarity, operator product expansion (OPE) and crossing symmetry to solve completely the theory. Despite the idea goes back to the '70, only recent progresses in the conformal block decomposition and numerical techniques have led to quantitative results.

In the first lecture we will review the main features of CFTs, introduce the conformal bootstrap idea and present a few applications. Topics covered in the first part include:

- review of conformal symmetry in D dimensions and its representations

- Operator product expansion

- structure of two, three and four point functions

- crossing symmetry constraints

- conformal bootstrap and numerical algorithms

- the Ising model in D<4 dimension

The second part extends the discussion to SuperConformal Field Theories (SCFT's), outlining the main differences from the non-supersymmetric case and presenting a few applications. The second lecture will cover:

- N=1 and N=4 Super conformal algebra

- protected operators

- super conformal blocks

- future directions

**References**

*CFT Review:*

- S. Ferrara, R. Gatto and A. F. Grillo, “Conformal algebra in space-time and operator product expansion”, Springer Tracts Mod. Phys. 67, 1 (1973).

- I. T. Todorov, M. C. Mintchev and V. B. Petkova, “Conformal Invariance In Quantum Field Theory”, Pisa, Italy: Sc. Norm. Sup. (1978)

- E. S. Fradkin and M. Y. Palchik, “Conformal quantum field theory in D-dimensions,” Dordrecht, Netherlands: Kluwer (1996)

- P. Di Francesco, P. Mathieu and D. Senechal, “Conformal Field Theory”, New York, USA: Springer (1997)

*Conformal blocks:*

- F. A. Dolan and H. Osborn, “Conformal partial waves and the operator product expansion”, Nucl. Phys. B 678, 491 (2004) [arXiv:hep-th/0309180]

- F. A. Dolan and H. Osborn, “Superconformal symmetry, correlation functions and the operator product expansion”, Nucl. Phys. B 629, 3-73 (2002) [arXiv:hep-th/0112251]

- F. A. Dolan and H. Osborn, “Conformal partial waves Further Mathematical Results”, Nucl. Phys. B 678, 491 (2004) [arXiv:1108.6194]

*Conformal bootstrap:*

- R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, “Bounding scalar operator dimensions in 4D CFT”, JHEP 12 (2008) 031, arXiv:0807.0004

- D. Poland, D. Simmons-Duffin, and A. Vichi, “Carving Out the Space of 4D CFTs”, JHEP 1205 (2012) 110, arXiv:1109.5176

S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, “Solving the 3D Ising Model with the Conformal Bootstrap”, Phys.Rev. D86 (2012) 025022, arXiv:1203.6064

*Super Conformal bootstrap:*

- D. Poland and D. Simmons-Duffin, “Bounds on 4D Conformal and SuperconformalField Theories”, JHEP 1105 (2011) 017, arXiv:1009.2087

- C. Beem, L. Rastelli, B. C. van Rees, “The N=4 Superconformal bootstrap”, Phys. Rev. Lett. 111 (2013) 071601, arXv:1304.1803

- L. F. Alday and A. Bissi, “Generalized boostrap equations for N=4 SCFT”, arXv:1404.5864

- M. Nirschl, H. Osborn, “Superconformal Ward identities and their solution“, Nucl.Phys. B711 (2005) 409-479, [arXiv:0407060]

- F.A. Dolan, H. Osborn , “Conformal partial wave expansions for N=4 chiral four point functions“, Annals Phys. 321 (2006) 581-626, [arXiv:0412335]

11.00 - 11.30 Coffee break

11.30 - 12.30

12:30 - 14:30 Lunch break

14.30 - 16.30

**Fabio Finelli **(INAF-IASF Bologna and INFN)

**Angiola Orlando** (Rome U. Sapienza)

**David Pirtskhalava** (SNS Pisa)

*Cosmology in the Planck/Bicep era*

### Abstract and references

**Abstract**

The theory of Cosmic Inflation postulates that our entire observable universe was spawned from a quantum fluctuation in an incredibly brief burst of hyper expansion. Inflation makes several predictions which appear to match cosmological observations. In these lectures we will present the implications for inflation from the measurement of the Cosmic Microwave Background (CMB) anisotropies pattern from two different experiments: BICEP2 and Planck.

One key prediction for inflation is a background of gravitational waves which may produce a specific observable feature in the CMB polarization anisotropies pattern - the long sought B-mode polarization. Using data from a specialized radio telescope called BICEP2 operating from the South Pole in Antarctica our collaboration recently reported a high significance detection of B-modes at 150GHz and few degree angular scales. The BICEP2 experiment and results will be presented, as well as the evolving situation as regards astrophysical polarized emission from dust grains in our own galaxy, which the Planck space mission suggests to be non-negligible contaminant of our cosmological interpretation of B-mode polarization signal.

The Planck space mission has measured the CMB anisotropies pattern scanning the full sky in nine frequency channels covering from 27 GHz to 1 THz between 2009 and 2013. Since the first cosmological results from the nominal mission data in 2013, Planck has been a formidable probe of the predictions of the simplest inflationary models. The main implications of the Planck measurements for the physics of inflation and cosmology will be presented, including a preview of the 2014 results.

In the following we suggest references which are already in the public domain. Others will appear soon on the topics discussed.

**References:**

- BICEP2 Collaboration, “Detection of B-mode Polarization at Degree Angular Scales by BICEP2”, Phys.Rev.Lett., 112, 24 (2014).
- Planck Collaboration, “Planck Intermediate Results. XXX. The angular power spectrum of polarized dust emission at intermediate and high Galactic latitudes”, arXiv: 1409.5738.
- Planck Collaboration, “Planck 2013 results. XXII. Constraints on Inflation”, Astronomy and Astrophysics, 571 (2014) A22
- Planck Collaboration, “Planck 2013 results. XVI. Cosmological parameters”, Astronomy and Astrophysics, 571 (2014) A16

16.30 - 17.00 Coffee break

17.00 - 18.00

20.00 *
Social dinner
*

### Friday, December 19 - Aula Bianchi:

9.00 - 11.00 **
Daniele Dorigoni** (Cambridge Univeristy)

*Resurgence: a bridge between perturbative and non perturbative physics*

### Abstract and references

**Abstract**

Resurgence provides a systematic unification of semiclassical analysis for the perturbative and non-perturbative sectors. It can be applied to resolve fundamental problems in quantum theories with degenerate minima. Expansions about different saddle points are quantitatively related to one another in a precise manner. I will illustrate the general framework by discussing various examples, from the double-well potential in QM, to asymptotically free QFTs such as the CP^N and the principal chiral model and Yang-Mills, where this resurgent approach yields a new semiclassical interpretation of IR renormalons.

**References**

Aniceto and Schiappa, ``Nonperturbative Ambiguities and the Reality of Resurgent Transseries’'

http://arxiv.org/abs/1308.1115

Dunne and Unsal, ``Uniform WKB, Multi-instantons, and Resurgent Trans-Series’'

http://arxiv.org/abs/1401.5202

Cherman, Dorigoni and Unsal, `Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles’'

http://arxiv.org/abs/1403.1277

Dorigoni, ``An Introduction to Resurgence, Trans-Series and Alien Calculus''

http://arxiv.org/abs/1411.3585

11.00 - 11.30 Coffee break

11.30 - 12.30

12:30 - 12:45