# XIII AVOGADRO MEETING <br> on Strings, Supergravity and Gauge Theories

Europe/Rome
Aula Magna (University of Padova, Scuola Galileiana)

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova
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Description

### The Avogadro Meetings started in 2005 as an occasion for young Italian theoretical physicists to share their ideas and results in an informal atmosphere. The meeting is named after the University of Piemonte Orientale that hosted its first three editions.The meeting is traditionally scheduled just before the Christmas break to facilitate the participation of Italian postdocs and PhD students working abroad who can take the chance of their travel back home for Christmas to meet young colleagues and exchange ideas.In order to stress the pedagogical aim of the meeting, preference is given to extended presentations on general themes rather than to conventional seminars on specific works, possibly organised and illustrated by more than one speaker and followed by a long discussion session.Consistently with the original spirit of the event, the invited speakers are usually Italian. However participation is open to anybody and non-Italians are more than welcome to join. For this reason the seminars are presented in English.

Participants
• Alberto Merlano
• Alessandro Bombini
• Alessandro Georgoudis
• Alessandro Pini
• Alessandro Sfondrini
• Alessandro Torrielli
• Alessio Marrani
• Andrea Campoleoni
• Andrea Dei
• Andrea Marzolla
• Andrea Orta
• Anton Nedelin
• Antonio Amariti
• Antonio Pittelli
• Antonio Sciarappa
• Carlo Heissenberg
• Carlo Maccaferri
• Chiara Toldo
• Céline Zwikel
• Daniele Dorigoni
• Dario Benedetti
• Dario Francia
• Dario Rosa
• Davide Cassani
• Diego Bombardelli
• Domenico Seminara
• Edoardo Vescovi
• Enrico Olivucci
• Erik Tonni
• Fabio Apruzzi
• Fabrizio Del Monte
• Fabrizio Nieri
• Fabrizio Rompineve
• Federico Bonetti
• Federico Carta
• Francesca Ferrari
• Francesco Benini
• Francesco D'Eramo
• Francesco Galvagno
• Gianguido Dall'Agata
• Gianluca Zoccarato
• Gianmassimo Tasinato
• Giuseppe Dibitetto
• Guglielmo Lockhart
• Igor Pesando
• Jacopo Sisti
• Lorenzo Bianchi
• Lorenzo Casarin
• Lorenzo Papini
• Lorenzo Ruggeri
• Luca Cassia
• Luca Griguolo
• Luca Martucci
• Luigi Tizzano
• Marco Bianchi
• Marco Fazzi
• Marco Matone
• Massimiliano Ronzani
• Matteo Lotito
• Matteo Poggi
• Michelangelo Preti
• Michele Cicoli
• Natalia Pinzani Fokeeva
• Niccolò Cribiori
• Nicolo' Piazzalunga
• Nicolò Petri
• Oscar de Felice
• Paolo Benincasa
• Paolo Glorioso
• Paolo Gregori
• Paolo Milan
• Pieralberto Marchetti
• Pierpaolo Mastrolia
• Praxitelis Ntokos
• Raffaele Savelli
• Riccardo Borsato
• Roberto Valandro
• Roberto Volpato
• Sergio Benvenuti
• SEYED MORTEZA HOSSEINI
• Simone Giacomelli
• Stefano Cremonesi
• Stefano Giusto
• Stefano Lanza
• Stefano Speziali
• Sukruti Bansal
• Valentin Reys
• Valentina Giangreco Puletti
• Wednesday, 20 December
• 08:45 09:00
Opening Aula Magna

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova
Convener: Davide Cassani (PD)
• 09:00 11:00
Weak Gravity and Swampland Conjectures in String Inflation Aula Magna

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova

The amount of gravitational waves produced during inflation is crucially linked to the distances in field space traversed by the inflaton field. In particular, large primordial tensor modes require field displacements larger than the Planck scale. This is the scenario of Large Field Inflation (LFI). As we await the final observational verdict on this class of realizations, it is important to understand whether transplanckian field values violate general properties of Quantum Gravity. The Weak Gravity Conjecture (WGC) aims at capturing implications of Quantum Gravity for Effective Field Theory (EFT), thus represents a promising tool to address the theoretical consistency of LFI.

In the first part we will give an introduction to the swampland program which tries to identify the criteria that low energy effective field theories ought to satisfy in order to couple consistently with quantum gravity. We will enumerate the known swampland conjectures offering arguments to support them as well as some examples drawn from string theory where these conjectures are satisfied. A lot of attention will be devoted to the Weak Gravity Conjecture and its various formulations.

In the second part we focus on the consequences of the WGC for models of Large Field Inflation. We start by introducing the basic problems of Large Field Inflation in EFT and thus motivate axions as inflaton candidates. We then review general constraints on axion inflation based on an appropriately generalized WGC. We also discuss constraints on setups based on string compactifications. We then specify to two subclasses of string-motivated models: Axion Alignement and Axion Monodromy. We focus on stringy realizations and discuss interpretations of the WGC in terms of the geometry of the extra dimensions and moduli stabilization.

Conveners: Fabrizio Rompineve (IFAE Barcelona), Gianluca Zoccarato (Hong Kong IAS and UW Madison)
• 11:00 11:30
Coffee break 30m Aula Magna

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova
• 11:30 12:30
Weak Gravity and Swampland Conjectures in String Inflation: discussion session Aula Magna

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova

The amount of gravitational waves produced during inflation is crucially linked to the distances in field space traversed by the inflaton field. In particular, large primordial tensor modes require field displacements larger than the Planck scale. This is the scenario of Large Field Inflation (LFI). As we await the final observational verdict on this class of realizations, it is important to understand whether transplanckian field values violate general properties of Quantum Gravity. The Weak Gravity Conjecture (WGC) aims at capturing implications of Quantum Gravity for Effective Field Theory (EFT), thus represents a promising tool to address the theoretical consistency of LFI.

In the first part we will give an introduction to the swampland program which tries to identify the criteria that low energy effective field theories ought to satisfy in order to couple consistently with quantum gravity. We will enumerate the known swampland conjectures offering arguments to support them as well as some examples drawn from string theory where these conjectures are satisfied. A lot of attention will be devoted to the Weak Gravity Conjecture and its various formulations.

In the second part we focus on the consequences of the WGC for models of Large Field Inflation. We start by introducing the basic problems of Large Field Inflation in EFT and thus motivate axions as inflaton candidates. We then review general constraints on axion inflation based on an appropriately generalized WGC. We also discuss constraints on setups based on string compactifications. We then specify to two subclasses of string-motivated models: Axion Alignement and Axion Monodromy. We focus on stringy realizations and discuss interpretations of the WGC in terms of the geometry of the extra dimensions and moduli stabilization.

• 12:30 14:30
Lunch break 2h
• 14:30 16:30
Strings and Number Theory Aula Magna

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova

In this session, we review three topics that lie at the intersection between string theory and number theory.

(1) Moonshine.
The name "moonshine" refers to a series of mysterious coincidences relating modular forms and finite groups. The classical example, Monstrous moonshine, led to the discovery of new unexpected connections between string theory, conformal field theory, algebra and geometry. After a brief description of Monstrous moonshine, we discuss the more recent and still unexplained examples of Mathieu and umbral moonshine, and their possible relations with string theory on K3 surfaces.

(2) Mock modular forms and Black holes.
We review the connection between mock Jacobi forms and generating functions for "immortal" 1/4-BPS dyons in N=4 string models, as described by Dabholkar-Murthy-Zagier.

(3) 3-Manifold invariants and Quantum modular forms.
We introduce the concept of quantum modular form in relation to Chern-Simons theory, as first analyzed by Lawrence and Zagier. This is the starting point to investigate the appearence of quantum, false and mock modular forms in the context of 3-manifolds.

Conveners: Francesca Ferrari (Amsterdam U.), Roberto Volpato (Univ. Padova)
• 16:30 17:00
Coffee break 30m Aula Magna

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova
• 17:00 18:00
Strings and Number Theory: discussion session Aula Magna

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova

In this session, we review three topics that lie at the intersection between string theory and number theory.

(1) Moonshine.
The name "moonshine" refers to a series of mysterious coincidences relating modular forms and finite groups. The classical example, Monstrous moonshine, led to the discovery of new unexpected connections between string theory, conformal field theory, algebra and geometry. After a brief description of Monstrous moonshine, we discuss the more recent and still unexplained examples of Mathieu and umbral moonshine, and their possible relations with string theory on K3 surfaces.

(2) Mock modular forms and Black holes.
We review the connection between mock Jacobi forms and generating functions for "immortal" 1/4-BPS dyons in N=4 string models, as described by Dabholkar-Murthy-Zagier.

(3) 3-Manifold invariants and Quantum modular forms.
We introduce the concept of quantum modular form in relation to Chern-Simons theory, as first analyzed by Lawrence and Zagier. This is the starting point to investigate the appearence of quantum, false and mock modular forms in the context of 3-manifolds.

• Thursday, 21 December
• 09:00 11:00
Effective Theories for Hydrodynamics Aula Magna

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova

In these lectures we describe a first principle formulation of hydrodynamics where, to incorporate effects of dissipation and fluctuations, the Schwinger-Keldysh formalism plays a crucial role. Using an effective field theory approach we first establish what are the necessary low-energy degrees of freedom and the symmetries that a Schwinger-Keldysh effective action should have to describe hydrodynamic behaviour. Subsequently, we show how this formalism can be efficiently implemented using superspace techniques to obtain simple effective actions. Finally, we give a derivation of the local second law of thermodynamics by applying a Noether-like procedure to the effective action.

Conveners: Natalia Pinzani Fokeeva (KU Leuven), Paolo Glorioso (Chicago U.)
• 11:00 11:30
Coffee break 30m Aula Magna

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova
• 11:30 12:30
Effective Theories for Hydrodynamics: discussion session Aula Magna

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova

In these lectures we describe a first principle formulation of hydrodynamics where, to incorporate effects of dissipation and fluctuations, the Schwinger-Keldysh formalism plays a crucial role. Using an effective field theory approach we first establish what are the necessary low-energy degrees of freedom and the symmetries that a Schwinger-Keldysh effective action should have to describe hydrodynamic behaviour. Subsequently, we show how this formalism can be efficiently implemented using superspace techniques to obtain simple effective actions. Finally, we give a derivation of the local second law of thermodynamics by applying a Noether-like procedure to the effective action.

• 12:30 14:30
Lunch break 2h
• 14:30 16:30
From SYK to Tensor Models Aula Magna

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova

The Sachdev-Ye-Kitaev (SYK) model and the tensor models have recently attracted a considerable amount of attention. Their popularity is mainly due to the fact that they are examples of quantum mechanical models which are exactly solvable in the large N limit but nevertheless they have interesting and non-trivial physical properties. They are also believed to be toy models to study in a quantitative way important aspects of quantum black holes.

In the first part of the lecture, after a brief overview of the key properties of quantum black holes which these models are able to reproduce, we will provide an overview to the SYK model and its flavored generalization, introduced by Gross and Rosenhaus. To be more precise, we will review the computations of the two-point function and of the four-point function. As we will see, the computation is greatly simplified by the fact that, in the large N limit, only a very particular set of diagrams survives (the so-called “melonic” diagrams). This simplification is the key ingredient
that makes the SYK model solvable. From the four-point function we will be able to compute the Lyapunov exponent and we will see that SYK is maximally chaotic. We will also make some comments about the main drawback of the SYK model (and of its flavored generalization): they are defined in terms of couplings which are random gaussian variables. This property makes problematic the interpretation of the SYK a standard, quantum mechanical system.

For this reason, in the second part of the talk, we will discuss tensor models. They are a natural generalization of matrix models, and it has been known since a few years that they have a large-N limit dominated by melonic diagrams. Therefore, quantum mechanical models of fermions with tensor indices are natural candidates for models with the same interesting features of the SYK model, without the drawbacks of quenched disorder. In the second half of the lecture we will review the basics of tensor models and their large-N limit, we will discuss the construction of tensorial analogues of the SYK model and some key differences between the two, and we will present some of their most recent developments.

Conveners: Dario Benedetti (LPT Orsay), Dario Rosa (KIAS Seoul)
• 16:30 17:00
Coffee break 30m Aula Magna

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova
• 17:00 18:00
From SYK to Tensor Models: discussion session Aula Magna

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova

The Sachdev-Ye-Kitaev (SYK) model and the tensor models have recently attracted a considerable amount of attention. Their popularity is mainly due to the fact that they are examples of quantum mechanical models which are exactly solvable in the large N limit but nevertheless they have interesting and non-trivial physical properties. They are also believed to be toy models to study in a quantitative way important aspects of quantum black holes.

In the first part of the lecture, after a brief overview of the key properties of quantum black holes which these models are able to reproduce, we will provide an overview to the SYK model and its flavored generalization, introduced by Gross and Rosenhaus. To be more precise, we will review the computations of the two-point function and of the four-point function. As we will see, the computation is greatly simplified by the fact that, in the large N limit, only a very particular set of diagrams survives (the so-called “melonic” diagrams). This simplification is the key ingredient
that makes the SYK model solvable. From the four-point function we will be able to compute the Lyapunov exponent and we will see that SYK is maximally chaotic. We will also make some comments about the main drawback of the SYK model (and of its flavored generalization): they are defined in terms of couplings which are random gaussian variables. This property makes problematic the interpretation of the SYK a standard, quantum mechanical system.

For this reason, in the second part of the talk, we will discuss tensor models. They are a natural generalization of matrix models, and it has been known since a few years that they have a large-N limit dominated by melonic diagrams. Therefore, quantum mechanical models of fermions with tensor indices are natural candidates for models with the same interesting features of the SYK model, without the drawbacks of quenched disorder. In the second half of the lecture we will review the basics of tensor models and their large-N limit, we will discuss the construction of tensorial analogues of the SYK model and some key differences between the two, and we will present some of their most recent developments.

• 20:00 22:30
Conference dinner 2h 30m Sala Rossa (Caffe Pedrocchi)

### Sala Rossa

#### Caffe Pedrocchi

• Friday, 22 December
• 09:00 11:00
Integrability and String Sigma Models Aula Magna

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova

Certain conformal field theories and dual string theories appearing in the AdS/CFT correspondence turn out to possess (in the planar limit) hidden symmetries. These allow us to apply the exact methods of integrability to physical questions such as the spectral problem. For the gauge theory this possibility leads to exact results which go beyond the usual perturbation theory. In this seminar we will concentrate on the string theory side of the correspondence. As a first step we will outline the derivation of the Green-Schwarz string action for the AdS_5 × S^5 background through the coset procedure. After that, we will briefly mention two possible light-cone gauge choices and describe the spectrum of elementary worldsheet excitations for the two resulting backgrounds. The scattering matrix of such excitations is one of the quantities which can be fixed using the symmetry of the problem and some additional physical inputs, such as unitarity, crossing and agreement with perturbation theory. We will focus on the latter and sketch the procedure for computing the worldsheet S-matrix at strong coupling. We conclude the first part of the talk with a summary of the available perturbative results and a list of possible future directions. In the second part we will review some concepts of classical integrability of two-dimensional sigma-models which appear also for the string on AdS_5 × S^5 . We will further present some methods to generate deformations of the worldsheet theories: they correspond to deformations (e.g. squashing) of the target space, and although they (partially) break the original global symmetries they do not spoil the integrable structure. Some new lines of research motivated by the study of deformations of the superstring will be also briefly discussed, including the so called “generalised supergravity equations”. We will conclude by mentioning proposals for the interpretation of these deformations from the gauge theory point of view, and some of the open problems.

Conveners: Lorenzo Bianchi (Hamburg U.), Riccardo Borsato (Nordita Stockholm)
• 11:00 11:30
Coffee break 30m Aula Magna

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova
• 11:30 12:30
Integrability and String Sigma Models: discussion session Aula Magna

### Aula Magna

#### University of Padova, Scuola Galileiana

via San Massimo 33, 35121 Padova

Certain conformal field theories and dual string theories appearing in the AdS/CFT correspondence turn out to possess (in the planar limit) hidden symmetries. These allow us to apply the exact methods of integrability to physical questions such as the spectral problem. For the gauge theory this possibility leads to exact results which go beyond the usual perturbation theory. In this seminar we will concentrate on the string theory side of the correspondence. As a first step we will outline the derivation of the Green-Schwarz string action for the AdS_5 × S^5 background through the coset procedure. After that, we will briefly mention two possible light-cone gauge choices and describe the spectrum of elementary worldsheet excitations for the two resulting backgrounds. The scattering matrix of such excitations is one of the quantities which can be fixed using the symmetry of the problem and some additional physical inputs, such as unitarity, crossing and agreement with perturbation theory. We will focus on the latter and sketch the procedure for computing the worldsheet S-matrix at strong coupling. We conclude the first part of the talk with a summary of the available perturbative results and a list of possible future directions. In the second part we will review some concepts of classical integrability of two-dimensional sigma-models which appear also for the string on AdS_5 × S^5 . We will further present some methods to generate deformations of the worldsheet theories: they correspond to deformations (e.g. squashing) of the target space, and although they (partially) break the original global symmetries they do not spoil the integrable structure. Some new lines of research motivated by the study of deformations of the superstring will be also briefly discussed, including the so called “generalised supergravity equations”. We will conclude by mentioning proposals for the interpretation of these deformations from the gauge theory point of view, and some of the open problems.