Lecture Topics and Speakers
1. Holographic RG
Speaker: Pavan Dharanipragada (IIT Madras)
Abstract/Plan:
In these lectures, I will develop the holographic RG correspondence. I will first describe the formalism of holographic renormalisation. In AdS/CFT, the bulk partition function is equal to the boundary generating functional of correlators. This also means a UV/IR connection. The IR divergence on the bulk side resulting from the infinite volume of AdS, and the UV divergences of the boundary theory are related. A subtraction procedure must be done so that these are consistently removed. This also implies that the radial cutoff of the bulk theory is dual to the UV cutoff of the boundary theory. Then moving the cutoff in the radial direction in the bulk is the same as performing RG in the boundary theory. I will describe how the bulk equations of motion can be seen as RG equations of the boundary theory, and then further develop the correspondence between integrating out the bulk degrees of freedom and Wilsonian RG of the boundary. In this context, I will briefly describe some of my work in which a precise regularisation scheme was found for the RG of the boundary theory that generates the bulk. Finally, I will briefly describe some recent developments in the field like the relation between holographic RG and TT bar deformation that connects the RG flow of TT bar deformed CFT2 with the bulk radial direction.
2. Algebraic field theory and Von-Neumann algebra
Speaker: Mohd. Ali (IISER Pune) and Abhinava Bhattacharjee (IISER TVM)
Abstract/Plan:
This pedagogical lecture series explores critical themes and advanced concepts in the algebraic formulation of quantum theory(AQT), focusing on its implications and applications in Quantum Field Theory and the mathematical structure of von Neumann algebras.
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The first part of the series will begin by introducing AQT, emphasizing the role of C* algebras, GNS representation theorem and unitarily inequivalent representations- a key phenomenon in QFT. The lectures will further delve into the structure of Algebraic Quantum Field Theory (AQFT), illustrating Haag-Kastler axioms, spectrum condition, Reeh–Schlieder theorem and split property.
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The Second part of the lecture series will cover the integral aspects of Von-Neumann algebras focusing on the Takesaki Modular theory, which is pivotal for understanding the classification of Von-Neumann algebras. Particularly the use of Type III factors in describing local observables will be emphasized. The lectures will conclude with applications in quantum field theory in curved spacetime and recent advancements in this area.
3. Complexity
Speaker: Budhaditya Bhattacharjee (Institute for Basic Science)
Abstract/Plan:
In this series of lectures, we will discuss various aspects of quantum complexity and demonstrate the methods and associated physics. The main focus of these lectures will be on various aspects of Krylov complexity and Out-of-time-ordered correlators. We will discuss these from both the holographic as well as quantum mechanical perspectives, wherever applicable. The approach of these lectures will be of a hands-on kind. The emphasis will be towards explicitly working out relevant examples, analytically and numerically. A proportionate amount of time will also be spent on discussing the open problems involving quantum complexities, including the state-of-the-art research progress in these directions. The overall contents of the lectures will be as follows:
We will discuss the (historical) motivations behind the study of complexity as a probe of the dynamical properties of both quantum and holographic systems. We will introduce the three main notions of complexity: circuit complexity, holographic complexity and Krylov complexity, and discuss each in some detail. (1 lecture)
We will turn our focus to Krylov complexity, and establish the general formalism. We shall discuss both the historical and recent progress regarding this. The discussion will mainly focus on the quantum mechanical aspect, with some foray into the relevant holographic notions. Important theorems (specifically regarding appropriate bounds) will be introduced and proved. (1-2 lecture)
We will turn our attention to specific examples. The following examples are planned for now (although that can be changed depending on the interest of the audience): the Sachdev-ye-kitaev model, conformal field theories, scalar quantum field theories and some quantum mechanical systems. Through these examples, we will introduce important aspects such as integrability breaking, coherent state formalism and quantum speed limits. We will also work out explicit examples in Random Matrix models (in the 3 Wigner Dyson classes). We shall demonstrate the numerical methods required to study these systems and spend some time on the advantages and limitations of the analytical and numerical methods. (2 lectures)
Finally, we will discuss some of the current questions regarding complexity that remain unanswered (or partially answered). Time permitting, we will also introduce some open quantum systems. We will establish the formalism in such open quantum systems and work out the properties of these probes. (1 lecture)
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The goal of these lectures is to acquaint the audience with the current state-of-the-art research regarding quantum complexity and enable them to tackle open problems without much further instruction.
4. Quantum Topology
Speaker: Visakh Narayanan (IISER Mohali)
Abstract/Plan:
Starting from Witten's 1985 paper, interpreting the Jones polynomial in the language of QFT, there has been an explosion of ideas intersecting many areas of Mathematics and Physics. I would be concentrating on a particular class of these, which are related to Topology and Chern-Simons theory. Using the Chern-Simons Lagrangian, one can produce several Topological Quantum Field theories and construct several invariants for manifolds. We will discuss some of these TQFT's such as Jones-Witten theory, Khovanov homology and Instanton Floer homology.
It is a famous problem in Physics (and now also in Mathematics) to interpret the Path integral as a Mathematically precise object. The usefulness of such an interpretation is going to be tremendous and may even be a trend-setter for new revolutions in both Physics and Mathematics! Hence through these discussions I wish to give a glimpse of how Mathematicians have been dealing with the path integral. It may be interesting that the tools involved are quite different from the ones in physics and there are no diverging sums anywhere!
5. Non-invertible Symmetries
Speaker: Sachin Grover (IIT Kanpur)
Abstract/Plan:
In these lectures, we will discover various aspects of non-invertible symmetries implemented by topological defects through examples in 2-dimensional CFTs and 3+1 dimensional abelian gauge theory. Additionally, we will discuss certain applications of non-invertible symmetries to anomalies and constraints on renormalisation group flows.
6. SCFT and Index
Speaker: Sujoy Mahato and Julius (HRI)
Abstract/Plan:
This set of lectures will cover an introduction to superconformal field theories (SCFT), superconformal indices and some advanced topics like 4d SCFT/ 2d chiral algebra correspondence.
Lecture 1: The first lecture will cover the basics of SCFTs including superconformal symmetry in four dimensions, unitary representations of SCFT algebra and building up to multiplet structures, we will also start to talk in greater detail about N=2 SCFT which will be the main class of theories relevant for us in later lectures.
Lecture 2: We will continue our discussion about N=2 SCFT. After finishing, we will move on to introducing the Witten index for supersymmetric quantum mechanics. Then we generalize it to define the superconformal index. We will mainly focus on a specific version of this index namely the Schur index. The other version of the indices will also be discussed very briefly.
Lecture 3: We will learn the Racah-Speiser algorithm as an efficient method to build supermultiplets bypassing sometimes messy honest supersymmetry calculations.
Lecture 4: We will then delve into the plethystic program, as a method to explicitly build partition functions in a theory. We will discuss some basic examples, before moving to the more complicated case of supersymmetric gauge theories.
Lecture 5: We will learn how to implement what we have learned in Mathematica. This will enable us to write our own Mathematica package, which we can potentially use in our own research, whenever we come across the need to count things.
Lecture 6: The last lecture of the series will be dedicated to something more advanced. We will discuss the 4d/2d correspondence in SCFs. And 4d SCFT with N=2 or more amount of supersymmetry possesses a protected subsector which satisfies a 2d chiral algebra. Constructing this chiral algebra sector starting from the 4d SCFT will be the larger part of this lecture. Toward the end, we will discuss the consequence of the 2d chiral algebra on four-dimensional physics.
Prerequisite: Familiarity with conformal symmetry and its representations will be highly beneficial to understanding the lectures. A lightning review of the same will be given for completeness.
Short Talks
(1) Aakash Kumar
Title: Classification of Dirac couplings under CRT transformations
Abstract: Massive higher spin particles are the bosonic irreducible representations of the Lorentz group with more than two Lorentz spacetime indices. Their study has been of great importance in the context of gravity at small length scales and the beyond-standard model of particle physics. Unlike massless particles, there is no proof that massive higher spin particles do not interact with particles of spin less than equal to 2. Recently, there has been a surge in understanding the interaction of the MHS particles with the standard model spectrum. In this work, we construct all possible kinematically allowed three-point interactions of two massless Dirac spinors with massive higher-spin bosons. In any $D$ spacetime, the interactions have been constructed using the projections of the massive higher spin representations of $Spin(D-1)$ over the massless complex spinor representations of $Spin(D-2)\times Spin(D-2)$. Based on this analysis, we have further classified the space of theories involving one massless Dirac spinor and a single (or multiple) MHS based on the discrete symmetries: $C, R,$ and $ T$. We found that in any $D=2m+1/2m$, the interacting theories of a single MHS have a $m$ mod $2$ classification.(2) Speaker:
(2) Aditya Singh
Title: Thermodynamic curvature of charged black holes with AdS_2 horizons.
Abstract: Sign and magnitude of the thermodynamic curvature provides empirical information about the nature of microstructures of a general thermodynamic system. For charged black holes in anti-de Sitter (AdS), thermodynamic curvature is positive for large charge or chemical potential, and diverges for extremal black holes, indicating strongly repulsive nature. We compute the thermodynamic curvature at low temperatures, for charged black holes with AdS_2 near horizon geometry, and containing a zero temperature horizon radius r_h, in a spacetime which asymptotically approaches AdS_D (for D > 3). In the semiclassical analysis at low temperatures, the curvature shows a novel crossover from negative to positive side, indicating the shift from attraction to repulsion dominated regime near T=0 before before diverging as 1/(γT), where γ is the coefficient of leading low temperature correction to entropy. Accounting for quantum fluctuations, the curvature computed in the canonical ensemble is positive, whereas the one in the grand canonical ensemble, shows a crossover from negative to positive side in the Schwarzian region. Moreover, the divergence of curvature at T=0 is cured irrespective of the ensemble used, resulting in a universal constant, inversely related to the number of symmetry generators of vacuum AdS_2.
(3) Arpit Maurya
Title: Differential Representation for Carrolian Correlators
Abstract: Recently, flat space limit of AdS Witten diagrams were used to develop Carrollian Witten diagrams. We consider the differential representation of AdS correlators andtake the appropriate flat space limit to consider differential representation of Carrolliancorrelators.
(4) Banashree Baishya
Title: A study of three butterflies
Abstract: In this talk, I will talk about the connection between two salient chaotic features, namely Lyapunov exponent and butterfly velocity, in an asymptotically Lifshitz black hole background with arbitrary critical exponent. These are calculated by using three methods - entanglement wedge method, out-of-time-ordered correlator computation and pole-skipping. I will present a comparative study of the results obtained from all these methods. So, we can rely on any one of these methods to compute the Lyapunov exponent and butterfly velocity in a holographic background. I will further study the chaos in the classical regime by computing the eikonal phase and Lyapunov exponent from the bulk gravity using a direct approach. Matching the results with those from the aforementioned methods yields a nontrivial dependence of the temperature of the chosen black hole with a critical exponent.
(5) Bhasker Shukla
Title: Impact of Chemical Potential on the Frame-Dependent Chaos of Suspended Strings in a Dynamical Holographic QCD Model with Magnetic Field.
Abstract: We extend our previous analysis of chaos in QCD confining string under a magnetic field by incorporating the effects of chemical potential. Utilizing a bottom-up holographic QCD model based on the Einstein-Maxwell-Dilaton action, we explore how varying chemical potential influences chaos from a frame-dependent perspective. Our results reveal distinct behaviours in the string and Einstein frames. Increasing the chemical potential with a fixed magnetic field in the string frame suppresses chaos in both parallel and perpendicular orientations. Conversely, in the Einstein frame, higher chemical potential enhances chaos in both orientations. These findings underscore the significant impact of chemical potential on the chaotic dynamics of confining strings and highlight the intricate interplay between magnetic fields, chemical potential, and frame dependence in holographic QCD models.
(6) Godwin Martin
Title: Exterior EFT for Hawking Radiation
Abstract: The gravitational Schwinger-Keldysh geometry allows one to derive effective field theories (EFTs) in the exterior of a black hole. These EFTs account for the effects of Hawking radiation. I will discuss the salient features of such EFTs in the case of scalar and fermionic fields. The talk will be based on the following aXiv preprints: 2403.10654 and 2403.10604.
(7) Kaustubh
Title: Proper time to singularity and thermal correlators
Abstract: We study certain higher point thermal correlators of heavy and light scalar primaries in a holographic CFT. Assuming simple self-interactions and couplings of the scalars in the bulk theory, we show that the thermal correlators contain a signature of the proper time to singularity from the horizon. The key idea is to use WKB approximations for the propagators and evaluate the bulk integrals using saddle point method.
(8) Raju Mandal
Title: An Infinite Family of S Invariant Theories on the Celestial Sphere
Abstract: We will start the talk with a discussion on the recent discovery of infinite dimensional symmetry algebra of non-abelian gauge theories in 4-D asymptotically flat spacetime. This algebra is known as S algebra and it was recently discovered by Strominger and collaborators in [1]. For gravity the asymptotic symmetry algebra is the wedge subalgebra of $w_{1+\infty}$ [2], which is different from \say{S algebra}. Recently for the gravity case, it was shown by Shamik Banerjee and collaborators that there exists a discrete infinite family of w-invariant theories on the celestial sphere [3]. Here we will discuss the similar analysis for gauge theories and will see that our analysis also hints at the existence of an infinite family of S-invariant gauge theories on the celestial sphere [4]. We will discuss all S-invariant celestial OPEs of the two positive helicity outgoing gluons and will also discuss the Knizhnik-Zamolodchikov type null states of these theories. Interestingly the bulk descriptions of these infinite number of theories are not known except for the tree-level MHV-sector and the self-dual Yang-Mills theory.[1]. arxiv.org/pdf/2103.03961[2]. arxiv.org/pdf/2105.14346[3]. arxiv.org/pdf/2301.13225[4]. arxiv.org/pdf/2311.16796
(9) Shivam Sharma
Title: Schwinger-Dyson Equations in Holography
Abstract: The AdS/CFT correspondence has significantly advanced our understanding of quantum gravity and strongly coupled systems. Real-time holography has proven to be an invaluable tool for studying non-equilibrium phenomena. While most studies have focused on the large N limit in the boundary or the classical regime in the bulk, there is still a need to examine finite N corrections in the boundary or loops in the bulk. This talk will introduce a systematic method to extend this framework to an arbitrary number of loops using Schwinger-Dyson equations in the bulk.
(10) Siddhi Swarupa Jena
Title: Holographic confining-deconfining gauge theories and entanglement measures with a magnetic field.
Abstract: We study various holographic pure and mixed-state entanglement measures in the confined/deconfined phases of a bottom-up AdS/QCD model in the presence of a background magnetic field. We analyze the entanglement entropy and entanglement wedge cross section and investigate how a background magnetic field leaves its imprints on the entanglement structure of these measures. Due to the anisotropy introduced by the magnetic field, we find that the behavior of these measures depends nontrivially on the relative orientation of the strip with respect to the field. In the confining phase, the entanglement entropy undergoes a phase transition at the same critical strip length, the magnitude of which increases/decreases for parallel/perpendicular orientation of the magnetic field. The entanglement wedge cross section similarly displays discontinuous behavior each time a phase transition between different entangling surfaces occurs. We further find that the magnetic field also introduces substantial changes in the entanglement measures of the deconfined phase; however, these changes remain qualitatively similar for all orientations of the magnetic field.
(11) Soumya Adhikari
Title: N=1 Euclidean Supergravity In Five Dimensions
Abstract: In this talk, we will discuss the systematic construction of N=1 Euclidean supergravity in five dimensions at the full non-linear level via an off-shell reduction scheme from N=(1,0) conformal supergravity in six dimensions, which is a Minkowskian theory. To get a Euclidean theory, the time-like coordinate of the six-dimensional theory have been compactified. The dimensional reduction of the six-dimensional N=(1,0) Weyl multiplet (40+40) gives a multiplet that decouples to the standard Weyl multiplet (32+32) and a vector multiplet (8+8). We will also discuss the construction of several Euclidean matter multiplets from the six-dimensional matter multiples. We will end the talk with some future directions.
(12) Vatsal
Title: Fermionic Chern--Simons theory on $S^2 \times S^1$ at large-$N$ in the `temporal' gauge
Abstract: Most of the computational evidence for the Bose--Fermi duality of fundamental fields coupled to $U(N)$ Chern--Simons theories originates in the large-$N$ calculations performed in the light-cone gauge. The evaluation of the thermal free energy on a finite-sized sphere is elusive in the light-cone gauge but more natural in another gauge, the `temporal' gauge. We use it to evaluate the finite-temperature partition function of $U(N)$ coupled fermions on $\mathbb{S}^2$ at large $N$, which is a novel physical setting for the problem. We set up the finite-temperature gap equations, solve them, and evaluate the partition function, closely following the tricks explored in \cite{ref1}, where we work out the results on $\mathbb{R}^2$ at large $N$, which demonstrate perfect agreement with the earlier light-cone gauge results.
