The bootstrap program for large N QCD, part 1: Overview and form factors

• Leonardo Rastelli

Room: Oliver Thompson Lecture Theatre 101 In this talk, I will first give a general overview of the S-matrix bootstrap program for large N QCD. I will then describe how to import the microscopic dynamics of the theory by incorporating the two-point function and pion form factor of the simplest local gauge invariant operator — the vector current.

The bootstrap program for large N QCD, part 2: Rho meson scattering

• Alessandro Vichi

Room: Oliver Thompson Lecture Theatre 101 In this talk I will consider the elastic scattering of Rho mesons in large N QCD . I will first present the bootstrap framework for scattering of spin one particles and discuss the constraints and expectations for this system. I will then present bounds on on-shell couplings based on various spectral assumptions.

S-matrix bootstrap with unequal particles

• Marco Serone

Room: Oliver Thompson Lecture Theatre 101 "We study 2-2 scattering amplitudes of two unequal scalars using the S-matrix bootstrap. Maximal analyticity does not hold, yet general bounds can be derived for arbitrary mass ratio between the two external particles. We discuss the main challenges one has to face in this set-up, such as the existence of pseudo-thresholds."

Imprints of asymptotic freedom on confining strings

• Alexandre Homrich

Room: Oliver Thompson Lecture Theatre 101 There is a longstanding dream of realizing large-N Yang-Mills theory in D = 3,4 as a theory of confining strings. So far, our knowledge of this putative string is restricted to low energies, where one can use an effective field theory to describe small deformations of a long flux tube. In this talk, I will describe what asymptotic freedom of the underlying gauge theory tells us about the high-energy behavior of the string. First, by comparing the partition function of a short cylinder with a perturbative computation, we will extract the high-energy asymptotics of the spectral density of closed-string states sourced by a Polyakov loop. We will then consider an integrable toy model for the worldsheet that realizes such high-energy behavior.

No particle-production in AdS2?

• Antonio Antunes

Room: Oliver Thompson Lecture Theatre 101 Integrable 2d QFTs with exact S-matrices and no particle production are one of the few examples of solvable interacting QFTs, especially in the absence of supersymmetry or conformal invariance. We discuss evidence that a similar structure can persist for QFTs in AdS2, manifesting itself in the absence of multi-twist operators in the OPE of the single-particle state in the dual CFT1. In particular, we present numerical solutions of the multi-correlator bootstrap which exhibit this property, discuss how they can be understood analytically and contrast them with previous no-go theorems on integrable QFTs in AdS2.

Where is tree-level heterotic string theory?

• Waltraut Knop

Room: Oliver Thompson Lecture Theatre 101 In this talk, we continue the tree-level S-matrix bootstrap program for quantum gravity in ten-dimensional theories with half-maximal supersymmetry. Imposing analyticity, crossing symmetry, unitarity, and Regge boundedness, we constrain weakly coupled UV completions of supergravity and derive bounds on effective field theory coefficients. We show how the heterotic and closed string theories arise as consistent UV completions within the allowed region. Under mild spectral assumptions, much of the boundary is explained by extremal amplitudes supported on a single linear Regge trajectory, providing further evidence that linear Regge trajectories are a robust feature of the tree-level quantum-gravity bootstrap.

Non-renormalizability properties of extremal correlators of 1/4 BPS operators.

• Agnese Bissi

Room: Oliver Thompson Lecture Theatre 101 I will discuss how to deal with extremal correlators involving at least one 1/4 BPS operator. In particular, I will present an analysis of the different multiplets appearing in the operator product expansion and their consequences for the renormalizability of the correlators, using analytic superspace techniques. This talk is based on work in progress with Giulia Fardelli and Hector Puerta-Ramisa.

Dynamics of Heavy Operators and Holography

• Francesco Aprile

Room: Oliver Thompson Lecture Theatre 101 I will describe the computation of a class of Heavy-Heavy-Light-Light (HHLL) correlators in which the heavy operator is a coherent state built from multi-graviton states. In N=4 SYM, this problem can be approached from several complementary perspectives, connecting gravity, string theory, and beautiful mathematics. Perturbatively, the leading-order contribution to the HHLL correlator is computed explicitly by combining a bootstrap approach based on the consistency of the OPE in the dual CFT with a gravity approach that maps the problem of determining the Green's function of the dilaton field in the heavy background to the connection problem for a Heun equation. Non-perturbatively, an integrated version of the HHLL correlator is computed exactly as a function of the relevant parameters using supersymmetric localization and modular resurgence analysis. Taken together, these results provide non-trivial predictions for the string theory dual and I will discuss how this information is encoded holographically.

Graded Unitarity

• Chris Beem

Room: Oliver Thompson Lecture Theatre 101 I will describe the variant of unitarity for vertex algebras/chiral algebras that holds for those algebras related to four-dimensional N=2 SCFTs via the SCFT/VOA correspondence. (A weakened version also applies to a large class of boundary VOAs for three-dimensional N=4 theories.) I will survey some (very partial) classification results for graded unitary VOAs under strong technical assumptions. Time allowing, I will outline some neat consequences of graded unitarity for certain BRST complexes arising from three- and four- dimensional gauge theories.

Real evolution from imaginary spin

• George Yuzhe Zhou

Room: Oliver Thompson Lecture Theatre 101 It is useful to characterize a high energy state in a massive QFT by its particle number densities as a function of particle momenta. We describe how, in massive RG flows starting from a UV CFT, complex spin CFT data defines evolution equations for these densities as a function of the energy of the state. This picture naturally arises when considering the idea of detector matching. Detector operators are non-local operators that provide the analytic continuation of operators to complex spin, and detector matching is the statement that detectors in an IR theory can be matched onto a sum of detectors in the UV. As an explicit example, we present results showing how density evolution and matching works in the 2D Ising model and its deformations. We also present an independent derivation of evolution and matching in the case of integrable models by treating states as a classical gas and employing thermodynamic arguments.

Bootstrapping the Maldacena-Polyakov loop in N=4 SYM at finite temperature

• Nika Sokolova

Room: Oliver Thompson Lecture Theatre 101 I discuss a thermal bootstrap setup for the Maldacena-Polyakov loop in the planar limit of N=4 Super Yang-Mills theory. In this limit, the spectrum of low-lying operators is accessible via integrability, specifically through the Quantum Spectral Curve. Our primary observables are two-point functions of operators inserted on the loop, which in this context is a thermal circle. Following recent advances in thermal bootstrap, we extract thermal structure constants for the first low-lying operators by combining the integrability-based spectrum with thermal bootstrap constraints.

Conformal collider bounds for interacting CFTs

• Cyuan-Han Chang

Room: Oliver Thompson Lecture Theatre 101 We consider the average null energy condition (ANEC) and its higher-spin generalization and derive bounds on the TTT OPE coefficients in general CFTs, generalizing the Hofman-Maldacena conformal collider bounds. In particular, we consider ANEC and higher-spin ANEC in a mixed system of the stress tensor and the leading spin-4 operator. We obtain bounds on the TTT coefficients in 3D and 4D CFTs as a function of the twist of the leading spin-4 operator and its OPE coefficients.

Two Applications of SSB in AdS

• Lorenzo Di Pietro

Room: Oliver Thompson Lecture Theatre 101 I will discuss the definition and the consequences of Spontaneous Symmetry Breaking (SSB) for a quantum field theory on a rigid Anti-de Sitter (AdS) background. In particular, for continuous symmetries, I will review the existence of a (trivial) conformal manifold, and explain how to compute its metric from the two-point function of the bulk current. I will then discuss two applications: (1) to quantum electrodynamics in 3d, where the possible pattern of symmetry breaking involving the flavor and magnetic symmetries can be studied reliably in perturbation theory, when the AdS radius is small; (2) to the subleading strong coupling expansion of the Bremsstrahlung function of the ½ BPS Wilson-Maldacena line in N=4 SYM, which can be computed from the two-point function of the tilt operator, using a judicious treatment of SSB on the AdS_2 worldsheet to fix the normalization and get a finite result.

Bootstrapping gravitational EFTs at low impact parameter

• Aninda Sinha

Room: Oliver Thompson Lecture Theatre 101 I will review the stringy dispersion relation and use it to study gravitational EFTs at low impact parameter. In particular, we will find that there is a remarkable organizational spectral structure that shows up at low impact parameter for a wide class of S-matrices. After feeding in the Eikonal support, we find Regge-like structures as well as black-hole-like unitarity capped support that respect the Giddings-Porto black hole support curves. However, due to the capping, it is unclear if the support structures favour black-holes over fuzzballs or more exotic stars.

New mechanism to solve fine-tuning in $d>2$?

• Yu Nakayama

Room: Oliver Thompson Lecture Theatre 101 The Ising model in $d>2$ is characterized by the OPEs $\sigma \times \sigma = 1 + \epsilon + \cdots$ and $\epsilon \times \epsilon = 1 + \epsilon + \cdots$. In $d=2$, the OPE coefficient $C_{\epsilon\epsilon\epsilon}= 0$, and this is explained by duality, leading to solvability and the absence of fine-tuning. We explore the possibility of demanding $C_{\epsilon\epsilon\epsilon} = 0$ in higher dimensions using the conformal bootstrap. Such a theory, if any, will solve the fine-tuning problem, offering a new mechanism for self-organized criticality. We show that in $d=2$ with this ansatz, the Ising model can be isolated to the tip of a very thin nose, and you will see what happens in $d>2$. During the search, you will also encounter an enigmatic, potentially spurious feature from the phenomenon called "trivial mix problem" by Ning Su.

Dynamics from the OPE

• Joseph Lap

Room: Oliver Thompson Lecture Theatre 101 In the conformal bootstrap, one typically focuses on static data: the scaling dimensions and OPE coefficients that characterize a given CFT. In this talk, I argue that the bootstrap philosophy can be fruitfully extended to real-time dynamics. I will study dynamical phase transitions, defined as non-analytic behavior of observables under Lorentzian time evolution. As a concrete example, I consider the mutual information—a UV-finite measure of entanglement between spatial subregions—and demonstrate that it exhibits sharp transitions as a function of time. These transitions separate regimes with qualitatively distinct patterns of entanglement propagation. I show that this phenomenon admits a natural bootstrap interpretation: the phase of the mutual information is determined by the conformal block that dominates in a given OPE channel. More broadly, these results suggest a framework in which Lorentzian dynamics can be analyzed using bootstrap techniques.

Gradient Flow in Scalar-Fermion Theories

• William Pannell

Room: Oliver Thompson Lecture Theatre 101 Complete generalizations of Zamolodchikov's c-theorem to higher dimensions remains the holy grail in the study of renormalization group flows. While in the intervening years there has been success in proving weak and strong monotonicity theorems in a variety of dimensions and contexts, the strongest version, gradient flow, remains much more elusive. After a brief review of monotonicity theorems in general and gradient flow in particular, I will detail recent advances in the study of gradient flow in general scalar-fermion systems in d=4-ϵ. Perturbatively, gradient flow can be reduced to a set of constraint equations on the coefficients appearing in the beta function, and using recent results for the generic beta function I will show that these constraints are satisfied through all known loop orders only once one replaces beta by the so-called B function. Unlike in previously studied purely scalar systems, there exist fixed points at which the difference between beta and B is non-zero, and B must be introduced to correctly deal with fictitious limit cycles. The monotonic quantity appearing in this solution is connected to a weak monotonicity conjecture by Fei, Giombi, Klebanov and Tarnopolsky, suggesting that this conjecture can be strengthened.

Classifying and bootstrapping 3d GNY-like models

• Matthew Mitchell

Room: Oliver Thompson Lecture Theatre 101 We perform a systematic classification of QFTs that contain Dirac fermions coupled to scalars, and that exhibit perturbative fixed points in the 4-ε expansion. These theories, which arise as quantum critical points of fermionic materials with scalar order parameters, exhibit nontrivial symmetry enhancements in 2+1 dimensions because Dirac spinors factorize to Majorana spinors. Our classification includes the Gross--Neveu--Yukawa, chiral Ising, chiral XY, and chiral Heisenberg CFTs, as well as two new theories which are not well-characterized. We then present numerical bootstrap bounds on the scaling dimensions of these CFTs for various N_f, which we compare to perturbative results.

Some general properties of conformal correlation functions

• Balt van Rees

Room: Oliver Thompson Lecture Theatre 101 I will discuss the several things we know, believe or would like to be true about conformal correlation functions. I will focus in particular on the large ∆ limit corresponding to the flat-space limit of a gapped QFT in AdS. Examples are correlator bounds and their connection to Haag-Ruelle theory, the conformal dispersion relation, and Mellin space expressions for multi-point correlation functions.

Bootstrapping Massive Spin-2 Amplitudes

• Simon Caron-Huot

Room: Oliver Thompson Lecture Theatre 101 Massive particles with spin two appear naturally in scenarios where spacetime has extra dimensions. We analyze the possible interactions between such particles from the perspective of causality and unitarity of the S-matrix, assuming a separation between their mass and the higher-spin ($J\geq 3$) scale $\Lambda$ that signals a more dramatic breakdown of effective field theory. We tame the explosion in tensor contractions using two techniques: superconvergence sum rules and novel "dual" amplitudes. Using a handful of these, for example, we show analytically that all three-point vertices must be proportional to the Einstein-Hilbert structure up to $1/\Lambda$ corrections. We investigate the resulting constraints on low-lying spectra and couplings.

No Shift, Sherlock

• Shota Komatsu

Room: Oliver Thompson Lecture Theatre 101 It is believed that global symmetry cannot exist in UV-complete quantum gravity; even if there appears to be global symmetry at low energy, it is always secretly broken or gauged. I will present a non-perturbative argument for this in quantum gravity in AdS. The argument is for the shift symmetry or more generally spontaneously broken continuous symmetry, but applies beyond the semiclassical regime of gravity unlike the previous results in the literature. I will also present the generalization to higher-form symmetries and provide quantitative estimate of the symmetry breaking using modular bootstrap in AdS3/CFT2.

Multi-particle S-matrix analyticity from Mellin amplitudes

• Jeremy Mann

Room: Oliver Thompson Lecture Theatre 101 In conformal field theory, the maximal Mellin analyticity conjecture states that all singularities of Mellin amplitudes arise from poles at the twists exchanged in the operator product expansion. For CFTs dual to quantum field theories in Anti de Sitter space, we explore the implications of this conjecture for multi-particle scattering amplitudes in the flat space limit. After motivating the analyticity conjecture from the non-perturbative Mellin transform of multi-point correlation functions, we find spectrum-dependent conditions for analyticity near the physical region. This provides a new perspective on the longstanding problem of local analyticity in S-matrix theory; that is, whether the amplitude at a physical point is the boundary value of a single analytic function. We also comment on deriving crossing symmetry using this method. Based on work in progress with Filip Herček and Balt van Rees.

A Bootstrap Study of Confinement in AdS

• Stefanos Kousvos

Room: Oliver Thompson Lecture Theatre 101 Yang-Mills theory in AdS$_4$ with Dirichlet boundary conditions is expected to undergo a transition as the AdS radius varies, since the boundary data is incompatible with confinement in flat space. Various mechanisms have been proposed for the disappearance of the Dirichlet boundary condition, which we discuss. From the boundary viewpoint, the associated CFT is a deformation of a generalised free theory of non-Abelian conserved currents. We test the various proposed scenarios by deriving non-perturbative constraints from the numerical conformal bootstrap of the four-point function of non-Abelian conserved currents. We rule out the scenario in which the boundary current decouples, and find favourable evidence that the Dirichlet boundary condition disappears via fixed point annihilation.

Region bounds, error margins and certified positivity in the numerical conformal bootstrap, or the first mathematically rigorous bounds on the 3d Ising CFT critical exponents.

• Marten Reehorst

Room: Oliver Thompson Lecture Theatre 101 Numerical conformal bootstrap bounds are commonly described as “rigorous,” yet as published they are not theorems: they hold at sampled points rather than over regions of parameter space, for a truncated rational approximation of the conformal blocks rather than the blocks themselves, for a finite list of spins rather than all spins, and only up to floating-point rounding. We address all four gaps. I show how to efficiently search for functionals that are positive over regions, including regions in external-dimension space; how positivity on the large-spin crossing vectors can be imposed as a region constraint; how new analytic bounds on conformal-block recursion relations allow us to efficiently find functionals guaranteed to be positive on the true, untruncated crossing vectors; and how to mathematically certify the positivity of the resulting functionals using exact arithmetic. As a flagship application, we certify that any Z₂-symmetric CFT with a Z₂-odd scalar of dimension Δσ ∈ [0.505, 0.5405], at most one relevant Z₂-even scalar ε, a stress tensor, and a gap ΔT′ ≥ 11/2 in the spin-2 sector must satisfy (Δσ, Δε) ∈ (0.5055, 0.540) × (1.23, 1.55) The exact certification pipeline is based on Bernstein-basis positivity certificates in up to three variables; exact arithmetic over the number field ℚ(√2); explicit bounds on the scalar block truncation errors. Finally, I emphasize that the innovations introduced in this talk are useful not only for formalizing bounds. They can prevent genuine practical failure modes routinely encountered in the numerical conformal bootstrap and can, perhaps surprisingly, also speed up numerical computations.

Bootstrapping 2d unitary rational CFT

• Ning Su

Room: Oliver Thompson Lecture Theatre 101 I will discuss the reconstruction program for 2d unitary rational CFTs. A key step is the “character level realizability problem”: given S, T, c, how to find all possible solutions to the holomorphic modular bootstrap equations. We invented a specialized integer scan algorithms to solve the problem at a very high S rank.

Black Holes, Random Variables and the Bootstrap

• Eric Perlmutter

Room: Oliver Thompson Lecture Theatre 101 Based on 2607.02233 and work in progress

3d Ising Field Theory from the Fuzzy Sphere and TCSA

• Liam Fitzpatrick

Room: Oliver Thompson Lecture Theatre 101 The 3d Ising CFT deformed by its Z2 odd sigma operator is one of the simplest examples in d>2 of a QFT built from a nontrivial UV fixed point. I will review the Fuzzy Sphere regulator approach to critical systems and describe results using the Fuzzy Sphere to obtain the spectrum of 3d Ising Field theory at finite and infinite volume, mainly focusing on the vacuum energy, lightest one-particle mass, the existence of a bound state in the spectrum, and dynamical consistency between sectors carrying different spin. I will also present comparisons with the system studied using the Truncated Conformal Space Approach.

Zamolodchikov recursion relations in 4 dimensions.

• Colum Flynn

Room: Oliver Thompson Lecture Theatre 101 The Zamolodchikov recursion relations are widely used in numerical conformal bootstrap, especially in three dimensions, as they provide an efficient way to compute conformal blocks. In even dimensions, however, they are not well understood because conformal blocks can develop double poles. We develop the general theory for conformal blocks with double poles by using representation theory of the conformal group and considering logarithmic extensions of parabolic Verma modules.

New Developments in Five-Point Bootstrap

• Petar Tadić

Room: Oliver Thompson Lecture Theatre 101 I will present recent progress on the five-point bootstrap of the critical Ising model in three dimensions. A central idea is to use disconnected five-point correlators as an approximation to the truncated part of the spectrum in the interacting theory. I will explain the method in detail and show how it can be used to extract OPE coefficients from five-point data. I will then compare the resulting estimates with those obtained from other approaches, in particular the four-point bootstrap of spinning operators and the fuzzy sphere regularization.

Multi-trace operators in \mathcal{N}=4 SYM

• Giulia Fardelli

Room: Oliver Thompson Lecture Theatre 101 In this talk, I will discuss recent progress toward understanding the spectrum of multi-trace operators in $\mathcal{N}=4$ SYM at large 't Hooft coupling and at tree level in the large-N expansion, focusing on triple-trace operators in the R-symmetry singlet sector. While partial information about these operators can be extracted from four-point functions, a complete understanding remains elusive, in part because of their large degeneracy at fixed spin. To extract the anomalous dimensions of these operators, we develop an AdS Hamiltonian approach, i.e., we diagonalize an effective Hamiltonian built from interactions involving the Lagrangian densities $L_p$, which are superdescendants of the stress-tensor superprimary and its Kaluza–Klein modes. We show that this Hamiltonian reproduces the known spectrum of double-trace operators and makes the hidden 10d symmetry manifest. We then compute its action on triple-trace operators built from three $L_p$'s, determining their anomalous dimensions as a function of spin. If time permits, I will also present preliminary results on higher-trace operators, exploring their eigenvalue statistics and possible signatures of chaotic behavior.

Moments in the CFT Landscape

• David Poland

Room: Lecture Theatre 1 I'll describe recent explorations of the idea of reorganizing CFT data into "moment" variables, describing statistical properties of the OPE measure for a given 4-point function. These moments capture coarse-grained information about the CFT data and are tightly constrained by crossing symmetry even at large external dimensions, giving rise to a powerful new way to map out theory space.

CFTs are lazy

• Vasileios Niarchos

Room: Lecture Theatre 1 How much information is needed to reconstruct a correlation function in conformal field theory? Surprisingly, very little. I will introduce a novel approach based on neural networks that produces a remarkably accurate approximation of scalar correlation functions in arbitrary CFTs in any spacetime dimension from a few input data. I will present a wide variety of examples and discuss the underlying mechanisms behind the approach highlighting potential future applications

Giant gravitons and zero-dimensional defects

• Xinan Zhou

Room: Lecture Theatre 1 In this talk, I will explain why correlation functions involving giant gravitons are naturally viewed as correlators of local operators in the presence of zero-dimensional defects. I will present strong-coupling results obtained by combining this defect perspective with analytic bootstrap techniques. I will also show that the defect formulation reveals hidden higher-dimensional structures, which are present at both strong and weak coupling. Finally, I will argue that zero-dimensional defects provide a natural framework for general light-light-heavy-heavy correlators beyond supersymmetric settings, and I will comment on a possible extension of the analytic conformal bootstrap related to analytic functionals.

Strings attached: the confining flux tube as a conformal defect

• Barak Gabai

Room: Lecture Theatre 1 We recently initiated the study of flux tubes in confining gauge theories placed in a rigid Andi-de Sitter (AdS) background, which serves as an infrared regulator. This setup provides a controlled arena to probe a central question in confinement: how the effective string worldsheet degrees of freedom emerge from the underlying UV gauge theory. Varying the AdS radius from large to small interpolates between a flat-space confining string and a weakly coupled string-like object bound by the AdS gravitational potential. In practice, the flux tube is realized as a Wilson loop inserted at the AdS boundary, which behaves as a conformal defect. In this talk, I will explain the setup and (some of) what we have learned so far. See also companion talk by Philine van Vliet.

More strings attached: bootstrapping the confining flux tube

• Philine van Vliet

Room: Lecture Theatre 1 Conformal defects break part of the symmetry of a bulk CFT. The broken Ward identities lead to very general sum rules on the defect CFT data as well as on the data of bulk operators in the presence of a defect. These sum rules have many applications e.g. in perturbation theory, and can also be used in combination with the numerical bootstrap. In this talk I will use them to bootstrap a confining flux tube in pure Yang-Mills in a rigid Anti-de Sitter (AdS) background numerically. The flux tube corresponds to the insertion of a Wilson loop at the boundary, resulting in a 1d defect CFT. In AdS, one has perturbative control at both large and small AdS radius, where the setup can be described by a flat-space confining string and a weakly coupled string-like object respectively. With the bootstrap we can additionally probe the intermediate regime, and compare with perturbative predictions. See also the companion talk by Barak Gabai.

Carving Out the Space of 3d CFTs with an Analytic-Functional Bootstrap

• Zechuan Zheng

Room: Lecture Theatre 1 Analytic functionals provide a basis for the crossing equation in which the functional coefficients correspond directly to the expansion of conformal blocks in terms of AdS Witten diagrams. This perspective offers a more natural formulation of the conformal bootstrap, both analytically and numerically. Growing evidence suggests that the analytic functional basis is significantly more efficient than the conventional derivative basis implemented by SDPB, opening new avenues for addressing several long-standing challenges in the bootstrap program, including the study of critical gauge theories. In this talk, I will discuss the construction and implementation of analytic functionals in arbitrary spacetime dimensions, with a particular focus on three-dimensional conformal field theories. I will show how this framework can sharpen numerical bootstrap analyses and provide evidence for previously unexplored or potentially new CFTs.

Do We Live on a Rubber Sheet?

• Denis Karateev

Room: Lecture Theatre 1 I will discuss whether the rubber-sheet picture of gravity can be made into a consistent quantum field theoretic model. The setup is a vibrating four-dimensional brane embedded in a higher-dimensional flat space-time, with matter confined to the brane. The induced metric is built from the brane fluctuations (phonons), which mediate forces between matter sources. I will explain why the generic force is not Newtonian, how an apparent Einstein-gravity regime could arise at intermediate distances, and how S-matrix bootstrap constraints on phonon scattering strongly constrain this possibility.

Defects and protected operators at the boundary of AdS

• Marco Meineri

Room: Lecture Theatre 1 I will present a mechanism that ensures that conformal defects placed at the boundary of Anti-de Sitter spacetime support operators whose dimension is protected. Some consequences of this fact on the physics of confinement in AdS and on the moduli of theories of quantum gravity will be discussed as well.

From Z to a & everything in between

• Sridip Pal

Room: Lecture Theatre 1 Using hydrodynamics, r.k.a. thermal effective field theory, I will show that locality and Weyl invariance constrain the coefficients of various angular-velocity structures appearing in the high-temperature expansion of the thermal partition function (**Z**) of even-dimensional CFTs. This leads to an infinite hierarchy of relations among these coefficients. Remarkably, the first of these relations determines the (**a**)-type Weyl anomaly. I will then reinterpret these relations in the large-spin regime. For local operators in CFT with large spin (J), allowing the twist to be large but parametrically smaller than J, the leading entropy is fixed up to a theory-dependent function of a universal scaling variable involving twist and spin—a phenomenon we identified and termed “**Semi-universality**” in arXiv:2512.00158 (in contrast to the “super-universality” we found in 2D CFT in arXiv:2505.02897 with van Rees and Qiao). The new results reflect the fact that the subleading semi-universal functions are themselves constrained by differential relations. If time permits, I will conclude with a sketch of the derivation of **stronger-than-unitarity** spectral bounds from this setup and extension of semi-universality to the averaged OPE data at large spin and twist. Based on arXiv:2512.00158 with Anand, Benjamin, Kumar, Minwalla, Mukherjee, Rahaman; and ongoing work with Advant, Anand, Benjamin, Kumar, Minwalla, Mukherjee, Rahaman, Ray.

Long Heavy Objects in CFTs: Defects meet Effective Field Theory

• Thekla Lepper

Room: Lecture Theatre 1 Defect conformal field theories (DCFTs) provide a universal framework for describing the low energy interaction of heavy degrees of freedom with light excitations. In a DCFT part of the Poincaré symmetry is broken explicitly. However, the fundamental laws of our world are Poincaré invariant and any breaking of space-time symmetries must be spontaneous. In particular, extended and thin objets which spontaneously break Poincaré invariance support the propagation of Goldstone bosons on their worldvolume. Their action is invariant under a non-linear realization of the Poincaré symmetry. Since the Goldstone bosons are massless, a natural question is, how DCFTs can describe the infrared limit of real-world heavy objects embedded in a massless bulk. The work I am going to present aims to answer this question by constructing an effective field theory that describes the coupling of the Goldstone bosons to a generic DCFT.

Accurate boundary bootstrap for the three-dimensional O(N) normal universality class

• Wenliang Li

Room: Lecture Theatre 1 The three-dimensional classical O(N) model with a boundary has received renewed interest due to the discovery of the extraordinary-log boundary universality class for 2≤N<Nc. The critical value Nc and the exponent of the boundary correlation function are related to certain amplitudes in the normal universality class. To determine their precise values, we revisit the 3d O(N) boundary conformal field theory for N=1,2,3,4,5. After substantially improving the accuracy of the boundary bootstrap, our determinations are in excellent agreement with the Monte Carlo results, resolving the previous discrepancies due to low truncation orders. We also use the recent bulk bootstrap results to deduce highly accurate Ising data. Many bulk and boundary predictions are obtained for the first time. Our results demonstrate the great potential of the η minimization method for many unexplored bootstrap problems in which positivity constraints are absent.

Lorentzian universality of heavy CFT correlators

• Gordon Rogelberg

Room: Lecture Theatre 1 I will discuss a class of universal dynamics that arise from the worldline effective action of a free massive scalar in AdS_{d+1}. Given a "heavying" sequence of CFT four-point functions of identical operators with increasing scaling dimension, I will provide necessary and sufficient conditions for their limits to be described universally in various Lorentzian kinematic regimes. These results extend the classic theorems of Hartman-Keller-Stoica and Dey-Pal-Qiao to the setting of CFT correlators in d>2, and provide new tests of bulk locality in the heavy regime.

Towards a quantum lattice bootstrap

• Sean Hartnoll

Room: Lecture Theatre 1 I will explain how the analyticity and boundedness properties underpinning the S-matrix bootstrap can be adapted to constrain the dynamics of quantum mechanical lattice models. The basic quantity will be the retarded Green’s function, for which we obtain strong analyticity and boundedness properties using the Lieb-Robinson bound, a fundamental result for local lattice dynamics that I will review. I will explain how these properties (bounds and analyticity) can be used to place constraints on transport coefficients and on non-Fermi liquid coupling constants.

Non-singlet physics from the matrix bootstrap

• Henry Lin

Room: Lecture Theatre 1 I will present new results on the thermal phase diagram of the ungauged matrix quantum mechanics, including tests of new and old conjectures using matrix bootstrap data. I will also discuss the adjoint sector of BFSS at small and large R-charge.

What is the simplest holographic matrix model?

• Pedro Vieira

Room: Lecture Theatre 1 We study a generalization of a two matrix model, proposed by J. Hoppe in '89. At strong coupling, we unveil an emergent area preserving diffeomorphism symmetry of this model, which allows us to completely solve it at large N. The solution for all n-point connected correlators takes the form of a Witten diagram-like expansion in an emergent two-dimensional dual spacetime. We conclude with speculation about the quantization of this dual model and comment, its extensions to higher dimensions and relations to holography more broadly. (Based on work to appear with Harish Murali, also on 2507.21207 from us and Andrea Guerieri.)

On the strong coupling limit of Yang-Mills matrix models

• Adrien Martina

Room: Lecture Theatre 1 Yang–Mills matrix models are simple yet rich laboratories for studying emergent geometry and holography. Some of them, such as IKKT and polarised IKKT, are proposed matrix descriptions of type IIB string theory. In this talk, I will focus on mass-deformed Yang–Mills matrix models in the strong ’t Hooft coupling limit, where a dual supergravity description may be expected. The central question is whether the matrices effectively commute in this regime, so that their eigenvalues can be interpreted as target-space coordinates. We will see that the answer depends sharply on the number of fermionic degrees of freedom.

Bootstrapping two-point correlators

• Jessica Yeh

Room: Lecture Theatre 1 We develop a bootstrap approach to Euclidean and Lorentzian two-point correlators in quantum mechanical systems. Using positivity constraints together with the Heisenberg equations of motion, the problem can be formulated as a semidefinite program. Solving this problem in its primal form naively requires time discretization and hence non-rigorous. We show that by going to the dual formulation, the Heisenberg equations of motion become "inequalities of motion" on the Lagrange multipliers enforcing the constraints, allowing one to obtain rigorous bounds on two-point correlators from a finite-dimensional semidefinite or polynomial matrix program. We illustrate the method by bootstrapping the Euclidean and Lorentzian two-point correlators in the ungauged one-matrix quantum mechanics in both the ground state and thermal state, from which we extract the spectrum and matrix elements of the low-lying adjoint states. This work is based on 2511.08560 and ongoing work with Henry Lin and Zechuan Zheng.

Bootstrapping Holographic Theories in the Planar Limit

• Daniele Pavarini

Room: Lecture Theatre 1 We apply the conformal bootstrap to holographic four-dimensional N=2 superconformal field theories in the planar limit. We employ a dispersion relation that reconstructs the correlator from its double discontinuity, which naturally isolates the single-trace spectrum. Supersymmetric localization provides an additional exact constraint that makes the bootstrap sensitive to the gauge coupling. Combining these two ingredients, we obtain rigorous bounds on the scaling dimension of single-trace operators at any value of the coupling. As examples we consider the stress tensor correlator in N=4 SYM dual to closed string scattering, and the flavour multiplet correlator in a certain N=2 theory dual to open string scattering. Our bounds are saturated at both weak and strong coupling and smoothly interpolate between the two.

Worldsheet bootstrap for strings on AdS

• Fernando Alday

Room: Lecture Theatre 1 Understanding string theory in curved spacetime is an outstanding problem, as the standard RNS formalism has difficulty handling Ramond-Ramond (RR) flux, which is often present in curved spacetimes. For anti-de Sitter (AdS) spacetime, a new bottom-up formalism for string scattering is the Worldsheet Bootstrap. This method combines the constraints of conformal symmetry with an ansatz for the worldsheet integral to compute string scattering as an expansion around flat space.I will explain how to apply this method to several backgrounds, including open and closed strings.

Bootstrapping Open and Closed Strings with Maximal Supersymmetry and Peculiar Parity

• Aidan Herderschee

Room: Lecture Theatre 1 Tree-level string amplitudes may be fixed by a remarkably small set of consistency conditions. I will discuss bootstrap approaches to open and closed strings in four dimensions, focusing on planar N=4 super-Yang-Mills theory and N=8 supergravity deformed by higher-derivative operators. In both cases, we impose parity invariance on selected six-scalar amplitudes. We combine this condition with maximal supersymmetry, R-symmetry, tree-level factorization through six points, and positivity. For planar N=4 super-Yang-Mills theory with SU(4) R-symmetry, the resulting bounds converge to the Veneziano amplitude, providing strong evidence for uniqueness. For supergravity with SU(4)xSU(4) R-symmetry, the allowed region develops two sharp corners corresponding to the Virasoro-Shapiro amplitude and an infinite-spin-tower amplitude. Requiring finite spin content at the lowest massive level excludes the latter. Assuming the observed nonlinear relations persist to all orders, the Virasoro-Shapiro amplitude is then the unique solution.

Heavy Conformal Data from Thermal Correlators

• Volker Schomerus

Room: Lecture Theatre 1 Thermal correlation functions provide a powerful window into the heavy sector of conformal field theories, complementing the information accessible through conventional bootstrap techniques. In this talk I will review recent progress on the thermal bootstrap, focusing on the extraction of spectral densities and heavy-heavy-light OPE coefficients from thermal one-point functions. After introducing thermal conformal blocks and their computation through Casimir equations, I will describe an emerging integrability framework that unifies these constructions and extends to more general conformal blocks. I will then explain how thermal inversion formulas lead to remarkably precise asymptotic predictions for heavy CFT data, which remain accurate well beyond the asymptotic regime. Applications include gapped CFTs whose high-temperature behavior is well captured by a thermal effective action, as well as CFTs dual to weakly coupled scalar field theories in AdS.

Beyond the Bootstrap: Gluons in AdS

• Rishi Mouland

Room: Lecture Theatre 1 We consider AdS/CFT in 3 to 6 CFT dimensions with half-maximal SUSY. Such setups feature flavour branes, whose gluon modes we can scatter to probe certain 4-point functions. The analytic bootstrap has successfully fixed several of the corrections to these 4-point functions in a large N expansion; we perform a bulk calculation of the first that is not, which involves the exchange of gravitons. These corrections have a novel feature not present with maximal SUSY: extremal couplings which—contrary to common lore—are perfectly harmless, and induce a mixing between single and double traces. We finally introduce a new way to solve half-maximal superconformal Ward identities in Mellin space, paving the way for new tools for the numerical bootstrap in this context.

On the existence of extremal solutions

• Nat Levine

Room: Lecture Theatre 1 Extremal solutions and functionals play an important role in the conformal bootstrap. One may wonder if it is possible to rigorously prove that they exist, and to deduce their properties. This is a question about what happens when numerics have fully converged, i.e. at Lambda=infinity. In the simplest setting of OPE maximisation in 1d CFT, we prove that extremal solutions exist using the Fenchel-Rockafellar theorem. These are interacting, unitary, exact solutions of the crossing equation that maximise an OPE coefficient. This approach may have interesting implications for (i) whether 3d Ising can be an extremal solution and (ii) potential 'hybrid' bootstrap approaches.

Bootstrapping de Sitter boundary

• Kamran Salehi Vaziri

Room: Lecture Theatre 1 In this talk, we will review some recent progress in QFT in de Sitter spacetime. We will use the Hilbert-space construction and representation theory of the de Sitter symmetry group, i.e. the Euclidean conformal group, to derive the spectral decomposition of the bulk two-point function and the conformal partial wave expansion of the boundary four-point function. In both cases, unitarity of the bulk theory implies positivity conditions that lead to nontrivial bounds on physical observables. Then, with the goal of bootstrapping the conformal boundary of de Sitter spacetime, I will propose a de Sitter-specific bulk-to-boundary expansion involving a continuous family of boundary operators. Finally, I will derive an inversion formula for the bulk-to-boundary expansion, in which, for a given bulk theory, the boundary operator content is constructed as an integral of the bulk operator multiplied by the bulk-to-boundary propagator. These boundary operators have two-point functions that include contact terms alongside standard CFT two-point functions.

Regge’s Inferno

• Zohar Komargodski

Room: Lecture Theatre 1 I will describe how placing CFTs on pp-wave backgrounds can teach us about large-spin operators. Heisenberg symmetries and locality control both light-cone/Regge and large-twist regimes, imply a new (3+1)-dimensional unitarity bound, and yield applications to gauge theories.

The two-flavor Schwinger model at 50: Solving Coleman’s puzzles

• Silviu Pufu

Room: Lecture Theatre 1 50 years ago, Coleman introduced two-dimensional quantum electrodynamics coupled to two charged massive fermions. While he elucidated much of the physics of this model, he listed three puzzles at the end of his paper. In this talk, I will present new analytical and numerical calculations that help solve these three puzzles. The analytical computations use two-loop RG, integrability, and effective field theory methods. The numerical computations use infinite matrix product states, and a new kind of matrix product operators called LEMPOs with which one can simulate two-dimensional gauge theories more efficiently.

Remarks on the Metric on 4D QFTs

• Tom Hartman

Room: Lecture Theatre 1 A natural metric on four-dimensional QFTs was introduced by Jack and Osborn in their work on the local renormalization group. This metric determines the flow of the a-anomaly and is closely tied to the a-theorem, but it is not manifestly positive. In this talk I will show that the Jack-Osborn metric is related to the ANEC operator, and discuss more broadly the connections among three different probes of RG flows: dilaton scattering amplitudes, light-ray sum rules, and perturbative RG.

From Partons to Strings: Scattering on the Coulomb branch of N = 4 SYM

• Elisabetta Armanini

Room: Lecture Theatre 1 We study scattering on the Coulomb branch of planar N=4 SYM. At weak 't Hooft coupling scattering amplitudes can be computed using Feynman diagrams and the relevant degrees of freedom are quarks and gluons. At strong 't Hooft coupling the dynamics is captured by string scattering in flat space. Dual conformal symmetry makes it possible to constrain the unitarity properties and the spectrum of the amplitude: bound states are encoded in the cusp anomalous dimension of a Maldacena-Wilson line with operators inserted at the cusp, which is fixed by integrability. In this talk, we show how to combine dual conformal invariance, integrability, worldsheet bootstrap and the S-matrix bootstrap to explore scattering at finite 't Hooft coupling. [Based on 2510.19909 + work in progress]

Unitarity bounds on cusp anomalous dimensions

• Ryan Lanzetta

Room: Lecture Theatre 1 In lattice studies of line defects, such as entanglement cuts or monodromy defects in 2+1d, or line defects in general dimensions, non-trivial geometries of the defect are typically forced to contain sharp corners or cusps. Each cusp contributes to the overall finite size scaling of the observable via its associated cusp anomalous dimension. I will present simple bounds on the cusp anomalous dimension that follow from unitarity and locality constraints obeyed by scale-invariant line defects in the continuum. One bound in particular relates the cusp anomalous dimension of a right angle cusp to the universal defect mass of a pair of defects in the fusion limit, complementing known universal asymptotics of the cusp anomalous dimension when the cusp angle is large or small. I will mention refinements of the analysis in the presence of global symmetries, which lead to new bounds on charged cusps.

Twisting and reducing the 6d (2,0) superconformal bootstrap

• Mitchell Woolley

Room: Lecture Theatre 1 We describe progress toward the superconformal bootstrap of the 6d (2,0) SCFTs. The first step was achieved in [2506.08094], where we used the twist of 6d (2,0) SCFTs to W-algebras to extract an infinite set of protected CFT data. As an application, we demonstrated the consistency of this information with protected higher derivative corrections in the holographic dual on AdS_7 x S^4/Z_o. A second advance was made in [2601.15407], where we derived a reduced superblock formalism to more efficiently constrain unprotected CFT data in the analytic and/or numerical bootstrap. We outline an application of these tools to the analysis of composite 1/2-BPS operators, which are naturally defined in the W-algebra and are dual to bound states in M-theory.