Week 07.10.2024 – 13.10.2024

These lectures will review recent developments surrounding the infrared sector of gravity in (3+1)-dimensional asymptotically flat spacetimes (AFS). In the first part of the course we will introduce soft theorems which govern the low-energy scattering of massless particles such as photons and gravitons. We will explain how these are related to classical observables known as memory effects and discuss their application to computing infrared-finite collider observables and gravitational waveforms. In the second part, we will introduce the notion of asymptotic or large-gauge symmetries and use it to derive the infinite-dimensional asymptotic symmetry algebra of (3+1)-dimensional AFS, also known as the BMS algebra. We will show that the conservation laws associated with these symmetries are equivalent to the Weinberg soft graviton theorem. Time-permitting, we will discuss some implications of these ideas for non-AdS holography.

Non-invertible symmetries are refined notions of symmetries intensively studied recently. I will show that non-invertible symmetries sometimes lead to a surprising consequence on scattering amplitudes --- a modification of crossing symmetry. I will demonstrate this using example of integrable field theories in 1+1 dimensions although the argument holds more generally; also for non-integrable theories. I will also present the results of S-matrix bootstrap, which constrains the space of physically consistent scattering amplitudes with categorical symmetries.

We demonstrate that crossing symmetry of S-matrices can be violated in theories with non-invertible symmetries. Focusing on integrable flows to gapped phases in two dimensions, we show that S-matrices derived previously from the bootstrap approach are incompatible with non-invertible symmetries along the flow. We present consistent alternatives, which however violate crossing symmetry and obey modified rules dictated by fusion categories. We also show how these modified crossing rules can be used to constrain the space of amplitudes with a given categorical symmetry.

Fundamental questions such as emergence of geometry and gravitational dynamics from QFT amplitudes, barring specific examples, remain unanswered at the full stringy level in gauge-gravity duality. In this talk I will discuss recent progress toward a microscopic approach based on the worldline formulation of QFT. In particular, I will consider large-loop quantum corrections in holographic QFTs where internal propagators of Feynman diagrams are characterized by the Schwinger parameters and argue that embedding of string in the holographic coordinate emerges from the continuum limit of these Schwinger parameters at infinite loop limit. I will demonstrate, employing the techniques of Strebel differentials and discrete exterior calculus, how a worldsheet action for a bosonic string embedded in asymptotically AdS space-time could emerge from multi-loop Feynman graphs in a class of bosonic QFTs. I will end with a discussion of possible loopholes in this approach.

We derive a universal inequality on the unitary 2D CFT partition function with general central charge $c\geqslant 0$, using analytical modular bootstrap. We derive an iterative equation for the domain of validity of the bound on the mixed-temperature plane. The infinite iteration of this equation gives the boundary of maximal-validity domain of our inequality. In the $c\to\infty$ limit, with additional assumption of having a sparse spectrum below the scaling dimension $\frac{c}{12}+\varepsilon$ and below the twist $\frac{\alpha c}{12}$ (with $\alpha\in(0,1]$ fixed), our inequality implies that the grand-canonical free energy has universal large-c behavior in the maximal-validity domain, which does not encompass the entire mixed-temperature phase diagram, except in the case of $\alpha = 1$. In particular, we prove the conjecture proposed by Hartman, Keller and Stoica [1405.5137] (the $\alpha=1$ case): the free energy is universal in the large c limit for all $\beta_L\beta_R \neq 4\pi^2$.