Quantum Fields and Strings

The Virasoro conformal blocks are very interesting since they have many connections to other areas of math and physics. For example, when $c=1$, they are related to tau functions of integrable systems of Painlev\'{e} equations. They are also closely related to non-perturbative completions in the topological string theories. I will first explain what Virasoro conformal blocks are. Then I will describe a new way to construct Virasoro blocks at $c=1$ on $C$ by using the "abelian" Heisenberg conformal blocks on a branched double cover of C. The main new idea in our work is to use a spectral network and I will show the advantages of this construction. This nonabelianization construction enables us to compute the harder-to-get Virasoro blocks using the simpler abelian objects. It is closely related to the idea of nonabelianization of the flat connections in the work of Gaiotto-Moore-Neitzke and Neitzke-Hollands. This is based on a joint work with Andrew Neitzke.

Thin enough black strings are unstable to rippling along their length, and the instability threshold indicates that static inhomogeneous black strings exist. These have indeed been constructed with increasing inhomogeneity until a high-curvature singular pinch appears. We study the string-scale version of this phenomenon: “string-ball strings”, which are linearly extended, self-gravitating configurations of string balls obtained within the Horowitz- Polchinski (HP) approach to near-Hagedorn string states. We construct inhomogeneous HP strings in spatial dimension d ≤ 6, and show that, as the inhomogeneity increases, they approach localized HP balls when d ≤ 5 or cease to exist when d = 6. We then discuss how string theory can smooth out the naked singularities that appear in the Kaluza-Klein black hole/black string transition, and we propose scenarios for the final stage of the evolution of the black string instability after string theory takes over.

The study of string scattering in general curved spacetimes presents a stark contrast to the relatively well-understood framework in flat space, where perturbative techniques and worldsheet methods can be used. However, the AdS/CFT correspondence offers a powerful indirect method to compute string amplitudes in AdS. In this talk, I will present recent developments in the AdS Virasoro-Shapiro program. The tree-level amplitude of four gravitons in AdS^5 x S^5 is mapped to a four-point correlator in N=4 SYM at large central charge, and studied by CFT methods combined with single-valuedness, which echoes its importance in flat space. The amplitude itself is defined and constructed as an expansion around flat space. While the goal is to compute it to all orders, attention can be placed on more tractable limits. I will present our results for the High Energy limit, with dual insights from the spacetime and worldsheet perspectives, and the richer case of the Regge limit, which encodes full information on the intermediate operators in the leading Regge trajectory.

We construct the phase space of a spherically symmetric causal diamond in (d+2)-dimensional Minkowski spacetime. Utilizing the covariant phase space formalism, we identify the relevant degrees of freedom that localize to the d-dimensional bifurcate horizon and, upon canonical quantization, determine their commutators. On this phase space, we find two Iyer-Wald charges. The first of these charges, proportional to the area of the causal diamond, is responsible for shifting the null time along the horizon and has been well-documented in the literature. The second charge is much less understood, being integrable for d ≥ 2 only if we allow for field-dependent diffeomorphisms and is responsible for changing the size of the causal diamond. 

"Null Polygons" in N=4 SYM theory describe the multi-point correlators of 1/2-BPS local operators with large R-charge, when they approach the vertices of a light-like polygon. The leading UV divergences of null polygons is conjectured to satisfy a hierarchy of coupled Toda field theory equations [E.O., Vieira ’22]. I will present some progress towards the prediction of Null Polygons beyond leading logarithms via the hexagons technique, appropriately truncated in the light-cone regime. The method, still conjectural, relies on a series of weak-coupling derivations performed in the Fishnet limit of the theory, where the hexagon representation is derived in the basis of eigenfunctions of a conformal Heisenberg magnet in the principal series. I will present a number of worked-out examples for multi-point multi-loop Fishnet Feynman integrals and Null Polygons.

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In this talk I will describe the correspondence between the double scaling limit of the SYK model and infinite temperature and de Sitter gravity in 2 and 3 dimensions. The dictionary involves a precise match between the chord rules that govern the SYK correlation functions and the braid interactions between line operators in the gravity theory.

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Topological strings are well understood in perturbation theory, but their non-perturbative structure has been the subject of much work and speculation. A useful approach to this problem is to unveil the non-perturbative sectors of the theory by looking at the large order behavior of the perturbative series. This approach can be formulated in a mathematically rigorous way by using the theory of resurgence. In this talk I review the basic ideas behind this approach and I show that it can be successfully applied to topological string theory on arbitrary Calabi-Yau threefolds. Thsee methods lead, not only to explicit non-perturbative topological string amplitudes, but also to a conjecture relating the “invariants” of the theory of resurgence (also known as Stokes constants) to BPS invariants. Physically, this conjecture implies that the large order behavior of the topological string perturbative series contains information about the spectrum of stable D-brane sectors.

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