# Research Area Quantum Theories

The three main questions of the cluster – the quantum origin of mass, the nature of dark matter, and the relation of gravity to the quantum world – all raise major theoretical challenges. They require a coordinated effort and concrete tools at the interface of mathematics and physics, which will trigger advances in both theoretical physics and mathematics.

Researchers in the quantum theories area are working at the interface between string theory, particle physics and cosmology to solve fundamental theoretical questions. One aspect of their work is to improve our understanding of traditional quantum field theories by the study of new observables that provide quantitative access to non- perturbative phenomena. This also includes observables from higher categorical structures and from loop and surface operators of defects. Another aspect is to find a mathematically adequate description of the holographic principle in general, and in particular the correspondence between gauge and gravitational theories.

## Key Questions

- What can string theory tell us about gravitational waves and Higgs models?
- How can we enter the realm of quantum gravity?
- What can we actually measure in gauge theory and quantum gravity?

**String-motivated models for new physics **

Researchers focus on the interface between string theory, particle physics and cosmology in the low-energy limit of string theory. This work covers mathematical aspects of the low-energy limit, cosmology related to the early Universe and elements of particle physics. The mathematical and physical implications of string theoretic models are manifested by the fact that underlying moduli spaces come equipped with highly constrained geometric structures. These constraints determine to a large extent the structure of perturbative and non-perturbative quantum corrections. This motivates the study of hypermultiplet moduli spaces from a combined mathematical and physical perspective and the construction of new classes of quaternionic Kähler (QK) manifolds. The determination of precise asymptotic properties is required for consistency with quantum gravity and their consequences for string cosmology and particle physics. Another aim is the description of positivity constraints obtained purely from unitarity in quantum field theory (QFT) on effective field theories (EFT) including the standard model EFT and the comparison with constraints involving quantum gravity.

Mathematical studies of perturbative quantum corrections of QK manifolds describe how continuous symmetries and curvature tensors transform under the hyper-Kähler/QK correspondence. Based on deformation of homogenous QK manifolds, they contribute to scalar geometry studies of manifolds with negative scalar curvature. Geometric aspects of supergravity and their equations of motion, including their associated Killing spinor equations, have also been investigated. Further, researchers have constructed non-supersymmetric solutions of heterotic supergravity on compact four-manifolds as well as classified infinite distance limits within Kähler moduli spaces with implications for the weak gravity conjecture.

Another set of works have connected string cosmology to weakly interacting light particles, inflation and gravitational waves. A new class of ‘harmonic hybrid inflation’ models in type IIB string theory has been established, arising from a 2-axion sector with inherent mechanisms which can drive oscillons and source gravitational waves. In a study of quantum corrected Calabi-Yau moduli spaces applied to cosmology, it has been found that topological invariants can help achieve the desired inflationary regime as well as uplift the vacuum to a positive cosmological constant as evidenced in the late-time Universe. Interest has been raised in the topic of the study of positivity bounds on Wilson coefficients of the dimension-8 fermionic operators. The flavour structure of the bounds has been mapped onto bounds of the Yukawa coefficients and/or the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix elements

**New concepts and observables for quantum fields **

In this area, researchers we study new observables that provide quantitative access to non-perturbative phenomena. The goal is to substantially improve our understanding of traditional quantum field theories. The new observables from higher categorical structures and from loop and surface operators of defects. New concepts and techniques enable the computation of traditional quantum field theory observables relevant to LHC and Higgs physics as well as gravitational waves from binaries.

New observables from higher categorical structures have included surface observables going beyond just Wilson lines leading to new theoretic constructions within 3-dimensional topological quantum field theories (TQFT), such as a generalisation of the orbifolding procedure, utilising a ‘quantum symmetry’ obtained from defects. Higher structures also appear when observables take values in complexes of vector spaces rather than just vector spaces. Higher degree representations yield genuinely new observables in chiral conformal field theory leading to proofs for the differential graded Verlinde formula for the derived conformal blocks and a new and explicit proof of Deligne’s conjecture for modular tensor categories. A new integral theory for non-semisimple Hopf algebras has been developed, setting the stage for construction of Kitaev models based on non-semisimple data.

Defects i.e. their loops and surface operators have also been investigated as probes of field theoretic dualities. Conformal blocks for defect correlators were constructed involving multiple defects of arbitrary dimension in terms of multivariable Hypergeometrics, and applications worked out to strongly coupled field theories and in particular the twist defect of the 3D conformal Ising model. Bootstrap consistency constraints for boundaries and defects in supersymmetric conformal field theories have also been analysed. Instantons on C3 in the presence of the most general intersecting codimension-two supersymmetric defects have been captured via partition function computations. A rigorous description of non- perturbative corrections to N = 2 supergravity theories at the level of quaternionic Kähler metrics was also given. It was shown that defect observables make integrable structures of N = 2 SUSY QFTs manifest. Partition functions of topological string theory are indirectly determined through the differential equations called the quantum curve equations. This has led to a proposal that topological string partition function are local sections of a canonical holomorphic line bundle.

Scattering amplitudes and correlation functions have been researched extensively as well. In exactly solvable spin magnets, they can encode the symmetries of local or non-local operators in quantum field theory. This has provided non- perturbative results like the scattering spectrum of high-energy gluons in quantum chromodynamics (QCD) and further computation of conformal Feynman integrals of the fishnet theory at finite coupling. Several results have been derived for scattering amplitudes in the planar limit of N=4 supersymmetric Yang-Mills theory, with some techniques shown to be applicable to Feynman integrals in ordinary QCD. In the area of superconformal bootstrap, a general theory of conformal blocks for 4-point functions of local fields has been developed. This lays the grounds for a long multiplet bootstrap, in particular in all 4-dimensional superconformal field theories (SCFT). Constraints imposed by crossing symmetry on mixed correlators in 4d N = 1 SCFT were established and a bootstrap analysis for short multiplets in 4-dimensional N = 2 superconformal field theories has been carried out.

**Mathematical structures for quantum spacetime and gravitational theories **

From a starting point of holographically related theories, researchers of the cluster are trying to find a mathematically adequate description of the general holographic principle, in particular the correspondence between gauge and gravitational theories.

The study of two-dimensional integrable sigma models like the Euclidean black hole equilibrium density matrix serves as a guide to understanding string dynamics in a holographic context. Integrity preserving deformations and their relation to semi-holographic Chern-Simons theory has been investigated. The general framework of affine Gaudin models has also been used to construct new integrable sigma models. Renormalisation group equations have been formulated in terms of sigma model twist function. A study of spin chains capturing the spectral problem of 4-dimensional N = 2 SCFTs in the planar limit has been done finding these chains to be dynamical.

A surprising application of 2d/3d holography to tensor network models for two-dimensional spin systems was found whereby tensor network states are described by a three-dimensional TQFT on manifolds with physical boundaries. Modular functors — the mathematical object that describes mapping class group actions on boundary states of a 3d TQFT —have been generalised to a non-semisimple and bicategorical setting, with a by-product for representation theory. In a similar transfer of techniques, equivariant Frobenius–Schur indicators have been explained in terms of three-dimensional topological field theory with boundaries. Non-semisimple generalisations of 3d TQFTs were defined and will be important in the investigation of holographic duality for logarithmic CFTs.

Progress in an important problem of combinatorial quantization for holonomies has been made using the algebraic input datum of a non-semisimple ribbon Hopf algebra to construct skein algebras for surfaces. A generalization of string nets beyond Turaev-Viro models has been developed. String-nets have been used to describe the correlators of bulk fields in the Cardy case in a way that is manifestly invariant under the mapping class group. This approach has been naturally combined with a novel description of the field content of two-dimensional conformal field theories in terms of internal natural transformations. In another direction, line defects in 2d CFT have been used to describe bulk and boundary correlators when the world sheet is equipped with a spin structure. In the 2d/3d holographic correspondence it has been found that for a 3d TQFT of Reshetikhin–Turaev type, if the mapping class group acts irreducibly, averaging over the group leads to a consistent 2d conformal field theory on the boundary.

With the goal of developing highly efficient new approximation schemes for local correlation functions in conformal field theories, an integrability based study of multi-point conformal blocks was initiated. It has uncovered a relation with Gaudin integrable models that exists for arbitrary dimension d and, using separation of variables techniques, holds promise for novel factorized representations of CFT correlation functions that are reminiscent of tensor network formulations.

## People Involved

**Area Coordinator**: Christoph Schweigert

**Principal Investigators**: Gleb Arutyunov, Tobias Dyckerhoff, Melanie Graf, Christophe Grojean, Julian Holstein, Jan Louis, Elli Pomoni, Andreas Ringwald, Ingo Runkel, Volker Schomerus, Christoph Schweigert, Géraldine Servant, Jörg Teschner, Timo Weigand, Paul Wedrich, Alexander Westphal

**Key Researchers:** Till Bargheer, Craig Lawrie, Sven-Olaf Moch, Sven Möller, Georgios Papathanasiou, David Reutter, Birgit Richter