YOUNGST@RS - Combinatorics in Fundamental Physics

26 Nov 2024, 09:15 → 29 Nov 2024, 18:00 Europe/Berlin

Mainz Institute for Theoretical Physics, Johannes Gutenberg University

Combinatorics, a branch of mathematics with powerful applications in modern physics, plays a significant role in fundamental areas such as enumerating Feynman diagrams and addressing discreteness in various approaches to Quantum Gravity. This workshop will highlight connections across different fields, encourage collaboration, and contribute to setting new scientific targets for the community.

The workshop is designed to build new bridges across adjacent disciplines, encouraging the exchange of ideas and the transfer of advanced mathematical tools from pure combinatorics to physics. By bringing together experts and young researchers, it aims to inspire mathematicians to explore topics relevant to physicists and facilitate interdisciplinary collaboration.

In addition to fostering high-level scientific exchange, this workshop offers a platform for early-career scientists to present their work to a global audience, helping them build connections and advance their careers in a dynamic research environment.

The first 3 days will be devoted to a particular topic at the intersection of fundamental physics and combinatorics:

Day 1: Random Geometry for Quantum Gravity

Day 2: Causal Set Theory

Day 3: Combinatorics in Perturbative QFT 

The 4th day will be dedicated to cross-disciplinary discussion. 

The talks will be aimed at an audience with expertise in some area of combinatorics or theoretical physics in the broad sense – in particular, they will be accessible to those who work outside that specific topic. 

 

Tuesday 26 November

Welcome Greeting by MITP Directors

Random Geometry for Quantum Gravity: Timothy Budd

1 Timothy Budd: Random geometry and the enumerative combinatorics of maps

In this talk I will give an introduction to the study of the random metrics on surfaces via the combinatorics of planar maps, a topic with a long history in math and physics. I will highlight the use of bijective methods, that have played an important role in establishing scaling limits of random maps towards two-dimensional quantum gravity.

mitp_2024_tbudd.pdf

Random Geometry for Quantum Gravity: Reiko Toriumi

2 Reiko Toriumi: Random tensor models for quantum gravity (a look at tensor models with a mixed U(N) and O(D) symmetry)

I will discuss random tensor models for quantum gravity, of a particular case with mixed U(N) and O(D) symmetry.

A certain U(N)^2 \otimes O(D) order-3 tensor model can be viewed as a complex multi-matrix model with D copies of complex matrices. This model admits an expansion in two parameters owing to the presence of N and D, and yields a more refined classification of Feynman graphs generated by the model. Such a refined expansion has piqued our curiosity in the possibility of finding a new universality class for this tensor model. In the end, we obtain universality classes of branched polymers and also the Brownian sphere (Liouville 2-dimensional quantum gravity).

As a dessert, I will also discuss enumerations of invariants for tensors of arbitrary order with mixed unitary and orthogonal symmetry, and unveil correspondence with the topological field theory.

online youngstar Reiko TORIUMI.pdf

Random Geometry for Quantum Gravity: Luca Lionni

3 Luca Lionni: Current challenges in the random geometry approach to quantum gravity

Wednesday 27 November

Causal Set Theory: Nosiphiwo Zwane

4 Nosiphiwo Zwane: Closeness of Lorentzian manifolds

In some theories of quantum gravity spacetime is assumed to have an underlying structure from which spacetimes arise as an approximation. In these theories it is important to show that the spacetime arising from the underlying structure is approximately isometric to General relativity spacetime in the classical limit. We will discuss how one can measure how close two spacetimes are from being isometric. The motivation for the discussed measure comes from causal set theory (an approach to quantum gravity).

Causal Set Theory: Yasaman Yazdi

5 Yasaman Yazdi: Can One Hear the Shape of a Causal Set?

Spectral geometry offers a powerful means to describe the geometric information of manifolds using the spectra of operators defined on them. While much work has been done on the spectral geometry of Riemannian manifolds, comparatively little work has been done on Lorentzian manifolds. Causal sets, being a discrete realization of Lorentzian manifolds, provide an ideal setting for such a study. I will review spectral geometry in causal set theory and discuss what geometric information can be obtained from the spectra of various operators on causal sets. I will show results indicating that one can indeed “hear” some physical aspects of a causal set.

From a physical point of view, these studies are motivated by their application in quantum gravity. If one were to uniquely distinguish causal sets through a set of spectra, one could sum over these spectra in a path sum. Spectra are relabeling invariant and lower dimensional compared to the adjacency matrices (such as the link and causal matrices) that are typically used to describe a causal set, making them beneficial to work with both in principle and in practice. Eigenvalues and eigenvectors of certain operators have also been central to defining quantum fields on causal sets. I will highlight some of these applications and comment on future directions of this area of research.

Causal Set Theory: Jane Gao

6 Jane Gao: Evolution of random graph orders and their dimensions

A poset P is a set X equipped with a partial order. A causal set is a locally finite poset, which was introduced to model the space-time universe in physics. In this talk I will briefly review the literature of different models on random orders and their relations to causal set theory. Then we focus on an open problem proposed by Erdos on the evolution of the dimensions of random graph orders. The random graph order is a type of classical sequential growth model that physicists use to generate a random causal set.
The dimension of a poset P is the minimum number of linear orderings required whose intersection is P. The dimensions of the random graph orders have been studied by Albert and Frieze, and by Erdos, Kierstad and Trotter around 1990. Better bounds on the dimensions were obtained by Bollobas and Brightwell in 1997, for “non-sparse” random graph orders. We complete the last piece of the puzzle by determining their asymptotic behaviours in the sparse regime.
This talk is based on a collaborated work with Arnav Kumar.

Thursday 28 November

Combinatorics in Perturbative QFT: Alessandra Frabetti

7 Alessandra Frabetti: Algebraic view on renormalization

Renormalization Hopf algebras represent a pro-algebraic group linking the BPHZ and Dyson renormalization formulas, where Feynman graphs are "virtual" coordinates for correlation functions and renormalization factors. In this talk I clarify the mathematical background of this statement and show how it can help improve the BPHZ formula. Time permitting, I also show how this naturally leads to a non-associative (perturbative) renormalization group for non-scalar theories.

Combinatorics in Perturbative QFT: Michael Borinsky

8 Michael Borinsky: Bivariate exponential integrals and edge-bicolored graphs

We show that specific exponential bivariate integrals serve as
generating functions of labeled edge-bicolored graphs. Based on this, we
prove an asymptotic formula for the number of regular edge-bicolored
graphs with arbitrary weights assigned to different vertex structures.
The asymptotic behavior is governed by the critical points of a
polynomial. As an application, we discuss the Ising model on a random
4-regular graph and show how its phase transitions arise from our formula.

Joint work with Chiara Meroni and Maximilian Wiesmann
Reference: arXiv:2409.18607

Combinatorics in Perturbative QFT: Simone Hu

9 Simone Hu: Orientation forms on the odd graph complex

In this talk we will introduce Kontsevich's odd commutative graph complex and describe a program to detect its non-trivial (co)homology classes via associating convergent integrals to combinatorial graphs, in the spirit of Feynman integrals. In particular, we study closed differential forms constructed using the Pfaffian of skew-symmetric matrices and whose integrals give rise to cocycles in the odd graph complex, two of which we can show are non-trivial.
This is based on joint work with Francis Brown and Erik Panzer.

Combinatorics in Perturbative QFT: Johannes Thürigen

10 Johannes Thürigen: Tensor combinatorics in the renormalization group

Generalizations of vector field theories to tensors allow to similarly apply large-N techniques but find a richer though often still tractable structure. Generating discrete geometries via their perturbative series, they are furthermore candidates for Quantum Gravity. However, the potential of such tensor theories has not been fully exploited since only a symmetry-reduced ``isotropic'' part of their phase space has been studied so far.
Here I will show how applying the renormalization group to tensor fields of rank r in the full, anisotropic cyclic-melonic potential approximation unveils a plethora of new non-Gaussian fixed points. From the Quantum-Gravity perspective, these fixed points correspond to continuum limits of distinguished ensembles of triangulations raising hope to find new classes of continuum geometry in this way.
This talk is based on arxiv.org/abs/2406.01368 with Leonardo Juliano.

Combinatorics in Perturbative QFT: Nathan Pagliaroli

11 Nathan Pagliaroli: Map enumeration from path integrals over noncommutative geometries

In this talk we will explore path integrals over finite spectral triples as models of Euclidean Quantum Gravity, first proposed by J. W. Barrett. Such integrals can be expressed as bi-tracial multi-matrix integrals. The Feynman diagrams of these integrals are decorated maps, and they satisfy their own Schwinger-Dyson equations which are generalizations of Tutte's recursion. I will outline how recent work has used map enumeration techniques to solve these models explicitly.

Friday 29 November

Panel discussion: Presentations

Panel discussion

Panel discussion: Open discussion

Panel discussion: Networking session