## Final Report Summary - QFT-2-MOT & 3-FOLDS (From Quantum Field Theory to Motives and 3-manifolds)

The initial objectives of the project QFT-2-Mot & 3-folds were the following:

(1) To investigate the appearance of multiple zeta values in Feynman integral computations (§1.1.1).

(2) To provide a combinatorial construction of 3-dimensional manifolds which gives an “algebraization” of three-dimensional geometric objects (§1.1.2).

In Project 1, PI and his collaborator Matilde Marcolli completed their investigation on the motivic aspects of Feynman integrals in configuration spaces. PI later developed a new formulation of the motivic approach on Feynman integrals by introducing and studying Feynman arrangements that are the loci of divergences of the position space Feynman integrals for Euclidean massless scalar quantum field theories. In the process, PI proved that these arrangements of singular quadrics define objects in category of mixed Tate motives for specific QFTs.

The results of Project 2 revealed that the combinatorial realisation of 3-dimensional manifolds as in its original form contains certain ambiguities. As a result, the relationships between classical handlebody decomposition, the Heegard splitting and the thickened graphs were revised accordingly.

In addition to the proposed projects, the PI and his collaborators recently discovered a new connection with Maslov dequantization (a.k.a. tropical geometry) and artificial neural networks. This collaboration showed that it is possible to accelerate the computations via suitable tropicalisations.

Dr. Afif Siddiki and PI developed a new approach on topological insulators via cut-paste topology however this project was stalled due to the ongoing unfortunate developments in Turkey.

During this period PI completed work on 3 research paper and the extensive revision of an earlier paper, and brought 3 more articles to preprint stage. Moreover, PI is currently preparing a patent application on tropical neural network algorithms. PI participated in workshops and conferences which provided him with ample opportunities to present his results. He used the funds to visit several research institutes and deliver lectures in Germany, Austria, Spain, Switzerland, USA, Colombia, Chile and Turkey. QFT- 2-Mot & 3-folds also financed the PI’s guests’ visits to Luxembourg.

Research progress

- All major objectives of Project 1 have been carried within the first period including a detailed study of motives and periods of configuration spaces and a description of renormalization based on our formulations on Feynman integrals. PI continued his research on the subject by introducing and studying new algebro-geometric objects such as arrangements of singular quadrics. PI verified that the arrangements of these singular quadrics associated with the Feynman integrals define objects in the category of mixed Tate motives for massless QFTs with cubic and quartic potentials.

- In Project 2, a combinatorial construction of 3-dimensional manifolds has been realized by using Morse and Cerf theories as proposed in §1.2.2. A further investigation on the subject revealed that there was an overlooked ambiguity in the original formulation of the 3-dimensional thickened graphs which necessitated an extensive rewriting of the initial draft. Due to the time conflict between the PI and his collaborator, they decided that PI finalise this paper alone in the following months.

- In addition to the previously proposed projects, the PI and his collaborators Atabey Kaygun and Peter Roenne recently discovered an interesting approach to artificial neural networks based on Maslov dequantization. During a joint attempt to understand deep learning algorithms, PI and his collaborators noticed that the need for computational power can be drastically reduced by a reformulation in tropical algebraic setting. This approach not only promises a drastic increase in computational efficiency, but also promises to achieve this goal by a simple coding adaptation.

Advancement beyond the state of the art in the field

- In Project 1, while the main problem of re-interpretation of Feynman integrals as periods seem to be well-defined at the first glance, it hides an ambiguity between the lines. As periods, Feynman integrals must be integrals over semialgebraic sets which are defined on real numbers. However, in their original form these semialgebraic sets do not admit canonical complexifications in general. PI developed a new approach using algebro-geometric properties of arrangements of singular quadrics after PI and his collaborator Matilde Marcolli completed their investigation on Feynman integrals in configuration spaces and their algebro-geometric and motivic aspects. Contrary to the configuration space approach, these arrangements provide Feynman integrals as periods directly. PI proved that their motives are computable and they define mixed Tate objects despite their singular nature.

- The complexity of the computations make practical implementations of neural network algorithms challenging. PI and A. Kaygun recently noticed that tropicalisation can reduce the time complexity drastically. As a result it would be significantly easier to implement neural network algorithms assuming one already has the necessary code for real minimisation at hand.

Impact

Each part of the project QFT-2-Mot & 3-folds has produced highly non-trivial, some of which were unexpected and counterintuitive. It would be natural to expect them to have further impacts in corresponding areas which will become visible over time.

- The outcomes of Project 1 produced new problems in addition to establishing a new perspective on the arithmetic nature of quantum field theory. This is a rapidly developing new subject which is expected be a well-tested field in mathematical physics in the future. The results from this approach find significances in divergences coming from earlier dimensional regularization studies produced, and are likely to point to a deeper arithmetic phenomena in quantum field theory.

- Project 2 connects two very different areas: three-dimensional topology and bi-algebras. As such, it has a big potential for transferring knowledge and techniques between these areas. It also promises further generalizations to higher dimensions and to connect these results to homotopy theory of moduli spaces of curves. These will be the focus of the investigations for the future.

- Our (currently seized) project on topological insulators has a potential to provide a new perspective. Unfortunately, Prof. A. Siddiki’s laboratory has been shut down during this period and the critical experimental studies now can not be completed. The timing is most unfortunate as studies on topological matters recently became a hot area of interest as the Nobel physics prize this year was given to the developments around the subject.

- Any technique that would reduce computational complexity of the artificial neural networks have a potential to make drastic changes. Tropicalisation of error functions in the implementations of artificial neural networks has a great promise.

(1) To investigate the appearance of multiple zeta values in Feynman integral computations (§1.1.1).

(2) To provide a combinatorial construction of 3-dimensional manifolds which gives an “algebraization” of three-dimensional geometric objects (§1.1.2).

In Project 1, PI and his collaborator Matilde Marcolli completed their investigation on the motivic aspects of Feynman integrals in configuration spaces. PI later developed a new formulation of the motivic approach on Feynman integrals by introducing and studying Feynman arrangements that are the loci of divergences of the position space Feynman integrals for Euclidean massless scalar quantum field theories. In the process, PI proved that these arrangements of singular quadrics define objects in category of mixed Tate motives for specific QFTs.

The results of Project 2 revealed that the combinatorial realisation of 3-dimensional manifolds as in its original form contains certain ambiguities. As a result, the relationships between classical handlebody decomposition, the Heegard splitting and the thickened graphs were revised accordingly.

In addition to the proposed projects, the PI and his collaborators recently discovered a new connection with Maslov dequantization (a.k.a. tropical geometry) and artificial neural networks. This collaboration showed that it is possible to accelerate the computations via suitable tropicalisations.

Dr. Afif Siddiki and PI developed a new approach on topological insulators via cut-paste topology however this project was stalled due to the ongoing unfortunate developments in Turkey.

During this period PI completed work on 3 research paper and the extensive revision of an earlier paper, and brought 3 more articles to preprint stage. Moreover, PI is currently preparing a patent application on tropical neural network algorithms. PI participated in workshops and conferences which provided him with ample opportunities to present his results. He used the funds to visit several research institutes and deliver lectures in Germany, Austria, Spain, Switzerland, USA, Colombia, Chile and Turkey. QFT- 2-Mot & 3-folds also financed the PI’s guests’ visits to Luxembourg.

Research progress

- All major objectives of Project 1 have been carried within the first period including a detailed study of motives and periods of configuration spaces and a description of renormalization based on our formulations on Feynman integrals. PI continued his research on the subject by introducing and studying new algebro-geometric objects such as arrangements of singular quadrics. PI verified that the arrangements of these singular quadrics associated with the Feynman integrals define objects in the category of mixed Tate motives for massless QFTs with cubic and quartic potentials.

- In Project 2, a combinatorial construction of 3-dimensional manifolds has been realized by using Morse and Cerf theories as proposed in §1.2.2. A further investigation on the subject revealed that there was an overlooked ambiguity in the original formulation of the 3-dimensional thickened graphs which necessitated an extensive rewriting of the initial draft. Due to the time conflict between the PI and his collaborator, they decided that PI finalise this paper alone in the following months.

- In addition to the previously proposed projects, the PI and his collaborators Atabey Kaygun and Peter Roenne recently discovered an interesting approach to artificial neural networks based on Maslov dequantization. During a joint attempt to understand deep learning algorithms, PI and his collaborators noticed that the need for computational power can be drastically reduced by a reformulation in tropical algebraic setting. This approach not only promises a drastic increase in computational efficiency, but also promises to achieve this goal by a simple coding adaptation.

Advancement beyond the state of the art in the field

- In Project 1, while the main problem of re-interpretation of Feynman integrals as periods seem to be well-defined at the first glance, it hides an ambiguity between the lines. As periods, Feynman integrals must be integrals over semialgebraic sets which are defined on real numbers. However, in their original form these semialgebraic sets do not admit canonical complexifications in general. PI developed a new approach using algebro-geometric properties of arrangements of singular quadrics after PI and his collaborator Matilde Marcolli completed their investigation on Feynman integrals in configuration spaces and their algebro-geometric and motivic aspects. Contrary to the configuration space approach, these arrangements provide Feynman integrals as periods directly. PI proved that their motives are computable and they define mixed Tate objects despite their singular nature.

- The complexity of the computations make practical implementations of neural network algorithms challenging. PI and A. Kaygun recently noticed that tropicalisation can reduce the time complexity drastically. As a result it would be significantly easier to implement neural network algorithms assuming one already has the necessary code for real minimisation at hand.

Impact

Each part of the project QFT-2-Mot & 3-folds has produced highly non-trivial, some of which were unexpected and counterintuitive. It would be natural to expect them to have further impacts in corresponding areas which will become visible over time.

- The outcomes of Project 1 produced new problems in addition to establishing a new perspective on the arithmetic nature of quantum field theory. This is a rapidly developing new subject which is expected be a well-tested field in mathematical physics in the future. The results from this approach find significances in divergences coming from earlier dimensional regularization studies produced, and are likely to point to a deeper arithmetic phenomena in quantum field theory.

- Project 2 connects two very different areas: three-dimensional topology and bi-algebras. As such, it has a big potential for transferring knowledge and techniques between these areas. It also promises further generalizations to higher dimensions and to connect these results to homotopy theory of moduli spaces of curves. These will be the focus of the investigations for the future.

- Our (currently seized) project on topological insulators has a potential to provide a new perspective. Unfortunately, Prof. A. Siddiki’s laboratory has been shut down during this period and the critical experimental studies now can not be completed. The timing is most unfortunate as studies on topological matters recently became a hot area of interest as the Nobel physics prize this year was given to the developments around the subject.

- Any technique that would reduce computational complexity of the artificial neural networks have a potential to make drastic changes. Tropicalisation of error functions in the implementations of artificial neural networks has a great promise.