Periodic Reporting for period 4 - GALOP (Galois theory of periods and applications.)
Okres sprawozdawczy: 2021-09-01 do 2022-08-31
The focus of the project GALOP was to study certain numbers which are ubiquitous in mathematics and physics and are known as `periods'. Examples include the number Pi, log 2, the period of a pendulum, values of zeta functions, and many other transcendental quantities given by integration. The idea of the project was to study these from a completely new perspective, namely the theory of groups. It is a very recently understood and highly surprising fact that there is a huge group of symmetries which transforms all (`motivic') periods in a consistent manner, in other words, in such a way that all equations between them are preserved. This changes the way to think about mathematical equations: the number pi for example is no longer seen as a static quantity, but one which is allowed to be rescaled by multiplying it by any rational number. The upshot is that periods acquire new features which are not possessed by arbitrary numbers, including a weight (e.g. pi has weight 2), a dimension (pi has dimension 1, but log 2 is 2-dimensional), and so on. In short, the Galois theory of periods allows one to classify periods according to different types, and to deduce new equations from old. It also allows one to solve seemingly intractable algebraic and combinatorial problems using completely different transcendental methods.
A major application of this theory is to high-energy physics, where one studies interaction of fundamental particles. It is a slowly emerging fact that all the different theories of fundamental particles (quantum field theories, or string theories, etc) all admit a unifying underlying mathematical framework, which is the focus of much of this project. In order to make physical predictions, physicists must assign a probability or `amplitude' to certain types of interactions between elementary particles, and this probability is invariably a period or more specifically a period of a moduli space (which is a number or function assigned to the space of all possible particle configurations). These special geometric spaces not only generate the important periods relevant for particle collider experiments, but also generate entire families of periods of given `types' in mathematics. Much of the project was devoted to studying these spaces and the structure of the periods they generate.
Combining the methods of GALOP the PI was led to deduce the existence of a completely new and vast group of symmetries which acts on amplitudes for particle interactions. This is the `cosmic' Galois group whose existence was posited many years ago but which remained elusive until very recently. It leads to a vast over-arching principle which organises the amplitudes in many different physical theories. It can be thought of as a new universal law which dictates how particle interactions constrain one another. It has very recently been verified by different research groups, in a wide variety of situations including: string theory, N=4 super Yang Mills theory, massless phi^4 theory, and even for the anomalous dipole moment of the electron.
The purpose of the GALOP project was to develop ideas from the Galois theory of periods to deepen our understanding of mathematical structures (applications include: the classification of mixed `motives', construction of invariants in geometric group theory, generate new classes of mathematical objects such as modular forms) and to apply these ideas to solve problems in high-energy physics. Conversely, new ideas arising from examples from physics problems were reformulated as general mathematical principles which in future may inspire and develop new mathematical ideas.
A major application of this theory is to high-energy physics, where one studies interaction of fundamental particles. It is a slowly emerging fact that all the different theories of fundamental particles (quantum field theories, or string theories, etc) all admit a unifying underlying mathematical framework, which is the focus of much of this project. In order to make physical predictions, physicists must assign a probability or `amplitude' to certain types of interactions between elementary particles, and this probability is invariably a period or more specifically a period of a moduli space (which is a number or function assigned to the space of all possible particle configurations). These special geometric spaces not only generate the important periods relevant for particle collider experiments, but also generate entire families of periods of given `types' in mathematics. Much of the project was devoted to studying these spaces and the structure of the periods they generate.
Combining the methods of GALOP the PI was led to deduce the existence of a completely new and vast group of symmetries which acts on amplitudes for particle interactions. This is the `cosmic' Galois group whose existence was posited many years ago but which remained elusive until very recently. It leads to a vast over-arching principle which organises the amplitudes in many different physical theories. It can be thought of as a new universal law which dictates how particle interactions constrain one another. It has very recently been verified by different research groups, in a wide variety of situations including: string theory, N=4 super Yang Mills theory, massless phi^4 theory, and even for the anomalous dipole moment of the electron.
The purpose of the GALOP project was to develop ideas from the Galois theory of periods to deepen our understanding of mathematical structures (applications include: the classification of mixed `motives', construction of invariants in geometric group theory, generate new classes of mathematical objects such as modular forms) and to apply these ideas to solve problems in high-energy physics. Conversely, new ideas arising from examples from physics problems were reformulated as general mathematical principles which in future may inspire and develop new mathematical ideas.
The results of the GALOP project were widespread, ranging from foundational results for new classes of mathematical objects, to the resolution of a number of conjectures and previously unsolved open problems in mathematics and physics.
- A complete Galois theory of multiple modular values which are periods of the relative completion of the moduli space of curves of genus 1.
- The development of a new theory of `non-abelian' or `mixed' modular forms. These form a huge class of non-holomorphic modular forms which satisfy many good properties analogous to the classical theory of holomorphic modular forms.
- Development of new mathematical objects including elliptic iterated integrals, periods of the motivic fundamental groups on moduli spaces, iterated modular and quasi-modular forms.
- Applications to amplitudes has led to new organisational principles coming from motivic Galois theory, and new techniques for their practical calculation.
- New geometric structures have been introduced during the GALOP project which are moduli spaces for combinatorial objects, objects in tropical geometry, or of amplitudes.
- A complete Galois theory of multiple modular values which are periods of the relative completion of the moduli space of curves of genus 1.
- The development of a new theory of `non-abelian' or `mixed' modular forms. These form a huge class of non-holomorphic modular forms which satisfy many good properties analogous to the classical theory of holomorphic modular forms.
- Development of new mathematical objects including elliptic iterated integrals, periods of the motivic fundamental groups on moduli spaces, iterated modular and quasi-modular forms.
- Applications to amplitudes has led to new organisational principles coming from motivic Galois theory, and new techniques for their practical calculation.
- New geometric structures have been introduced during the GALOP project which are moduli spaces for combinatorial objects, objects in tropical geometry, or of amplitudes.
Several new and unexpected results emerged providing novel directions for future research.
- Singe-valued integration. Classical theories of integration involve integrating differential forms over domains. When they depend on parameters this leads to multi-valued functions. On the other hand, many important quantities in physics are given by integrals, but are nonetheless single-valued: the reason being that a well-defined question has a well-defined answer. The solution to this paradox is the existence of a theory of single-valued integration which has combines the best of both worlds. GALOP studied new features of the theory of single-valued integration, which involves pairing differential forms with their duals. It led to the resolution of several open problems, and to a general `double-copy’ formalism inspired by physics. It is steadily, but extensively, being adopted in a very wide range of disciplines, to the extent that the single-valuedness can now be imposed as an a priori constraint to find solutions to hitherto intractable problems.
- Motives associated to Mellin transforms. The results of GALOP, namely the resolution of a conjecture on the Galois coaction associated to hypergeometric functions, points conclusively to the existence of a Tannakian category of motives associated to Mellin transforms, with its own Galois theory, which massively generalises the theory of motives. Such a theory promises to have many interesting applications in the theory of amplitudes by proving, at last, the existence of a Cosmic Galois group within dimensional regularisation. Applications include string theory, zeta functions, and irrationality proofs. It implies a Galois theory for power series involving infinitely many periods simultaneously. This theory is very surprising, were it not for the 1-dimensional cases established in the GALOP project. With hindsight, it explains the uniformity of formulae for Galois actions on periods.
- Mixed L-functions. The GALOP project unexpectedly answers a question which naturally extends the programme on special values of L-functions in a novel way. In brief, the usual conjectures due to Deligne, Beilinson and others express values of motivic L-functions as very specific entries in the period matrices of simple extensions. The elephant in the room was the question: how to describe the remaining entries? The GALOP project proposes the existence of a new kind of multiple or mixed L-function which provides the missing entries, and was proven in a range of situations. This represents the first steps in a new direction of travel for the theory of L-functions.
- Singe-valued integration. Classical theories of integration involve integrating differential forms over domains. When they depend on parameters this leads to multi-valued functions. On the other hand, many important quantities in physics are given by integrals, but are nonetheless single-valued: the reason being that a well-defined question has a well-defined answer. The solution to this paradox is the existence of a theory of single-valued integration which has combines the best of both worlds. GALOP studied new features of the theory of single-valued integration, which involves pairing differential forms with their duals. It led to the resolution of several open problems, and to a general `double-copy’ formalism inspired by physics. It is steadily, but extensively, being adopted in a very wide range of disciplines, to the extent that the single-valuedness can now be imposed as an a priori constraint to find solutions to hitherto intractable problems.
- Motives associated to Mellin transforms. The results of GALOP, namely the resolution of a conjecture on the Galois coaction associated to hypergeometric functions, points conclusively to the existence of a Tannakian category of motives associated to Mellin transforms, with its own Galois theory, which massively generalises the theory of motives. Such a theory promises to have many interesting applications in the theory of amplitudes by proving, at last, the existence of a Cosmic Galois group within dimensional regularisation. Applications include string theory, zeta functions, and irrationality proofs. It implies a Galois theory for power series involving infinitely many periods simultaneously. This theory is very surprising, were it not for the 1-dimensional cases established in the GALOP project. With hindsight, it explains the uniformity of formulae for Galois actions on periods.
- Mixed L-functions. The GALOP project unexpectedly answers a question which naturally extends the programme on special values of L-functions in a novel way. In brief, the usual conjectures due to Deligne, Beilinson and others express values of motivic L-functions as very specific entries in the period matrices of simple extensions. The elephant in the room was the question: how to describe the remaining entries? The GALOP project proposes the existence of a new kind of multiple or mixed L-function which provides the missing entries, and was proven in a range of situations. This represents the first steps in a new direction of travel for the theory of L-functions.