Periodic Reporting for period 3 - GALOP (Galois theory of periods and applications.)
Okres sprawozdawczy: 2020-03-01 do 2021-08-31
The project GALOP studies 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 is 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 orthogonal transcendental methods.
A major application of this theory is to high-energy physics, where one studies interaction of fundamental particles. It is slowly emerging 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, of a certain special type, namely 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 is 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 by Cartier 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 many 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 is 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 are reformulated as general mathematical principles which can in turn 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 slowly emerging 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, of a certain special type, namely 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 is 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 by Cartier 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 many 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 is 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 are reformulated as general mathematical principles which can in turn inspire and develop new mathematical ideas.
The main results achieved in the project so far include:
1). A complete Galois theory of `multiple modular values' which are the periods of the relative completion of the moduli space of curves of genus 1.
2). The development of a new theory of `non-abelian' or `mixed' modular forms. These form a huge class of non-holomorphic modular forms on the upper half plane which satisfy many good properties analogous to the classical theory of holomorphic modular forms. Applications involve solving a long-standing problem of describing the mathematical objects arising from genus 1 closed string perturbation theory, as well as settling a question about the Fourier coefficients of weak harmonic Maass forms.
3). A new theory of single-valued integration and double-copy formalism inspired by KLT relations in high-energy physics. A hot topic in high-energy physics is the revelation that certain quantum field theories, such as gravity, should conjecturally be obtained as a double-copy of others (such as Yang-Mills theory). With Dupont we have shown that this is underpinned by a very general mathematical double-copy formalism, and in the process settled some conjectures in string theory (e.g. Stieberger's conjecture on the single-valued projection).
4). The first computations of the Galois coaction on hypergeometric motives. This strongly suggests the existence of a theory of motives for cohomology with coefficients which is, most crucially, consistent with the classical theory of motives and periods. This opens the door to a new area of research. In joint work with Dupont, we in the process settled a recent conjecture on motivic coactions for hypergeometric functions.
1). A complete Galois theory of `multiple modular values' which are the periods of the relative completion of the moduli space of curves of genus 1.
2). The development of a new theory of `non-abelian' or `mixed' modular forms. These form a huge class of non-holomorphic modular forms on the upper half plane which satisfy many good properties analogous to the classical theory of holomorphic modular forms. Applications involve solving a long-standing problem of describing the mathematical objects arising from genus 1 closed string perturbation theory, as well as settling a question about the Fourier coefficients of weak harmonic Maass forms.
3). A new theory of single-valued integration and double-copy formalism inspired by KLT relations in high-energy physics. A hot topic in high-energy physics is the revelation that certain quantum field theories, such as gravity, should conjecturally be obtained as a double-copy of others (such as Yang-Mills theory). With Dupont we have shown that this is underpinned by a very general mathematical double-copy formalism, and in the process settled some conjectures in string theory (e.g. Stieberger's conjecture on the single-valued projection).
4). The first computations of the Galois coaction on hypergeometric motives. This strongly suggests the existence of a theory of motives for cohomology with coefficients which is, most crucially, consistent with the classical theory of motives and periods. This opens the door to a new area of research. In joint work with Dupont, we in the process settled a recent conjecture on motivic coactions for hypergeometric functions.
The results mentioned above all go beyond the state of the art. Whilst it is hard to predict the results expected in the future, we can hope for:
- a general theory of motives and periods`with coefficients' generalising the hypergeometric examples.
- further evidence for a new conjecture formulated by the PI relating `mixed L-functions' to periods of mixed motives.
- development of the Galois theory of amplitudes and Cosmic Galois group.
- a general theory of motives and periods`with coefficients' generalising the hypergeometric examples.
- further evidence for a new conjecture formulated by the PI relating `mixed L-functions' to periods of mixed motives.
- development of the Galois theory of amplitudes and Cosmic Galois group.