## Periodic Reporting for period 3 - BG-BB-AS (Birational Geometry, B-branes and Artin Stacks)

Reporting period: 2020-09-01 to 2022-02-28

This project lies somewhere between pure mathematics and theoretical ‘high-energy’ physics, which means the physics of very small things, millions of times smaller than atoms.

Physical theories have to be expressed in the language of mathematics, so mathematical tools and concepts have to be developed before a physicist’s ideas can be made precise. For example when Newton developed his theories of gravity and motion, he first had to invent the mathematical technique of calculus. Without the idea of the derivative of a function, his equations can’t even be written down properly, much less solved.

Sometimes in history the maths has been ahead of the physics. This was true for example when Einstein was inventing General Relativity, all the maths he needed had been developed a few decades earlier, mostly by the great German geometers Gauss and Riemann. Today however, the physics is ahead of the maths - physicists have ideas, and some calculational techniques, that are beyond what mathematicians can currently understand and make precise. This is great news for mathematicians, because studying the physicists’ work leads to lots of interesting new mathematics.

Most of this new maths is linked to a particular physical idea called String Theory. String Theory is controversial amongst physicists because despite having been around for about thirty years and having lots of people working on it, it hasn’t yet developed into a proper theory that can be tested against experiments. This is mostly because the maths is still lacking! So what String theorists (and mathematicians like myself) are still doing is developing the mathematical language with which the theory can be described. Fortunately even if String Theory turns out to be wrong, then whatever the correct theory is will undoubtedly still use this same mathematical language. And for mathematicians this language is very interesting and beautiful in its own right.

The fundamental idea of String theory is that instead of thinking of electrons, protons, etc. as ‘particles’, like little tennis balls, we should think of them as little vibrating pieces of ‘string’. The advantage of this is that a single piece of string can vibrate in lots of different ways, like a string on a violin playing different notes. This means that we don’t need lots of different kinds of particles, we just need one thing: strings. When we see an electron it’s a string playing one kind of note. When we see any other particle it’s the same kind of string, it’s just playing a different note.

One controversial aspect of String Theory involves dimension. The ordinary space we live in seems to be three-dimensional, or four-dimensional if you include time. But (for a technical reason) String Theory works more nicely if you assume that the strings are moving around in a space that has ten dimensions. String Theorists like to do this, and to explain away the missing six dimensions they assume they are curled up into some kind of complicated shape which is too small to detect. Some physicists think this is dubious, but mathematicians love it because we know lots of complicated six-dimensional shapes, and String Theory gives us new tools for understanding them. The physics of strings moving around in these spaces turns out to be deeply related to pure mathematics - including to lots of areas that previously had nothing to do with physics, such as abstract algebra and algebraic geometry.

Strings can be formed into loops, like rubber bands, but if they’re not then the ends of the string have to be attached to something. What they attach to are objects called ‘branes’, short for ‘membranes’, which are a bit like sheets of rubber (although they might really have many more dimensions). The branes themselves can also move around and this makes the theory much more complicated and interesting.

This project is about studying the mathematics of a particular kind of brane called a ‘B-brane’. B-branes turn out to be closely related to things called `derived categories', which are a highly-abstract algebraic structure that pure mathematicians have been studying for decades. By studying the physicists' predictions for B-branes we are going to learn deeper properties of derived categories and the spaces on which they live. By bringing in this other perspective we are anticipating making progress on some of the hardest and most fascinating problems in this field of maths; we are also expecting to use our mathematical approach to teach physicists new things about B-branes.

Physical theories have to be expressed in the language of mathematics, so mathematical tools and concepts have to be developed before a physicist’s ideas can be made precise. For example when Newton developed his theories of gravity and motion, he first had to invent the mathematical technique of calculus. Without the idea of the derivative of a function, his equations can’t even be written down properly, much less solved.

Sometimes in history the maths has been ahead of the physics. This was true for example when Einstein was inventing General Relativity, all the maths he needed had been developed a few decades earlier, mostly by the great German geometers Gauss and Riemann. Today however, the physics is ahead of the maths - physicists have ideas, and some calculational techniques, that are beyond what mathematicians can currently understand and make precise. This is great news for mathematicians, because studying the physicists’ work leads to lots of interesting new mathematics.

Most of this new maths is linked to a particular physical idea called String Theory. String Theory is controversial amongst physicists because despite having been around for about thirty years and having lots of people working on it, it hasn’t yet developed into a proper theory that can be tested against experiments. This is mostly because the maths is still lacking! So what String theorists (and mathematicians like myself) are still doing is developing the mathematical language with which the theory can be described. Fortunately even if String Theory turns out to be wrong, then whatever the correct theory is will undoubtedly still use this same mathematical language. And for mathematicians this language is very interesting and beautiful in its own right.

The fundamental idea of String theory is that instead of thinking of electrons, protons, etc. as ‘particles’, like little tennis balls, we should think of them as little vibrating pieces of ‘string’. The advantage of this is that a single piece of string can vibrate in lots of different ways, like a string on a violin playing different notes. This means that we don’t need lots of different kinds of particles, we just need one thing: strings. When we see an electron it’s a string playing one kind of note. When we see any other particle it’s the same kind of string, it’s just playing a different note.

One controversial aspect of String Theory involves dimension. The ordinary space we live in seems to be three-dimensional, or four-dimensional if you include time. But (for a technical reason) String Theory works more nicely if you assume that the strings are moving around in a space that has ten dimensions. String Theorists like to do this, and to explain away the missing six dimensions they assume they are curled up into some kind of complicated shape which is too small to detect. Some physicists think this is dubious, but mathematicians love it because we know lots of complicated six-dimensional shapes, and String Theory gives us new tools for understanding them. The physics of strings moving around in these spaces turns out to be deeply related to pure mathematics - including to lots of areas that previously had nothing to do with physics, such as abstract algebra and algebraic geometry.

Strings can be formed into loops, like rubber bands, but if they’re not then the ends of the string have to be attached to something. What they attach to are objects called ‘branes’, short for ‘membranes’, which are a bit like sheets of rubber (although they might really have many more dimensions). The branes themselves can also move around and this makes the theory much more complicated and interesting.

This project is about studying the mathematics of a particular kind of brane called a ‘B-brane’. B-branes turn out to be closely related to things called `derived categories', which are a highly-abstract algebraic structure that pure mathematicians have been studying for decades. By studying the physicists' predictions for B-branes we are going to learn deeper properties of derived categories and the spaces on which they live. By bringing in this other perspective we are anticipating making progress on some of the hardest and most fascinating problems in this field of maths; we are also expecting to use our mathematical approach to teach physicists new things about B-branes.

The mathematical spaces studied in the project are called `algebraic varieties', which means they are the space of solutions to polynomial equations. Algebraic varieties have been intensely studied for over a hundred years. There is a procedure for turning one algebraic variety into another called a `birational transformation', and one of the main themes of this project has been to understand what this procedure does to the derived categories (or B-branes) on the varieties. Our main tools have been `Gauged Linear Sigma Models' (GLSMs) – these are a kind of physical model widely used by string theorists, and they are closely related to birational transformations of algebraic varieties. We have been using physicists' predictions of the behaviour of B-branes in GLSMs to discover new aspects of derived categories.

Here is a brief summary of some of the main results achieved by the group members and their collaborators:

The PI (in collaboration with Joergen Rennemo) has given a mathematical interpretation and proof to a subtle new duality of GLSMs predicted by the string theorist Kentaro Hori. This was published in the prestigious Duke Mathematical Journal.

The PI has proved a very general result about the structure of symmetries of derived categories. This was published in International Mathematics Research Notices.

The PI (in collaboration with Alex Kite) has studied the behaviour of derived categories in abelian GLSMs, proving new results about their symmetries.

Graduate student Federico Barbacovi has discovered a new general result about symmetries of derived categories related to birational transformations.

The PI (in collaboration with Bradley Doyle) has used GLSMs to understand `Homological Projective Duality' of derived categories for certain algebraic varieties called `Grassmannians'. This can also be seen as another instance of Hori's duality.

The PI (in collaboration with Tarig Abdelgadir) has discovered some new GLSMs that fill in the missing parts of the McKay correspondence.

Here is a brief summary of some of the main results achieved by the group members and their collaborators:

The PI (in collaboration with Joergen Rennemo) has given a mathematical interpretation and proof to a subtle new duality of GLSMs predicted by the string theorist Kentaro Hori. This was published in the prestigious Duke Mathematical Journal.

The PI has proved a very general result about the structure of symmetries of derived categories. This was published in International Mathematics Research Notices.

The PI (in collaboration with Alex Kite) has studied the behaviour of derived categories in abelian GLSMs, proving new results about their symmetries.

Graduate student Federico Barbacovi has discovered a new general result about symmetries of derived categories related to birational transformations.

The PI (in collaboration with Bradley Doyle) has used GLSMs to understand `Homological Projective Duality' of derived categories for certain algebraic varieties called `Grassmannians'. This can also be seen as another instance of Hori's duality.

The PI (in collaboration with Tarig Abdelgadir) has discovered some new GLSMs that fill in the missing parts of the McKay correspondence.

The structure of Homological Projective Duality for Grassmannians is completely new. We expect to be able to connect it to geometric constructions in certain dimensions.

We expect to be able to prove the remaining case of Hori's duality (the case of orthogonal groups).

We have discovered new features of `non-commutative-resolutions', which are a modified version of derived categories essential for describing B-branes on singular algebraic varieties.

The PI and Kite have formulated a new conjecture about the role of discriminants for the derived categories of toric varieties.

We expect to be able to prove the remaining case of Hori's duality (the case of orthogonal groups).

We have discovered new features of `non-commutative-resolutions', which are a modified version of derived categories essential for describing B-branes on singular algebraic varieties.

The PI and Kite have formulated a new conjecture about the role of discriminants for the derived categories of toric varieties.