Periodic Reporting for period 1 - Modular Functors (Non-semisimple differential graded modular functors)
Période du rapport: 2022-01-01 au 2023-12-31
The mathematical aspects of certain types of field theories (more precisely: topological field theories in dimension three and conformal field theories in dimension two) can be described by rather complicated algebraic entities that are called modular categories. These play an important role in topological quantum computing.
A modular category is built from elementary building blocks, the so-called simple objects. These simple objects can be added and multiplied (via the tensor product), but they can also be braided, which means that the position of two objects in a tensor product is changed in a certain way. If the modular category has a property that is called semisimplicity, then all objects can be built from elementary building blocks in a rather naive way. If this is not the case, then the modular category is called non-semisimple.
Generally, determining the tensor product of two objects can be a very hard problem. One of the pleasant properties of a semisimple modular category is that this tensor product can be directly computed via the famous Verlinde formula. Superficially, this formula is somewhat surprising, but there is a clear topological picture behind it: The vector space generated by the simple objects can be thought of as the result of applying a certain algebro-topological prescription (a so-called modular functor) to a torus (a space shaped like a doughnut). This has the consequence that the symmetries of the torus act on the vector space spanned by the simple objects of our modular category. There is now a certain symmetry operation (commonly referred to as S-transformation) that transforms the tensor multiplication (the one that we want to compute), into a much simpler multiplication. As a result, the tensor multiplication can be recovered from a very simple multiplication and the S-transformation. This is exactly the origin of the Verlinde formula!
So far, this applies to the semisimple case. In the very interesting non-semisimple case, the vector space of simple objects can be replaced by a chain complex, more specifically the so-called Hochschild (co)chain complex of the modular category. Compared to a vector space, this chain complex now contains not only one layer of information, but infinitely many. Technically speaking, the modular functor must be adjusted and replaced with a chain complex valued version. As we know from previous work of Schweigert and myself, building on works of Lentner-Mierach-Schweigert-Sommerhäuser, the symmetries of the torus still correctly act on this chain complex. Therefore, one could hope that it is still true that the multiplication coming from the tensor product is transformed into a much simpler multiplication, i.e. that there is indeed a differential graded Verlinde formula. Establishing such a formula is the main objective of the project.
The technical obstacles are immediate: For the traditional Verlinde formula, having a second multiplication was key. In the chain complex valued case, it is not clear where this multiplication should even come from. The proposal gives a candidate for such a second multiplicative structure, namely the so-called Deligne structure on the Hochschild cochain complex. The project requires a deep understanding of the Deligne structure for modular categories. The reason for the technical difficulty is that multiplicative structures on chain complexes are very hard to describe since information on infinitely layers must be specified. Even worse, commutativity, which for ordinary products is a property, may become additional structure that one needs to keep track of using a gadget called an operad (for us more specifically the little disks operad).
A modular category is built from elementary building blocks, the so-called simple objects. These simple objects can be added and multiplied (via the tensor product), but they can also be braided, which means that the position of two objects in a tensor product is changed in a certain way. If the modular category has a property that is called semisimplicity, then all objects can be built from elementary building blocks in a rather naive way. If this is not the case, then the modular category is called non-semisimple.
Generally, determining the tensor product of two objects can be a very hard problem. One of the pleasant properties of a semisimple modular category is that this tensor product can be directly computed via the famous Verlinde formula. Superficially, this formula is somewhat surprising, but there is a clear topological picture behind it: The vector space generated by the simple objects can be thought of as the result of applying a certain algebro-topological prescription (a so-called modular functor) to a torus (a space shaped like a doughnut). This has the consequence that the symmetries of the torus act on the vector space spanned by the simple objects of our modular category. There is now a certain symmetry operation (commonly referred to as S-transformation) that transforms the tensor multiplication (the one that we want to compute), into a much simpler multiplication. As a result, the tensor multiplication can be recovered from a very simple multiplication and the S-transformation. This is exactly the origin of the Verlinde formula!
So far, this applies to the semisimple case. In the very interesting non-semisimple case, the vector space of simple objects can be replaced by a chain complex, more specifically the so-called Hochschild (co)chain complex of the modular category. Compared to a vector space, this chain complex now contains not only one layer of information, but infinitely many. Technically speaking, the modular functor must be adjusted and replaced with a chain complex valued version. As we know from previous work of Schweigert and myself, building on works of Lentner-Mierach-Schweigert-Sommerhäuser, the symmetries of the torus still correctly act on this chain complex. Therefore, one could hope that it is still true that the multiplication coming from the tensor product is transformed into a much simpler multiplication, i.e. that there is indeed a differential graded Verlinde formula. Establishing such a formula is the main objective of the project.
The technical obstacles are immediate: For the traditional Verlinde formula, having a second multiplication was key. In the chain complex valued case, it is not clear where this multiplication should even come from. The proposal gives a candidate for such a second multiplicative structure, namely the so-called Deligne structure on the Hochschild cochain complex. The project requires a deep understanding of the Deligne structure for modular categories. The reason for the technical difficulty is that multiplicative structures on chain complexes are very hard to describe since information on infinitely layers must be specified. Even worse, commutativity, which for ordinary products is a property, may become additional structure that one needs to keep track of using a gadget called an operad (for us more specifically the little disks operad).
The most important goal of the proposal was achieved. For this, it was proven that the multiplicative structures relevant for the differential graded generalization of the Verlinde formula arise from certain braided commutative algebras inside the modular category (or rather its Drinfeld center) via taking homotopy invariants. The crucial part was to prove that Deligne's E_2-structure can be described this way. Together with previous results, this implies the differential graded Verlinde formula, as a statement about E_2-algebras. The proof involves to a large extent braided operads and the multiplicative structures on the so-called canonical end of a modular category.
The insights that were gained cannot only be applied to the main goal of the proposal, but go beyond that. They were also used to endow the Ext algebra of a unimodular pivotal finite tensor category with a framed E_2-structure, thereby improving a result of Menichi.
These results are contained in the article "Homotopy Invariants of Braided Commutative Algebras and the Deligne Conjecture for Finite Tensor Categories" with C. Schweigert (arXiv preprint, submitted).
The insights that were gained cannot only be applied to the main goal of the proposal, but go beyond that. They were also used to endow the Ext algebra of a unimodular pivotal finite tensor category with a framed E_2-structure, thereby improving a result of Menichi.
These results are contained in the article "Homotopy Invariants of Braided Commutative Algebras and the Deligne Conjecture for Finite Tensor Categories" with C. Schweigert (arXiv preprint, submitted).
Our main results do not only produce a Verlinde formula at a previously unavailable level of generality, but also provide sophisticated and effective tools to investigate and compare a large class of higher multiplicative structures that naturally appear in quantum algebra. This goes significantly beyond the state of the art and is likely to not only form the basis for my own further research, but to also be used as a tool by other researchers.
There are, for the moment, no socio-economic or societal implications.
There are, for the moment, no socio-economic or societal implications.