Invertible bimodule categories and generalized Schur orthogonality
Jacob Bridgeman, Laurens Lootens, and Frank Verstraete

arXiv:2211.01947

EQuAL — February 2024
Overview

  1. An elementary question

  2. Problem statement

  3. Motivation

  4. Skeletal fusion categories and their (bi)module categories

  1. Main result

  2. The annular algebra

  4. Schur orthogonality

  5. A familiar example:

  6. Summary

An elementary question

Given an matrix :

Is there a simple condition to check invertibility?




Given a bimodule category :

Is there a simple condition to check invertibility?



  2. This work
This work

Given a bimodule category, specified by its skeletal data, determine whether it's invertible

Aside : Tensor networks
  • Quantum states on qudits reside in a -dimensional complex Hilbert space
  • Tensor networks allow for efficient representation of states with low entanglement



Motivation : MPO injectivity
States of topologically ordered phases can be described by tensor networks with a particular type of symmetry

arXiv:1409.2150
arXiv:2008.11187

Motivation : Morita equivalence and anyon models

  • Fusion categories are the input to the Levin-Wen construction

    • Excitations (anyons) are described by the Drinfel'd center

    • If an invertible bimodule exists between two fusion categories, they lead to the same excitations

  • Levin-Wen construction can be extended to include bimodules over the fusion categories

    • These describe gapped boundaries and interfaces

    • Invertible bimodules give rise to anyon-permuting interfaces

    • These can be used to enact logical gates in topological quantum computing

Skeletal fusion categories

  1. Finite set of simple objects

  2. For each triple , a finite dimensional vector space

  3. For each 4-tuple, isomorphisms

obeying the pentagon equation. After choosing bases for all , the category can be specified by a finite set of matrices. We will use the string-diagram notation:

Skeletal (bi)module categories

  1. Finite set of simple objects

  2. For each triple , a vector space

  3. For each 4-tuple, isomorphisms

obeying the module pentagon equation

Skeletal bimodule categories

A skeletal bimodule category is specified by providing 5 sets of matrices:

obeying 6 couples pentagon equations.

Given this data, how can we determine whether is invertible?

Main result

Theorem 1:

Let be unitary, skeletal fusion categories, and an indecomposable, unitary, finitely semisimple, skeletal bimodule category.

Then is invertible if and only if and

Proof sketch:

  • Given , there is an associated category which makes invertible
  • is equivalent to the category of representations of an algebra
  • is equivalent to exactly when the conditions hold
The annular algebra

Associated to any module category is an algebra , with basis

and product
The Morita dual

Associated to any module category , there is a fusion category called the Morita dual, denoted

  • This is the unique fusion category such that is invertible

  • Typically, is defined as the category of -module endofunctors on

Here, we make use of a different characterisation:
is (equivalent to) the category , of representations of the annular algebra

Our task becomes verifying that the given is equivalent to

  • Unfortunately, it is computationally challenging to do this by computing directly
Representations of the annular algebra

We can associate a representation of to each object as follows:

Define be the vector space with the action of given by:

which remains in .

Representations of the annular algebra

In the case that is equivalent to , the simple objects of correspond to the irreducible representations of

We can therefore verify invertibility by checking that:

  1. Simple objects of label irreducible representations of
  2. Distinct simples label to distinct irreducibles
  3. All irreducibles of are labelled by simples of
A natural inner product

We can verify the above conditions by utilizing a generalization of Schur orthogonality to weak Hopf algebras.

Recall that for a finite group , the irreducible characters satisfy

We can generalize this to weak Hopf algebras, and in particular to the annular algebra , using the natural inner product

where is the Haar integral on

Verifying invertibility

We can verify invertibility by checking that:

  1. Simple objects of label irreducible representations of
  2. Distinct simples label to distinct irreducibles
  3. All irreducibles of are labelled by simples of

Using the Schur relations:

  1. Check that for all
  2. Check that for distinct
  3. Combined with (1) and (2), the dimension condition verifies (3)

Using the definition of the annular algebra action, we have

Example

Given a finite group, we can always define two fusion categories:

  • , the category of -graded vector spaces
  • , the category of finite dimensional representations of

The category of finite dimensional vector spaces is a bimodule category between and , with skeletal data

Inserting this definition into the main result, it becomes

MPO injectivity

Recall from the introduction that we wanted to understand MPO-injectivity

The data used to define these tensors is exactly the skelatal data of a bimodule category

As an applicaiton of our main result, we can now understand MPO-injectivity as the requirement that the bimodule category is invertible

Summary and open questions

  • Extending classical results to quantum symmetries

    • Schur orthogonality
    • Wigner-Eckart theorem: Constraints on symmetric tensors
    • Applications to tensor networks

  • Open questions

    • Going beyond finite case?
    • Higher dimensions?
      • Fusion n-categories
      • Higher module module categories
      • Generalizing weak Hopf algebras?

Any questions?

arXiv:2211.01947

Slides available at jcbridgeman.github.io

Product of bimodule categories

Given bimodule categories , , the product is the category , with objects given by:

Objects , together with a collection of isomorphisms which must be compatible with all associators



More details:

arXiv:0909.3140
arXiv:1406.4204

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