# Unitary Howe dualities in fermionic and bosonic algebras and related Dirac operators

### Submission summary

 Authors (as Contributors): Guner Muarem
Submission information
Date submitted: 2023-01-05 12:27
Submitted by: Muarem, Guner
Submitted to: SciPost Physics Proceedings
Proceedings issue: 34th International Colloquium on Group Theoretical Methods in Physics
Ontological classification
Specialties:
• Mathematical Physics
Approach: Theoretical

### Abstract

In this paper we use the canonical complex structure $\boldsymbol{\mathbb{J}}$ on $\boldsymbol{\mathbb{R}^{2n}}$ to introduce a twist of the symplectic Dirac operator. This can be interpreted as the bosonic analogue of the Dirac operators on a Hermitian manifold. Moreover, we prove that the algebra of these Dirac operators is isomorphic to the Lie algebra $\boldsymbol{\mathfrak{su}(1,2)}$ which leads to the Howe dual pair $(\boldsymbol{\operatorname{U}(n)},\boldsymbol{\mathfrak{su}(1,2))}$.

In voting

### Author comments upon resubmission

I would like to thank the reviewer for carefully reviewing the paper.

### List of changes

1) "As a matter of fact" is used too often, and can usually be removed (for example, remove it in the Abstract) => Adapted
2) Abstract: analogon -> analogue. => fixed
3) First line of Introduction: put the meaning of CAR just behind it (as you did for CCR) => fixed
4) Section 3.3: you use already ∂zj and ∂¯¯¯zj, but define it only in the next subsession. => moved above
5) Last paragraph of 3.4: ... with H an holomorphic function in several variables (i.e. is -> in) => fixed
6) Section 4, after the table: 2. We have three copies of ... (i.e. two -> three). => fixed
7) Sentence -3 of Section 4: Recall that these are ... (i.e. this -> these). => fixed
8) List in Section 5: replace m by n => fixed