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Unitary Howe dualities in fermionic and bosonic algebras and related Dirac operators
by Guner Muarem
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Submission summary
| Authors (as registered SciPost users): | Guner Muarem |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202212_00038v1 (pdf) |
| Date submitted: | 2022-12-17 11:01 |
| Submitted by: | Muarem, Guner |
| Submitted to: | SciPost Physics Proceedings |
| Proceedings issue: | 34th International Colloquium on Group Theoretical Methods in Physics (GROUP2022) |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| 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. As a matter of fact, this can be interpreted as the bosonic analogon 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 $(\boldsymbol{\operatorname{U}(n)},\boldsymbol{\mathfrak{su}(1,2))}$.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2023-1-4 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202212_00038v1, delivered 2023-01-04, doi: 10.21468/SciPost.Report.6452
Strengths
1) Well written introduction and conclusion.
2) Logical structure of the paper.
3) Explicit operator realizations of the relevant algebras.
Weaknesses
1) Some of the main results (e.g. tensor products in Section 4) are just mentioned, and will be published in a forthcoming paper (Ref. [7]).
Report
The author investigated a new Howe dual pair occurring in symplectic Clifford analysis: $(U(n), \mathfrak{su}(1,2))$. Herein, $\mathfrak{su}(1,2)$ is a non-compact real form of $\mathfrak{sl}(3)$. It arises as the algebra of symplectic Dirac operators and their duals.
As far as I know, this is an original result, worth publishing.
Requested changes
Mostly typing errors or small mistakes:
1) "As a matter of fact" is used too often, and can usually be removed (for example, remove it in the Abstract).
2) Abstract: analogon -> analogue.
3) First line of Introduction: put the meaning of CAR just behind it (as you did for CCR).
4) Section 3.3: you use already $\partial_{z_j}$ and $\partial_{\overline z_j}$, but define it only in the next subsession.
5) Last paragraph of 3.4: ... with H an holomorphic function in several variables (i.e. is -> in).
6) Section 4, after the table: 2. We have three copies of ... (i.e. two -> three).
7) Sentence -3 of Section 4: Recall that these are ... (i.e. this -> these).
8) List in Section 5: replace m by n.