# Adjoint affine fusion and tadpoles

## Abstract

We study affine fusion with the adjoint representation. For simple Lie algebras, elementary and universal formulas determine the decomposition of a tensor product of an integrable highest-weight representation with the adjoint representation. Using the (refined) affine depth rule, we prove that equally striking results apply to adjoint affine fusion. For diagonal fusion, a coefficient equals the number of nonzero Dynkin labels of the relevant affine highest weight, minus 1. A nice lattice-polytope interpretation follows and allows the straightforward calculation of the genus-1 1-point adjoint Verlinde dimension, the adjoint affine fusion tadpole. Explicit formulas, (piecewise) polynomial in the level, are written for the adjoint tadpoles of all classical Lie algebras. We show that off-diagonal adjoint affine fusion is obtained from the corresponding tensor product by simply dropping non-dominant representations.

- Authors:

- Physics and Astronomy Department, University of Lethbridge, Lethbridge, Alberta T1K 3M4 (Canada)
- (SISSA), via Bonomea 265, 34136 Trieste (Italy)

- Publication Date:

- OSTI Identifier:
- 22596683

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 57; Journal Issue: 6; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; INTEGRAL CALCULUS; LIE GROUPS; POLYNOMIALS; TENSORS

### Citation Formats

```
Urichuk, Andrew, E-mail: andrew.urichuk@uleth.ca, Walton, Mark A., E-mail: walton@uleth.ca, and International School for Advanced Studies.
```*Adjoint affine fusion and tadpoles*. United States: N. p., 2016.
Web. doi:10.1063/1.4954909.

```
Urichuk, Andrew, E-mail: andrew.urichuk@uleth.ca, Walton, Mark A., E-mail: walton@uleth.ca, & International School for Advanced Studies.
```*Adjoint affine fusion and tadpoles*. United States. doi:10.1063/1.4954909.

```
Urichuk, Andrew, E-mail: andrew.urichuk@uleth.ca, Walton, Mark A., E-mail: walton@uleth.ca, and International School for Advanced Studies. Wed .
"Adjoint affine fusion and tadpoles". United States.
doi:10.1063/1.4954909.
```

```
@article{osti_22596683,
```

title = {Adjoint affine fusion and tadpoles},

author = {Urichuk, Andrew, E-mail: andrew.urichuk@uleth.ca and Walton, Mark A., E-mail: walton@uleth.ca and International School for Advanced Studies},

abstractNote = {We study affine fusion with the adjoint representation. For simple Lie algebras, elementary and universal formulas determine the decomposition of a tensor product of an integrable highest-weight representation with the adjoint representation. Using the (refined) affine depth rule, we prove that equally striking results apply to adjoint affine fusion. For diagonal fusion, a coefficient equals the number of nonzero Dynkin labels of the relevant affine highest weight, minus 1. A nice lattice-polytope interpretation follows and allows the straightforward calculation of the genus-1 1-point adjoint Verlinde dimension, the adjoint affine fusion tadpole. Explicit formulas, (piecewise) polynomial in the level, are written for the adjoint tadpoles of all classical Lie algebras. We show that off-diagonal adjoint affine fusion is obtained from the corresponding tensor product by simply dropping non-dominant representations.},

doi = {10.1063/1.4954909},

journal = {Journal of Mathematical Physics},

number = 6,

volume = 57,

place = {United States},

year = {Wed Jun 15 00:00:00 EDT 2016},

month = {Wed Jun 15 00:00:00 EDT 2016}

}