- #26

- 986

- 174

They exist only for even dimensions, and for D dimensions, they have (D/2-1) vertices. Thus, in 4D, they are triangular, and in 10D, they are hexagonal.

For gauge field i operating on the fermion rep, the gauge operator is L

_{i}. For each chirality c, one has to calculate

L

_{c,ijk}= Tr(L

_{i}.L

_{j}.L

_{k})

_{symmetric}

for gauge fields i,j,k in 4D, and likewise for other numbers of dimensions. The overall result is

L

_{ijk}= L

_{L,ijk}- L

_{R,ijk}

and it must vanish for the anomaly to disappear. That's a constraint on the Standard Model and extensions of it like Grand Unified Theories.

I have been valiantly searching for general formulas for the likes of Tr(L

_{i}.L

_{j}.L

_{k})

_{symmetric}, without success. However, it's easy to calculate for an algebra's Cartan subalgebra, and I'm able to do that with my Lie-algebra code.

One can construct scalar invariants from a Lie algebra, starting with the commutator formula, [L

_{i},L

_{j}] = f

_{ij}

^{k}L

_{k}

One first gets a metric, g

_{ij}= f

_{ia}

^{b}f

_{jb}

^{a}, and if the metric can be inverted, then the algebra is semisimple. So one can find "Casimir invariants", with the quadratic one given by

C = g

^{ab}L

_{a}L

_{b}

To extend into higher powers, one takes F

_{i}

^{j}= f

_{ia}

^{j}g

^{ab}L

_{b}and finds

C(p) = Tr(F

^{p})

I've found it hard to find general formulas for those, though I've found some in Francesco Iachello's book

*Lie Algebras and Applications*and A. M. Perelomov, V. S. Popov, “Casimir operators for semisimple Lie groups”,

*Izv. Akad. Nauk SSSR Ser. Mat.*, 32:6 (1968), 1368–1390. They are for A(n), B(n), C(n), D(n), and G2, with possible extension to E6 and E7, leaving F4 and E8 remaining. I've verified them in the quadratic case, though I've had difficulty doing so for higher powers.

C(1) = 0, C(2) is proportional to the earlier C, and C(p) is a degree-p polynomial in the highest weights. For rank n, there are n independent ones. Here are the lowest ones that form independent sets:

A(n): 2, 3, ..., n+1

B(n), C(n), D(n): 2, 4, ..., 2n

G2: 2, 6

F4: 2, 6, 8, 12

E6: 2, 5, 6, 8, 9, 12

E7: 2, 6, 8, 10, 12, 14, 18

E8: 2, 8, 12, 14, 18, 20, 24, 30

D(n) has the complication that one can form a degree-n polynomial in the highest weights from the independent C(p)'s, a sort of C'(n) that can substitute for C(2n).