Witten zeta function
In mathematics, the Witten zeta function, introduced by Witten (1991), is a function associated to a root system that encodes the degrees of the irreducible representations of the corresponding Lie group. It is a special case of the Shintani zeta function.
Definition
Witten's original definition of the zeta function of a semisimple Lie group was
where the sum is over equivalence classes of irreducible representations R.
If Δ of rank r is a root system with n positive roots in Δ+ and with simple roots λi, the Witten zeta function of several variables is given by
The original zeta function studied by Witten differed from this slightly, in that all the numbers sα are equal, and the function is multiplied by a constant.
References
- Witten, Edward (1991), "On quantum gauge theories in two dimensions", Communications in Mathematical Physics 141 (1): 153–209, doi:10.1007/bf02100009, ISSN 0010-3616, MR 1133264, Zbl 0762.53063
- Zagier, Don (1994), "Values of zeta functions and their applications", First European Congress of Mathematics, Vol. II (Paris, 1992), Progr. Math. 120, Basel, Boston, Berlin: Birkhäuser, pp. 497–512, MR 1341859, Zbl 0822.11001
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