Funció zeta de Witten

En matemàtiques, la funció zeta de Witten, introduïda per Witten (1991), és una funció associada a un sistema d'arrels que codifica els graus de les representacions irreductibles del corresponent grup de Lie.

És un cas especial de la funció zeta de Shintani.

Definició

La definició original de Witten de la funció zeta d'un grup semisimple de Lie era

\sum _{R}{\frac {1}{\dim(R)^{s}}}

on la suma és sobre classes d'equivalència de representacions irreductibles de R.

Si Δ de rang r és un sistema d'arrels amb n arrels positives en Δ⁺ i amb arrels simples λ_i_, la funció zeta de Witten de diverses variables està donada per

\zeta _{W}(s_{1},\dots ,s_{n})=\sum _{m_{1},\dots ,m_{r}>0}\prod _{\alpha \in \Delta ^{+}}{\frac {1}{(\alpha ^{\lor },m_{1}\lambda _{1}+\cdots +m_{r}\lambda _{r})^{s_{\alpha }}}},

La funció zeta original estudiada per Witten diferia lleugerament d'aquesta, en què tots els nombres s_α són iguals, i la funció és multiplicada per una constant.

Exemples

$\zeta _{SU(2)}(s)=\zeta (s)$ , la funció zeta de Riemann.
$\zeta _{SU(3)}(s)=\sum _{x=1}^{\infty }\sum _{y=1}^{\infty }{\frac {1}{(xy(x+y)/2)^{s}}}.$

Referències

Witten, Edward «On quantum gauge theories in two dimensions». Communications in Mathematical Physics, 141, 1, 1991, p. 153–209. DOI: 10.1007/bf02100009. ISSN: 0010-3616.
Zagier, Don. «Values of zeta functions and their applications». A: First European Congress of Mathematics, Vol. II (Paris, 1992). 120. Basel, Boston, Berlin: Birkhäuser, 1994, p. 497–512.

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