In mathematics, the Witten zeta function, is a function associated to a root system that encodes the degrees of the irreducible representations of the corresponding Lie group. These zeta functions were introduced by Don Zagier who named them after Edward Witten's study of their special values (among other things).[1][2] Note that in,[2] Witten zeta functions do not appear as explicit objects in their own right.

Definition

edit

If   is a compact semisimple Lie group, the associated Witten zeta function is (the meromorphic continuation of) the series

 

where the sum is over equivalence classes of irreducible representations of  .

In the case where   is connected and simply connected, the correspondence between representations of   and of its Lie algebra, together with the Weyl dimension formula, implies that   can be written as

 

where   denotes the set of positive roots,   is a set of simple roots and   is the rank.

Examples

edit
  •  , the Riemann zeta function.
  •  

Abscissa of convergence

edit

If   is simple and simply connected, the abscissa of convergence of   is  , where   is the rank and  . This is a theorem due to Alex Lubotzky and Michael Larsen.[3] A new proof is given by Jokke Häsä and Alexander Stasinski [4] which yields a more general result, namely it gives an explicit value (in terms of simple combinatorics) of the abscissa of convergence of any "Mellin zeta function" of the form

 

where   is a product of linear polynomials with non-negative real coefficients.

Singularities and values of the Witten zeta function associated to SU(3)

edit

  is absolutely convergent in  , and it can be extended meromorphicaly in  . Its singularities are in   and all of those singularities are simple poles.[5] In particular, the values of   are well defined at all integers, and have been computed by Kazuhiro Onodera.[6]

At  , we have   and  

Let   be a positive integer. We have

 

If a is odd, then   has a simple zero at   and

 

If a is even, then   has a zero of order   at   and

 

References

edit
  1. ^ Zagier, Don (1994), "Values of Zeta Functions and Their Applications", First European Congress of Mathematics Paris, July 6–10, 1992, Birkhäuser Basel, pp. 497–512, doi:10.1007/978-3-0348-9112-7_23, ISBN 9783034899123
  2. ^ a b Witten, Edward (October 1991). "On quantum gauge theories in two dimensions". Communications in Mathematical Physics. 141 (1): 153–209. doi:10.1007/bf02100009. ISSN 0010-3616. S2CID 121994550.
  3. ^ Larsen, Michael; Lubotzky, Alexander (2008). "Representation growth of linear groups". Journal of the European Mathematical Society. 10 (2): 351–390. arXiv:math/0607369. doi:10.4171/JEMS/113. ISSN 1435-9855. S2CID 9322647.
  4. ^ Häsä, Jokke; Stasinski, Alexander (2019). "Representation growth of compact linear groups". Transactions of the American Mathematical Society. 372 (2): 925–980. arXiv:1710.09112. doi:10.1090/tran/7618.
  5. ^ Romik, Dan (2017). "On the number of $n$-dimensional representations of $\operatorname{SU}(3)$, the Bernoulli numbers, and the Witten zeta function". Acta Arithmetica. 180 (2): 111–159. doi:10.4064/aa8455-3-2017. ISSN 0065-1036.
  6. ^ Onodera, Kazuhiro (2014). "A functional relation for Tornheim's double zeta functions". Acta Arithmetica. 162 (4): 337–354. arXiv:1211.1480. doi:10.4064/aa162-4-2. ISSN 0065-1036. S2CID 119636956.