My copy of the proceedings of the conference in honor of Graeme Segal’s 60th birthday finally arrived and I’ve been spending some enjoyable time reading parts of it. To me, the most interesting contributions were the ones by Ben-Zvi and Frenkel, Dijkgraaf, Moore, Stolz and Teichner, Teleman and Witten. Unfortunately, Dijkgraaf’s beautiful paper about how matrix integrals give you Gromov-Witten invariants of Calabi-Yau manifolds doesn’t seem to be available on-line. Neither is Witten’s very interesting paper, which is about explaining the SL(2,Z) symmetry seen in N=4 SSYM in four dimensions in terms of the existence of a six-dimensional superconformal theory.

The Stolz and Teichner paper is quite interesting . They are pursuing the idea that conformal field theories provide geometrical representatives of elliptic cohomology classes. Segal and Mike Hopkins worked on this a bit in the late 80s, with no conclusive results. Recently Hopkins has reformulated the whole elliptic cohomology story in terms of a new cohomology theory he calls “topological modular forms”. He gave a beautiful series of talks about this at the Segal Conference; this isn’t written up in the proceedings, but was for the 2002 ICM. For a more expository version of the ideas of Stolz and Teichner, see Teichner’s survey talk at a conference in Santa Barbara last summer.

Finally, the proceedings volume contains Segal’s wonderful unfinished manuscript “The Definition of Conformal Field Theory”, together with nine pages of very interesting comments about what he was trying to do then, what he would do differently now, and what had kept him from finishing the manuscript. The main problem seems to have been that he was unable by his methods to explicitly construct the “modular functor” that one should get out of WZW models, so for this reason the crucial chapter 11 on WZW models remains unwritten.

His comments begin with:

“The manuscript that follows was written fifteen years ago. On balance, though, conformal field theory has evolved less quickly than I expected, and to my mind the difficulties that kept me from finishing the paper are still not altogether elucidated.”

Re Witten’s paper:

Lie algebra of SO(3,3):

Let indices go 1..4 (spacetime) with Pm=Jm5, Qm=Jm6, and M=J56.

[Jmn, Jrs] = -i(gmr Jns – gms Jnr + gns Jmr – gnr Jms)

[Jmn, Pr] = -i(gmr Pn – gnr Pm)

[Jmn, Qr] = -i(gmr Qn – gnr Qm)

[Pm, Pn] = [Qm, Qn] = -iJmn

[Pm, Qn] = -[Qm, Pn] = -iM gmn

[Pm, M] = iQm

[Qm, M] = -iPm

[Jmn, M] = 0

Let Rm = Pm+iQm, Sm = Pm-iQm. Then Rm and Sm separately combined with Jmn, are Poincare algebras, implying finite translations (two are needed to get a real translation on spacetime). The appearence of algebraic translations is reminiscent of the appearence of translations as repeated supersymmetries.

Is this related to what Witten is talking about?

Note also that

[Rm, M] = Rm

[Sm, M] = -Sm

[Rm, Sn] = -2(M gmn + iJmn)

so the elementary translations are distinguished with respect to the “mass” M=J56 and they don’t commute. One can imagine this as a primitive distinction between matter and antimatter on the lowest level. (sorry in advance for sign errors).

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