Witten tutorial addendum

PiTP 2006 7-26-06Addendum To Tutorial Of E. WittenI want to explain a few points that I didn’t have time for at the end of the tutorial.(But see also the paper hep-th/9505186 for more.) I am writing this note because it wouldtake too much time to explain all these details in the next lecture. I am probably goingto go into more detail here than most students would flnd relevant, but having gotten thisfar I would like to tidy up a few loose ends.In computing the partition function of free electrodynamics, the lattice sum comesout to beX1 1(¡(x;x)+(x;?x) (x;x)+(x;?x)4 4q q„x2⁄2The lattice ⁄ is H (M;Z) modulo torsion, that is, it is the second cohomology group ofthe four-manifold M, modulo torsion. This type of lattice sum, as I mentioned, also arisesas the partition function in genus 1 of a toroidally compactifled string theory. (As such itiscalledthe Narainthetafunction; in mathematics, it isattributedto C.L.Siegel.) Inthe2sum, q = exp(2…i¿) where ¿ = µ=2…+4…i=g . Also (x;x) is the intersection pairing, i.e.R(x;x)= x[x, where[ is the cup product (if you think of x as a difierential form thenMyoucanusethewedgeproduct). Finally,tocompute(x;?x),werepresentxbyaharmonictwo-form of the right periods (which I called F (x)=2… in the lecture), and apply to it the0RHodge ? operator (the duality operator, for physicists), and then (x;?x) = x^?x. IMwrote this formula using the wedge product, since here it is most natural to think in termsof difierential forms.In ...