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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 22nd 2018

I have scratched the rough Idea-section that I had here previously, and started afresh, now with mentioning of the K-theoretic McKay correspondence. Also added references. But it’s still just a stub entry

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeSep 22nd 2018
• (edited Sep 22nd 2018)

added an Idea-subsection on the way that the equivalence between fractional D-branes stuck at orbifold singularities, and wrapped D-branes on the blow-up resolution (hence the K-theoretic incarnation of the McKay correspondence), is supposed to appear in the worldvolume gauge theory as passage between Higgs branch and Coulomb branch. Now here.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeSep 28th 2018

added an Idea-section making more explicit the central special case At global linear orbifold singularities

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeSep 29th 2018

Is Langlands duality lurking nearby?

Concerning finite groups, David Ben-Zvi once told us:

the finite groups of Lie type - eg matrix groups GL,SL,SO,Sp etc over finite fields.. this class of groups contains almost all finite simple groups (other than cyclic, where the same holds since it’s easy for abelian groups, symmetric groups where analogous stories hold – the “F_1 case” – and some of the exceptional groups). Lusztig classified the reps of this enormous class of finite groups (that certainly should have earned him a Fields medal!), and they’re classified by conjugacy classes in the dual groups. I won’t give the precise statement but I think this is not a particularly misleading simplification.. it’s a pretty amazing piece of math!

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeSep 29th 2018

Dunno. But the self-duality of the representations in the image of $\beta \;\colon\; A(G) \to R(G)$ is what gives rise to the Hecke operations.

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeSep 29th 2018

’Hecke operator’ appears twice on the nLab. Are these the operators mentioned towards the end of Hecke algebra?

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeSep 29th 2018

Yes! You see the self-duality of permutation representations used in the second step of the chain of equivalences shown there.

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeSep 29th 2018

Is it worth us importing some of John Baez’s account of Hecke algebras/operators from the groupoidification project? It seems difficult to find exposition at that straightforward level:

we prefer the case where $G$ is finite, to avoid technicalities — and also because it’s good to consider a pair of subgroups $H,K \subseteq G$ and build intertwining operators $C[G/H] \to [G/K]$ from double cosets in $H\G/K$ and to groupoidify this construction.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeOct 11th 2018
• (edited Oct 11th 2018)

made explicit the proportionality of the RR-charge in the $g$-twisted sector with the value of the corresponding character at $g$, and added a list of references for that expression (here)

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeOct 11th 2018
• (edited Oct 11th 2018)

So does that tell us anything (physical) that for two fractional D-branes corresponding to different irreps, the ’inner product’ of the charges is zero: $\sum_g Q_V(g) \cdot Q_{V'}(g) = 0$?

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeOct 11th 2018

There may be a better answer than the following, but from equation (4.70) to (4.71) in

• M. Billó, B. Craps, F. Roose, Orbifold boundary states from Cardy’s condition, JHEP 0101:038, 2001 (arXiv:hep-th/0011060)

the authors point out that the orthogonality relation of the characters is equivalently the consistency condition that – after passing through the K-theoretic McKay correspondence from the wrapped D-branes on the resolution of the orbifold singularity to fractional branes at the actual orbifold singularity – the orbifold RR-fields in different twisted sectors are orthogonal to each other, as it must be (since closed strings cannot change their twisted sector).

• CommentRowNumber12.
• CommentAuthorDavid_Corfield
• CommentTimeOct 11th 2018

Interesting. I like these physical interpretations of long-known mathematical facts.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeOct 20th 2018
• (edited Oct 20th 2018)

I have added pointer (here and in related entries) that around (137) of

it is argued that the full equivariant K-theory group $K_G(\ast)$ is in fact too large for the classification of physical D-branes, that only elements determined by low degree Chern classes should be kept.

Which brings me back to the, meanwhile, widely appreciated but still unanswered question (here) how to compute the Chern classes of the bundle corresponding to a representation in terms of its characters.