This entry provides some hyperlinks for the articles
Twisted K-theory and loop group representations (arXiv:math/0312155)
Loop groups and twisted K-theory I, J. Topology, 4 (2011), 737-789 (arXiv:0711.1906)
Loop Groups and Twisted K-Theory II, J. Amer. Math. Soc. 26 (2013), 595-644 (arXiv:0511232)
about the twisted equivariant K-theory of suitable Lie groups, and the representation theory of their loop groups. Part II also deals with Dirac induction and the orbit method for producting representations by the geometric quantization of coadjoint orbits.
This construction is (FHT, part II, cor. 3.39).
The FHT-theorem asserts that this construction gives an equivalence of the category of loop group representations at some level (Verlinde ring) and the -twisted K-theory, equivariant with respect to , on .
This is (FHT, part II, theorem. 3.43).
Some raw notes from a Teleman talk, to be polished
Group actions on categories and Langlands duality
I. Index formula on moduli of G_bundles and TQFT
revolves around the Verlinde formula
formulation in twisted K-theory
For line bundles, relation between loop groups reps on twisted K-theory
construction of twisted K-theory classes from families of Dirac operators
called the “level”
the determinant line bundle for
If is a semi-simple algebra , where
the isogeny of being defined by the level
the projectors range over ( a Weyl orbit)
Where is order of conjugacy class of
Substantial work went into proving various casesy of the formula
turned out to be immaterial for holomorphic sections (and higher cohomology of line bundles, which vanishes in both cases)
But for more general vector bundles it became cleat that the Verline-style formulas apply to the moduli stack and not the space
Recall that Narasimhan-Sechadri identity the moduli space of semi-stable algebraic bundles (modulo grade invariance)
The moduli stack of all holomorphic -bundles
Has an atractive complex analytic presentation due to Graeme Segal
choose a disk with smooth parameterized boundary . Then the stack is the double coset
where the holomorphic maps have smooth boundary values
is a complex Kähler homogeneous space for analogous in many ways to a flag variety of complex semsimple Lie groups
As a symplectic manifold it can be realized as
The symplectic structure here is independent of the complex structure omn
The action is projective Hamiltonian, being a central extension lift to the prequantum line bundle
So holomorphic and cohomological questions about the stack of all -bundles on is equivalently
-equivariant holomorphic and cohomological on
Example: What is as an -representation? Turns out the multiplicity of a certain “vacuum” representation inside is equal to
While complex analysis on such infinite-dimensional manifolds is still out of reach, there exists an algebraic model for for which we can ask and answer analogous question
Example: “Kodeira vanishing”
So we have an analytic index theorem for these varieties but without a topological side (this was “paradoxical”, because usually the topological side is easier.)
But this requires finding a receptable fot the topological index (Riemann-Roch theorem)
Whet acts on R%X%, RR takes values in
Here is not “topological”, not got equivariant K-theory for non-compact groups!, so that map
But it turns out that both problems have a simultaneous solution.
Instead consider for the gauge action of inb tge space of flat -connections on the circle
Instead of projecting to a point, use the flat connection model and restrict to the boundary
This is a proper map!
The based loop group acts freely and we can
which leads to a well-defined map
we wanted projective representations of with cocycle
So we should map to the twisted K-theory group
actually there is an extra shift by the dual Coxeter number ( for ) coming from the spinors on , and the key theorem is
is a free abelian group generated by the positive energey irreps of at level .
generalization by Teleman and Woodward:
analytical index = topological index for
for essentially all K-theory clases, at least after inverting
II. From loop group representations to K-theory
Preparation: Compact groups:
Recall the Kirillov correspondence between
irreducilble representations and co-adjoint orbits (+ line bundles)
In the connected case it can be summarized by saying that both of those correspond to the set of dominant integral regular wheights.
Example: -dimensional rep of : sphere of radius in
for , odd radii,
but one can describe a canonical correspondence without directs reference to the classification of irreps
For a vector bundle on the orbit of , which is a sum of line bundles with curvature and which carries a lifted -action.
The Dirac index is a representation of and this establishes the Kirillov correspondence
Remarks: For connected , carries no information beyond its rank
Wehn the -action on should be projective and cancel the spinor projective cocycle
The Dirac family consruction (Freed, Hopkins, Teleman) provides an inverse to this, assigning an orbit,
Input: irreducible representation of
highest weight invariant metric on
Uee Kostant’s “cubic Dirac operator” on , which is the Dirac operator on for the metric connection with canonical torsion coming from .
Two key properties of this Dirac operator
The Dirac family is the -graded vector bundle with fiber over and odd operator
Theorem (FHT): The kernel of is supported on the orbit of and equals In fact is a model for the Atiyah-Bott-Shapiro