nLab Loop Groups and Twisted K-Theory

This entry provides some hyperlinks for the articles

about the twisted ad-equivariant K-theory of compact Lie groups, and the positive energy representations of their loop groups (forming the Verlinde ring). Part II also deals with Dirac induction and the orbit method for producing representations by the geometric quantization of coadjoint orbits.


This identification enhanced to an equivalence of categories:

We identify the category of integrable lowest-weight representations of the loop group LGL G of a compact Lie group GG with the linear category of twisted, conjugation-equivariant curved Fredholm complexes on the group GG: namely, the twisted, equivariant matrix factorizations of a super-potential built from the loop rotation action on LGL G. This lifts the isomorphism of K-groups of [FHT1,2, 3] to an equivalence of categories. The construction uses families of Dirac operators.

Statement of the theorem

Every 𝔤\mathfrak{g}-valued differential form AΩ 1(S 1,𝔤)A \in \Omega^1(S^1, \mathfrak{g}) on the circle defines an element in the free loop group. Also it can be understood as a connection on a GG-principal bundle over the circle.

Given a Dirac operator DD on an associated spinor bundle over the circle, and using a loop group representation VV one can hence shift its defining connection by AA and form the associated Dirac operator D AD_A.

By remembering for each AA its holonomy hol(A)Ghol(A) \in G, this construction defines a GG-parameterized family of Fredholm operators over GG, one for each loop group representation.

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 τ\tau (Verlinde ring) and the τ\tau-twisted K-theory, equivariant with respect to Ad GAd_G, on GG.

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

construction of twisted K-theory classes from families of Dirac operators

  • puzzling appearance of Langlands dual group in relation to Lie group action on categories


GG a compact Lie group connected simply connected

TGT \subset G maximal torus, T T^\vee the dual torus, WW Weyl group

kH 4(BG,)k \in \mathbb{Z} \simeq H^4(B G, \mathbb{Z}) called the “level”

Σ\Sigma a closed oriented Riemann surface

K G(Σ)=FlatGBund(Σ)Hom(π 1Σ,G)/GK_G(\Sigma) = Flat G Bund(\Sigma) \simeq Hom(\pi_1 \Sigma, G)/G the moduli space of flat GG-principal bundles on Σ\Sigma. This inherits a complex structure from Σ\Sigma

𝒪(k)M G(Σ) \mathcal{O}(k) \to M_G(\Sigma)

the determinant line bundle for G=SU(n)G = SU(n) det kH (Σ,standardvectorbundle)det^{-k} H^\bullet(\Sigma, standard vector bundle)

Intense activity was generated in 1990s by the Verlinde formula suggested by CFT

d=dimH 0(M G(Σ),𝒪(k)) d = dim H^0 ( M_G(\Sigma), \mathcal{O}(k))

which is the partition function of a CFT level-kk line bundle

Digression recall that 2d TQFTs correspond to Frobenius algebras.

If AA is a semi-simple algebra P i\simeq \oplus \mathbb{C} P_i, where

θ(P i)=θ i ×\theta(P_i) = \theta_i \in \mathbb{C}^\times and Z(Σ)=θ i 1g(Σ)Z(\Sigma) = \sum \theta_i^{1-g(\Sigma)}

In our case, the Verlinde ring is a Frobenius ring / \mathbb{Z}. It is a quotient of the ring of representations R GR_G by the ideal of characters vanishing at the regular points F=ker(Tk+cT )F = ker(T \stackrel{k+c}{\to}T^\vee), where cc is the dual Coxeter number

the isogeny of kk being defined by the level

kH 4(BG,)H 4(BT,)=Sym 2π 1(T) k \in H^4(B G, \mathbb{Z}) \to H^4(B T, \mathbb{Z}) = Sym^2 \pi_1(T)^\vee

the projectors P fP_f range over fF reg/Wf \in F^{reg}/W (FF a Weyl orbit)

The traces θ(P f)=vol(c f)|F|=Δ(f) 2|F|\theta(P_f) = \frac{vol(c_f)}{|F|} = \frac{\Delta(f)^2}{|F|}

Where c Fc_F is order of conjugacy class of FF

Substantial work went into proving various casesy of the formula…

A distinction aroise between the moduli space topological space and the moduli stack of all algebraic G G_{\mathbb{C}}-bundles;

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 G G_{\mathbb{C}} bundles (modulo grade invariance)

The moduli stack of all holomorphic G G_{\mathbb{C}}-bundles

Has an atractive complex analytic presentation due to Graeme Segal

choose a disk ΔΣ\Delta \subset \Sigma with smooth parameterized boundary Δ\partial \Delta. Then the stack G(Σ)\mathcal{M}_G(\Sigma) is the double coset Hol(Δ,G )\LG /Hol(Σ\Δ,G ) Hol(\Delta, G_{\mathbb{C}})\backslash L G_{\mathbb{C}} / Hol(\Sigma \backslash \Delta, G_{\mathbb{C}})

where the holomorphic maps have smooth boundary values

The variety

X Σ\Δ=LG/Hol(Σ\Δ,G ) X_{\Sigma \backslash \Delta} = L G / Hol(\Sigma \backslash \Delta, G_{\mathbb{C}})

is a complex Kähler homogeneous space for LG L G_{\mathbb{C}} analogous in many ways to a flag variety of complex semsimple Lie groups

As a symplectic manifold it can be realized as

(flatGbundlesinΣ\Δ)/gaugeequivalencestrivialonΔ (flat G bundles in \Sigma \backslash \Delta) / {gauge equivalences trivial on \partial \Delta}

The symplectic structure here is independent of the complex structure omn Σ\Sigma

The LGL G action is projective Hamiltonian, being a central extension lift to the prequantum line bundle

So holomorphic and cohomological questions about the stack of all G G_{\mathbb{C}}-bundles on Σ\Sigma is equivalently

Hol(Δ,G)Hol(\Delta, G)-equivariant holomorphic and cohomological on X Σ\ΔX_{\Sigma \backslash \Delta}

Example: What is H 0(X Σ\Δ,𝒪(k))H^0(X_{\Sigma \backslash \Delta}, \mathcal{O}(k)) as an LGL G-representation? Turns out the multiplicity of a certain “vacuum” representation inside is equal to dimH 0(K G(Σ),𝒪(k))dim H^0(K_G(\Sigma), \mathcal{O}(k))

While complex analysis on such infinite-dimensional manifolds is still out of reach, there exists an algebraic model for X Σ\ΔX_{\Sigma \backslash \Delta} for which we can ask and answer analogous question

Example: H >0(X Σ\Δ ab,𝒪(k))=0H^{\gt 0}(X^{ab}_{\Sigma \backslash \Delta}, \mathcal{O}(k)) = 0 “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 𝒢\mathcal{G} acts on R%X%, RR takes values in

Rep(𝒢)=K 𝒢(point):K 𝒢(X)p *K 𝒢(point) Rep(\mathcal{G}) = K_{\mathcal{G}}(point) : K_{\mathcal{G}}(X) \stackrel{p_*}{\to} K_{\mathcal{G}}(point)

fiber integration in generalized cohomology theory

Here K LG(point)K_{L G}(point) is not “topological”, not got equivariant K-theory for non-compact groups!, so that map p *p_*

But it turns out that both problems have a simultaneous solution.

Instead consider K LG(𝒜 S 1)K_{L G}(\mathcal{A}_{S^1}) for the gauge action of LGL G inb tge space of flat GG-connections on the circle

Instead of projecting X Σ\ΔX_{\Sigma \backslash \Delta} to a point, use the flat connection model and restrict to the boundary Δ:X Σ\Δo𝒜 Δ\partial \Delta : X_{\Sigma \backslash \Delta} \o \mathcal{A}_{\partial \Delta}

This is a proper map!

The based loop group ΩG\Omega G acts freely and we can

which leads to a well-defined map

K G(G 2g)p *K G(G) K_G(G^{2 g}) \stackrel{p_*}{\to} K_G(G)


we wanted projective representations of LGL G with cocycle

kH LG 2(𝒪 ×)(H G 3(G,)) k \in H^2_{L G}(\mathcal{O}^\times)(\stackrel{\sim}{\to} H^3_G(G, \mathbb{Z}))

So we should map to the twisted K-theory group

K G k(G) K_G^{k}(G)

actually there is an extra shift by the dual Coxeter number cc (=n= n for SU(n)SU(n)) coming from the spinors K\sqrt K on X Σ\ΔX_{\Sigma \backslash \Delta}, and the key theorem is

Freed-Hopkins-Teleman theorem:

k+cK G dimG(G) {}^{k + c}K_G^{dim G}(G)

is a free abelian group generated by the positive energey irreps of LGL G at level kk.

generalization by Teleman and Woodward:

analytical index = topological index for X Σ\Δ algX_{\Sigma \backslash \Delta}^alg

for essentially all K-theory clases, at least after inverting k+ck+ c

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: nn-dimensional rep of SU(2)SU(2): sphere of radius nn in 𝔰𝔲(2) 3\mathfrak{su}(2) \simeq \mathbb{R}^3

for SO(3)SO(3), odd radii,

but one can describe a canonical correspondence without directs reference to the classification of irreps

Each coadjoint orbit has a symplectic form (see at orbit method)

ω 2(ad ξ *(λ),ad η *(λ))=λ|[ξ,η]] \omega_2(ad^*_\xi(\lambda), ad^*_\eta(\lambda)) = \langle \lambda | [\xi,\eta]]\rangle

For a vector bundle VV on the orbit O λO_\lambda of λ\lambda, which is a sum of line bundles with curvature ω λ\omega_\lambda and which carries a lifted GG-action.

The Dirac index DInd(O λ,V)D Ind(O_\lambda, V) is a representation of GG and this establishes the Kirillov correspondence

Remarks: For connected GG, VV carries no information beyond its rank

Wehn π 1(G)0\pi_1(G) \neq 0 the GG-action on VV 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 V λV_\lambda of GG

ω\omega highest weight λ\lambda invariant metric on 𝔤\mathfrak{g}

Use Kostant’s “cubic Dirac operator” on GG, which is the Dirac operator on GG for the metric connection with canonical torsion coming from H 3(G,)H^3(G, \mathbb{Z}) \simeq \mathbb{Z}.

Two key properties of this Dirac operator

[D,ψ(ξ)]=2T(ξ) [D, \psi(\xi)] = 2 T(\xi)


D 2=(λ+p) 2 D^2 = - (\lambda + p)^2

on V λS ±V_\lambda \otimes S^{\pm}

The Dirac family is the /2\mathbb{Z}/2-graded vector bundle with fiber VS ±V \otimes S^{\pm} over 𝔤\mathfrak{g} and odd operator

D ξ V:=D V+ψ(ξ) D_\xi^V := D^V + \psi(\xi)

Theorem (FHT): The kernel of D ξ VD^V_\xi is supported on the orbit of (λ+p)(\lambda + p) and equals (Kirillovlinebundle)(Spinorstonormalbundle)(Kirillov line bundle) \otimes (Spinors to normal bundle) In fact D ξ VD^V_\xi is a model for the Atiyah-Bott-Shapiro…

category: reference

Last revised on October 31, 2020 at 16:06:19. See the history of this page for a list of all contributions to it.