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Loop Groups and Twisted K-Theory

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• Dan Freed, Mike Hopkins, Constantin Teleman,

  Loop groups and twisted K-theory I, J. Topology, 4 (2011), 737-789 (arXiv:0711.1906)

  Loop Groups and Twisted K-Theory II (arXiv:0511232)

about loop groups and twisted K-theory

Statement of the theorem

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

Given a Dirac operator $D$ on an associated spinor bundle over the circle, and using a loop group representation $V$ one can hence shift its defining connection by $A$ and form the associated Dirac operator $D_A$.

By remembering for each $A$ its holonomy $\mathrm{hol}(A)\in G$, this construction defines a $G$-parameterized family of Fredholm operators over $G$, 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 $\tau$ (Verlinde ring) and the $\tau$-twisted K-theory, equivariant with respect to ${\mathrm{Ad}}_G$, on $G$.

This is (FHT, part II, theorem. 3.43).

Notes

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

• generalization from line bundles to general vector bundles

• For line bundles, relation between loop groups reps on twisted K-theory

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

start

$G$ a compact Lie group connected simply connected

$T\subset G$ maximal torus, $T^{\vee }$ the dual torus, $W$ Weyl group

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

$\Sigma$ a closed oriented Riemann surface

$K_G(\Sigma )=\mathrm{Flat}G\mathrm{Bund}(\Sigma )\simeq \mathrm{Hom}({\pi }_1\Sigma ,G)/G$ the moduli space of flat $G$-principal bundles on $\Sigma$. This inherits a complex structure from $\Sigma$

$$𝒪(k)\to M_G(\Sigma )$$\mathcal{O}(k) \to M_G(\Sigma)

the determinant line bundle for $G=\mathrm{SU}(n)$ ${\det }^{-k}{H}^{•}(\Sigma ,\mathrm{standard}\mathrm{vector}\mathrm{bundle})$

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

$$d=\dim H^0(M_G(\Sigma ),𝒪(k))$$d = dim H^0 ( M_G(\Sigma), \mathcal{O}(k))

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

Digression recall that 2d TQFTs correspond to Frobenius algebras.

If $A$ is a semi-simple algebra $\simeq \oplus ℂP_i$, where

$\theta (P_i)={\theta }_i\in {ℂ}^{\times }$ and $Z(\Sigma )=\sum {\theta }_i^{1-g(\Sigma )}$

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

the isogeny? of $k$ being defined by the level

$$k\in H^4(BG,\mathbb{Z})\to H^4(BT,\mathbb{Z})={\mathrm{Sym}}^2{\pi }_1(T{)}^{\vee }$$k \in H^4(B G, \mathbb{Z}) \to H^4(B T, \mathbb{Z}) = Sym^2 \pi_1(T)^\vee

the projectors $P_f$ range over $f\in F^{\mathrm{reg}}/W$ ($F$ a Weyl orbit)

The traces $\theta (P_f)=\frac{\mathrm{vol}(c_f)}{\mid F\mid }=\frac{\Delta (f{)}^2}{\mid F\mid }$

Where $c_F$ is order of conjugacy class of $F$

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 $G_{ℂ}$-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 $G_{ℂ}$ bundles (modulo grade invariance)

The moduli stack of all holomorphic $G_{ℂ}$-bundles

Has an atractive complex analytic presentation due to Graeme Segal

choose a disk $\Delta \subset \Sigma$ with smooth parameterized boundary $\partial \Delta$. Then the stack ${ℳ}_G(\Sigma )$ is the double coset $\mathrm{Hol}(\Delta ,G_{ℂ})\backslash L{G}_{ℂ}/\mathrm{Hol}(\Sigma \backslash \Delta ,G_{ℂ})$

where the holomorphic maps have smooth boundary values

The variety

$$X_{\Sigma \backslash \Delta }=LG/\mathrm{Hol}(\Sigma \backslash \Delta ,G_{ℂ})$$X_{\Sigma \backslash \Delta} = L G / Hol(\Sigma \backslash \Delta, G_{\mathbb{C}})

is a complex Kähler homogeneous space for $L{G}_{ℂ}$ analogous in many ways to a flag variety of complex semsimple Lie groups

As a symplectic manifold it can be realized as

$$(\mathrm{flat}G\mathrm{bundles}\mathrm{in}\Sigma \backslash \Delta )/\mathrm{gauge}\mathrm{equivalences}\mathrm{trivial}\mathrm{on}\partial \Delta$$(flat G bundles in \Sigma \backslash \Delta) / {gauge equivalences trivial on \partial \Delta}

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

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

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

$\mathrm{Hol}(\Delta ,G)$-equivariant holomorphic and cohomological on $X_{\Sigma \backslash \Delta }$

Example: What is $H^0(X_{\Sigma \backslash \Delta },𝒪(k))$ as an $LG$-representation? Turns out the multiplicity of a certain “vacuum” representation inside is equal to $\dim H^0(K_G(\Sigma ),𝒪(k))$

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

Example: $H^{>0}(X_{\Sigma \backslash \Delta }^{\mathrm{ab}},𝒪(k))=0$ “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 $𝒢$ acts on R%X%, RR takes values in

$$\mathrm{Rep}(𝒢)=K_{𝒢}(\mathrm{point}):K_{𝒢}(X)\stackrel{p_{*}}{\to }{K}_{𝒢}(\mathrm{point})$$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 $K_{LG}(\mathrm{point})$ is not “topological”, not got equivariant K-theory for non-compact groups!, so that map $p_{*}$

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

Instead consider $K_{LG}({𝒜}_{S^1})$ for the gauge action of $LG$ inb tge space of flat $G$-connections on the circle

Instead of projecting $X_{\Sigma \backslash \Delta }$ to a point, use the flat connection model and restrict to the boundary $\partial \Delta :X_{\Sigma \backslash \Delta }o{𝒜}_{\partial \Delta }$

This is a proper map!

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

which leads to a well-defined map

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

Amendments;

we wanted projective representations of $LG$ with cocycle

$$k\in H_{LG}^2({𝒪}^{\times })(\stackrel{\sim }{\to }{H}_G^3(G,\mathbb{Z}))$$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 $c$ ($=n$ for $\mathrm{SU}(n)$) coming from the spinors $\sqrt{K}$ on $X_{\Sigma \backslash \Delta }$, and the key theorem is

Freed-Hopkins-Teleman theorem:

$${}^{k+c}{K}_G^{\dim G}(G)$${}^{k + c}K_G^{dim G}(G)

is a free abelian group generated by the positive energey irreps of $LG$ at level $k$.

generalization by Teleman and Woodward:

analytical index = topological index for $X_{\Sigma \backslash \Delta }^{\mathrm{alg}}$

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

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: $n$-dimensional rep of $\mathrm{SU}(2)$: sphere of radius $n$ in $\mathrm{𝔰𝔲}(2)\simeq {ℝ}^3$

for $\mathrm{SO}(3)$, odd radii,

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

Each coadjoint orbit has a symplectic form

$${\omega }_2({\mathrm{ad}}_{\xi }^{*}(\lambda ),{\mathrm{ad}}_{\eta }^{*}(\lambda ))=⟨\lambda \mid [\xi ,\eta ]]⟩$$\omega_2(ad^*_\xi(\lambda), ad^*_\eta(\lambda)) = \langle \lambda | [\xi,\eta]]\rangle

For a vector bundle $V$ on the orbit $O_{\lambda }$ of $\lambda$, which is a sum of line bundles with curvature ${\omega }_{\lambda }$ and which carries a lifted $G$-action.

The Dirac index $D\mathrm{Ind}(O_{\lambda },V)$ is a representation of $G$ and this establishes the Kirillov correspondence

Remarks: For connected $G$, $V$ carries no information beyond its rank

Wehn ${\pi }_1(G)\ne 0$ the $G$-action on $V$ 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 $V_{\lambda }$ of $G$

$\omega$ highest weight $\lambda$ invariant metric on $𝔤$

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

Two key properties of this Dirac operator

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

(translation)

$$D^2=-(\lambda +p{)}^2$$D^2 = - (\lambda + p)^2

on $V_{\lambda }\otimes S^{\pm }$

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

$$D_{\xi }^V:=D^V+\psi (\xi )$$D_\xi^V := D^V + \psi(\xi)

Theorem (FHT): The kernel of $D_{\xi }^V$ is supported on the orbit of $(\lambda +p)$ and equals $(\mathrm{Kirillov}\mathrm{line}\mathrm{bundle})\otimes (\mathrm{Spinors}\mathrm{to}\mathrm{normal}\mathrm{bundle})$ In fact $D_{\xi }^V$ is a model for the Atiyah-Bott-Shapiro…

category: reference