nLab extended geometric quantization of 2d Chern-Simons theory

under construction, notes related to material that appears in detail elsewhere, see the References.

Local 2d prequantum boundary field theory and KK-Quantization

1) Local 2d prequantum field theory

ambient (∞,1)-topos $\mathbf{H} =$ Smooth∞Grpd.

We use in there mostly just the full sub-(∞,1)-category

$$DiffStack \hookrightarrow SmoothGrpd \hookrightarrow Smooth \infty Grpd$$

of differentiable stacks, i.e. the full subcategory on object that are presented by Lie groupoids but as coefficients at least we also need smooth 2-groupoids.

In particular, write $\mathbf{B}^2 U(1)$ for the smooth 2-groupoid which is the circle 3-group.

For $\mathbf{Fields} \in \mathbf{H}$ a moduli stack of fields, we say that a map

$$\array{ \mathbf{Fields} \\ \downarrow^{\mathrlap{exp(i S)}} \\ \mathbf{B}^2 U(1) }$$

is a local action functional.

(More generally we consider smooth super ∞-groupoids and replace $\mathbf{B}^2 U(1)$ by the moduli stack of super 2-line bundles.)

Since this is necessarily a dualizable objects in the (∞,2)-category of correspondences $Corr_2(\mathbf{H}, \mathbf{B}^2 U(1))$ it induces by the cobordism theorem a local prequantum field theory given by a monoidal (∞,2)-functor

$$\array{ && Corr_2(\mathbf{H},\mathbf{B}^2 U(1)) &\to& Corr_2(\mathbf{H}, KU Mod) \\ & {}^{\mathllap{\exp(i S)}}\nearrow& \downarrow \\ Bord_n &\stackrel{\mathbf{Fields}}{\to}& Corr_2(\mathbf{H}) }$$

Here we postcomposed with geometric realization

$$\mathbf{B}^2 U(1) \to {\vert \mathbf{B}^2 U(1)\vert} \simeq K(\mathbb{Z},3) \to B gl_1(KU) \to KU Mod \,.$$

2) Local boundary 2d prequantum field theory

Considering one marked boundary condition we have boundary field theory defined in addition to the data of $\exp(i S) \colon \mathbf{Fields} \to KU Mod$ by a morphisms in $Corr_2(\mathbf{H}, KU Mod)$ which as a diagram in $\mathbf{H}$ is of the form

$$\array{ && Q \\ & \swarrow && \searrow^{\mathrlap{i}} \\ \ast && \swArrow_{\mathrlap{\xi}} && \mathbf{Fields} \\ & {}_{\mathllap{\mathbb{I}}}\searrow && \swarrow_{\mathrlap{\exp(i S)}} } \,.$$

This is the boundary condition.

3) Quantization in KK-Theory

We quantize by interpreting the path integral as push-forward in generalized cohomology in complex topological K-theory $KU$.

Specifically we quantize a correspondence as above by applying the following procedure

1. Decompose the correspondence explicitly as

   $$\array{ && Q \\ & \swarrow && \searrow^{\mathrlap{i}} \\ \ast &\swArrow_{\mathrlap{\xi}}& \downarrow^{\mathrlap{i^\ast \chi}} && \mathbf{Fields} \\ & {}_{\mathllap{\mathbb{I}}}\searrow && \swarrow_{\mathrlap{\chi}} \\ && KU Mod } \,.$$

2. Form twisted groupoid convolution algebras to produce a co-span of Hilbert bimodules

   $$\mathbb{C} \stackrel{\xi}{\to} C_{i^\ast \chi}(Q) \stackrel{i^\ast}{\leftarrow} C_\chi(X) \,.$$

   Regard this as a cospan in KK-theory.

3. Assume that the middle and right objects are dualizable objects. Choose a map in KK-theory

   $$Th(Q) \colon \left(C_{i^\ast \chi}\left(Q\right)\right) \to \left(C_{i^\ast \chi}\left(Q\right)\right)^{\vee}$$

   to the dual C*-algebra. (If this is an KK-equivalence then this is the Thom isomorphism in this context.)

4. Consider the dual morphism to $i^\ast$, to be denoted

   $$i_! \;\colon\; \left(C_{i^\ast}\left(Q\right)\right)^{\vee} \to \left(C_\chi\left(X\right)\right)^\vee$$

   and then turn the above cospan into the following consecutive composite in KK-theory

   $$i_! Th\left(Q_{i^\ast \chi}\right) \xi \colon \mathbb{C} \stackrel{\xi}{\to} C_{i^\ast \chi}(Q) \stackrel{Th}{\to} (C_{i^\ast \chi}(Q))^\vee \stackrel{i_!}{\to} (C_{\chi}(X))^\vee \,.$$

   This we regard as the quantization of the boundary correspondence.

(…)

2d Poisson Chern-Simons theory – Idea

General

We discuss aspects of the extended geometric quantization of one of the simplest and yet interesting examples of ∞-Chern-Simons theory, namely 2d Chern-Simons theory and here specifically the case induced by a binary and non-degenerate invariant polynomial, namely the Lie integrated version of the Poisson sigma-model.

It turns out that several aspects of the extended geometric quantization of this 2d theory are known in the literature already, albeit in disguise: the 2-plectic moduli stack of fields is equivalently what in the literature is known as a symplectic groupoid and its higher geometric quantization is essentially what is known as the geometric quantization of symplectic groupoids.

The suggestion that there should be such a relation is contained already in (Cattaneo-Felder 01), but at the time of that writing the geometric quantization of symplectic groupoids had not been performed yet.

To some extent in the following we review the traditional geometric quantization of symplectic groupoids while showing at the same time how its ingredients are (more) naturally interpreted in higher symplectic geometry. This makes quantization of symplectic groupoids a good test case against which to check notions of higher geometric quantization.

Specifically, we discuss what should be the first nontrivial case of the Chern-Simons-type holographic principle realized in higher geometric quantization: The moduli stack of the 2d Chern-Simons theory is the Lie integration of the Poisson Lie algebroid associated to a Poisson manifold. The latter we may think of as defining a quantum mechanical system, hence a $(2-1) = 1$-dimensional quantum field theory. The higher geometric quantization of the 2-d theory yields a 2-vector space of quantum 2-states (assigned to the point n codimension 2). Under the identification of 2-vector spaces with categories of modules over an associative algebra, this space of quantum 2-states identifies (the Morita equivalence-class of) an algebra. Suitably re-interpreting traditional results about the quantization of symplectic groupods shows that this algebra serves as a kind of strict deformation quantization of that Poisson manifold [Hawkins (2008)].

Notice that at the level of just infinitesimal (“formal”) deformation quantization a similar holographic relation between quantization of a Poisson manifold and of its associated 2d sigma-model QFT has famously been shown by Alberto Cattaneo and Giovanni Felder to underly Kontsevich‘s construction of deformation quantization (see at Poisson sigma-model). We suggest that the discussion here provides the refinement of this relation to strict deformation quantization.

All of this should be a blueprint for an analogous situaton in one dimension higher, where the analogous procedure should reproduce the famous holographic quantization of the 2d Wess-Zumino-Witten theory in terms of that of a 3d Chern-Simons theory.

Identifying “symplectic groupoids” as 2-plectic groupoids

We briefly indicate the basis for re-interpreting traditional symplectic groupoid-theory in terms of higher symplectic geometry.

The identification of the traditional notion of “symplectic groupoid” as really a 2-plectic structure is evident as soon as one translates the traditional definition of a symplectic groupoid to more intrinsic language of higher differential geometry. We discuss this in detail below, but in brief it works as follows:

A symplectic groupoid $(\mathbf{X},\omega)$ is traditionally defined to be a Lie groupoid $\mathbf{X}$ which is equipped with a (non-degenerate) differential 2-form $\omega \in \Omega^2(\mathbf{X}_1)$ on its smooth manifold of morphisms, such that it is annihilated by the de Rham differential as well as by the operator $\delta$ that sums the pullback of $\omega$ to the space $\mathbf{X}_2$ of composable morphisms along the source and target maps and minus that along the composition map. But together this just means that regarded as a triple $(0, \omega, 0) \in \underset{k = 0,1,2}{\oplus} \Omega^{3-k}(\mathbf{X}_k)$ the symplectic form is a cocycle in the simplicial de Rham complex over the simplicial manifold which is the nerve of the Lie groupoid. And this finally means fully intrinsically that $\omega$ is a degree-3 cocycle in the intrinsic de Rham cohomology of smooth ∞-groupoids over $\mathbf{X}$, which we denote by

$$\omega \;\colon\; \mathbf{X} \to \flat_{dR} \mathbf{B}^3 U(1) \,.$$

This simple observation shows that symplectic groupoids, which in traditional literature are treated as a topic in symplectic geometry, are really objects in higher symplectic geometry, namely in 2-plectic geometry.

More specifically, if $(X,\pi)$ is a Poisson manifold, there is canonically associated with it a Lie algebroid $\mathfrak{P}$, the Poisson Lie algebroid. This is an example of a symplectic Lie n-algebroid and as such it carries a canonical binary invariant polynomial. Together this serves as the target space and background gauge field of what is called the Poisson sigma-model. But moreover, the Lie integration of this data is the extended Lagrangian of an ∞-Chern-Simons theory, namely a map

$$\mathbf{L} \;\colon\; \tau_1\exp(\mathfrak{P})_{conn} \to \mathbf{B}^2 (\mathbb{R}/\Gamma)_{conn}$$

from the moduli stack of fields of the 2d Chern-Simons theory to that of circle 2-bundles with connection.

If we concretify? the moduli stack by forgetting the connection data here and just consider the underlying instanton sectors, then this is precisely the symplectic groupoid data above

$$\array{ \tau_1\exp(\mathfrak{P})_{conn} &\stackrel{\mathbf{L}}{\to}& \mathbf{B}^2 (\mathbb{R}/\Gamma)_{conn} \\ {}^{\mathllap{concretify}}\downarrow && \downarrow \\ \mathbf{X} &\stackrel{\omega}{\to}& \flat_{dR} \mathbf{B}^3 U(1) } \,.$$

It is in this way that we may identify the symplectic groupoid of a Poisson manifold

$$\mathbf{X} \simeq \tau_1 \exp(\mathfrak{P})$$

as the moduli stack of the extended prequantum 2d Chern-Simons theory which refines the Poisson sigma-model.

Hence we identify the geometric quantization of $(\mathbf{X}, \omega)$ as the extended geometric quantization of the 2d Chern-Simons theory. But traditional literature shows (and this was motivation for introducing symplectic groupoids in the first place) that this encodes the quantization of the Poisson manifold $(X, \pi)$ itself, regarded as a quantum mechanical system. This is the traditional topic of geometric quantization of symplectic groupoids.

Taken together this says: the extended geometric quantization of the extended prequantum Poisson sigma-model computes the ordinary geometric quantization of the underlying Poisson manifold.

This statement we recognize as a geometric quantization analog of a famous relation in algebraic deformation quantization. As discussed there, the construction by Kontsevich of the algebraic deformation quantization of any Poisson manifold was identified by Cattaneo and Felder as a limiting case of the 3-point function in the perturbative quantization of the corresponding 2d Poisson $\sigma$-model.

These relations to traditional theory we use in the following to explore aspects of the extended geometric quantization of 2d Chern-Simons theory.

Summary and survey

∞-Chern-Simons theory from binary and non-degenerate invariant polynomial

$n \in \mathbb{N}$	symplectic Lie n-algebroid	Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $(n+1)$-d sigma-model	higher symplectic geometry	$(n+1)$d sigma-model	dg-Lagrangian submanifold/ real polarization leaf	= brane	(n+1)-module of quantum states in codimension $(n+1)$	discussed in:
0	symplectic manifold	symplectic manifold	symplectic geometry		Lagrangian submanifold	–	ordinary space of states (in geometric quantization)	geometric quantization
1	Poisson Lie algebroid	symplectic groupoid	2-plectic geometry	Poisson sigma-model	coisotropic submanifold (of underlying Poisson manifold)	brane of Poisson sigma-model	2-module = category of modules over strict deformation quantiized algebra of observables	extended geometric quantization of 2d Chern-Simons theory
2	Courant Lie 2-algebroid	symplectic 2-groupoid	3-plectic geometry	Courant sigma-model	Dirac structure	D-brane in type II geometry
$n$	symplectic Lie n-algebroid	symplectic n-groupoid	(n+1)-plectic geometry	$d = n+1$ AKSZ sigma-model

(adapted from Ševera 00)

General theory

We use the general tools for formulating notions in physics in terms of higher differential geometry as discussed at geometry of physics. Here we just collect some of the main ingredients needed below.

Definition

We write

$$DL \;\colon\; [CartSp^{op}, Ch_\bullet(Ab)] \stackrel{\simeq}{\to} [CartSp^{op}, Ab^{\Delta^{op}}] \stackrel{forget}{\to} [CartSp^{op}, KanCplx] \stackrel{}{\to} L_{lhe} [CartSp^{op}, KanCplx] \simeq Smooth\infty Grpd$$

where the first equivalence is the Dold-Kan correspondence and the last map is the map to the simplicial localization.

Definition

For $k,n \in \mathbb{N}$, $k \leq n$ we write

$$\mathbf{B}^n U(1)_{conn^k} \coloneqq DK\left[ \underline{U}(1) \stackrel{\mathbf{d}}{\to} \Omega^1 \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \Omega^k \stackrel{}{\to} 0 \stackrel{}{\to} \cdots \stackrel{}{\to} 0 \right]$$

with $\Omega^k$ in degree $(n-k)$ (hence with $(n-k)$ vanishing entries on the right).

In particular

• $\mathbf{B}^n U(1) \simeq \mathbf{B}^n U(1)_{conn^0}$

• $\mathbf{B}^n U(1)_{conn} \simeq \mathbf{B}^n U(1)_{conn^n}$.

Example

For $n,k \in \mathbb{N}$, $k \lt n$ we have

1. $\Omega (\mathbf{B} U(1)_{conn}) \simeq \flat \mathbf{B}^{n-1}$

2. $\Omega (\mathbf{B}^n U(1)_{conn}^k) \simeq \mathbf{B}^{n-1} U(1)_{conn}^k$.

(…)

$$\array{ field space && \tau_n \exp(\mathfrak{a})_{conn} &\to& \mathbf{B}^n U(1)_{conn} \\ && \downarrow && \downarrow \\ delooped phase space && \tau_n \exp(\mathfrak{a}) &\to& \mathbf{B}^n U(1)_{conn^{n-1}} }$$

$$\array{ C_0 &\to& * \\ \downarrow && \downarrow \\ \tau_n \exp(\mathfrak{a}) &\to& \mathbf{B}^n U(1)_{conn^{n-1}} \\ \uparrow && \uparrow \\ C_1 &\to& * }$$

under homotopy fiber product this yields the phase space prequantum bundle

$$Phase(C_0, C_1) \to \mathbf{B}^{n-1}U(1)_{conn} \,.$$

(…)

The setup

Let $(X, \pi)$ be a Poisson manifold, which we tend to think of as defining a quantum mechanical system. This canonically induces the structure of Lie algebroid over $X$, the Poisson Lie algebroid $(\mathfrak{P}, \mathbf{\omega})$ over $X$. This is a symplectic Lie algebroid, with graded symplectic form (binary invariant polynomial) $\mathbf{\omega}$ which is in transgression with the Poisson tensor $\pi$, regarded as a Lie algebroid cocycle. The transgression is witnessed by a Chern-Simons element $cs_\pi$.

Moduli stack of fields

By the general construction of infinity-Chern-Simons theory this means that there is a universal differential characteristic map

$$\mathbf{L} \;\colon\; \exp(\mathfrak{P})_{conn} \to \mathbf{B}^2 \mathbb{R}_{conn}$$

and its truncation (Lie integration)

$$\mathbf{L} \;\colon\; \tau_1 \exp(\mathfrak{P})_{conn} \to \mathbf{B}^2 (\mathbb{R}/\Gamma)_{conn} \,.$$

Here $\tau_1 \exp(\mathfrak{P})_{conn}$ is the smooth moduli stack of fields of the 2d Poisson-Chern-Simons theory and $\mathbf{H}$ is the extended Lagrangian of a 2d Chern-Simons theory (FRS). Infinitesimally this yields the Poisson sigma-model.

We discuss here the higher geometric quantization of this theory defined by $\mathbf{L}$.

The Phase space stack

There is the canonical universal forgetful map

$$\tau_1 \exp(\mathfrak{P})_{conn} \to \tau_1 \exp(\mathfrak{P})$$

which forgets the conection data of fields and only retains their “instanton sector”. The smooth groupoid $\tau_1 \exp(\mathfrak{P})$ is the symplectic groupoid which is the Lie integration of the Poisson Lie algebroid $\mathfrak{P}$. (In the traditional literature this is called the symplectic groupoid only if it happens to be representable by a Lie groupoid, hence if $\mathfrak{P}$ is integrable in the traditional sense of Lie groupoid theory.)

Notice that this is equivalently differential concretification over the point: $\tau_1 \exp(\mathfrak{P})$ may be understood as the smooth moduli stack of $\mathfrak{P}$-valued forms on the point.

But moreover, we may think of $\tau_1 \exp(\mathfrak{P})$ as being the phase space of the open string 2d Poisson-Chern-Simons sigma-model for the space-filling D-brane boundary condition: in this interpretation the objects of $\tau_1 \exp(\mathfrak{P})$ are the endpoints of open strings, the morphisms are solutions to the Euler-Lagrange equations of motion along an interval (an initial spatial slice of string) and composition is string concatenation.

This is the point of view suggested in (Cattaneo-Felder 01), there argued for by the observation that the space of morphisms of $\tau_1 \exp(\mathfrak{P})$ is naturally the symplectic reduction of (in our language) of that of $\tau_1 \exp(\mathfrak{P})$ for the moment map that characterizes the equations of motion and gauge symmetries of the Poisson sigma-model.

D-branes

Let $\mathfrak{C}_0, \mathfrak{C}_1 \hookrightarrow \mathfrak{P}$ be two Lagrangian sub-Lie algebroids of the given Poisson Lie algebroid. These correspond to coisotropic submanifolds of the underlying Poisson manifold.

Accoring to (Cattaneo-Felder 03) we are two think of these as being two D-brane inclusions for the open string Poisson-Chern-Simons model.

Their Lie integration induces two Lagrangian sub-groupoids of the symplectic groupoid

$$tau_1 \exp(\mathfrak{C}_0) \to \tau_1 \exp(\mathfrak{P}) \leftarrow \tau_1 \exp(\mathfrak{C}_1) \,.$$

This means that the moduli stack of open strings with endpoints restricted to these two D-branes is the homotopy fiber product of smooth groupoids

$$\array{ & & \mathbf{Fields}_{\mathfrak{C}_0, \mathfrak{C}_1} \\ & \swarrow && \searrow \\ \tau_1 \exp(\mathfrak{C}_0) && && \tau_1 \exp(\mathfrak{C}_1) \\ & \searrow && \swarrow \\ && \tau_1 \exp(\mathfrak{P}) } \,.$$

One finds that this homotopy pullback is equivalently the symplectic reduction considered in (Cattaneo-Felder 03, page 6, 7).

That for space-filling branes $\mathfrak{C}_0, \mathfrak{C}_1 \coloneqq \mathfrak{P}$ one recovers the original moduli stack

$$\mathbf{Fields}_{\mathfrak{P}, \mathfrak{P}} \simeq \tau_1 \exp(\mathfrak{P})$$

is just the statement that the homotopy pullback of an equivalence is an equivalence.

Constructions

Moduli of 2d CS fields and symplectic groupoids

We discuss how the moduli stack of the 2d Chern-Simons theory obtained by Lie integration of the Poisson sigma-model is a symplectic groupoid.

Let $\mathfrak{P}$ be the Poisson Lie algebroid corresponding to a Poisson manifold that comes from a symplectic manifold $(X,\omega)$.

The symplectic groupoid associated to this is (by the discussion there) supposed to be the fundamental groupoid $\Pi_1(X)$ of $X$ equipped on its space of morphisms with the differential form $p_1^* \omega - p_2^* \omega$, where $p_1,p_2$ are the two endpoint projections from paths in $X$ to $X$.

We demonstrate in the following how this is indeed the result of applying the ∞-Chern-Weil homomorphism to this situation.

For simplicity we shall start with the simple situation where $(X,\omega)$ has a global Darboux coordinate chart $\{x^i\}$. Write $\{\omega_{i j}\}$ for the components of the symplectic form in these coordinates, and $\{\omega^{i j}\}$ for the components of the inverse.

Then the Chevalley-Eilenberg algebra $CE(\mathfrak{P})$ is generated from $\{x^i\}$ in degree 0 and $\{\partial_i\}$ in degree 1, with differential given by

$$d_{CE} x^i = - \omega^{i j} \partial_j$$

$$d_{CE} \partial_i = \frac{\partial \pi^{j k}}{\partial x^i} \partial_j \wedge \partial_k = 0 \,.$$

The differential in the corresponding Weil algebra is hence

$$d_{W} x^i = - \omega^{i j} \partial_j + \mathbf{d}x^i$$

$$d_{W} \partial_i = \mathbf{d} \partial_i \,.$$

By the discussion at Poisson Lie algebroid, the symplectic invariant polynomial is

$$\mathbf{\omega} = \mathbf{d} x^i \wedge \mathbf{d} \partial_i \in W(\mathfrak{P}) \,.$$

Clearly it is useful to introduce a new basis of generators with

$$\partial^i := -\omega^{i j} \partial_j \,.$$

In this new basis we have a manifest isomorphism

$$CE(\mathfrak{P}) = CE(\mathfrak{T}X)$$

with the Chevalley-Eilenberg algebra of the tangent Lie algebroid of $X$.

Therefore the Lie integration of $\mathfrak{P}$ is the fundamental groupoid of $X$, which, since we have assumed global Darboux oordinates and hence contractible $X$, is just the pair groupoid:

$$\tau_1 \exp(\mathfrak{P}) = \Pi_1(X) = (X \times X \stackrel{\overset{p_2}{\to}}{\underset{p_1}{\to}} X) \,.$$

It remains to show that the symplectic form on $\mathfrak{P}$ makes this a symplectic groupoid.

Notice that in the new basis the invariant polynomial reads

$$\begin{aligned} \mathbf{\omega} &= - \omega_{i j} \mathbf{d}x^i \wedge \mathbf{d} \partial^j \\ & = \mathbf{d}( \omega_{i j} \partial^i \wedge \mathbf{d}x^j) \end{aligned}$$

and that we may regard this as a morphism of $L_\infty$-algebroids

$$\mathbf{\omega} : \mathfrak{T}\mathfrak{P} \to \mathfrak{T}b^3 \mathbb{R}$$

The corresponding infinity-Chern-Weil homomorphism that we need to compute is given by the ∞-anafunctor

$$\array{ \exp(\mathfrak{P})_{diff} &\stackrel{\exp(\mathbf{\omega})}{\to}& \exp(b \mathbb{R})_{dR} &\stackrel{\int_{\Delta^\bullet}}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^3 \mathbb{R} \\ \downarrow^{\mathrlap{\simeq}} \\ \exp(\mathfrak{P}) } \,.$$

Over a test space $U$ in degree 1 an element in $\exp(\mathfrak{P})_{diff}$ is a pair $(X^i, \eta^i)$

$$X^i \in C^\infty(U \times \Delta^1)$$

$$\eta^i \in \Omega^1_{vert}(U \times \Delta^1)$$

subject to the verticality constraint, which says that along $\Delta^1$ we have

$$d_{\Delta^1} X^i + \eta^i_{\Delta^1} = 0 \,.$$

The vertical morphism $\exp(\mathfrak{P})_{diff} \to \exp(\mathfrak{P})$ has in fact a section whose image is given by those pairs for which $\eta^i$ has no leg along $U$. We therefore find the desired form on $\exp(\mathfrak{P})$ by evaluating the top morphism on pairs of this form.

Such a pair is taken by the top morphism to

$$\begin{aligned} (X^i, \eta^j) & \mapsto \int_{\Delta^1} \omega_{i j} F_{X^i} \wedge F_{\partial^j} \\ & = \int_{\Delta^1} \omega_{i j} (d_{dR} X^i + \eta^i) \wedge d_{dR} \eta^j \in \Omega^3(U) \end{aligned} \,.$$

Using the above verticality constraint and the condition that $\eta^i$ has no leg along $U$, this becomes

$$\cdots = \int_{\Delta^1} \omega_{i j} d_U X^i \wedge d_U d_{\Delta^1} X^j \,.$$

By the Stokes theorem the integration over $\Delta^1$ yields

$$\cdots = \omega_{i j} d_{dR} x^i \wedge \eta^j|_{0} - \omega_{i j} d_{dR} x^i \wedge \eta^j|_{1} \,.$$

This completes the proof.

$$\array{ X &\to& * \\ \downarrow && \downarrow \\ \tau_1 \exp(\mathfrak{P}) &\stackrel{\omega}{\to}& \flat_{dR} \mathbf{B}^3 U(1) } \,.$$

Prequantum 2-states

Generally, given a prequantum line 2-bundle the corresponding (local) prequantum n-states are the (local) sections of this line 2-bundle, and these in turn are equivalently the twisted unitary bundles (maybe best known for the WZW model where these are the Chan-Paton gauge fields). These 2-sections/twisted bundles form a 2-vector space of prequantum 2-states. This is equivalently a category of modules over some associative algebra, well defined up to Morita equivalence.

A canonical way of constructing this algebra is as follows: present the line 2-bundle as a bundle 2-gerbe, hence as a multiplicative line bundle over the morphisms of a Lie groupoid, then the algebra is the groupoid algebra (see there for details) of sections of this line bundle. A module over this is manifestly a bundle gerbe module, which in turn is equivalently a unitary bundle twisted by the line 2-bundle.

More in detail:

Let $\mathbf{X}$ be a smooth groupoid and $\mathbf{X} \to \mathbf{B}^2 U(1)$ the map modulating a circle 2-group-principal 2-bundle $P \to \mathbf{X}$.

Let $(\coprod_n \mathbf{B}U(n))//\mathbf{B}U(1) \to \mathbf{B}^2 U(1)$ the canonical 2-representation, the sections of the associated 2-bundle are unitary twisted bundles equivariant on $\mathbf{X}$.

If we present the 2-bundle by a bundle gerbe exhibited as a multiplicative line bundle over the space of morphisms $\mathbf{X}_1$, then there is the convolution algabra $\mathcal{A}$ of sections of this line bundle. Twisted unitary vector bundles are equivalently projective modules over this algebra.

This means that under the identification

$$\phi \colon Alg_k \stackrel{\simeq}{\to} 2 Vect_k$$

of the 2-category of 2-vector spaces with that of algebras, bimodules and intertwiners (see at n-vector space), the convolution algebra $\mathcal{A}$ is the 2-vector space of sections

$$\phi\left(\mathcal{A}\right) \simeq \Gamma_{\mathbf{X}}\left(P \times_{\mathbf{B}U\left(1\right)} \coprod_n \mathbf{B}U\left(n\right) \right) \,.$$

Polarizations and branes

The full generalization of the notion of polarization from traditional geometric quantization to higher geometric quantization may need more thinking, but in the case of 2d Chern-Simons theory we can apply the following plausible shortcut.

A Poisson Lie algebroid $(\mathfrak{P}, \mathbf{\omega})$ is a symplectic Lie n-algebroid for $n = 1$. This means that if we regard it as a dg-manifold (the dg-manifold whose dg-algebra of functions is the Chevalley-Eilenberg algebra $(CE(\mathfrak{P})$) then the invariant polynomial $\mathbf{\omega}$ constitutes a graded symplectic form on $\mathfrak{P}$. Since a real polarization of an ordinary symplectic manifold is equivalently a foliation by Lagrangian submanifolds, it hence makes sense to take a real polarization of $(\mathfrak{P}, \mathbf{\omega})$ to consist of Lagrangian dg-submanifolds. As discussed there, for the Poisson Lie algebroid these corespond to the coisotropic submanifolds of the underlying Poisson manifold $(X, \pi)$. The same definition of polarization has been used/obtained in the study of geometric quantization of symplectic groupoids (see there).

The branes of the 2d Chern-Simons theory should be those leaves of the foliation which satisfy an extra intrability condition.

Indeed, according to branes of the Poisson sigma-model are supposed to be coisotropic submanifolds, see at Poisson sigma-model – Properties – Branes.

Quantum 2-states

Summing up, the convolution subalgebra $\mathcal{A}_q \hookrightarrow \mathcal{A}$ of polarized sections is under $\phi$ the actual 2-vector space of states.

In geometric quantization of symplectic groupoids it is show that this is the algebra of observables of the quantum mechanical system of the underlying Poisson manifold (its strict deformation quantization).

In fact we have to quantize the whole atlas of the symplectic groupoid

$$X \to \tau_1 \exp(\mathfrak{P}) \,.$$

By the discussion at groupoid convolution algebra this yields a bimodule

Symplectic case

Consider the simple case where $(X,\omega)$ is a symplectic manifold and in fact a symplectic vector space.

Here the symplectic groupoid is the pair groupoid $Pair(X)_\bullet$ carrying the trivial twist. The atlas is the object inclusion

$$X \to Pair(X) ,.$$

Notice that this is equivalent (Morita equivalent) to just the terminal map

$$X \to * \,.$$

Accordingly, the groupoid convolution algebra of $Pair(X)_\bullet$ is the C-star-algebra of compact operators $\mathcal{K}(X)$. A Morita equivalence bimodule from there to the ground field is (…)

The groupoid-principal-bibundle corresponding to this atlas regarded as a Morita morphism is just the projection $X \times X \stackrel{p_1}{\to} X$. Hence the $C^\ast$-bimodule here is that generated from integral kernels.

In conclusion, up to equivalence the bimodule is $L^2(X)$ regarded as a $C(X)-\mathbb{C}$-bimodule.

(…)

References

The above discussion was inspired by

• Eli Hawkins, A Groupoid Approach to Quantization, J. Symplectic Geom. 6 1 (2008) 61-125 [arXiv:math/0612363, euclid]

and details have been written out in these two theses:

• Stefan Bongers, Geometric quantization of symplectic and Poisson manifolds, master thesis, Utrecht (2013)

• Joost Nuiten, Cohomological quantization of local prequantum boundary field theory, master thesis, Utrecht (2013)

Here section 4 of Bongers (2013) recalls and makes explicit relevant details in the construction of the symplectic groupoid, and section 5.2.2 of Nuiten (2013) essentially provides a re-casting of the quantization construction in Hawkins (2008) which gives it an invariant home and brings out the above holographic picture more explicitly.

This reveals a general abstract nature of the construction which was meant to be further brought out in:

• Urs Schreiber, Quantization via Linear homotopy types [arXiv:1402.7041]

where the above situation corresponds to “running example a)”, Example 7.3 (p. 70).