nLab
higher geometric quantization

under construction, for a more coherent account see (hpqg).

Context

Geometric quantization

Higher geometry

Contents

Idea

Higher geometric quantization is meant to complete this table:

classical mechanicsquantization\toquantum mechanics
symplectic geometrygeometric quantization\toquantum field theory
higher symplectic geometry–higher geometric quantization\toextended quantum field theory

Being a concept in higher geometry, higher geometric quantization is formulated naturally in (∞,1)-topos theory. More precisely, since it involves not just cohomology but differential cohomology, it is formulated in cohesive (∞,1)-topos theory (cohesive homotopy type theory).

In this context, write B n𝔾 connH\mathbf{B}^n \mathbb{G}_{conn} \in \mathbf{H} for the cohesive moduli ∞-stack of circle n-bundles with connection, in the ambient cohesive (∞,1)-topos H\mathbf{H}. Then for XHX \in \mathbf{H} any object to be thought of as the moduli ∞-stack of fields or as the target space for a sigma-model, a morphism

c conn:XB n𝔾 conn \mathbf{c}_{conn} : X \to \mathbf{B}^n \mathbb{G}_{conn}

modulates a circle n-bundle with connection on XX. We regard this as a extended action functional in that for Σ kH\Sigma_{k} \in \mathbf{H} of cohomological dimension knk \leq n and sufficiently compact so that fiber integration in ordinary differential cohomology exp(2πi Σk())\exp(2 \pi i \int_{\Sigma}_k(-)) applies, the transgression of c conn\mathbf{c}_{conn} to low codimension reproduces the traditional ingredients

k=k = transgression of c conn\mathbf{c}_{conn} to [Σ n1,X][\Sigma_{n-1},X]meaning in geometric quantization
nnexp(2πiS()):[Σ n,X][Σ n,c conn][Σ n,B n𝔾 conn]exp(2πi Σ n())𝔾\exp(2 \pi i S(-)) : [\Sigma_n, X] \stackrel{[\Sigma_n, \mathbf{c}_{conn}]}{\to} [\Sigma_n, \mathbf{B}^n \mathbb{G}_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_n}(-))}{\to} \mathbb{G} action functional
n1n-1exp(2πiS()):[Σ n1,X][Σ n1,c conn][Σ n1,B n𝔾 conn]exp(2πi Σ n1())B𝔾 conn\exp(2 \pi i S(-)) : [\Sigma_{n-1}, X] \stackrel{[\Sigma_{n-1}, \mathbf{c}_{conn}]}{\to} [\Sigma_{n-1}, \mathbf{B}^n \mathbb{G}_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_{n-1}}(-))}{\to} \mathbf{B}\mathbb{G}_{conn} \,ordinary (off-shell) prequantum circle bundle

The idea is to consider the higher geometric quantization not just of the low codimension transgressions, but of all transgressions of c conn\mathbf{c}_{conn}. The basic constructions that higher geometric quantization is concerned with are indicated in the following table. All of them have also a fundamental interpretation in twisted cohomology (independent of any interpretation in the context of quantization) this is indicated in the right column of the table:

higher geometric quantizationcohesive homotopy type theorytwisted cohomology
n-plectic ∞-groupoidXωΩ cl n+1(,𝔾)X \stackrel{\omega}{\to} \Omega^{n+1}_{cl}(-,\mathbb{G})twisting cocycle in de Rham cohomology
symplectomorphism groupAut /Ω n+1(,𝔾)(ω)={X X ω ω Ω cl n+1(,𝔾)}\mathbf{Aut}_{/\Omega^{n+1}(-,\mathbb{G})}(\omega) = \left\{ \array{ X &&\stackrel{\simeq}{\to}&& X \\ & {}_{\mathllap{\omega}}\searrow && \swarrow_{\mathrlap{\omega}} \\ && \Omega^{n+1}_{cl}(-,\mathbb{G}) } \right\}
prequantum circle n-bundle B n𝔾 conn c conn curv X ω Ω n+1(,𝔾)\array{ && \mathbf{B}^n \mathbb{G}_{conn} \\ & {}^{\mathllap{\mathbf{c}_{conn}}}\nearrow & \downarrow^{\mathrlap{curv}} \\ X &\stackrel{\omega}{\to}& \Omega^{n+1}(-,\mathbb{G})}twisting cocycle in differential cohomology
Planck's constant \hbar1c conn:XB n𝔾 conn\tfrac{1}{\hbar}\mathbf{c}_{conn} : X \to \mathbf{B}^n \mathbb{G}_{conn}divisibility of twisting class
quantomorphism group superset\superset Heisenberg groupAut /B n𝔾 conn(c conn)={X X c conn c conn B n𝔾 conn}\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}) = \left\{ \array{ X &&\stackrel{\simeq}{\to}&& X \\ & {}_{\mathllap{\mathbf{c}_{conn}}}\searrow &\swArrow_\simeq& \swarrow_{\mathrlap{\mathbf{c}_{conn}}} \\ && \mathbf{B}^n \mathbb{G}_{conn} } \right\}twist automorphism ∞-group
Hamiltonian quantum observables with Poisson bracketLie(Aut /B n𝔾 conn(c conn))Lie(\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}))infinitesimal twist automorphisms
Hamiltonian actions of a smooth ∞-group GG / dual moment mapsμ:BGBAut /B n𝔾 conn(c conn) \mu : \mathbf{B}G \to \mathbf{B}\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn})GG-∞-action on the twisting
gauge reductionc conn//G:X//GB n𝔾 conn\mathbf{c}_{conn}//G \,:\, X//G \to \mathbf{B}^n \mathbb{G}_{conn}GG-∞-quotient of the twisting
Hamiltonian symplectomorphisms∞-image of Aut /B n𝔾 conn(c conn)Aut /Ω cl n+1(,𝔾)(ω)\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}) \to \mathbf{Aut}_{/\Omega^{n+1}_{cl}(-,\mathbb{G})}(\omega)twists in de Rham cohomology that lift to differential cohomology
∞-representation of n-group B n1𝔾\mathbf{B}^{n-1}\mathbb{G} on V nV_nV n V n//B n1𝔾 p B n𝔾\array{ V_n &\to& V_n//\mathbf{B}^{n-1}\mathbb{G} \\ && \downarrow^{\mathbf{p}} \\ && \mathbf{B}^n \mathbb{G} }local coefficient bundle
prequantum space of statesΓ X(E):=[c,p] /B n𝔾={X σ V//B n1𝔾 c p B n𝔾}\mathbf{\Gamma}_X(E) := [\mathbf{c},\mathbf{p}]_{/\mathbf{B}^n \mathbb{G}} = \left\{ \array{ X &&\stackrel{\sigma}{\to}&& V//\mathbf{B}^{n-1}\mathbb{G} \\ & {}_{\mathllap{\mathbf{c}}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{\mathbf{p}}} \\ && \mathbf{B}^n \mathbb{G} } \right\} cocycles in [c][\mathbf{c}]-twisted V-cohomology
prequantum operator()^:Γ X(E)×Aut /B n𝔾 conn(c conn)Γ X(E)\widehat{(-)} : \mathbf{\Gamma}_X(E) \times \mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}) \to \mathbf{\Gamma}_X(E)∞-action of twist automorphisms on twisted cocycles
trace to higher dimension[S 1,V n//B n1𝔾 conn] trhol S 1 V n1//B n2𝔾 conn p conn V n p conn V n1 B n𝔾 conn exp(2πi S 1()) B n1𝔾 conn\array{ [S^1, V_n//\mathbf{B}^{n-1}\mathbb{G}_{conn}] &\stackrel{tr\,hol_{S^1}}{\to}& V_{n-1}//\mathbf{B}^{n-2}\mathbb{G}_{conn} \\ \downarrow^{\mathrlap{\mathbf{p}^{V_n}_{conn}}} && \downarrow^{\mathrlap{\mathbf{p}^{V_{n-1}}_{conn}}} \\ \mathbf{B}^n \mathbb{G}_{conn} &\stackrel{\exp(2 \pi i \int_{S^1}(-))}{\to}& \mathbf{B}^{n-1} \mathbb{G}_{conn} }fiber integration in ordinary differential cohomology adjoined with one in nonabelian differential cohomology

Examples

Ordinary symplectic manifolds

Of non-integral 2-forms

While only integral presymplectic forms have a prequantization to a prequantum circle bundle with connection, hence to a ()(\mathbb{Z} \to \mathbb{R})-principal 2-bundle, a general 2-form has a higer prequantization given by a connection on a 2-bundle on a principal 2-bundle with structure-2-group that coming from the crossed module (Γ)(\Gamma \hookrightarrow \mathbb{R}), where Γ\Gamma is the discrete group of periods of the 2-form.

This is discussed further at prequantization of non-integral 2-forms.

Of 2-plectic \infty-groupoids

In codimension 2

In codimension 1

Proposition There is a lift of coefficient bundles to loop space

[S 1,(BU(n))//(BU(1)) conn] trhol S 1 //U(1) conn p BU p [S 1,B 2U(1) conn] exp(2πi S 1()) BU(1) conn \array{ [S^1,(\mathbf{B}U(n))//(\mathbf{B}U(1))_{conn}] &\stackrel{tr hol_{S^1}}{\to}& \mathbb{C}//U(1)_{conn} \\ \downarrow^{\mathrlap{\mathbf{p}^{\mathbf{B}U}}} && \downarrow^{\mathrlap{\mathbf{p}^{\mathbb{C}}}} \\ [S^1,\mathbf{B}^2 U(1)_{conn}] &\stackrel{\exp(2 \pi i \int_{S^1}(-))}{\to}& \mathbf{B}U(1)_{conn} }

where on the left we have loop space objects formed in H\mathbf{H} and on the bottom we have fiber integration in ordinary differential cohomology.

Forming the pasting composite with this sends 2-states and 2-operators in codimension 2 to ordinary states and operators in codimension 1.

In particular it sends twisted bundles to sections of a line bundle.

For XX a D-brane and c conn\mathbf{c}_{conn} the B-field, this reproduces Freed-Witten anomaly cancellation mechanism.

\infty-Chern-Simons theory

∞-Chern-Simons theory

GG a smooth ∞-group,

c conn:BG connB nU(1) conn \mathbf{c}_{conn} : \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}

a universal differential characteristic map.

The following examples are of this form.

Extended (4k+3)d(4k+3)d abelian Chern-Simons theory

higher dimensional Chern-Simons theory

prequantum circle (4k+3)-bundle from Beilinson-Deligne cup product

B 2k+1U(1) conn()()B 4k+3U(1) conn \mathbf{B}^{2k+1}U(1)_{conn} \stackrel{(-)\cup (-)}{\to} \mathbf{B}^{4k+3}U(1)_{conn}

The quantomorphism \infty-group of this should be

2Aut(U(1)). \mathbb{Z}_2 \simeq Aut(U(1)) \,.

For there is, up to equivalence, a unique autoequivalence

B 2k+1U(1) connB 2k+1U(1) conn, \mathbf{B}^{2k+1}U(1)_{conn} \stackrel{\simeq}{\to} \mathbf{B}^{2k+1}U(1)_{conn} \,,

the one induced by the nontrivial automorphism of U(1)U(1). Since the cup-product is strictly invariant under this, this extends to

B 2k+1U(1) conn B 2k+1U(1) conn ()() ()() B 4k+3U(1) conn. \array{ \mathbf{B}^{2k+1}U(1)_{conn} &&\stackrel{\simeq}{\to}&& \mathbf{B}^{2k+1}U(1)_{conn} \\ & {}_{\mathllap{(-)\cup(-)}}\searrow &\swArrow_\simeq& \swarrow_{\mathrlap{(-)\cup(-)}} \\ && \mathbf{B}^{4k+3}U(1)_\conn } \,.

But for any further nontrivial such autoequivalence in the slice we would need in particular a gauge transformation parameterized by (2k+1)(2k+1)-forms over test manifolds from CdCC \wedge d C to itself. But the only closed 2k2k-forms that we can produce naturally from CC are multiples of CCC \wedge C. But these all vanish since CC is of odd degree 2k+12k+1.

For k=1k = 1 the total space of the prequantum circle 3-bundle of U(1)U(1)-Chern-Simons theory over the point is the smooth moduli 2-stack of differential T-duality structures.

Extended 3d Spin\mathrm{Spin}-Chern-Simons theory

So Planck's constant here is =2\hbar = 2 (relative to the naive multiple).

The total space of the prequantum 3-bundle is

BString conn Ω 13 * BSpin conn 12p^ 1 B 3U(1) conn B 3U(1) \array{ \mathbf{B}String_{conn'} &\to& \Omega^{1 \leq \bullet \leq 3} &\to& * \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}Spin_{conn} &\stackrel{\tfrac{1}{2}\hat \mathbf{p}_1}{\to}& \mathbf{B}^3 U(1)_{conn} &\to& \mathbf{B}^3 U(1) }

as it appears in The moduli 3-stack of the C-field.

But the quantomorphism group of this will be small, as the Chern-Simons form is far from being gauge invariant.

See the discussion at Chern-Simons theory – Geometric quantization – In higher codimension.

Extended 3d G×GG \times G-Chern-Simons theory

However, when we consider G×GG \times G CS theory given by

B(G×G) connc conn 1c conn 2B 3U(1) conn \mathbf{B}(G \times G)_{conn} \stackrel{\mathbf{c}^1_{conn}- \mathbf{c}^2_{conn}}{\to} \mathbf{B}^3 U(1)_{conn}

then diagonal gauge transformations B(G×G) connB(G×G) conn\mathbf{B}(G \times G)_{conn} \to \mathbf{B}(G \times G)_{conn} have interesting extensions to quantomorphisms, because for g:UGg : U \to G the given gauge transformation at stage of definition UU, the Chern-Simons form transforms by an exact term

CS(A 1 g,A 2 g)=CS(A 1,A 2)+dA 1A 2,g *θ. CS(A_1^g,A_2^g) = CS(A_1,A_2) + d \langle A_1 - A_2, g^* \theta\rangle \,.

Extended 7d String\mathrm{String}-Chern-Simons theory

So Planck's constant here is =6\hbar = 6 (relative to the naive multiple).

The total space of the prequantum 7-bundle is

BFivebrane conn Ω 17 * BString conn 16p^ 2 B 7U(1) conn B 7U(1) \array{ \mathbf{B}Fivebrane_{conn'} &\to& \Omega^{1 \leq \bullet \leq 7} &\to& * \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}String_{conn} &\stackrel{\tfrac{1}{6}\hat \mathbf{p}_2}{\to}& \mathbf{B}^7 U(1)_{conn} &\to& \mathbf{B}^7 U(1) }

\infty Wess-Zumino-Witten theory

Differentially twisted looping of \infty-Chern-Simons theory

Ωc:GB n1𝔾 \Omega \mathbf{c} : G \to \mathbf{B}^{n-1}\mathbb{G}

Ordinary GG-WZW model

Ω˜12p 1:GB 2U(1) conn \tilde\Omega \tfrac{1}{2}\mathbf{p}_1 : G \to \mathbf{B}^2 U(1)_{conn}

studied in (Rogers PhD, section 4.2).

StringString-WZW model

(…)

FivebraneFivebrane-WZW model

(…)

duality between algebra and geometry in physics:

algebrageometry
Poisson algebraPoisson manifold
deformation quantizationgeometric quantization
algebra of observablesspace of states
Heisenberg pictureSchrödinger picture
AQFTFQFT
higher algebrahigher geometry
Poisson n-algebran-plectic manifold
En-algebrashigher symplectic geometry
BD-BV quantizationhigher geometric quantization
factorization algebra of observablesextended quantum field theory
factorization homologycobordism representation

References

2-geometric quantization over smooth manifolds is discussed in section 6 and section 7 of

with further indications in

  • Chris Rogers, Higher geometric quantization, at Higher Structures 2011 (pdf)

The special case of geometric quantization over infinitesimal action groupoids can be described in terms of BRST complexes. For references on this see Geometric quantization – References – Geometric BRST quantization.

Higher geometric quantization in a cohesive (∞,1)-topos over smooth ∞-groupoids is discussed in

and the examples of higher Chern-Simons theories in

Revised on April 17, 2013 14:56:17 by Urs Schreiber (89.204.137.137)