Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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

A is an element of H. Such states are said to be

‘‘cylindrical’’ with respect to the graph  and their

space is denoted by Cyl



. These are ‘‘typical states’’

in the sense that Cyl := [



Cyl



is dense in H.

Finally, as ensured by the Gel’fand theory, the

holonomy (or configuration) operators

act just

by multiplication. The momentum operators

S, f

act

as Lie derivatives:

S, f

=ihL

S, f

.

Remark Given any graph  in M, and a labeling of

each of its edges by a nontrivial irreducible represen-

tation of SU(2) (i.e., by a nonzero half integer j), one

can construct a finite-dimensional Hilbert space H

, j

which can be thought of as the state space of a spin

system ‘‘living on’’ the graph . The full Hilbert space

admits a simple decomposition: H= 

, j

. This

is called the spin-network decomposition. The geo-

metric operators discussed in the next section leave

each H

, j

invariant. Therefore, the availability of this

decomposition greatly simplifies the task of analyzing

their properties.

Geometric Operators

In the classical theory, E := 8GP has the inter-

pretation of an orthonormal triad field (or a

‘‘moving frame’’) on M (with density weight 1).

Here,  is a dimensionless, strictly positive number,

called the Barbero–Immirzi parameter, which arises

as follows. Because of emphasis on connections, in

the classical theory the first-order Palatini action is a

more natural starting point than the second-order

Einstein–Hilbert action. Now, there is a freedom to

add a term to the Palatini action which vanishes

when Bianchi identities are satisfied and therefore

does not change the equations of motion.  arises as

the coefficient of this term. In some respects  is

analogous to the  parameter of Yang–Mills theory.

Indeed, while theories corresponding to any permis-

sible values of  are related by a canonical

transformation classically, quantum mechanically

this transformation is not unitarily implementable.

Therefore, although there is a unique representation

of the algebra a (or W ), there is a one-parameter

family of inequivalent representations of the algebra

of geometric operators generated by suitable func-

tions of orthonormal triads E, each labeled by the

value of . This is a genuine quantization ambiguity.

As with the  ambiguity in QCD, the actual value of

 in nature has to be determined experimentally.

The current strategy in quantum geometry is to fix

its value through a thought experiment involving

black hole thermodynamics (see below).

The basic object in quantum Riemannian geome-

try is the triad flux operator

S, f

:= 8G

S, f

.Itis

self-adjoint and all its eigenvalues are discrete. To

define other geometric operators such as the area

operator

associated with a surface S or a volume

operator

associated with a region R, one first

expresses the corresponding phase-space functions in

terms of the ‘‘elementary’’ functions E

, f

using

suitable surfaces S

and test functions f

and then

promotes E

, f

to operators. Even though the

classical expressions are typically nonpolynomial

functions of E

, f

, the final operators are all well

defined, self-adjoint and with purely discrete eigen-

values. Therefore, in the sense of the word used in

elementary quantum mechanics (e.g., of the hydro-

gen atom), one says that geometry is quantized.

Because the theory has no background metric or

indeed any other background field, all geometric

operators transform covariantly under the action of

the Diff M. This diffeomorphism covariance makes

the final expressions of operators rather simple. In

the case of the area operator, for example, the

action of

on a state 



[4] depends entirely on

the points of intersection of the surface S and the

graph  and involves only right- and left-invariant

vector fields on copies of SU(2) associated with

edges of  which intersect S. In the case of the

volume operator

, the action depends on the

vertices of  contained in R and, at each vertex,

involves the right- and left-invariant vector fields on

copies of SU(2) associated with edges that meet at

each vertex.

To display the explicit expressions of these

operators, let us first define on Cyl



three basic

operators

(v, e)

, with j 2 {1, 2, 3}, associated with the

pair consisting of an edge e of  and a vertex v of e:

ðv;eÞ







AÞ¼

t¼0



ð...; U



AÞexpðt

Þ; ...Þ

if e begins at v

t¼0



ð...; expðt

ÞU



AÞ; ...Þ

if e ends at v

½5

where 

denotes a basis in su(2) and ‘‘...’’ stands for

the rest of the arguments of 



which remain

unaffected. The quantum area operator A

assigned to a finite two-dimensional submanifold S

in M. Given a cylindrical state we can always

represent it in the form [4] using a graph  adapted

to S, such that every edge e either intersects S at

exactly one endpoint, or is contained in the closure



S, or does not intersect



S. For each vertex v in S of

the graph , the family of edges intersecting v can be

divided into three classes: edges {e

, ..., e

} lying on

one side (say ‘‘above’’) S, edges {e

uþ1

, ..., e

uþd

} lying

232 Quantum Geometry and Its Applications

on the other side (say ‘‘below’’), and edges contained

in S. To each v we assign a generalized Laplace

operator



S;v

¼

I¼1

ðv;e



uþd

I¼uþ1

ðv;e



K¼1

ðv;e



uþd

K¼uþ1

ðv;e

½6

where 

stands for 1=2 the Killing form on su(2).

Now, the action of the quantum area operator





is defined as follows:





¼ 4‘

v2S

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



S;v





½7

The quantum area operator has played the most

important role in applications. Its complete spec-

trum is known in a closed form. Consider arbitrary

sets j

(u)

, j

(d)

, and j

(uþd)

of half-integers, subject to the

condition

ðuþdÞ

2fjj

ðuÞ

j

ðdÞ

j; jj

ðuÞ

j

ðdÞ

jþ1; ...; j

ðuÞ

þj

ðdÞ

g½8

where I runs over any finite number of integers. The

general eigenvalues of the area operator are given by:

¼ 4‘



ðuÞ

ðj

ðuÞ

þ 1Þþ2j

ðdÞ

ðj

ðdÞ

þ 1Þ

 j

ðuþdÞ

ðj

ðuþdÞ

þ 1Þ



1=2

½9

On the physically interesting sector of SU(2)-

gauge-invariant subspace H

inv

of H, the lowest

eigenvalue of

– ‘‘the area gap’’ – depends on

some global properties of S. Specifically, it ‘‘knows’’

whether the surface is open, or a 2-sphere, or, if M is

a 3-torus, a (nontrivial) 2-torus in M . Finally, on

inv

, one is often interested only in the subspace of

states 



, where  has no edges which lie within a

given surface S. Then, the expression of eigenvalues

simplifies considerably:

¼ 8‘

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðj

þ 1Þ

½10

To display the action of the quantum volume

operator

, for each vertex v of a given graph ,

let us first define an operator

on Cyl



¼ð8‘



e;e

ðe; e

; e

Þc

ijk

ðv;eÞ

ðv;e

½11

where e, e

, and e

run over the set of edges

intersecting v, (e, e

, e

) takes values 1or0

depending on the orientation of the half-lines

tangent to the edges at v,[

, 

] = c



and the

indices are raised by the tensor 

. The action of the

quantum volume operator on a cylindrical state [4]

is then given by





¼ 

v2R

ﬃﬃﬃﬃﬃﬃﬃﬃ

:



½12

Here, 

is an overall, independent of a graph,

constant resulting from an averaging.

The volume operator plays an unexpectedly

important role in the definition of both the gravita-

tional and matter contributions to the scalar

constraint operator which dictates dynamics.

Finally, a notable property of the volume operator

is the following. Let R(p, ) be a family of neighbor-

hoods of a point p 2 M. Then, as indicated above,

R(p, )





= 0if has no vertex in the neighborhood.

However, if  has a vertex at p

lim

!0

Rðx;Þ





exists but is not necessarily zero. This is a reflection

of the ‘‘distributional’’ nature of quantum geometry.

Remark States 



2 Cyl have support only on the

graph . In particular, they are simply annihilated

by geometric operators such as

and

if the

support of the surface S and the region R does not

intersect the support of . In this sense the

fundamental excitations of geometry are one dimen-

sional and geometry is polymer-like. States 



where  is just a ‘‘small graph,’’ are highly quantum

mechanical – like states in QED representing just a

few photons. Just as coherent states in QED require

an infinite superposition of such highly quantum

states, to obtain a semiclassical state approximating

a given classical geometry, one has to superpose a

very large number of such elementary states. More

precisely, in the Gel’fand triplet Cyl HCyl

semiclassical states belong to the dual Cyl

of Cyl.

Applications

Since quantum Riemannian geometry underlies loop

quantum gravity and spin-foam models, all results

obtained in these frameworks can be regarded as its

applications. Among these, there are two which

have led to resolutions of long-standing issues. The

first concerns black hole entropy, and the second,

quantum nature of the big bang.

Black Holes

Seminal advances in fundamentals of black hole

physics in the mid-1970s suggested that the entropy

of large black holes is given by S

= (a

hor

=4‘

Quantum Geometry and Its Applications 233

where a

hor

is the horizon area. This immediately

raised a challenge to potential quantum gravity

theories: give a statistical mechanical derivation of

this relation. For familiar thermodynamic systems, a

statistical mechanical derivation begins with an

identification the microscopic degrees of freedom.

For a classical gas, these are carried by molecules;

for the black body radiation, by photons; and for a

ferromagnet, by Heisenberg spins. What about black

holes? The microscopic building blocks cannot be

gravitons because the discussion involves stationary

black holes. Furthermore, the number of micro-

scopic states is absolutely huge: some exp 10

for a

solar mass black hole, a number that completely

dwarfs the number of states of systems one normally

encounters in statistical mechanics. Where does this

huge number come from? In loop quantum gravity,

this is the number of states of the ‘‘quantum horizon

geometry.’’

The idea behind the calculation can be heuristi-

cally explained using the ‘‘It from Bit’’ argument,

put forward by Wheeler in the 1990s. Divide the

black hole horizon into elementary cells, each with

one Planck unit of area, ‘

, and assign to each cell

two microstates. Then the total number of states N

is given by N = 2

, where n = (a

hor

=‘

) is the

number of elementary cells, whence entropy is

given by S = ln Na

hor

. Thus, apart from a

numerical coefficient, the entropy (It) is accounted

for by assigning two states (Bit) to each elementary

cell. This qualitative picture is simple and attractive.

However, the detailed derivation in quantum geo-

metry has several new features.

First, Wheeler’s argument would apply to any

2-surface, while in quantum geometry the surface

must represent a horizon in equilibrium. This

requirement is encoded in a certain boundary

condition that the canonically conjugate pair (A, P)

must satisfy at the surface and plays a crucial role in

the quantum theory. Second, the area of each

elementary cell is not a fixed multiple of ‘

but is

given by [10], where I labels the elementary cells

and j

can be any half-integer (such that the sum is

within a small neighborhood of the classical area of

the black hole under consideration). Finally, the

number of quantum states associated with an

elementary cell labeled by j

is not 2 but (2j

þ 1).

The detailed theory of the quantum horizon

geometry and the standard statistical mechanical

reasoning is then used to calculate the entropy and

the temperature. For large black holes, the leading

contribution to entropy is proportional to the

horizon area, in agreement with quantum field

theory in curved spacetimes. (The subleading term

(1=2) ln(a

hor

=‘

) is a quantum gravity correction

to Hawking’s semiclassical result. This correction,

with the 1=2 factor, is robust in the sense that it

also arises in other approaches.) However, as one

would expect, the proportionality factor depends on

the Barbero–Immirzi parameter  and so far loop

quantum gravity does not have an independent way

to determine its value. The current strategy is to

determine  by requiring that, for the Schwarzschild

black hole, the leading term agrees exactly with

Hawking’s semiclassical answer. This requirement

implies that  is the root of algebraic equation and

its value is given by   0.2735. Now, quantum

geometry theory is completely fixed. One can

calculate entropy of other black holes, with angular

momentum and distortion. A nontrivial check on the

strategy is that for all these cases, the coefficient in

the leading-order term again agrees with Hawking’s

semiclassical result.

The detailed analysis involves a number of

structures of interest to mathematical physics. First,

the intrinsic horizon geometry is described by a U(1)

Chern–Simons theory on a punctured 2-sphere (the

horizon), the level k of the theory being given by

k = a

hor

=4‘

. The punctures are simply the inter-

sections of the excitations of the polymer geometry

in the bulk with the horizon 2-surface. Second,

because of the horizon boundary conditions, in the

classical theory the gauge group SU(2) is reduced to

U(1) at the horizon. At each puncture, it is further

reduced to the discrete subgroup Z

of U(1),

sometimes referred to as a ‘‘quantum U(1) group.’’

Third, the ‘‘surface phase space’’ associated with the

horizon is represented by a noncommutative torus.

Finally, the surface Chern–Simons theory is entirely

unrelated to the bulk quantum geometry theory but

the quantum horizon boundary condition requires

that the spectrum of a certain operator in the

Chern–Simons theory must be identical to that of

another operator in the bulk theory. The surprising

fact is that there is an exact agreement. Without this

seamless matching, a coherent description of the

quantum horizon geometry would not have been

possible.

The main weakness of this approach to black hole

entropy stems from the Barbero–Immirzi ambiguity.

The argument would be much more compelling if

the value of  were determined by independent

considerations, without reference to black hole

entropy. (By contrast, for extremal black holes,

string theory provides the correct coefficient without

any adjustable parameter. The AdS/CFT duality

hypothesis (as well as other semiquantitative) argu-

ments have been used to encompass certain black

holes which are away from extremality. But in these

cases, it is not known if the numerical coefficient is

234 Quantum Geometry and Its Applications

1/4 as in Hawking’s analysis.) It’s primary strengths

are twofold. First, the calculation encompasses all

realistic black holes – not just extremal or near-

extremal – including the astrophysical ones, which

may be highly distorted. Hairy black holes of

mathematical physics and cosmological horizons

are also encompassed. Second, in contrast to other

approaches, one works directly with the physical,

curved geometry around black holes rather than

with a flat-space system which has the same number

of states as the black hole of interest.

The Big Bang

Most of the work in physical cosmology is carried

out using spatially homogeneous and isotropic

models and perturbations thereon. Therefore, to

explore the quantum nature of the big bang, it is

natural to begin by assuming these symmetries.

Then the spacetime metric is determined simply by

the scale factor a(t) and matter fields (t) which

depend only on time. Thus, because of symmetries,

one is left with only a finite number of degrees of

freedom. Therefore, field-theoretic difficulties are

bypassed and passage to quantum theory is simpli-

fied. This strategy was introduced already in the late

1960s and early 1970s by DeWitt and Misner.

Quantum Einstein’s equations now reduce to a

single differential equation of the type

ðf ðaÞ ða;ÞÞ ¼ const:



ða;Þ½13

on the wave function (a, ), where



is the matter

Hamiltonian and f (a) reflects the freedom in factor

ordering. Since the scale factor a vanishes at the big

bang, one has to analyze the equation and its

solutions near a = 0. Unfortunately, because of the

standard form of the matter Hamiltonian, coeffi-

cients in the equation diverge at a = 0 and the

evolution cannot be continued across the singularity

unless one introduces unphysical matter or a new

principle. A well-known example of new input is the

Hartle–Hawking boundary condition which posits

that the universe starts out without any boundary

and a metric with positive-definite signature and

later makes a transition to a Lorentzian metric.

Bojowald and others have shown that the situa-

tion is quite different in loop quantum cosmology

because quantum geometry effects make a qualita-

tive difference near the big bang. As in older

quantum cosmologies, one carries out a symmetry

reduction at the classical level. The final result

differs from older theories only in minor ways. In

the homogeneous, isotropic case, the freedom in the

choice of the connection is encoded in a single

function c(t) and, in that of the momentum/triad, in

another function p(t). The scale factor is given by

= jpj. (The variable p itself can assume both signs;

positive if the triad is left handed and negative if it is

right handed. p vanishes at degenerate triads which

are permissible in this approach.) The system again

has only a finite number of degrees of freedom.

However, quantum theory turns out to be inequi-

valent to that used in older quantum cosmologies.

This surprising result comes about as follows.

Recall that in quantum geometry, one has well-

defined holonomy operators

h but there is no

operator corresponding to the connection itself. In

quantum mechanics, the analog would be for

operators

U() corresponding to the classical func-

tions exp ix to exist but not be weakly continuous

in ; the operator

x would then not exist. Once the

requirement of weak continuity is dropped, von

Neumann’s uniqueness theorem no longer holds and

the Weyl algebra can have inequivalent irreducible

representations. The one used in loop quantum

cosmology is the direct analog of full quantum

geometry. While the space A of smooth connections

reduces just to the real line R, the space



A of

generalized connections reduces to the Bohr com-

pactification



Bohr

of the real line. (This space was

introduced by the mathematician Harold Bohr (Nils’

brother) in his theory of almost-periodic functions.

It arises in the present application because holo-

nomies turn out to be almost periodic functions

of c .) The Hilbert space of states is thus

H= L

(



Bohr

,d

) where 

is the Haar measure

on (the abelian group)



Bohr

. As in full quantum

geometry, the holonomies act by multiplication and

the triad/momentum operator

p via Lie derivatives.

To facilitate comparison with older quantum

cosmologies, it is convenient to use a representation

in which

p is diagonal. Then, quantum states are

functions (p, ). But the Wheeler–DeWitt equation

is now replaced by a difference equation:

ðpÞðp þ 4p

;ÞþC

ðpÞðp;Þ

þ C



ðpÞðp  4p

ÞðÞ¼const:



ðp;Þ½14

where p

is determined by the lowest eigenvalue of the

area operator (‘‘area gap’’) and the coefficients C



(p)

and C

(p) are functions of p. In a backward ‘‘evolu-

tion,’’ given  at p þ 4andp, such a ‘‘recursion

relation’’ determines  at p  4, provided C



does not

vanish at p  4. The coefficients are well behaved and

nowhere vanishing, whence the evolution does not stop

at any finite p, either in the past or in the future. Thus,

near p = 0 this equation is drastically different from the

Wheeler–DeWitt equation [13]. However, for large p –

that is, when the universe is large – it is well

Quantum Geometry and Its Applications 235

approximated by [13] and smooth solutions of [13] are

approximate solutions of the fundamental discrete

equation [14] in a precise sense.

To complete quantization, one has to introduce a

suitable Hilbert space structure on the space of

solutions to [14], identify physically interesting

operators and analyze their properties. For simple

matter fields, this program has been completed.

With this machinery at hand, one begins with

semiclassical states which are peaked at configura-

tions approximating the classical universe at late

times (e.g., now) and evolves backwards. Numerical

simulations show that the state remains peaked at

the classical solution till very early times when the

matter density becomes of the order of Planck

density. This provides, in particular, a justification,

from first principles, for the assumption that space-

time can be taken to be classical even at the onset of

the inflationary era, just a few Planck times after the

(classical) big bang. While one would expect a result

along these lines to hold on physical grounds,

technically it is nontrivial to obtain semiclassicality

over such huge domains. However, in the Planck

regime near the big bang, there are major deviations

from the classical behavior. Effectively, gravity

becomes repulsive, the collapse is halted and then

the universe re-expands. Thus, rather than modify-

ing spacetime structure just in a tiny region near the

singularity, quantum geometry effects open a bridge

to another large classical universe. These are

dramatic modifications of the classical theory.

For over three decades, hopes have been expressed

that quantum gravity would provide new insights

into the true nature of the big bang. Thanks to

quantum geometry effects, these hopes have been

realized and many of the long-standing questions

have been answered. While the final picture has

some similarities with other approaches, (e.g.,

‘‘cyclic universes,’’ or pre-big-bang cosmology),

only in loop quantum cosmology is there a fully

deterministic evolution across what was the classical

big-bang. However, so far, detailed results have

been obtained only in simple models. The major

open issue is the inclusion of perturbations and

subsequent comparison with observations.

See also: Algebraic Approach to Quantum Field Theory;

Black Hole Mechanics; Canonical General Relativity;

Knot Invariants and Quantum Gravity; Loop Quantum

Gravity; Quantum Cosmology; Quantum Dynamics in

Loop Quantum Gravity; Quantum Fields Theory in

Curved Spacetime; Spacetime Topology, Causal Structure

and Singularities; Spin Foams; Wheeler–De Witt Theory.

Further Reading

Ashtekar A (1987) New Hamiltonian formulation of general

relativity. Physical Review D 36: 1587–1602.

Ashtekar A and Krishnan B (2004) Isolated and dynamical

horizons and their applications. Living Reviews in Relativity

10: 1–78 (gr-qc/0407042).

Ashtekar A and Lewandowski L (2004) Background independent

quantum gravity: a status report. Classical Quantum Gravity

21: R53–R152.

Bojowald M and Morales-Tecotl HA (2004) Cosmological

applications of loop quantum gravity. Lecture Notes in

Physics 646: 421–462 (also available at gr-qc/0306008).

Gambini R and Pullin J (1996) Loops, Knots, Gauge Theories and

Quantum Gravity. Cambridge: Cambridge University Press.

Perez A (2003) Spin foam models for quantum gravity. Classical

Quantum Gravity 20: R43–R104.

Rovelli C (2004) Quantum Gravity. Cambridge: Cambridge

University Press.

Thiemann T (2005) Introduction to Modern Canonical Quantum

General Relativity, Cambridge: Cambridge University Press.

(draft available as gr-qc/0110034).

Quantum Group Differentials, Bundles and Gauge Theory

T Brzezin

ski, University of Wales Swansea,

Swansea, UK

Introduction

Mathematics of classical gauge theories is contained

in the theory of principal and associated vector

bundles. Principal bundles describe pure gauge

fields and their transformations, while the asso-

ciated bundles contain matter fields. A structure

group of a bundle has a meaning of a gauge group,

while the base manifold is a spacetime for the

theory. In this article, we review the theory of

bundles in which a structure group is a quantum

group and base space or spacetime might be

noncommutative. To fully deal with geometric

aspects, we first review differential geometry of

quantum groups. Then we describe the theory of

quantum principal bundles, connections on such

bundles, gauge transformations, associated vector

bundles and their sections. We indicate that, for a

certain class of quantum principal bundles, sections

of an associated bundle become vector bundles of

noncommutative geometry a`laConnes, that is,

236 Quantum Group Differentials, Bundles and Gauge Theory

finite projective modules. The theory is illustrated

by two explicit examples that can be viewed as

deformations of the classical magnetic monopole

and the instanton.

Differential Structures on Algebras

Algebraic Conventions

Throughout this article, A (P etc.,) will be an

associative unital complex algebra. To gain some

geometric intuition the reader can think of A as an

algebra of continuous complex functions on a

compact (Hausdorff) space X, C(X), with product

given by pointwise multiplication fg (x) = f (x)g( x),

and with the unit provided by a constant function

x 7!1. The algebra C(X) is commutative, but, in

what follows, we do not assume that A is a

commutative algebra. By an A-bimodule we mean

a vector space with mutually commuting left and

right actions of A. All modules are unital (i.e., the

unit element of A acts trivially). On elements, the

multiplication in an algebra or an action of A on a

module is denoted by juxtaposition.

Differential Calculus on an Algebra

A first-order differential calculus on A is a pa ir

(

(A), d), where 

(A)isanA-bimodule and

d:A !

(A) is a linear map such that:

1. for all a, b 2A,d(ab) = (da)b þ adb (the Leibniz

rule); and

2. every ! 2

(A) can be written as ! =

for

some a

, b

2A.

Elements of 

(A) are called differential 1-forms

and the map d is called an exterior deri vative. As a

motivating example, take A = C(X) and 

(A) the

space of 1-forms on X (sections of the cotangent

bundle T



X), and d the usual exterior differential.

Higher-differential forms corresponding to ( 

(A), d)

are defined as elements of a differential graded

algebra (A). This is an algebra which can be

decomposed into the direct sum of A-bimodules



(A), that is, (A) = A  

(A)  

(A) .In

addition to d : A !

(A), there are maps d

: 

(A) !

nþ1

(A) such that, for all !

2

(A),

2

(A),

1. d

d = 0 and d

nþ1

 d

= 0, n = 1, 2, ...;

2. !

2

nþk

(A); and

3. d

nþk

) = (d

þ (1)

Elements of 

(A) are known as ‘‘differential

n-forms .’’ 

(A) contains all linear c ombinations

of expressions a

da

with a

, ..., a

2A.

One says that (A) satisfies the ‘‘density

condition’’ if any element of 

(A)isofthe

above form, for any n. To simplify notation, one

writes d for d

As an exam ple of (A), take A = C(X)andthen

the exterior algebra (X)for(A). The e xterio r

algebra satisfies density condition as any n-f orm

can be written as f (x) ^ dg(x) ^ dh(x) ^.The

wedge product is anticommutative, but for a

noncommutative algebra A, the anticommutativity

of the product in (A) cannot be generally

required.

The Universal Differential Calculus

Any algebra A comes equipped with a universal

differential calculus denoted by (

A, d). 

A is def-

ined as the kernel of the multiplication map, that

is, 

A:= {

b

2A A j

= 0}  A A.

The derivative is defined by d(a)= 1 a a1. The

n-forms are defined as 

A=

A



A 







A (n-copies of 

A). 

A can be identified

with a subspace of A A  A (n þ1-copies of

A) consisting of all such elements that vanish upon

multiplication of any two consecutive factors. With

this identification, higher derivatives read



a

a



nþ1

k¼0

ð1Þ

a

a

k1

1 a

a

The universal differential calculus satisfies the

density condition.

This calculus captures very little (if any) of the

geometry of the underlying algebra A, but it has the

universality property, that is, any differential calcu-

lus on A can be obtained as a quotient of A.

In other words, any differential calculus (A)is

fully determined by a system of A-sub-bimodules

nþ1

(or homogeneous ideals in the algebra

A), so that 

(A) =

A=N

. The differentials d in

(A) are derived from universal differentials via the

canonical projections 

: 

A !

(A).

Typical examples of algebras in quantum geome-

try are given by generators and relations, that is,

A= Chx

,...,x

i=hR

,..., x

)i, where Chx

,...,x

is a free algebra on generators x

and R

,...,x

)

are polynomials, so that R

,...,x

)= 0inA.

Correspondingly, the modules 

(A) are given by

generators and relations. If (A) satisfies the density

condition, that the whol e of (A) must be generated

by some 1-forms. The sub-bimodu les N

contain

relations satisfied by these generators.

Quantum Group Differentials, Bundles and Gauge Theory 237

-Calculi

If A is a -algebra, then a calculus is called a

‘‘-calculus’’ provided (A) is a graded -algebra,

and d(



) = (d)



, for all  2(A).

Differential Structures on Quantum

Groups (Hopf Algebras)

Hopf Algebra Preliminaries

From now on, A is a Hopf algebra (quantum group),

with a coproduct  : A !A A, counit " : A !C

and antipode S. We use Sweedler’s notation

(a) =

(1)

a

(2)

. We also write A

= ker " (the

augmentation ideal).

For any algebra P, the convolution product of

linear maps f , g : A !P is a linear map f  g : A !P,

defined by f  g(a) =

f (a

(1)

)  g(a

(2)

). A map

f : A !P is said to be convolution invertible,

provided there exists f

1

: A !P such that

f  f

1

= f  f

1

= 1".

An A-coaction on a comodule V, % : V !V A,is

denoted by %( v ) =

(0)

v

(1)

. The right adjoint

coaction in A is a map

Ad : A !A A;

AdðaÞ¼

ð2Þ

ðSa

ð1Þ

Þa

ð3Þ

A subspace B of A is said to be ‘‘Ad-invariant’’

provided Ad(B)  B A. For example, A

is such a

space.

Covariant Differential Calculi

For Hopf algebras one can study calculi that are

covariant with respect to . For A = C[G] (an

algebra of functions on a Lie group), this corre-

sponds to the covariance of a differential structure

on G with respect to regular representations.

A first-order differential calculus 

(A)ona

quantum group A is said to be left-covari ant, if

there exists a linear map 

: 

(A) !A 

(A)

(called a left coaction) such that, for all a, b 2A,



ðadbÞ¼

ð1Þ

a

ð2Þ



(A) is called a right-covariant differential calculus

if there exists a linear map 

: 

(A) !

(A) A

(called a right coaction) such that, for all a , b 2A,



ðadbÞ¼

ð1Þ

a

ð2Þ

If 

(A) is both left- and right-covariant, it is called

a ‘‘bicovariant differential calculus.’’ A bicovariant



(A) has a structure of a Hopf A-bimodule, that is,

it is an A-bimodule and an A-bicomodule such that

the coactions are compatible with actions.

The universal calculus on A is bicovariant with

coactions





b



ð1Þ

b

ð1Þ

a

ð2Þ

;





b



ð1Þ

a

ð2Þ

b

ð2Þ

Since 

(A) =

A=N for an A-sub-bimodule

N 2

A, the calculus 

(A) is left (resp. right)

covariant if and only if 

(N)  A N (resp.



(N)  N A ).

The Woronowicz Theorems

A form ! in a left-covariant differential calculus



(A) is said to be left-invariant provided



(!) = 1 !. 

(A) is a free A-module with basis

given by left-invariant forms, that is, one can choose

a set of left-invariant forms !

such that any 1-form

 can be uniquely written as a finite sum

 =

, a

2A.

The first Woronowicz theorem states that there is

a one-to-one correspondence between left-covariant

calculi on A and right A -ideals Q  A

. The

correspondence is provided by the map

 : A Q !N; a q 7!

aSq

ð1Þ

q

ð2Þ

where N is such that 

(A) = (

A)=N. The inverse of

 reads 

1

(

b

) =

(1)

b

(2)

. The map

 induces the map



 : A

=Q !

(A), via

!([a]) = [(1 a )] where [ ] denotes cosets in

=Q and in 

(A) = (

A)=N. This establishes a

one-to-one correspondence between the space



= A

=Q and the space of left-invariant 1-forms in



(A). The dual space to 

, that is, the space of linear

functionals 

!C, is often termed a ‘‘quantum Lie

algebra’’ or a ‘‘quantum tangent space’’ corresponding

to a left-covariant calculus 

(A). The dimension of



is known as a dimension of 

(A).

The definitions and analysis of right-covariant

differential calculi are done in a symmetric manner.

For a bicovariant calculus, a form ! that is both left-

and right-invariant, is termed a ‘‘bi-invariant’’ form.

The second Woronowicz theorem states a one-to-

one correspondence between bicovariant differential

calculi and Ad-invar iant A-ideals Q  A

(cf. the

subsec tion ‘‘Hopf algebra prel iminaries ’’). The

correspondence is provided by the map  above.

For the universal calculus, Q is trivial, and hence



= A

= ker (").

Higher-order Bicovariant Calculi

Given a first-order bicovariant calculus 

(A), one

constructs a braiding operator, known as the

238 Quantum Group Differentials, Bundles and Gauge Theory

‘‘Woronowicz braiding’’  : 

(A) 



(A) !

(A)





(A)bysetting(a! 

) = a 

! for all a 2A,

and any left-invariant ! and right-invariant , and then

extending it A-linearly to the whole of



(A) 



(A). This operator satisfies the braid

relation (id 

) ( 

id) (id 

) = (  

id) 

(id

) ( 

id), and is invertible provided the

antipode S is invertible. The Woronowicz braiding is

used to define symmetric forms as those invariant

under . One then defines exterior 2-forms as elements

of 

(A) 



(A)=ker (id ), and introduces the

wedge product. The wedge product is not in general

anticommutative, but one does have ! ^ =  ^!

for bi-invariant !, . This construction is extended to

higher forms and leads to the definition of the exterior

algebra (A). To define exterior n-forms, one maps

any permutation on n-elements to the corresponding

element of the braid group generated by  and then

takes the quotient of the nth tensor power of 

(A)by

all elements corresponding to even permutations. The

differential d : A !

(A) is extended to an exterior

differential in the whole of (A) in the following way.

First, 

(A) is extended by a one-dimensional

A-bimodule generated by a form  that is required to

be bi-invariant. The resulting extended bimodule

(which, in general, is not a first-order differential

calculus, as  is not necessarily of the form

for some a

, b

2A) is then determined from the

relation da =  a a for all a 2A.Higherexterior

derivative is then defined by d=^ (  1)

^,

for any  2

(A).

The algebra (A)isaZ

-graded differential Hopf

algebra, that is, it has a coproduct such that

ð! ^ Þ¼

ð1Þ

ð2Þ

k

ð1Þ

^ 

ð1Þ

!

ð2Þ

^ 

ð2Þ

where j!

(2)

j etc., denotes the degree of a homo-

geneous component in the decomposition of (!).

Furthermore,

ðd!Þ¼



ð1Þ

!

ð2Þ

þð1Þ

ð1Þ

d!

ð2Þ



On the 1-forms this coproduct is simply the sum



þ

Classification

There is no unique covariant differential calculus on A,

so classification of covariant differential calculi is an

important problem. For example, it is known that the

quantum group SU

(2) admits a left-covariant three-

dimensional calculus, but there is no three-dimen-

sional bicovariant calculus. On the other hand, there

are two four-dimensional bicovariant calculi on

(2). Differential calculi are classified for standard

quantum groups such as SL

(N)orSp

(N).

General classification results are based on

the equivalence between the category of Hopf

bimodules of a finite-dimensional Hopf algebra

A and that of Yetter–Drinfeld or crossed modules

of A. These are the modules of the Drinfeld double

of A. As a result, in the case of a finite-dimensional

factorizable coquasitriangular Hopf algebra A with

a dual Hopf algebra H, the bicovariant 

(A) are

in one-to-one correspondence with two-sided ideals

in H

. If, in addition, A is semisimple, then

(coirreducible) calculi are in one-to-one correspon-

dence with nontrivial irreducible representations of

H. This can be extended to infinite-dimensional

algebras, provided one works over a field of formal

power series in the deformation parameter.

Quantum Group Principal Bundles

Quantum Principal Bundles

In classical geometry, a (topological) principal

bundle is a locally compac t Hausdorff space with a

(continuous) free and proper action of a locally

compact group (e.g., a Lie group). In terms of

algebras of functions this gives rise to the following

structure. A is a Hopf algebra (the model is

functions on a group G), P is a right A-comodule

algebra with a coaction 

: P !P A (the model

is functions on a total space X). Let

B = {b 2P j

(b) = b 1} be the coinvariant sub-

algebra (the model is functions on a base manifold

M = X=G). Fix a bicovariant calculus 

(A), with

the corresponding Q and 

= A

=Q as in the

subsec tion ‘‘The Woronow icz theore ms.’’ Take a

differential calculus 

(P) =

P=N

such that:

1. 



) N

A, where for all

q

2







q



ð0Þ

q

ð0Þ

p

ð1Þ

2

P A

(N

)  N

Q, where

 :

P !P A

;

q



ðq

Þ¼

ð0Þ

q

ð1Þ

3. N

= N

\ 

B gives rise to a differential struc-

ture 

(B) =

B=N

on B. Condition (1) ensures

that 



descends to a coaction 



(P)



(P) !

(P) A, while (2) allows for defining

a map

ver :

ðPÞ!P 

; verð½!Þ ¼ ½

ð!Þ

Quantum Group Differentials, Bundles and Gauge Theory 239

Since B is a subalgebra of P, the P-bimodule

P

ðBÞP :¼

ðdb

Þq

; q

2P; b

is a sub-bimodule of 

P, known as horizontal

forms. P is called a ‘‘quantum principal bundle’’

over B with quantum structure group A an d calculi



(A) and 

(P) provided the following sequence;

0 !P

ðBÞP ! 

ðPÞ!

ver

P 

! 0

is exact. This definition reflects the geometric

content of principal bundles, but is not restricted

to any specific differential calculus. The surjectivity

of ver corresponds to the freeness of the (co)action,

while the condition ker (ver) = P

(B)P corresponds

to identification of vertical vector fields as those that

are annihilated by horizontal form s.

The Universal Calculus Case

In the universal calculus case, both N

and Q in the

previous subsection are trivial, and ver =

. Uni-

versal horizontal forms P(

B)P coinc ide with the

kernel of the canonical projection P 

P !P P.

The exactness of the sequence in the last su bsection

is equivalent to the requirement that the map

can : P 

P !P A

p 

q 7!p

ðqÞ¼

ð0Þ

q

ð1Þ

be bijective. In algebra, such an inclusion of algebras

B  P is known a s a Hopf–Galois extension. Thus, a

geometric notion of a quantum principal bundle

with the universal calculus is the same as the

algebraic notion of a Hopf–Galois extension.

If (2) in the previous subsection is replaced by

stronger conditions

(N

) = N

Q and (N

ker

)  P(

B)P, then exactness of the sequence

in the previous subsection is equivalent to the

bijectivity of ‘‘can.’’ Thus, although defined in a

purely algebraic way, the notion of a Hopf–Galois

extension carries deep geometric meaning. It there-

fore makes sense to consider primarily Hopf–Galois

extensions and then specify differential structure in

such a way that this stronger version of (2) is

satisfied. Henceforth, unless specified otherwise, a

quantum principal bundle is taken with the uni-

versal differential calculus .

Quantum Homogeneous Bundles

Suppose that P is a Hopf algebra, and that there is a

Hopf algebra surjection  : P !A. This induces a

coaction of P on A via 

= (id )  , where now

 is a coproduct in P. P is a quantum principal

A-bundle over the coinvariants B, provided ker  

P, where B

= B \ P

. B is a left quantum

homogeneous space in the sense that (B)  P B,

and P is known as a quantum homogeneous bundle.

An example of this is the standard quantum

2-sphere – a quantum homogeneous space of

(2) (see the subsec tion ‘‘The Dirac q-monopo le’’).

This construction reflects the classical constr uction of

a principal bundle over a homogeneous space, since

every homogeneous space of a group G can be

identified with a quotient G=H, where H  G is a

subgroup. Not every quantum homogeneous space

can be obtained in this way (e.g., nonstandard

quantum 2-spher es), as quantum groups P do not

have sufficiently many quantum subgroups A (in a

sense of Hopf algebra projections  : P !A). To study

gauge theory on general quantum homogeneous

spaces, more general notion of a bundle needs to be

develo ped (see the subsec tion ‘‘Gener alizations of

quan tum pr incipal bundles ’’).

A general differential calculus on a quantum

homogeneous bundle is specifi ed by choosing a

left-covariant calculus on P with an ideal Q

such that (id )  Ad(Q

)  Q

A. A bicovariant

calculus on A is then given by Q

= (Q

Quantum Trivial Bundles

A quantum principal bundle (with the universal

differential calculus) is said to be ‘‘trivial’’ or ‘‘cleft’’

provided there exists a linear map  : A !P such that

1. (1) = 1 (unitality);

2. 

 =( id)   ( colinearity or covariance);

and

3.  is convolution invertible (cf. the subsection

‘‘Hop f algebra prelim inaries’’).

 is called a trivialization. In this case, P is

isomorphic to B A as a left B-module and right

A-comodule via the map B A !P, b a 7!b(a).

In particular, an A-covariant (i.e., colinear) algebra

map j : A !P is a triv ialization (the convolution

inverse of j is j  S).

Based on trivial bundles, locally trivial bundles

can be constructed by choosing a compatible cover-

ing of B (in terms of ideals).

At this point, the reader should be warned that

the notion of a trivial quantum principal bundle

includes bundles which are not trivial classically

(i.e., do not correspond to functions on the

Cartesian product of spaces). As an example,

consider the Mo¨ bius strip viewed as a Z

-principal

bundle over the circle S

. Obviously, this is not a

trivial bundle (the Mo¨ bius strip is not isomorphic to

240 Quantum Group Differentials, Bundles and Gauge Theory

 Z

). It can be shown, however, that the

quantum principal bundle corresponding to the

Mo¨ bius strip has a trivialization  in the above

sense.

Generalizations of Quantum Principal Bundles

In the case of majority of quantum homogeneous

spaces , the map  in the su bsection ‘‘Quantum

homogene ous bundles ’’ is a coalgebra and right

P-module map, but not an algebra map. Thus, the

induced coaction is not an algebra map either. To

cover examples like these, one needs to introduce

a generalization of quantum principal bundles.

Consider an algebra P that is also a right comodule

of a coalgebra C with coaction 

. Define

B :¼

b 2Pj8p 2P; 

ðbpÞ¼b

ðpÞ

ð0Þ

p

ð1Þ

B is a subalgebra of P. P is a principal coalgebra-

bundle over B or B  P is a coalgebra-Galois

extension provided the map

can : P 

P !P C

p 

q 7!p

ðqÞ¼

ð0Þ

q

ð1Þ

is bijective. This purely algebraic requirement

induces a rich symmetry structure on P, given in

terms of entwining, which allows one for developing

various differential geometric notions such as those

discussed in the next section. The lack of space does

not permit us to describe this theory here.

Connections, Gauge Transformations,

Matter Fields

Connections and Connection Forms

A ‘‘connection’’ in a quantum principal bundle with

calculi 

(P), 

(A) is a left P-linear map

 : 

(P) !

(P) such that:

1.   = ( is a projection);

2. ker =P

(B)P; and

3. 



(P)

 =( id)  



(P)

(colinearity or

covariance).

The exact sequence in the subsection ‘‘Quantum

principal bundles’’ implies that  is a left P-linear

projection if and only if there exists a left P-linear

map  : P  

!

(P) such that ver   = id. Since

 is left P-linear, it is fully specified by its action on



. This leads to the equivalent definition of a

connection as a connection form or a gauge field,

that is, a map ! : 

!

(P) such that:

1. for all  2

, ver( ! ()) = 1 ; and

2. 



(P)

 ! = (! id)  Ad



(Ad-covariance), where



1 is a projection of the adjoint coaction to 

that is, Ad



1 ([a]) = [Ad(a)] (well defined, because

Q is Ad-invariant for a bicovariant calculus, see

the subsection ‘‘The Woronowicz theorems’’).

The correspondence between connections and con-

nection 1-forms is given by the formula

ðpdqÞ¼

ð0Þ

!ð½q

ð1Þ

Þ

In the universal differential calculus case, 

= A

hence ! can be viewed as a map ! : A !

P, such

that !(1) = 0. The map F

: A !

P, given by

= d! þ !  ! is called a ‘‘curvature’’ of !. The

curvature satisfies the Bianchi identity, dF

 !  ! F

In the case of a trivial bundle with trivialization 

and universal calculus, any linear map  : A !

such that (1) = 0 defines a connection 1-form

! ¼ 

1

 d þ 

1

   

The corresponding curvature is F

=

1

F



 ,

where F



= d þ   .

In the case of a quantum homogeneous bundle

with calculus determined by Q

2 P

and

= (Q

) (cf. the sub section ‘‘Quan tum homo-

geneous bundl es’’), a canonic al con nection form can

be assigned to any algebra map i : A !P such that

1.   i = id (i-splits );

2. "

 i = "

(co-unitality);

3. (id )  Ad

 i = (i id)  Ad

(Ad-covariance);

and

4. i(Q

)  Q

(differentiability).

Explicitly, !([a]) =

(Si(a)

(1)

)di(a)

(2)

Covariant Derivative: Strong Connections

A covariant derivative associated to a connection 

is a map D : P !P

(B)P, p 7!dp  (dp). A covar-

iant derivative maps elements of P into horizontal

forms, since ker =P

(B)P, and satisfies the

Leibniz rule D( bp) = (db)p þ bDp, for all b 2B,

p 2P.

A connection is ‘‘strong’’ provided D(p) 2

(B)P.

A covariant derivative of a strong connection is a

connection on module P in the sense of Connes.

Furthermore, in the universal calculus case, and when

A has invertible antipode, the existence of strong

connections leads to rich gauge theory of associated

bundles (cf . the subsec tion ‘‘Assoc iated bundl es:

matter fields ’’). A connect ion in a trivia l bundle

Quantum Group Differentials, Bundles and Gauge Theory 241