Geometry and quantum physics

408 Pages·2000·1.946 MB·English

by  Gausterer H., Grosse H., Pittner L. (eds.)

Preface
In many areas of theoretical physics, geometrical and functional analytic
methods are used simultaneously. During the course of our studies and edu-
cation, these methods appeared to be extremely di(cid:11)erent. For classical (cid:12)eld
theories, a di(cid:11)erential complex over smooth manifolds, smooth sections of
vector bundles over manifolds, principal (cid:12)bre bundles, connections, curvatu-
res, etc. are introduced. For quantum (cid:12)eld theory, singular operator-valued
distributionsactonFockspaceandleadtoallthewell-knowndi(cid:14)culties.Af-
tertheworkofAlainConnesandthemanyapplicationsofnon-commutative
geometry to various subjects, we became familiar with, for example, the fact
that on any associative superalgebra at least one di(cid:11)erential calculus exists.
Butclearlytherearemanymoreaspects.Weselectedsevensubjectsandlec-
turerstocoversymplecticgeometry,quantumgravity,stringtheory,anintro-
duction to non-commutative geometry, and the application to fundamental
interactions, to the quantum Hall e(cid:11)ect as well as to physics on q-deformed
spaces.
In the (cid:12)rst contribution, Anton Alexseev uses techniques of symplec-
tic geometry to evaluate certain integrals over manifolds that have special
symmetries. For them the stationary phase approximation becomes exact.
Starting from simple examples, he introduces equivariant cohomology and
sketches the proof of localization formulas of Duistermaat - Heckman type.
The Weil model, well-known from the BRST formalism, is introduced, and a
non-commutative generalization is used to prove group-valued localization.
John Baez elucidates the present status of quantum geometry of space-
time. Since the implementation of constraints within the Hamiltonian ap-
proach to Einstein gravity is complicated, he explains in detail the simplify-
ing BF system connected to the Chern - Simons model. We learn about
spin net works and spin foams, become familiar with triangulations of four-
manifolds,andcalculatespectraofquantumtetrahedra.Hislecturenotesnot
only survey the subject, but also give an extensive list of references arranged
according to nine di(cid:11)erent subjects.
Cesar Gomez then gives an introduction to string theory. Starting from
themodeexpansionheexplainsT-duality,asymmetryrelatingclosedstrings
of radius R and 1=R. Next D-brane ideas are represented.
Both quantum gravity and string theory have their own way of treating
space-time. But there is a further approach that goes by the name of non-
commutative geometry. This is the subject of the chapter by Daniel Kastler.
John Madore introduces a di(cid:11)erential complex over an arbitrary associa-
tive algebra. We learn that, depending on the procedure, a regularization
e(cid:11)ect may result. Attempts to add a gravitational (cid:12)eld are also reviewed.
This introduction is useful to follow the lectures of Daniel Kastler. He
spoke about Connes’ approach to the standard model as well as recent ideas
forincludinggravityasanexternal(cid:12)eld.Hepresentstheinterestingideathat
VI Preface
additional dimensions (cid:12)nally lead via the Higgs e(cid:11)ect to masses of particles,
as well as the recent attempts to use SU (2) for q being a third root of unity
q
to \deduce" the gauge group of the standard model.
JuliusWessappliesideasofnon-commutativegeometrytotheq-deformed
Heisenberg algebra. The spectra of position and momentum are discrete.
Phase space gets a lattice-like structure. Higher-dimensional analogues are
investigated too.
Finally,RuediSeilerexplainsgeometricalpropertiesoftransportinquan-
tum Hall systems. The integer e(cid:11)ect is treated in detail.
Although we learn here about many di(cid:11)erent applications of non-com-
mutative geometry, the hope is that a unique picture of a quantum space-
timemay(cid:12)nallyresult.Quantumtheorychangesourideasongeometry,and
further surprises may come along soon.
We have tried to cover many subjects and were lucky to be supported by
so many excellent lecturers. We hope that the school helped young people
seekinganentrytothesesubjects.Theywillhopefullyremembertheexcellent
atmosphere of the Schladming Winter School 1999.
Atthispointwealsowanttoexpressourthankstothemainsponsorofthe
School, the Austrian Ministry for Science and Transportation. In addition,
we grateful acknowledge the generous support by the Government of Styria
and the Town of Schladming. Valuable help towards the organization was
received from the Wirtschaftskammer Steiermark (Sektion Industrie), Steyr-
Daimler-Puch AG, Ricoh-Austria, and Styria Online.
Organizing the 1999 Schladming Winter School, and making it the suc-
cessful event that it was, would not have been possible without the help of
a number of colleagues and graduate students from our institute. Without
naming them all we want to acknowledge the traditionally good coopera-
tion within the organizing committee and beyond, which once again guaran-
teed the smooth running of all organizational, technical, and social matters.
Thanks are due to Wolfgang Schweiger for his valuable technical assistance.
Finally we should like to express our sincere thanks to Miss Sabine Fuchs
for carrying out the secretarial work and for (cid:12)nalizing the text and layout of
these proceedings.
Graz, November 1999 H. Gausterer, H. Grosse, L. Pittner
Contents
Notes on Equivariant Localization
AntonAlekseev . . . . . . . . . . . . . . . . . . . . . . . . 1
An Introduction to Spin Foam Models
of BF Theory and Quantum Gravity
JohnC.Baez . . . . . . . . . . . . . . . . . . . . . . . . . 25
T-Duality and the Gravitational Description
of Gauge Theories
CesarGomez and PedroSilva . . . . . . . . . . . . . . . . . . 95
Noncommutative Geometry and Basic Physics
DanielKastler . . . . . . . . . . . . . . . . . . . . . . . . 131
An Introduction to Noncommutative Geometry
JohnMadore . . . . . . . . . . . . . . . . . . . . . . . . . 231
Geometric Properties of Transport in
Quantum Hall Systems
ThomasRichter and RuediSeiler . . . . . . . . . . . . . . . . 275
q-Deformed Heisenberg Algebras
JuliusWess . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Abstracts of the Seminars
(given by participants of the school) . . . . . . . . . . . . . . . 383
Notes on Equivariant Localization
AntonAlekseev
Institutionenf¨orTeoretiskFysik,UppsalaUniversitet,Box803,S-75108,Uppsala,
Sweden
Abstract. We review the localization formula due to Berline-Vergne and Atiyah-
Bott, with applications to the exact stationary phase phenomenon discovered by
Duistermaat-Heckman. We explain the Weil model of equivariant cohomology and
recall its relation to BRST. We show how to quantize the Weil model, and obtain
new localization formulas which, in particular, apply to Hamiltonian spaces with
group valued moment maps.
1 Introduction
The purpose of these lecture notes is to present the localization formulas
for equivariant cocycles. The localization phenomenon was (cid:12)rst discovered
by Duistermaat and Heckman in [DH], and then explained in the works of
Berline-Vergne[BV]andAtiyah-Bott[AB].Themainideaofthelocalization
formulas is similar to the residue formula: a multi-dimensional integral is
evaluated exactly by summing up a number of the (cid:12)xed point contributions.
In Section 2 we review the localization formula of [BV] and [AB]. We use
an elementary example of the sphere S2 as an illustration. Then, we outline
the relation between the localization formulas and Hamiltonian Mechanics,
and recover the Duistermaat-Heckman formula [DH].
In Section 3 we discuss the relations between the localization formulas
and the group actions. In the case of the Duistermaat-Heckman formula,
localization is intimately related to the symmetry group of the underlying
Hamiltoniansystem.Inparticular,wecomparetheequivariantdi(cid:11)erentialto
the BRST di(cid:11)erential.
InSection4weexplainhowtoquantizetheequivariantcohomology.This
Sectionisbasedonthepapers[AMM],[AM],[AMW1]and[AMW2].Weend
up by presenting the new localization formula which is derived in [AMW2].
Some simple applications of this new formula can be found in [P]. Section 4
is based on the joint works with A.Malkin, E.Meinrenken and C.Woodward.
These notes do not touch upon various applications of localization for-
mulas in Physics. Usually, one proceeds by extrapolating the localization
phenomenon to path integrals. Some of the most exciting examples of this
approach can be found in [MNP], [W2], [G], [BT], [MNS]. In fact, [W2] was
the original motivation for the formulas of Section 3.
I am grateful to the organizers and participants of the 38th Schladming
Winter School for the inspiring atmosphere!
H. Gausterer, H. Grosse, and L. Pittner (Eds.): Proceedings 1999, LNP 543, pp. 1−24, 2000.
(cid:211) Springer-Verlag Berlin Heidelberg 2000
2 AntonAlekseev
2 Localization Formulas
InthisSectionwereviewthelocalization formuladuetoBerline-Vergne[BV]
andAtiyah-Bott[AB].Itisthenusedtoderivetheexactstationaryphasefor-
mula due to Duistermaat and Heckman [DH]. The presentation is illustrated
at the elementary example of sphere S2.
2.1 Stationary Phase Method
In this section we recall the stationary phase method. It applies when one is
interested in the asymptotic behavior at large s of the integral
Z
1
I(s)= dxeisf(x)g(x): (1)
−1
Hereweassumethatfunctionsf(x)andg(x)arereal,andsu(cid:14)cientlysmooth.
Atlarges>0theleadingcontributionintotheintegral(1)isgivenbythe
neighborhoodofthecriticalpointsoff(x),whereitsderivativeinxvanishes.
Let x be such a critical point. Then, one can approximate f(x) near x by
0 0
the (cid:12)rst two terms of the Taylor series,
1
f(x)=f(x )+ f00(x )(x−x )2+:::;
0 2 0 0
where ::: stand for the higher order terms.
The leading contribution of the critical point x into the integral I(s) is
0
given by a simpler integral
Z
1
I0(s)=g(x0)eisf(x0) dxe2isf00(x0)(x−x0)2:
−1
This integral is Gaussian, and can be computed explicitly,
(cid:18) (cid:19)
I0(s)=g(x0)ei(sf(x0)+"(cid:25)4) sjf020((cid:25)x )j 12
0
Here " is the sign on the second derivative f00(x ).
0
A similar formula holds for multi-dimensional integrals,
Z
I(s)= dnxg(x)eisf(x): (2)
Again, the leading contribution into the asymptotics at large s is given by
thecriticalpointsoff(x),whereitsgradientvanishesrf =0.Atthecritical
point x one can expand f(x) into the Taylor series,
0
1 Xn
f(x)=f(x )+ f00(x )(x−x ) (x−x ) +:::; (3)
0 2 ij 0 0 i 0 j
i;j=1
Notes on Equivariant Localization 3
where
@2f
f00 = :
ij @x @x
i j
We assume that the critical point is non-degenerate, that is, the matrix f00
ij
is invertible. Then, the leading contribution of x into the integral I(s) is of
0
the form,
(cid:18) (cid:19)
I0(s)=g(x0)eisf(x0) 2s(cid:25) n2 jdet(fei0(cid:27)0((cid:25)4x ))j1: (4)
0 2
Here (cid:27) =(cid:27) −(cid:27) is the signature of the matrixf00, (cid:27) and (cid:27) are numbers
+ − ij + −
of positive and negative eigenvalues, respectively.
In general, one can have several critical points. Then, on can add the
leading contributions (4) to obtain the approximate answer for I(s),
(cid:18) (cid:19)
I(s)(cid:25) 2s(cid:25) n2 Xg(xi)eisf(xi) jdet(efi(cid:27)00i((cid:25)x4 ))j1 (5)
i i 2
Ofcourse,thereisnoreasonfortherighthandsidetobetheexactanswerfor
I(s).Butsometimesthisisthecase!Suchasituationiscalledexactstationary
phase, and will be studied in these notes.
Example: sphere S2 The simplest example of the exact stationary phase
phenomenon is the computation of the following integral. Consider the unit
sphere S2 de(cid:12)ned by equation x2+y2+z2 =1. We choose g(x;y;z)=1 and
f(x;y;z)=z. Then, the integral I(s) is of the form,
Z
I(s)= dA eisz; (6)
S2
R
where dA is the area element normalized in the standard way, dA=4(cid:25).
S2
The critical points of the function f(x;y;z) = z are the North and the
South poles of the sphere. At both points one can use x and y as local
coordinates to obtain,
1
z (cid:25)1− (x2+y2)
2
near the North pole, and
1
z (cid:25)−1+ (x2+y2)
2
near the South pole. Thus, for the stationary phase approximation one ob-
tains,
2(cid:25) sin(s)
I(s)(cid:25) (−ieis+ie−is)=4(cid:25) : (7)
s s
HerewehaveusedthatatbothNorthandSouthpolesdet(f )=1,andthat
ij
(cid:27) =−2 and (cid:27) =2.
N S
4 AntonAlekseev
Now we can compare the ‘approximate’ result (7) with the exact calcu-
lation. It is convenient to use polar angles 0 < (cid:18) < (cid:25);0 < (cid:30) < 2(cid:25). Then,
the coordinate functionz =cos((cid:18)), and the area element is dA=dcos((cid:18))d(cid:30).
The simple calculation gives,
Z
eis−e−is sin(s)
I(s)= dcos((cid:18))d(cid:30)eiscos((cid:18)) =2(cid:25) =4(cid:25) : (8)
is s
This expression coincides with the stationary phase result (7).
In the following sections we shall see that the equality of the exact and
approximate results (7) and (8) is not a coincidence. In fact, this is the
simplest example of equivariant localization.
2.2 Equivariant Cohomology
Stokes’s theorem and residue formula The main tool in proving the
localization formula will be the generalization of the Stokes’s integration for-
mula. The latter states that given an exact di(cid:11)erential form (cid:11), (cid:11) = d(cid:12), its
integral over the domain D can be expressed as an integral of (cid:12) over the
boundary of D, Z Z
d(cid:12) = (cid:12): (9)
D @D
As a warm up exercise we prove the standard residue formula using the
Stokes’s formula (9). Given a function f(z) analytic on the complex plane
with the exception of some (cid:12)nite number of poles, its integral over a closed
contour C is given by the sum of residues at the poles inside C,
Z
1 X
f(z)dz = res f: (10)
2(cid:25)i zi
C i
We naturally choose (cid:12) in the form,
1
(cid:12) = f(z)dz;
2(cid:25)i
thedomainD istheinteriorofC,anditsboundaryisC.Theform(cid:11)isgiven
by equation,
1 1
(cid:11)=d(cid:12) = d(f(z)dz)= (@(cid:22)f)dz(cid:22)^dz;
2(cid:25)i 2(cid:25)i
where @(cid:22)f is the partial derivative in z(cid:22). The function f(z) is analytic. Hence,
(cid:11) vanishes everywhere except for the poles. We conclude, that (cid:11) is a distri-
bution supported at some number of points. Such a distribution is a sum of
(cid:14)-functions and its derivatives. The only terms which contribute into the in-
tegralof(cid:11)overD are(cid:14)-functionsatthepoles.Theygiverisetotheresidues,
res f
@(cid:22) zi =(res f)(cid:14)(z−z):
2(cid:25)i(z−z ) zi i
i
Notes on Equivariant Localization 5
Now we use the Stokes’s formula,
Z Z
1 X X
f(z)dz = (res f)(cid:14)(z−z)= res f;
2(cid:25)i zi i zi
C D i i
and recover the residue formula.
S1-action and (cid:12)xed points The derivation of the localization formula re-
quires more structure on the integration domain. In particular, the notion of
symmetry plays an important role. We assume that our symmetry is conti-
nuous. In particular, this may be the action of the circle group S1, which is
our main example in this Section. For instance, in the case of S2 there is an
action of S1 by rotations around the z-axis.
We always choose the integration domain to be a compact manifold M
without boundary (as in the case of S2). The S1-action de(cid:12)nes a vector (cid:12)eld
v = @=@(cid:30) on M. Zeroes of v correspond to (cid:12)xed points of the circle action.
For simplicity, we assume that all the (cid:12)xed points are isolated. This is only
possible if the dimension of the manifold is even, n=2m. Given such a (cid:12)xed
point x one can write the action near this point as
0
xi((cid:30))=xi +Ri((cid:30))(x−x )j +:::;
0 j 0
where:::standforhigherordertermsinx−x .Itiseasytoseethatonecan
0
linearize the action (drop higher order terms). The matrix R gives a linear
representation of S1,
R((cid:30) )R((cid:30) )=R((cid:30) +(cid:30) );
1 2 1 2
and satis(cid:12)es condition R(2(cid:25)) = id. By the appropriate choice of the basis
suchamatrixcanalwaysberepresentedasadirectsumof2(cid:2)2blocks,each
block of the form (cid:18) (cid:19)
cos((cid:23)(cid:30)) sin((cid:23)(cid:30))
;
−sin((cid:23)(cid:30))cos((cid:23)(cid:30))
where (cid:23) is an integer. We can denote the corresponding local coordinates by
x ;y where i = 1:::m. In these local coordinates the vector (cid:12)eld v has the
i i
form,
(cid:18) (cid:19)
Xm @ @
v = (cid:23) x −y :
i i @y i @x
i i
i=1
The integers (cid:21) are called indices of the vector (cid:12)eld v at the point x .
i 0
In fact the indices (cid:23) are de(cid:12)ned up to a sign: the flip of coordinates
i
x and y changes the sign of the corresponding index. In what follows we
i i
shall need a product of all indices corresponding to the given (cid:12)xed point,
(cid:23) :::(cid:23) . It is well de(cid:12)ned is the tangent space at the (cid:12)xed point is oriented:
1 m
one should choose the coordinate system (x ;y ;:::;x ;y ) with positive
1 1 m m
orientation. This condition determines the product of indices in a unique
6 AntonAlekseev
way.Inparticular,ifthemanifoldisoriented,theproductsofindicesat(cid:12)xed
points are well de(cid:12)ned.
In the following we shall use that one can always choose an S1-invariant
metriconM.Indeed,givenanymetric,onecanalwaysaverageitovertheS1-
action. In particular, this metric at the (cid:12)xed point x can always be chosen
0
in the form,
Xm
g = (dx2+dy2): (11)
i i
i=1
Equivariant di(cid:11)erential Now we are ready to de(cid:12)ne the equivariant di(cid:11)e-
rential,andequivariantcohomology.Wede(cid:12)nethespaceofequivariantforms
on M as S1 invariant di(cid:11)erential forms with values in functions of one varia-
ble,whichwedenoteby(cid:24).Typically,suchadi(cid:11)erentialformisapolynomial,
XN
(cid:11)((cid:24))= (cid:11)s(cid:24)s;
s=0
where (cid:11)s are S1-invariant di(cid:11)erential forms. We shall also need equivariant
forms with more complicated (cid:24)-dependence.
Sometimes it is convenient to decompose equivariant forms according to
the degree,
Xn
(cid:11)((cid:24))= (cid:11) ((cid:24));
j
j=0
where (cid:11) ((cid:24)) is a form of degree j which takes values in functions of (cid:24).
j
The di(cid:11)erential on the space of equivariant forms is de(cid:12)ned by formula,
d =d+i(cid:24)(cid:19) ; (12)
S1 v
where (cid:19) is the contraction with respect to the vector (cid:12)eld v. One can assign
v
to parameter (cid:24) degree 2 in order to make the equivariant di(cid:11)erential homo-
geneous. Unfortunately, this arrangement is only meaningful for equivariant
di(cid:11)erential forms polynomial in (cid:24).
It is the basic property of the di(cid:11)erential (12) that it squares to zero on
the space of equivariant forms. Indeed,
d2 =(d+i(cid:24)(cid:19) )2 =i(cid:24)(d(cid:19) +(cid:19) d)=i(cid:24)L ;
S1 v v v v
where we have used Cartan’s formula for L . The Lie derivative L vanishes
v v
on equivariant forms, and so does d2 .
S1
Usingthedi(cid:11)erential(12)onecande(cid:12)neequivariantlyclosedforms,d (cid:11)=
S1
0, and equivariantly exact forms (cid:11) = d (cid:12). Because d2 = 0, equivariantly
S1 S1
exact forms are automatically equivariantly closed, and one can de(cid:12)ne equi-
variant cohomology H (M) as the quotient of the space of (equivariantly)
S1
closed forms by the space of (equivariantly) exact forms. If (cid:11)((cid:24)) is an equi-
variant cocycle, it satis(cid:12)es the closedness condition,
Notes on Equivariant Localization 7
(d+i(cid:24)(cid:19) )(cid:11)((cid:24))=0;
v
which implies a number of equations for the forms (cid:11) ((cid:24)),
k
d(cid:11) ((cid:24))+i(cid:24)(cid:19) (cid:11) ((cid:24))=0: (13)
k−2 v k
Note that this recurrence relation has step 2. Hence, odd and even degree
parts of an equivariant cocycle are also equivariant cocycles. If the manifold
M is even dimensional, the closedness condition relates the top degree com-
ponent (cid:11) ((cid:24)) and the function (cid:11) ((cid:24)). We shall see that exactly this relation
n 0
is used in the localization formula.
The Stokes’ integration formula generalizes to equivariantly exact forms.
Indeed,
Z Z
(d+i(cid:24)(cid:19) )(cid:12) = (cid:12);
v
D @D
R
where (cid:19) (cid:12) = 0 because the integrand has a vanishing top degree compo-
D v
nent (at least one degree is eaten up by (cid:19) ). In particular, if the integration
v
domain has no boundary, the integral of any equivariantly exact form vanis-
hes, and the integration map descends to equivariant cohomology. That is,
given a class [(cid:11)]2H (M) one can choose any representative (cid:11) in integrate
S1
it over M. The result is a function of (cid:24) which is independent of the repre-
sentative: the representatives di(cid:11)er by an exact form, and the integral of an
exact form vanishes.
The localization formula is a tool of computing integral of equivariant
cocycles in terms of (cid:12)xed points. This formula was discovered by Berline-
Vergne [BV] and by Atiyah-Bott [AB]. For an equivariant cocycle (cid:11)((cid:24)), its
integral over M is given by,
Z (cid:18) (cid:19)
(cid:11)((cid:24))= 2(cid:25) n2 X ((cid:11)0((cid:24)))(xp); (14)
i(cid:24) (cid:23)p:::(cid:23)p
M p 1 m
where the index p labels (cid:12)xed points of the circle action (we assume that all
of them are isolated), and (cid:23)p;:::;(cid:23)p are indices of the p’s (cid:12)xed point.
1 m
Note that the integral on the left hand side of (14) depends only on the
top degree component (cid:11) ((cid:24)) of the cocycle (cid:11)((cid:24)). At the same time the right
n
hand side is expressed in terms of the zero degree component (cid:11) ((cid:24)). This is
0
possiblebecause(cid:11) ((cid:24))and(cid:11) ((cid:24))arerelatedbytherecurrencerelations(13)
n 0
expressing closedness of (cid:11)((cid:24)).
Also note that even if the equivariant form (cid:11)((cid:24)) is smooth at (cid:24) = 0, the
right hand side of (14) contains the divergent factor (cid:24)−n=2. The singularity
at (cid:24) =0 is canceled by the sum of contributions of (cid:12)xed points, which has a
zero of degree n=2 at (cid:24) =0. This idea leads to reside formulas [JK].