Topological field theories

Nuclear Physics B (Proc. Suppl.) 15 (1990) 125-134 North-Holland

TOPOLOGICAL

FIELD THEORIES*

Antal Jevicki Department of Physics, Brown University, Providence, RI 02912

INTRODUCTION During the last year very interesting progress has

the quantum mechanical theory of spin and it serves as

been made in understanding the structure of certain

an illustrative example for quantization of systems with

three (2-1-1) dimensional field theories{ l] . These theories

finite phase space.

are based on actions of Chern-Simous typeM and turn out to be exactly soluble.

In Section HI general non-abelian Chern-Simons theories are discussed.

It is shown how their canonical

A Chern-Simons gauge theory based on the abelian

quantization on nontrivial Riemann surfaces leads to

U(1) group gives a model of fractional statistics and is of

a system with finite number of degrees of freedom. The

relevance to some novel theories of high Tc superconduc-

connection with conformal field theory is then explained

tivity[S]. Non-abelian C-S theories, when quantized on

using the abelian example. Implications to knot theory

a nontrivial Riemann surface, exhibit close connection

and quantum gravity are then discussed in Section IV.

with conformal field theories in 2d[4,5,6]. As such, they offer a framework for classification of 2d conformal field

II. SPIN - A SYSTEM WITH A COMPACT

theories.

PHASE SPACE

In general, a Chern-Simons theory can be shown to carry a finite number of quantum mechanical degrees of freedom which are of topological origin. The dynamics

The simplest nonabelian theory of Chern-Simons type is given by a Lagrangian for spin. In quantum mechanics spin is given by

of these theories is then given by the dynamics of topological excitations. In this way one achieves a connec= , ( , + 1)

tion with knot theory[ 1J]. Gravity in 3 dimensions belongs to these types of theories Is]. A canonical counting shows that it contains no degrees of freedom, nonetheless, in perturbation theory it looks nonrenormalizable and quite novtrivial.

The Chern-Simons formulation

renders this theory solvable [9]. The outline of this review is as follows. In Section

and the associated 2s + 1 dimensional Hilbert space )/e = {[s,m >

m = -s,...,+s}.

question of what is the classical system whose quantization gives this finite dimensional spin Hilbert space. A formal answer is often given in terms of anticommuting Grassman variables

II a simple non-abelian C-S theory is described. It is * Lectures given at the Vlth Adriatic Meeting on Particle Physics, Dubrovnik, Yugoslavia, June 12-22, 1989. 0920-5632/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holl~nd)

One can ask the

A. Jevicki/ Topological field theories

126 & = ~+n~.

rameters (u,t) such that ue(O,1) and te(O,2~r).

For

u - O, one has a point on the north pole, while for u - 1, Here ~ -

(~2) is a two dimensional Grassman vector

with the Poisson brackets {{~i,~1-} _. 6iy.

one is on the equator which represents the boundary. The Lagrangian then becomes

#

This answer, although formally correct, is not satisfactory. There actually exists a canonical description of

L-/

dt /

duff. COtff×cgnff)

0

spin in terms of standard bosonic variables. The only unusual feature of the bosonic phase space is that it is

and depends only on the values at the equator if(t).

compact and it is the compactness which induces a finite

Consequently one has a 1-dimensional theory. To obtain

number of states in the Hilbert space.

an explicit form for the Lagrang~an in terms of K(t), one will have to break manifest rotational invariance.

The phase space for spin is simply the two dimensional sphere ,q2. A point on the sphere is given by the

W e can use the stereograpbic variables f(t),~(t)

unit vector

nl + ins = i = (m,n2,n3)

i2--1.

2f

1 + I~l2

nl - in2 =

2~

1 + I~l 2

An infinitesimal area element defines a two form

1 -I~12 ,,8 = -1- -+i I~1

Onj On#, to = Eiykni bz Op = a .

(a,i

x o,i).

•

The 2-form becomes

,..

Here the coordinates (z, y) parametrize points on the

(

df A d~"

fd~

sphere and tgzl, tgy5 are the linearly independent tangent vectors. In general, a 2-form induces the Poisson

which is integrated to the Lagrangian

brackets according to fd~

s

/

"-- ._

ff - ff

{F, a } p s = ~, ( V F , v a ) . In the present case

In terms of the basic spin variable ff = s i the La-

{F, C} =

aF aa

grangian reads

eo'kni Oni Onl~

and the basic Poisson brackets are s+88

{ni, nj} = eoknt~ .

"

It is not manifestly rotational invariant, instead it trans-

To derive a Lagrangian for the canonical variables if(t) one integrates the 2-form

forms as a total derivative under rotations. This is a

L=fw

tion of the coupling constant (2s = integer) is one other feature of a C-S theory. The above form for the La-

Parametrize the upper hemisphere in terms of two pa-

grangian for spin was originally derived using coherent

typical feature of a Chern-Simons theory. A quantiza-

A. 3evicki/Topological field theories

127

states of SU(2). This formulation in terms of bosonic

then defines the 2$ + 1 dimensional Hilbert space of the

phase space variables was useful for semiclassical stud-

quantum mechanical spin.

ies (see ref. [11]). The path integral representation is

This procedure for quantisation is generally applicable to theories with x Kahler potential [zsi.

simply

z = / pg(t)~t6(g ~ - ,21ei"(~'~)

III. CHERN-SIMONS THEORIES A general non-AbeUan Chern-Simons theory is de-

and one can approximate it through classical trajecto-

fined by the Lagrangian

ries. 2

= k e~Tr(a~AvA~ + ~A~AvA~)

In the operator formalism, quantization can be performed following the method of geometric quantization. The problem of quantization is interesting and nontrivial since one is to produce a finite dimensional Hilbert

where euvx is the totally antisymmetric tensor in 2+1 dimensions aad A l, =

A~T"

is a gauge potential associ-

ated with a group G. The equation of motion following

space.

is In geometric quantization, one represents the wavefunctions as analytic functions @(f). The scalar product is specified as follows. The two-form can be written as

oJ = -iafa~h where

so that A~ is (locally) a pure gauge. The Lagrangian is not gauge invariant but transforms as

+ k-~Tr(g-la~g g-man9g-la~g). h = 2sin (1 +

3

tfl 9)

The last term is a topological charge density for a mapdefines the Kahler potential. In this case, one postulates

ping 83 -* G. In case of a compact non-Abelian group,

the scalar product

its integral is an integer. The exponential of the action

(~i,~2) = f dpeh~l~2 where

dp

is an invariant integration measure. For the

exp(i f d3z£) is gauge invariant, however, if the coupling constant is quantized 4~rk = integer. Quantization of general Chern-Simons theory on a nontrivial Riemannian manifold leads to an interesting

sphere

dfd~ d/~-

system with a finite number of degrees of freedom. Con-

(1 -I-Icl2) z '

sider M3 = M2 x R where the real coordinate R rep-

taking into account the contribution of the Kahler po-

resents the time and space is represented by a general

tential, one has

2-manifold M2. This can be taken as a Riemann surface with g handles. The canonical decomposition of

fdfd~ (,z,~,,z,2)= 2~ (1 +1 ~,),._,7},4, The following basic set of analytic functions

#.(~,) = N~f °+~

m =-8,...8

the action then reads

L=~

~2xtr M2

*

A. Jevicki/ Topological field theories

128

with the canonical commutators [A,(z), As(y)] - c i j 6 ( z -

theorem states that

y). A0 is a Lagrange multiplier leading to the constraint

t-f-v=2g-2 where f, l and v represent the number of triangles, links,

= O, Ai - OiA, + [A,, Ai] = 0 .

and vertices respectively. On each triangular simplex In flat space this would imply that ,4/is a pure gauge

the condition F~, = 0 implies a pure gauge configura-

Ai = UOiU+ and that there are no physical degrees of

tion

freedom. More precisely one considers the gauge invari-

A (") -- g.(z)cg~g~ 1 .

ant Wilson loops When crossing from one simplex to the neighboring one we have to allow a transition matrix on each link go = u ( o g o , . For them Fij = 0 implies that the contour can be arbitrarily deformed and possibly shrunk to zero. On a

This leads to group variables whose number is equal to

nontrivial Riemann surface however there are 2g topo-

the number of links = l. Gauge transformations are associated with simplices (vertices of the dual lattice)

logically distinct closed loops that cannot be deformed to zero (the ai and bi holonomies, i = 1 , . . . ,g). Conse-

u{l)

quently one has 2g group variables, Ui, Vi representing

g(f)u(og(f; + .

the nontrivial holonomies. They obey the constraint We still have curvature concentrated at the vertices. De= Ul V,

U2V2UQ vi

...

= 1

manding ~ j -- 0 everywhere gives the constraints

IIU=I

which signifies the fact that this composite loop is trivial. One also has global symmetry

for U's merging into a vertex. Consequently the number U~ -, gU~o + --, oV~g +

of degrees of freedom is l - f - v which by the GaussBonet theorem equals 2g - 2. We have found the same

which is a remainder of local gauge invariance. Conse-

number of degrees of freedom as in the Wilson loop

quently one finds 2g - 2 independent matrix degrees of

approach. The problem of writing down a lattice action

freedom. The quantization of this system represents a

for C-S theories is interesting and nontrivial [14].

problem of quantizing a compact phase space. This was completely done for the abelian theory but not yet for the most general case. Let us now describe a lattice model which is use. ful for exhibiting and counting these discrete degrees of freedom. Triangulate the Riemann surface in terms of

Let us now turn to the relationship of Chern-Simons theories and conformal field theory. The simplest way to exhibit such a relationship is to take the space M2 to be a disk M2 - D. In this case Ai = UcgiU-l everywhere. The Chern-Simons action simply reduces to s(u) =

triangular simplices which are glued on links with the curvature centered at the vertices. The Gauss-Bonet

+

k / Tr(U_IS÷U U_lcgtU)d~dt

./

TrlU-

OU)3

A. Jevicki / TopologicaI field theories where ~ p~,rametrizes a circle 8*. This action depends

129

The solutions are obtained as follows. One

only on the boundary values U(@, t). This two-dimensional

that Fzm generates gau~e transfom~tions tkrough the

Lagrangian is nothing but the chiral Wess-Zumino c-

unitary operator

model. The gauge fields A÷ are the currents generating

U(g) = exp{i[ A'F2~} J

a Kac-Mood7 algebra. Quantization of Chern-Simom theory on Riemann

w i t h g = e iA a group element. O n e h a s

surfaces leads to a correspondence with conformal field theory (of Wess-Zumino-Witten type).

U + ( g ) A z ( f (g) =- A~ = g-ZAzg - g-lcg=g .

More specifi-

cally, the Hilbert space of the 2+1 dimensional gauge

Because of the last term in

theory turns out to give generating functionals of currents in the nonlinear o-model. Consider the canonical

k

-

+

Am,

-

OIAz

quantization with complex coordinates (complex strucwe cannot say that

ture on M2):

[Aaz(z),Ab(z')] -- T2~ri 6 a b 6 ( z--

O+(g)~(A) = ~(Ag) .

zl ) • Rather

Now the Hilbert space is represented by (analytic) wavefunctionals

U(g)'~,(.,4z) = exp { _ ~_~k/ d a z T r ,z, =

(Azg-ZS,g)+ik/(g)}~(A¢)

,Z,[A,]

where f(g) is an Am-independent function. From the which are holomorphic functions of A=. One has the standard scalar product

composition law 0(Oa)0(gl) = 0(g2ot) there follows a co-cycle condition

f DA,Dizexp{f

f(gzg2) =/(gl) + f(g2")+ ITr(g[ZOgagllOgz) "

•

One can const ct the function/(g) since it is given by The physical wavefunctions have to obey the Gauss law. One has

Fzm = 8mAz - 8zAm + [A,, Am]

where AFt,mis the first term in F. Introducing a parameter y e(0, I) we find

and the equation =

0.

~-~ f(gy) = Tr (g-'Szgg-'Sig) _

e.-~Tr(g-Z8ugg-Z8=gg-ZS~g)

In holomorphic quantization Am -

21ri 6 k ~Az

and the G~uss law becomes a functional differential equation basically equaling the Ward identity of currents in the 2-dimensional W-Z-W oomodel [el.

where g~ = eiyA. An integration produces the WessZumino-Witten Lagrangian

/(g) = Swzw(g) = f Tr {g-,o, gg-,O,g + F(g)} where the second term denotes the integral f eaD~g-ZOagg-1

A. Jevicki / Topological 6eld theories

130

A.

O/~gg-zO,lg. We see that O(g)~b(Az) = eO~S(g'A')~b(Ag)

= o.~ +

i.o~(zm.)~%(z)

where $1(z,g,) represents the pure gauge degrees of freedom and {ai} denotes the topological holonomy vari-

where

ables. U(I) gauge transformations now read

S ( g , A , ) = S w z w ( g ) - ~1 / Tr(Azg_ZOmg )

~(.,~) -~ ~(~,~) + A(~,~)

In the second term we recognize an external coupling to the current

a i --~ c ~ i + m i + T i j n

i

where m, n are integers. The Kahhr measure becomes

j+ = g-t cg~g . Now we recall that general, gauge invariant wavefunctionals are obtained by averaging over all gauge trans-

exp { - ~=-~k/ OAO~A} exp { - k ~ ' , • ([mr)-I • (:z)

formations. Taking, a test functional

~bo = • ~Tr(?A) we have the corresponding gauge invariant functional

¢(a) = f Pg exp{ikS(g, Az, Am = 3)}

and the wavefunctions factorize

~(A) = Z(~)¢(~). One requires the combination

This is the functional integral of the 2d WZW a-model with external field Az, Am coupled to the currents. Deriva-

e-k,r~'U,,,,)-'.oC)(a)o(a)

tives with respect to the external fields give the current to be invariant under the discrete gauge transforma-

correlation functions. This analysis shows that the problem of solving a Chern-Simons theory is equivalent to the problem of constructing a conformal invariant current algebra and vice versa.

tions. This results in nontrivial transformation properties for the wavefunctions (I)((~). In the scalar product one recognizes the symphctic form

w = -iTrkda. (Irnr) -1 • d a .

For the case of a U(1) group, a solution can be written down in closed form[Sl. On the Riemann surface one

The Kahler potential h then equals

has a basis of holomorphic one=forms wi(z) such that

f

h = _Tr..(~/~ _ a ) . (T. d~ - T . d~) wi=6ii

ai

for ai holonomies and

2

T = ( I r n r ) - z . r . (lrnr) -1 One then shows that the corresponding Hilbert space is

fwi=rii bi

for the bi contours. One parametrizes the gauge field

given by the functions[ s]

[' I

@'¢,,,¢,~,,(a)--0 2rk_ ~+~ 27r

(k~[kr)exp

-k~ ~a . (l,'nr)-' . a

)

A. Jevicki/ Topologicalfield theories where one has the generalized O-functions

I31

This leads the 0-functions of level k + h (h is the dual Coexter number, the combination k + h is farnillar for

O [b] (zlv') = ~exp{i~r(n+a).l"(n+a)+27ri(v~+a)(z+b)}

IV. _QUANTUM GRAVITY, KNOT THEORY

and @l, ~b~ are arbitrary phases. We have argued before that the basic degrees of freedom in the C-S problem are the Wilson loops of closed curves on the Riemann surface. Indeed the wavefunctions and the Hilbert space can be seen to give a representation of the holonomy algebra. One finds that exp(i / A)~¢..~,.,(a)

the Kac-Mood:- algebra of the WZW model).

= exp (i(~b' ~( -" -21rr)' ) ~'4,,,4'2,r

The 2+1 dimensional gravity, when written in the Vierbein formalism, exhibits a Chern-Simons structure. Consider the spin connection ~

and the Vierbein 4 "

The Einstein-Hilbert action is then written as

£

116~G / ~"~e,, (a,,,,,,,~- a,,,,,,~ + [,,,,,,,,,,~]')

6 One can interpret the pair (e~, w~) as a connection for

the group ISO (2,1). The generators of SO(2,1) are

exp(i f AJ'qh,q~.,v.(a ) = exp ( ~ ) bi The phases

(~1, ~2)

"÷,,,.,r+l •

denoted by J ' , and for translations we have P" with the Lie algebra

[J,, J~] = ~,bcj"

define different representations of

the loop algebra while r's label the states of a fixed

[J,,A] = c~,P '

representation.

[P,, Phi = 0 .

On the torus, M2 = T 2, one can solve the non-

The gauge field is then

abelian problem in a similar fashion. One wants to find

A, = e~P, + w ~ J , .

a cor~plete set of gauge invariant wavefunctions: One easily evaluates the curvature tensor

~b[Az]-" eikO(g'A')~b[Agz] . By a gauge transformation, the general non-abelian field

Az can

+

be reduced to a constant A , = g - ~ ( a , + a)g .

Then the scalar product becomes (~1, ~ 2 ) =

+

To evalute the Chern-Simons action for the ISO(2,1) gauge group, one can evaluate the topological density F ~ A F#~a#

f DADAexp [-ikS(gO;a,-a)ik

!

~a~] ~, (~)~2(~) •

+

where ~ab is the invariant quadratic form on the Lie algebra. For ISO(2,1) "Y,b=

In addition, a Jaeobian

Det(O+a) appears in the charge

of variables from A to g and a. Effectively, the scalar product becomes

/ dadaexp{ i(k2~r + h) Tr(aa)}~,(a)~b~(a)

(J~Pb) = 6=bgive the only

nonzero contribution and one finds

,,-,

- a . . t . +...)

+

b

¢

This is a total derivative, and when integrated over a

132

A. Jev~cki/Topological ~eld theories

four manifold with a boundary M, it gives precisely the

theory goes as follows. In a general gauge theory, the

Einstein-Hilbert action. This demonstrates that Ein-

gauge invariant observables are Wilson loops

stein gravity is a C-S gauge theory of ISO (2,1). w(c)s

= TrR e i § . A ' ~ = .

Canonical quantization of gravity can then be performed as for a general Chern-Simons gauge theory. In

Here the matrices are taken in an arbitrary representa-

the space,time decomposition M -- ~ × R and

tion R (so is the trace) and C represents an arbitrary closed contour. Expectation values of Wilson loops con-

c "e ~ ~ w ]

+ eS~"

-

+ wiwi]

tain all the physical information of the theory. In C-S gauge theory, a special simplification takes place: be-

E

cause of F~, = 0 a contour can be continuously deformed with the expectation value remaining the same.

+ ecbc(tdibe,c+ eibW,©)] } •

Consequently the expectation values depend only on the topology of loops. Already for a single contour we have

One sees that e~ and t~ are canonically conjugate to

nontrivial expectation values if the contour is knot-ed.

each other

Witten has succeeded in giving a computation procedure (based on conformai field theory) which can be =

y)

used to explicitly evaluate the knot-ed contours. They •

The variables e~ and w~ serve as Lagrange multipliers, the constraints enforce ISO(2,1) invariance. We see in the present gravitational case, a simplication compared

2fl

depend (for an SU(N) group) on q -- ex+x and that is where the connection with the invariant polynomials of Jones and Kauffman enters.

with a general gauge group: we can take the spin con-

Let us outline shortly the procedure for evaluation of

nection to be the coordinate (with the Vierbeins serving

Wilson loop expectation values. One has the functional

. momenta) and write a general wavefunctional as a

integral

function of w~ : ~ = gl[w~]. The constraints are then

Z ( M , C ) = / DAW(C)e '~o. .

imposed as follows. First w~ are not independent but

M

obey ~y (w) -- 0. So again the real variables correspond

In the 2+1 decomposition the Lagrange multiplier (A0)

to the holonomies on the Riemann surface ~'~. On each

term reads

holonomy we have the SO(2,1) phase factor matrices Ui and V/ i = 1,2,...g.

The e~) constraint requires

invariance under global gauge transformations rJ~ - , a u r a -~

,

~ - , n ~ f ~ -~ .

Here P is the point where the contour C punctures the 2 manifold ~"~. Consequently the "Gauss law" constraint

One can show then that the wavefunctions ~(Ui, Y~) ba-

now has a source term at the puncture

sically depend only on the Teichmuller variables of the Riemann surface ~ which is a set of 2g - 2 coordinates. The dynamics of quantum gravity in 2+1 dimensions reduces to the dynamics in the Teichmuller space[9,1°].

T a denotes the matrices defining the representation R.

The connection of Chern-Simons theories with knot

Obviously for several Wilson loops one has several source

13-3

A. Jevicki / Topological field theories

terms. A canonical approach for evMuating Wilson loops

For the group SU(N), using the k n o w ~

is then based on constructing the Hilbert space on

real field theory on sphere with four punctures, one can

with punctures.

explicitly find the coefficients a, ~ and -y:

There are several factorization properties that one

f

2~ri

of ~ o

)

has to take note of before proceeding. First of all, if two manifolds MI and M~ are joined along a sphere S~ into

( i . ( 2 - N - N~) ~

M one has z(M)z(s3) = z ( M d z ( ~ , )

.

( i , , ( 2 + iV - ~ )

"I = exp ~ N ( N "-I-k) /

Here 83 is the three sphere filling in S¢. In general, erie This then gives the numerical coefficients for unlinking

denotes (c)=

z(~,c) z(M)

the contour. "

As a simple application consider the three closed For unlinked loops one has that

(c,c~...c,,)

contours in the figure below

= n,,(c,,).

This factorization property means that we can reduce such problem to evaluation of a single loop. Let us now consider a single loop C with M being

All the contours involved are topologically equiva-

the simplest three manifold, the three sphere $3. What

lent and we denote such a simple contour by C. Then

Witten has succeeded in doing is in giving rules for un-

we have

knoting the contour. Concentrate on a particular region where two pieces of the contour pass each other.

,,z(c) + ~z(c)z(c) +,yz(c) = o

The unknoting is achieved by relating the three different ways of crossing (denoted in the picture)

which gives the expectation value for the single contour ( C ) --

a + "7 _ qN/~ _ q-N/Z ql/Z _ q-l/Z

"

9wl

Here q = e~-~. The above procedure is iteratively generalized to more complicated contours. It turns out that the three distinct configurations In a very nice application of this formalism Carlip [Is] are linearly related

has computed the scattering amplitude J two particles in 2+1 quantum gravity. The trajectories of the parti-

a f t + ~F: + -/I'3.

cles are represented by two separate Wilson loops since these correctly describe the coupling of external par-

134

A. Jevicki/Topological field theories

ticles to the spin connection. Evaluating then the ex-

K. Gawedzki and A. Kupiainen, Helsinki preprint

pectation value of the Wilson loops he has recovered the

(1988).

earlier result of 't Hooft and Deser et al.{s] It is clear that

[7] V. F. R. Jones, Ann. Math. 126, 335 (1987);

the Chern-Simons approach will lead to further beauti-

L. H. Kauffman, Topology 26, 395 (1987).

ful applications.

[8] G. 't Hooft, Comm. Math. Phys. 117, 495 (1988); S. Deser, R. Jackiw and G. 't Hooft, Ann. &Phys.

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