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

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stability subgroup H of 

(the group of unbroken

symmetries in this ground state):

H ¼fg 2 G : Dð gÞ

¼ 

g½3

In terms of this subgroup, we can find a useful

characterization of the manifold M of degenerate

ground states. As noted above, for each g 2 G,

U(g)j0i is also a ground state. However, these are

not all distinct, because clearly

U(gh)j0i=

U(g)j0i

for all h 2 H. Hence, the distinct ground states are

in one-to-one correspondence with the left cosets gH

of H in G,andM may be identified with the

quotient space G/H, the space of left cosets.

For example, suppose G is the rotation group

SO(3), and

f belongs to the three-dimensional

vector representation. If f 6¼ 0 in the groun d state,

we may choose f

= (0, 0, v). Then, clearly,

H = SO(2), the group of rotations about the z-axis,

and M= SO(3 )=SO(2) = S

, the 2-sphere. It is useful

to think of M as the subset of the order-parameter

space comprising the possible expectation values

f = h

i for the various degenerate ground states. For

example, in this case, M= {f: f

= v

Defect Formation

It is often possible to characterize the dynamics at

finite temperature in terms of a function of the order

parameter, the effective potential V(), which is

necessarily invariant under G, and whose minima

define the equilibrium states. At low temperatures, it

has a form like V = (f

 v

)

, whose minima

occur at nonzero values of f. But above the critical

temperature T

, the only minimum is at f = 0, so the

equilibrium state is symmetric under G. In the high-

temperature phase, there may be large fluctuations

f, but its mean value will be zero.

Now, when the system is cooled through the

phase tra¨nsition,

 will acquire a nonzero expecta-

tion value, gradually approaching one of the

degenerate ground states characterized by a point

of M. But the choice of which one is unpredict-

able; the symmetry breaking is spontaneous.

Moreover, in a large system, there is no reason

why the same choice should be made everywhere.

For example, a ferromagnet cooling through its

Curie point may acquire a spontaneous magneti-

zation in differe nt direct ions in different parts of

the sample.

Of course, there is an energetic penalty to having

a spatially varying order parameter, so it will tend to

become more unifor m as the temperature is lowered.

But the question arises whether there may be any

topological obstruction to this process. It can

happen that if we choose points on M in a

continuous manner everywhere around the periph-

ery of some region, it is topologically impossible to

complete the process throughout its interior.

Continuity may require that there are points where

 leaves the surface M. For example, if our

ferromagnet has two opposite possible directions of

easy magnetization, described by f

and f

, then

M consists essentially of these two points. Regions

where f  f

and where f f

must be separated

by domain walls across which f varies smoothly

from one to the other.

Homotopy Groups

To classify the various possible types of defect, we

need to consider the homotopy groups of the

manifold M of degenerate ground states. In this

section, we briefly review the necessary definitions.

A path in M is a map  : I !Mfrom the unit

interval I = [0, 1]  R. We choose a base point m

M (which may be identified with 

), and consider

loops in M, paths such that (0) = (1) = m

.We

say that two loops are homotopic, and write   ,

if one can be continuously deformed into the other

within M, that is, if there exists a map  : I

such that

ð0; tÞ¼ðtÞ and ð1; tÞ¼ ðtÞ½4

for all t, and

ðs; 0Þ¼ðs; 1Þ¼m

½5

for all s. This is an equivalence relation. The set



(M) is the set of equivalence classes [] of loops

under this relation.

On the set of loops, we may define a product  ,

comprising the loop  followed by (see Figure 1).

Explicitly,

ð ÞðtÞ¼

ð2tÞ; 0  t 

ð2t  1Þ;

< t  1

(

½6

It is easy to show that if   

and 

, then

  

. Hence, this defines a product on 

(M),

by [][ ] = [ ]. So equipped, 

(M) becomes the

φψ

Figure 1 The product of loops.

258 Topological Defects and Their Homotopy Classification

fundamental group or first homotopy group of M.

Note that the identity is the equivalence class [

]of

the trivial loop with 

(t)  m

, while the inverse is

[]

1

= [

], where the map

 is the reverse of

:

(t) = (1  t).

Strictly speaking, we should write 

(M, m

)in

place of 

(M). However, for any path-connected

space, the groups 

(M, m

)and

(M, m

)are

always isomorphic, and, more importantly, the same

is true for any coset space M= G=H,whereG is a

Lie group and H a closed subgroup. For a general

manifold M, 

(M) is not necessarily abelian, but it

is so if M is a Lie group, or more generally a

Riemannian symmetric space. The space M is said to

be simply connected if 

(M) = 0, the group compris-

ing only the identity element, 0 = {[

]}. (Although



(M) is not always abelian, it is conventional for

homotopy groups to use an additive notation and

represent the trivial group by 0 rather than 1.)

The nth homotopy group 

(M) may be defined

similarly, as a set of equivalence classes of maps

 : I

!Msuch that  maps the entire boundary @I

to the base point m

. Two such maps are homotopic

(  )ifthereexistsamap : I

nþ1

!Msuch that

ð0; tÞ¼ðtÞ and ð1; tÞ¼ ðtÞ½7

for all t = (t

, ..., t

), and, for each s 2 I , (s, t ) = m

for all t 2 @I

. The product  is defined by

ð Þðt

; ...; t

ð2t

; t

; ...; t

Þ; 0  t



ð2t

 1; t

; ...; t

Þ;

< t

 1

(

½8

The choic e of t

rather than any other t

is arbitrary;

all choices yield homotopic product maps. The

product again defines a product on 

(M), which

thereby becom es a group, the nth homotopy group.

One new feature is that, for all n > 1, 

(M)is

always abelian.

Note that since the entire boundary of I

mapped to a single point, it is possible to collapse it,

and talk instead about maps from the n-sphere S

M, taking one designated point to m

. The fact that



(M) is nontrivial indicates the existence in M of

closed n-surfaces that cannot be smoothly shrunk to

a point. In particular, it is worth noting that, for any n,



) = Z, the additive group of integers, while



) = 0 for all m < n.

A special case is n = 0. Here, S

comprises two

points only, and since one of them is always mapped

to m

, we really have to consider maps from a single

point to M, that is, points in M. Two points are

homotopic if they can be joined by a path in M.

Thus, 

(M) may be identified with the set of path-

connected components of M. Note, however, that in

general no product can be defined on 

(M), so



(M) should be called the zeroth homotopy set

(not group). There is an important exception,

however: if G is a Lie group, and G

its connected

subgroup (the subset of elements joined by paths to

the identity e), then 

(M) may be identified with

the quotient group G=G

. Note, however, that this

group 

(M) = G=G

is not necessarily abelian.

Classification of Defects

We now turn to the classification of defects by

means of homotopy groups. It will be useful to start

with simple specific examples in three-dimensional

space, R

First, suppose again that f belongs to the vector

representation of G = SO(3). Then M= SO(3)/

SO(2) = S

may be identified with the sphere

M= {f: f

= v

}in space. Consider a closed surface

S, an embedding of a 2-sphere S

in R

. Assume

that everywhere on S the field f (r)hasoneofthe

ground-state values. In other words, we have a map

f : S!M, from one 2-sphere to another. The map f

can be extended to a map from the interior of S to M

only if it belongs to the trivial homotopy class [f

] 2



(M), where 

: I

!M:(t

, t

) 7!m

= eH.Inall

other cases, there must be at least one point where

f(r) = 0; this is a point defect. The second homotopy

group in this case is 

) = Z,sothepossible

point defects, or monopoles, are labeled by an integer

n 2 Z, the winding number. (An example of a map

with winding number n is (in spherical polars)

(r, , ’) 7!(v, , n’).)

More generally, point defects in R

are classified

by 

d1

(M). A map  from a closed (d  1)-

dimensional surface SR

to M can be extended

to the interior of S if and only if it belongs to the

trivial homotopy class [

] 2 

d1

(M). If this is not

the case, there must be at least one point around

which (r) leaves the surface M, although in general

it is not required to vanish anywhere.

Second, take the case wher e  is a single complex

field, and G is the phase symmetry group U(1). In

this case, H is the subgroup 1 = {1}  G. Thus,

M= U(1)=1 = S

; this manifold may be identified

with the circle {: jj= v} in the order-parameter

space. Now consider a closed loop C in space, an

embedding of S

in R

(see Figure 2). Suppose that

on C, (r) takes one of the ground-state values,

say (r) = v exp [i(r)]. If S is some surface with

boundary C, then the map  : C!M can be

extended to a map  : S!Mif and only if it

belongs to the trivial homotopy class [

] 2 

(M).

If it does not, then there must be at least one

point on S within C where  = 0. Moreover, this

Topological Defects and Their Homotopy Classification 259

must be true of every surface S spanning C, so there

must be a curve passing through C along which

 = 0. This is a linear defect, a string or vortex line.

In this case, the first homotopy group is 

) = Z,

so we see that the possible linear defects are

classified by an integer, the winding number n.An

example of a map with winding number n is

’ 7!ve

ni’

Again, this result can easily be generalized. Linear

defects in R

are classified by 

d2

(M). If, on a

(d  2)-dimensional surface C, (r) takes values in

M, and if it does not belong to the trivial homotopy

class, there must be a linear defect threading

through C, around which  leav es the surface M –

although again it need not necessarily vanish.

More generally yet, in the d-dimensional space R

defects of dimension p are classified by the homotopy

group 

dp1

(M). For example, in three dimensions,

planar defects – domain walls – are classified by



(M).

The Exact Sequence

There are mathematical theorems that greatly

facilitate the computation of the homotopy group

of homogeneous spaces, of the form M= G=H.

We begin with the maps relating these spaces to

each other. There is a canonical injective homo-

morphism i : H ! G : h 7!h, and a canonical pro-

jection associating each element of G with its coset:

p : G !M: g 7!gH. Moreover, it is clear that the

image of i, namely the subgroup H, is also the kernel

of p, the inverse image p

1

of the distinguished

element m

= eH of M. These statements can be

summarized by saying that

1 ! H !

G !

M!1

is an exact sequence: the image of each map is the

kernel of the following one (see Figure 3).

Next, we note that since any closed loops (or

n-surfaces) in H belongi ng to the same homotopy

class are also homotopic as loops (or n-surfaces) in

G, there is an induced homomorphism i



: 

(H) !



(G). Similarly, homotopic loops or n-surfaces in G

project to homotopic loops or n-surfaces in M,so

there is an induced homomorphism p



: 

(G) !



(M). Moreover, it is easy to see that although i



not necessarily injective and p



not necessarily

projective, it is true that the image of i



is the kernel

of p



. For example, any loop in G will be mapped to

a homotopically trivial loop in M if and only if it is

homotopic to the image of a loop in H.

In addition, there is a boundary map that

relates homotopy groups of different dimension:

@ : 

nþ1

(M) ! 

(H). To see this, it is useful to

think of G as a fiber bundle with base space M and

fiber H. Now consi der a map  :(I

nþ1

, @I

nþ1

) !

(M, m

). Since p is a projection,  can always be

lifted to a map

 :(I

nþ1

, @I

nþ1

) ! (G, H), that is, we

can find a (nonunique) map

 such that  = p 



(see Figure 4). However,

 does not necessarily map

the boundary to a single point; what is true is that



must map the boundary to a subset of H, and since

topologically @I

nþ1

’ S

, this defines a map

 : S

! H.

If we allow  to vary over some homotopy class

of maps, and

 to vary continuously, then

 will

Figure 3 An exact sequence.

φ (t )

(t )

∧

Figure 4 Lift of a loop.

φ = 0

Figure 2 A linear defect.

260 Topological Defects and Their Homotopy Classification

also remain in one homotopy class. Thus, we have

defined a map @ : 

nþ1

(M) ! 

(H):[] 7![

].

It is also easy to see that the image

of @ : 

nþ1

(M) ! 

(H) is the kernel of i



: 

(H) !



(G), because the n-surface in H defined by

 is

necessarily homotop ically trivial in G. Similarly, one

can see that the image of p



: 

nþ1

(G) ! 

nþ1

(M)is

the kernel of @ : 

nþ1

(M) ! 

(H).

Putting all these results together, we see that there

is a (semi-infinite) exact sequence connecting all the

homotopy groups:

!





nþ1

ðMÞ!



ðHÞ!





ðGÞ!





ðMÞ



n1

ðHÞ!





ðGÞ!





ðMÞ!



ðHÞ





ðGÞ!





ðMÞ

½9

This sequence makes it easy to compute most of

the low-dimensional homotopy groups of M. Let us

begin with 

(M), which mer ely labels its discon-

nected components. As noted earlier, for the Lie

group G, 

(G) is the quotient group 

(G) = G=G

where G

is the connected subgroup of G. Now the

image of 

(H) under i



is clearly the set of

connected components of G that contain ele ments

of H,soifG has m connected components, and n of

them contain elements of H, then 

(M) has m=n

elements (see Figure 5).

Next, we note that, for all the higher homotopy

groups, disconnected pieces are irrelevant. Since a

loop, for example, starting at m

must remain

within its connected component M

M,it

follows that 

(M) = 

), and similarly



(M) = 

) for all n > 1. So one can ignore

any disconnected parts of the symmetry group G,

and assume from now on that 

(G) = 0. Moreover,

it is always possible to replace G by its simply

connected covering group, replacing SO(3), for

example, by SU(2). Thus, we may also assume that



(G) = 0. Then the section of the exact sequence in

the second line of [9] becomes

0 !





ðMÞ!



ðHÞ!



which implies that the two groups in the center are

isomorphic:



ðMÞ ¼ 

ðHÞ½10

For example, if the symmetry group G = SO(3) is

completely broken, so that H = 1, then replacing G by

G = SU(2) requires replacing H by

H = { þ1, 1}

’ Z

, hence also 

(M) = 

(

H) = Z

; there is only

one nontrivial class of linear defects in this model.

To find 

(M), we need a standard theorem

about Lie groups, namely that the second homotopy

group of any Lie group is trivial: for any

G, 

(G) = 0. (No details of the proof are given

here. It derives from the fact that a generic element

g 2 G belongs to a unique one-parameter subgroup

{ exp (tX), t 2 R}  G, where X is an element of the

Lie algebra of G. Thus, all the points on a surface in

G may be joined by these paths to the identity, and

the surface may then be shrunk along the resulting

cone. There are exceptional elements for which

this is not true, but it can be shown that in a d-

dimensional group they lie on (d  3)-dimensional

surfaces, so any 2-surface can be smoothly deformed

to avoid them.)

It follows from this theorem that another section

of the exact sequence is

0 !





ðMÞ!



ðHÞ!



which again implies an isomorphism:



ðMÞ ¼ 

ðHÞ½11

For example, if G = SO(3) and H = SO(2), or

equivalently

G = SU(2) and

H = U(1) (a double

cover of the SO(2)), then 

(M) = 

(

H) = Z,so

point defects in this theory are labeled by an integer

winding number.

Examples

The simplest continuous symmetry is the U(1) phase

symmetry

 7!

e

i

of a complex field. In a weakly

interacting Bose gas, below the Bose–Einstein con-

densation temperature, or in superfluid helium-4,

a macroscopic fraction of the atoms occupies a

single quantum state, and

 acquires a nonzero

expectation value, h

i= , whose phase is arbitrary,

so the symmetry is completely broken to H = 1.

Thus, M= S

; we have a circle of equivalent

degenerate ground states. (This corresponds to

(G)

( )

(H )

Figure 5 The disconnected components of G are shaded, those

of H are cross-hatched. Here 

(M) has two elements.

Topological Defects and Their Homotopy Classification 261

spontaneous breaking of the particle-number sym-

metry. It is possible to describe the system in a U(1)-

invariant way, by projecting out a state of definite

particle number, a uniform superposition of all the

states in M, but it is generally less convenient to do

so.) In this case, the only nontrivial homotopy group

is 

(M) = Z, so the only defects are linear defects

classified by a winding number n 2 Z. The defects

with n = 1 are stable vortices. Those with jnj > 1

are in general unstable and tend to break up into jnj

single-quantum vortices.

Low-temperature superc onductors also have a

U(1) symmetry, although there are important differ-

ences. This is not a global symmetry but a local,

gauge symmetry, with coupling to the electromag-

netic field. Moreover, it is not single atoms that

condense but Cooper pairs, pairs of electrons of

equal and opposite momentum and spin. These

systems too exhibit linear defects, magnetic flux

tubes carrying a magnetic flux 4nh=e.

A less trivial example is a nematic liquid crystal.

These materials are composed of rod-shaped mole-

cules that tend, at low temperatures, to line up

parallel to one another. The nematic state is

characterized by a preferred orientation, described

by a unit vector n, the director. (Note that n and  n

are physically equivalent.) There is long-range

orientational order, with molecules preferentially

lining up parallel to n, but unlike a solid crystal

there is no long-range translational order – the

molecules move freely past each other as in a normal

liquid.

A convenient order parameter here is the mean

mass quadrupole tensor  of a molecule. In the

nematic state,  is proportional to (3nn  1); for

example, if n = (0, 0, 1), then  is diagonal with

diagonal elements proportional to (1, 1, 2). In

this case, the symmetry group is SO(3) (or, more

precisely, O(3); but the inversion symmetry is not

broken, so we can restrict our attention to the

connected part of the group). The subgrou p H that

leaves this  invariant is a semidirect product,

H = SO(2) n Z

(isomorphic to O(2)), composed of

rotations about the z-axis and rotations through 

about axes in the x–y plane. (If we enlarge G to its

simply connected covering group

G = SU(2), then H

becomes

H = [U(1) n Z

]=Z

, where U(1) is gener-

ated as before by J

. The essential difference is that

the square of any of the elements in the disconnected

piece of

H is not now the ident ity but the element

2iJ

= 1 2 U(1).) The manifold M of degenerate

ground states in this case is the projective space RP

(obtained by identifying opposite points of S

Since

H has disconnected pieces, we have



(M) = 

(

H) = Z

. Thus, there can be topologically

stable linear defects, here called disclination lines,

around which the director n rotates by  (see Figure 6).

The fact that these defects are classified by Z

rather

than Z means that a line around which n rotates by 2

is topologically trivial; indeed, n can be smoothly

rotated near the line to run parallel to it, leaving a

configuration with no defect.

There are also point defects; since 

(M) =



(

H) = Z, they are labeled by an integer winding

number n. In a defect with n = 1, the vector n points

radially outwards all round the defect position.

Helium-3

Finally, let us turn to helium-3, one of the most

fascinating and complex examples of spontaneous

symmetry breaking, which becomes a superfluid at a

temperature of a few millikelvin. Unlike helium-4, this

is, of course, a Fermi liquid, so it is not the atoms that

condense, but bound pairs of atoms, analogous to

Cooper pairs. In this case, however, the most attractive

channel is not the

S, but the

P, so the pairs have both

orbital and spin angular momentum, L = S = 1. There-

fore, the order parameter is not a single complex scalar

field but a 3  3 complex matrix 

,wherethetwo

indices label the orbital and spin angular momentum

states.

To a good approximation, the system is invar iant

under separate rotations of L and S (the effects of

the small spin–orbit coupling will be discussed

later), so the symmetry group is

G ¼ Uð1Þ

 SOð 3Þ

 SOð3Þ

½12

where the subscripts denote the generators and U(1)

represents multiplication by an overall phase factor,

iY

: 

7!

i

. This complicated symmetry allows

much scope for a large variety of defects. There are, in

fact, two distinct superfluid phases, A and B,with

different symmetries (and indeed in the presence of a

magnetic field there is a third, A1).

In the

He-A phase, the order parameter has the

form 

/ (m

þ in

, where m, n, d are unit

vectors, with m ? n; if we set l = m ^ n, then

Figure 6 Orientation of molecules around a disclination line.

262 Topological Defects and Their Homotopy Classification

l defines the orb ital angular momentum state by

l  L = 1, while d defin es the spin quantization axis,

such that d  S = 0. The manifold M

for this

phase is

¼½SOð3ÞS

=Z

½13

where the Z

is present because (m, n, d)and

(m, n, d) represent the same state . If, for

example, we take l and d in the z-direction, the

unbroken symmetry subgroup is

¼ SOð2Þ

þY

½SOð2Þ

n Z

½14

where the nontrivial element of Z

may be taken to

be e

i(S

þL

)

. The covering group of G is, of course,

G ¼ R

 SUð2Þ

½15

Correspondingly,

¼ R

þY

½Uð1Þ

n Z

½16

It follows that the homotopy groups are



ðM

Þ¼0;

ðM

Þ¼Z

;

ðM

Þ¼Z ½17

There are linear defects labeled by a mod-4 quantum

number and point defects labeled by an integer.

For the

He-B phase, by contrast, the order

parameter is of the form



/ R

i

½18

where R is a rotation matrix, R 2 SO(3). Her e then,

¼ SOð3ÞS

½19

with homotopy groups



ðM

Þ¼0;

ðM

Þ¼Z

 Z;



ðM

Þ¼0 ½20

In this phase, there are two distinct types of linear

defect, the mass vortices with an integer label, and

the spin vortices with a mod-2 label. (One can also

have a ‘‘spin–mass vortex’’ carry ing both quantum

numbers.)

Composite Defects

There are several cases, including in particular

helium-3, that exhibit symmetry breaking with

multiple length or energy scales. For example, there

may be two order parameters, say , , with

jjj j.Ifj j is negligible, the symmetry G is

broken by  to H, and the manifold of degenerate

ground states is M= G=H. However, these states

are not all exactly degenerate: breaks the

symmetry further to K  H, so the precisely degen-

erate ground states form a submanifold M

= G=K.

The c ase of h elium-3 is slightly different. Here it

is the small spin–orbit coupling, arising f rom long-

range dipole–dipole interactions, that introduces

the sec ond scale. Its effect is only significant over

large distances.

In the

He-A phase, at short range the l and d

vectors are uncorrelated but, over large distances,

they tend to be aligned parallel or antiparallel. We

can use the Z

symmetry mentioned earlier to

choose l = d. Hence, the mani fold M

of true

ground states is only a submanifold of M

, namely

= SO( 3), whose homotopy groups are



ðM

Þ¼0;

ðM

Þ¼Z

;

ðM

Þ¼0 ½21

Because of different behavior on different scales,

‘‘composite’’ defects can arise. For example, because



) = Z, there are short-range monopole con-

figurations. For the n = 1 monopole, we have a

configuration with uniform l, and with d pointing

outwards from the center. But, eventually the

misalignment of d with l is energetically disfavored,

and at large distances d tends to rotate to align with

l except around one particular direction where it is

oppositely aligned (see Figure 7). We have a

composite defect: a small monopole coupled to a

relatively fat string.

To see how the small- and large-scale structures fit

together, one has to look also at the relative

homotopy groups 

(M, M

), whose elements are

homotopy classes of maps from I

to M such that

one face of the boundary is mapped into M

, and the

remainder to the chosen base point m

. For example,



(M, M

) classifies paths that terminate at m

while

beginning at any point of M

. There is, in fact, a

long exact sequence, similar to [9], relating these

homotopy groups, of which a typical segment is

!



ðM

Þ!





ðMÞ!





ðM; M



n1

ðM

Þ!



 ½22

Figure 7 Cross-section of a short-range monopole attached to

a fat string.

Topological Defects and Their Homotopy Classification 263

The relevant groups in the present case are



ðM

; M

Þ¼Z

;

ðM

; M

Þ¼Z ½23

Because 

) = Z

, there are three distinct classes

of linear defects at small scales, but only those with

quantum number n = 2 (mod 4) survive unchanged to

large scales; they correspond to the nontrivial element

of 

) = Z

. On the other hand, the homotopy

classes n = 1 (mod 4) are mapped to nontrivial

elements of 

, M

) = Z

, which indicates that

the corresponding linear defects are coupled at long

range to fat domain walls, across which d rotates

through  with a compensating rotation through 

about l. Similarly, the nontrivial elements of



) = Z are mapped to nontrivial elements of



, M

), confirming that these short-range mono-

poles are coupled to fat strings, as in Figure 7.

For

He-B, the effect of the spin–orbit coupling

is to make the most energetically favorable

configurations those in which the rotation

matrix R in [18] represents a rotation about an

arbitrary axis n through the Leggett angle 

arccos(1/4) = 104



: R = exp (i

n  J).

Consequent ly,

¼ S

 S

½24

and so



ðM

Þ¼0;

ðM

Þ¼Z;

ðM

Þ¼Z ½25

The relative homotopy groups are



ðM

; M

Þ¼Z

;

ðM

; M

Þ¼0 ½26

Here the mass vortex persists at long range, but the

configuration around the spin vortex deforms so

that they become attached to fat domain walls. The

‘‘monopole’’ configurations corresponding to

nontrivial elements of 

) have no short-range

singularity at all.

See also: Abelian Higgs Vortices; Leray–Schauder

Theory and Mapping Degree; Liquid Crystals; Phase

Transition Dynamics; Quantum Field Theory: A Brief

Introduction; Quantum Fields with Topological Defects;

Solitons and Other Extended Field Configurations; String

Topology: Homotopy and Geometric Perspectives;

Symmetries and Conservation Laws; Symmetry Breaking

in Field Theory; Variational Techniques for

Ginzburg–Landau Energies.

Further Reading

Helgason S (2001) Differential Geometry, Lie Groups and Sym-

metric Spaces. Providence, RI: American Mathematical Society.

Hu S-T (1959) Homotopy Theory. New York: Academic Press.

Kibble TWB (1976) Topology of cosmic domains and strings. Journal

of Physics A: Mathematical and General 9: 1387–1398.

Kibble TWB (2000) Classification of topological defects and their

relevance to cosmology and elsewhere. In: Bunkov YM and

Godfrin H (eds.) Topological Defects and the Non-Equilibrium-

Dynamics of Symmetry Breaking Phase Transitions,NATO

Science Series C: Mathematical and Physical Sciences. vol. 249,

pp. 7–31. Dordrecht: Kluwer Academic Publishers.

Shellard EPS and Vilenkin A (1994) Cosmic Strings and

Other Topological Defects. Cambridge: Cambridge University

Press.

Tilley DR and Tilley J (1990) Superfluidity and Superconductiv-

ity, 3rd edn. Bristol: IoP Publishing.

Toulouse G and Kle´man M (1976) Principles of a classification of

defects in ordered media. Journal de Physique Lettres 37: 149.

Vollhardt D and Wo¨ lfle P (1990) The Superfluid Phases of

Helium 3. London: Taylor and Francis.

Volovik GE (1992) Exotic Properties of Superfluid

He.

Singapore: World Scientific.

Volovik GE and Mineev VP (1977) Investigation of singularities

in superfluid

He and in liquid crystals by the homotopic

topology methods. Zhurnal Eksperimentalnoi i Teoreticheskoi

Fiziki 72: 2256–2274 (Soviet Physics–JETP 24: 1186–1196).

Topological Gravity, Two-Dimensional

T Eguchi, University of Tokyo, Tokyo, Japan

Introduction

It is well known that large-N Hermitian matrix models

generate Feynman diagrams which represent the

triangulation of Riemann surfaces. For instance, if we

consider the integral of an N  N Hermitian matrix H

Z ¼

dH exp N

tr H



tr H



½1

we find that the free energy F = log Z has the 1=N

expansion

F ¼

g¼0

22g

ðÞ½2

Inspection of the Feynman diagrams shows that F

reproduces the sum over the triangulations of genus

g Riemann surfaces. The theory [1] is obviously well

defined for   0. In the large-N expansion, the

theory continues to exist also at negative values of

 down to the critical point 

= 1=12.

The double scaling limit of large-N matrix

models (Bre´zin and Kazakov 1990, Douglas and

264 Topological Gravity, Two-Dimensional

Shenker 1990, Gross and Migdal 1990) is given by

adjusting the coupling  to 

and at the same time

taking the limit N !1. In this limit, contributions of

all genera survive, and the theory describes the

dynamics of fluctuating surfaces of arbitrary topolo-

gies. Results obtained in this way do not, in fact,

depend on the detailed choice of the potential (

type

in [1]) and have a high degree of universality. Thus, it

provides an interesting model of two-dimensional (2D)

quantum gravity.

Soon after the discovery of double scaling limit of

matrix models, Witten observed that the correlation

functions of the 2D gravity theory may be given a

geometrical interpretation as topological invariants

of the moduli space of Riemann surfaces M, and

that the 2D gravity theory may be reformulated as a

topological field theory (Witten 1990). This refor-

mulation of the results of the 2D gravity theory is

called ‘‘2D topological gravity.’’

In fact, 2D gravity theories come in a family

parametrized by a pair of integers (p, q). The double

scaling limit of [1] gives the simplest example

(p = 2, q = 1). Models with a chain of p  1 Hermi-

tian matric es give the (p, q) 2D gravity theories. The

label q stands for the order of criticality of the

model, and higher values of q are achieved by fine-

tuning the parameters of the potential. At q = 1, 2D

gravity theories possess a topological interpretation.

The most basic case (p = 2, q = 1) is called pure

topological gravity, and in theories at higher values

of p, topo logical gravity is coupled to a matter

system, that is, topological minimal models. Topo-

logical minimal models are obtained by twisting

N = 2 superconformal field theories.

Let us first consider the case of pure gravity (p = 2, 1).

Let O

denote the observables in the theory and t

the

coupling constants to these operators. The correlation

functions of topological gravity are given by

...O

; n

¼ 1; 2; ... ½3

where hi

denotes the expectation value on a

surface with g handles. The precise significance of

eqn [3] as the intersection number on the moduli

space is discussed below. The string partition

function (t) is defined as the genera ting function

of all possible correlation functions

ð tÞ¼exp

g¼0

exp

½4

The most striking aspect of topological gravity is

the connection of the intersection theory on M to

the theory of completely integrable systems, that is,

Korteveg–de Vries (KdV) and KP hierarchies.

Witten conjectured that the generating function of

intersection numbers on moduli space (t) is the

-function of KdV hierarchy. KdV hierarchy is

obtained by generalizing the well-known KdV

equation

½5

Identification of the KdV equation with topological

gravity is give n by u = 2hO

i, x = t

, t = t

Witten’s conjecture was verified by Kontsevich

(1991) by an explicit construction of a new type of

matrix model which generates the triangulation of

the moduli space of Riemann surfaces.

In the general case of (p, 1) topological gravi ty,

the partition function of the theory obeys the

equations of pth generalized KdV hierarchy (p

reduction of KP hierarchy).

Intersection Theory

We now present some basic features of intersection

theory on the moduli space of Riemann surfa ces. It

is known that 2D oriented surfa ces  with g handles

and s marked points x

(i = 1, ..., s) possess a finite

number of inequivalent complex structures (comple x

structures are identified when they differ only by

diffeomorphism). The space of inequivalent complex

structures is called the moduli space M

g,s

of the

Riemann surface . Its dimensi on is given by

dim M

g;s

¼ 3g  3 þ s ½6

For a mathematically rigorous treatment, we have to

consider a compactification



g,s

of moduli space

g,s

by adding suitable boundary components

which arise due to va rious types of degenerations

of Riemann surfaces. In the Deli gne–Mumford or

stable compactification, one considers the following

three classes of singular Riemann surfaces :

1. Two points, x

and x

,on come close together. In

this case, an extra 2-sphere is pinched off from the

surface by forming a thin neck. The sphere contains

points x

and x

and also the point x

at the end of

the neck (see Figure 1a). Since the original surface

now has one point less and the 2-sphere with three

points has no moduli, the degenerate surface has

3g  4 þ s parameters and forms a boundary

divisor of the moduli space



g,s

2. If a cycle of nontrivial homology class shrinks to

a point, we have a surface with one less genus

and two extra marked points. Singular surface

has 3(g  1)  3 þ s þ 2 number of moduli and

this is again a complex codimension-1 compo-

nent (see Figure 1b).

Topological Gravity, Two-Dimensional 265

3. Similarly, if a dividing cycle pinches, one obtains

two disconnected surfaces of genus g

with s

þ 1

marked points (i = 1, 2; g

þ g

= g, s

þ s

= s).

This type of degeneration also has the same

number of parameters

(3g

 3) þ

þ 1) =

3g  4 þ s (see Figure 1c).

It is known that



g,s

is a compact and smooth

orbifold space, and observables of topological gravity

are given by the cohomology classes on



g,s

. There

exist special cohomology classes introduced by

Mumford and Morita, which are defined as follows:

There are natural line bundles L

, ..., L

on the

moduli space



g,s

. The fiber of the bundle L

at a

point  2



g,s

is the cotangent space T



 to the

point x

on the surface . These line bundles have the

first Chern classes c

) and by taking their exterior

power we can define 2n-dimensional classes



ðiÞ¼c

ðL

2 H



g;s

Þ½7

Correlation functions are defined by integrating

these classes over the moduli space:

h







g;s

ðL

^^c

ðL

½8

These integrals are topological invariants of



g,s

and are nonzero only when the degree of the

cohomology classes adds up to the dimension of

the moduli space

i¼1

¼ 3g  3 þ s ½9



(i) is known as the nth descendant of the puncture

operator 

(i), since it is associated with the marked

point x

The above correlation functions are evaluated

using various recursion relations. First, one has the

puncture equation

h





i¼1;n

6¼0

h



1



½10

which can be derived by considering a map

 :



g, sþ 1



g, s

where one forgets the position of

an extra point. Contributions arise when the for-

gotten point coincides with the other points. This

relation can be used to eliminate 

’s from correla-

tion functions when they are well defined. At g = 0,

less than three insertions are ill-defined and one has

h



¼ 1 ½11

Another basic relation is the dilaton equation for

the operator 

h





¼ð2g  2 þ sÞh



½12

The dilaton equation follows from the fact that since



is the first Chern class c

(L), it calculates the

degree of the canonical line bundle of genus g

surface with s punctures. At g = 1, one insertion is

required and one has

h

½13

By combining these recursion relations, one can

evaluate the correlation functions. For instance, at

g = 0 one finds

h



ðn

þþn

Þ!

! n

½14

A powerful way of computing correlation functions is

given by the KdV hierarchies and Virasoro conditions

as discussed below. In the context of integrable

systems, it is convenient to redefine the observables as

2nþ1

¼ð2n þ 1Þ!!  

; n  0 ½15

(a)

(b)

(c)

Figure 1 Degenerate Riemann surface obtained when (a) the

points x

and x

coincide; (b) a nontrivial cycle collapses, two

new points x

and x

are created; (c) a pinching cycle collapses,

two new points x

and x

are created.

266 Topological Gravity, Two-Dimensional

Topological Minimal Models

Standard intersection theory applies to the case of

pure topological gravity, p = 2. At higher values of

p, the theory is generalized as follows: one intro-

duces the coupling of topological gravity to the

topological matter sector which is obtained by

twisting the N = 2 superconformal theories.

We recall that N = 2 superconformal symmetry is

generated by the operators, stress tensor T(z), U(1)

current J(z), and two types of supersymmetry

generators G(z)



. (In the holomorphic sector of the

theory these operators depend on the holomorphic

coordinate z of the Riemann surface. In the antiholo-

morphic sector they depend on the antiholomorphic

variable

z.) Mode expansion of the stress tensor and

U(1) current is given by

TðzÞ¼

n2

; JðzÞ¼

n1

½16

generates the Virasoro algebra

½L

; L

¼ðm  nÞL

mþn

mðm

 1Þ

mþn;0

½17

where c denotes the central charge of the theory.

Commutators of J

and L

are given by

½J

; J

¼

m

mþn;0

; ½L

; J

¼nJ

mþn

½18

It is known that there is a continuum of unitary

N = 2 conformal theories in the range c  3;

however, only discrete values of the central charge

c = 3k=(k þ 2), k = 1, 2, ... are allowed in the

region 3  c  1. These are the N = 2 m inimal

models labeled by the level k. Only a finite

number of primary fields exist in these theories.

In N = 2 theory, primary fields 



are characterized

by their conformal dimension and U(1) charge:

j



i¼hj



i; J

j



i¼qj



i½19

There exists a special set of primary operators, chiral

primary fields 

‘

(‘ = 0, ..., k), which are annihilated

by the supercharge operator G

dzG

ðzÞj

‘

i¼0 ½20



‘

has the dimension and U(1) charge

qð

‘

Þ¼

‘

k þ2

; hð

‘

Þ¼

qð

‘

Þ;‘¼0; 1; ...; k ½21

By considering primary fields annihilated by G



we can also define antichiral fields. Antichiral fields

have U(1) charge opposite to those of chiral fields.

If one defines the twisted stress tensor by

ðzÞ¼TðzÞþ

@JðzÞ½22

then T

(z) has a vanishing central charge. Further-

more, the conformal dimensions of the supersym-

metry operators G



become shifted from 3/2 to

h(G

) = 1andh(G



) = 2. It is then possible to

integrate G

on the Riemann surface and define a

fermionic scalar operator G

dzG

(z). From the

N = 2 algebra, one has

ðG

¼ 0; fG

; G



ðzÞg ¼ 2 T

ðzÞ½23

If we identify G

as the Becchi–Rouet–Stora–Tyupin

(BRST) operator of the theory, then the twisted

stress tensor becomes BRST trivial, which is the

characteristic feature of topological field theory.

Thus, we obtain a topological field theory by twisting

N = 2 conformal theory (Eguchi and Yang 1990).

These are topological minimal models. BRST-invar-

iant observables are given by the chiral primary fields

[20]. (To be precise, when we take account of the

antiholomorphic sector, we may define either Q =

or Q = G



as the BRST operator.

Thus, in general, we obtain two different topological

field theories. This is the origin of the mirror

symmetry. In the context of topological gravity, one

takes the convention Q = G

Now, we consider the coupling of topological

gravity to topological minimal models. We identify

k = p  2. Making use of chiral fields 

‘

(‘ = 0, ...,

p  2), observables are constructed:



n;‘

¼ 

 

‘

½24

N = 2U(1) charge is identified as the degree of

differential form of the moduli space. Thus, the

degree of 

n, ‘

is n þ ‘= p. Correlation functions

i = 1



,‘

are nonzero if the selection rule

i¼1

‘



¼ 3 

p  2



ðg  1Þþs ½25

is obeyed.

We may assemble 

n, ‘

into operators with one

index O

npþ‘þ1

r¼0

ðrp þ ‘ þ 1Þ

n;‘

½26

where one introduces a convenient normalization

factor. Note that the operators O

do not exist

when m  0 mod p and the corresponding paramters

are ab sent. This is a characteristic feature of p

reduced KP hierarchy.

Topological Gravity, Two-Dimensional 267