The Erwin Schr odinger International Boltzmanngasse
ESI Institute for Mathematical Physics A Wien Austria
Conformal Boundary Conditions
and Three Dimensional
Top ological Field Theory
Giovanni Felder
Jurg Fr ohlich
Jurgen Fuchs
Christoph Schweigert
Vienna Preprint ESI Septemb er
Supp orted by Federal Ministry of Science and Transp ort Austria
Available via http www esi ac at
hep th
ETH TH
Septemb er
CONFORMAL BOUNDARY CONDITIONS
AND THREE DIMENSIONAL
TOPOLOGICAL FIELD THEORY
Giovanni Felder Jurg Fr ohlich Jurgen Fuchs and Christoph Schweigert
ETH Zuric h
CH Zuric h
Abstract
We present a general construction of all correlation functions of a two dimensional
rational conformal eld theory for an arbitrary numb er of bulk and b oundary elds
and arbitrary top ologies The correlators are expressed in terms of Wilson graphs
in a certain three manifold the connecting manifold The amplitudes constructed
this way can b e shown to b e mo dular invariant and to ob ey the correct factorization
rules
Two dimensional conformal eld theory plays a fundamental role in the theory of two
dimensional critical systems of classical statistical mechanics in quasi one dimensional con
densed matter physics and in string theory The study of defects in systems of condensed
matter physics of p ercolation probabilities and of op en string p erturbation theory
in the background of certain string solitons the so called D branes forces one to analyze
conformal eld theories on surfaces that may have b oundaries and or can b e non orientable
In this letter we present a new description of correlation functions of an arbitrary numb er
of bulk and b oundary elds on general surfaces We also show how to compute various typ es
of op erator pro duct co e cients from our formulas For simplicity in this letter we restrict
our attention to b oundary conditions that preserve all bulk symmetries Moreover we take
the mo dular invariant torus partition function that enco des the sp ectrum of bulk elds of the
theory to b e given by charge conjugation Technical details and complete pro ofs will app ear in
a separate publication
Given a chiral conformal eld theory such as a chiral free b oson our aim is to compute
correlation functions on a two dimensional surface X that may b e non orientable and can have
a b oundary To this end we rst construct the so called double X of the surface X This
is an oriented surface on which an orientation reversing map of order two acts in such a
way that X is obtained as the quotient of X by Thus X is a two fold cover of X but this
cover is branched over the b oundary p oints which corresp ond to xed p oints of the map
For example when X is the disk D then X is the two sphere and is the re ection ab out
its equatorial plane For X the cross cap i e the real pro jective plane RP X is again the
two sphere but is now the antip o dal map Finally when X is closed and orientable the
X t X double X consists of two disconnected copies of X with opp osite orientation X
Quite generally correlation functions on a surface X can b e constructed from conformal
blo cks on its double X As a rst step one has to nd the pre images on X of all insertion
p oints on X and asso ciate a primary eld of the chiral conformal eld theory to each of them
Since bulk p oints have two pre images for a bulk eld two chiral lab els j and j are needed
corresp onding to left and right movers Boundary elds in contrast carry a single lab el k yet
they should not b e thought of as chiral ob jects
Having asso ciated these lab els to the geometric data we can assign a vector space of confor
mal blo cks not necessarily of dimension one to every collection of bulk and b oundary elds on
X The correlation function is one sp eci c element in this space This element must ob ey mo d
ular invariance and factorization prop erties The conformal b o otstrap programme allows to
determine the correlation function by imp osing these prop erties as constraints Fortunately
the connection b etween conformal eld theory in two dimensions and top ological eld theory
in three dimensions supplies us with a most direct way to construct concrete elements in the
spaces of conformal blo cks We brie y describ e this construction for WZW mo dels where
Chern Simons theory can b e used But these metho ds can b e generalized to arbitrary
rational conformal eld theories For details which are based on the axiomatization in we
refer to What one must do is to nd a three manifold M whose b oundary is X
X
M X
X
as well as a Wilson graph W in M that ends at the marked p oints on X Performing the path
X
Z Z
2 k
A A A Tr A dA DAW exp i
4 3
M
X
of the Chern Simons theory with appropriate parab olic conditions at the insertion p oints then
sp eci es a de nite element in the space of conformal blo cks
Thus to obtain a correlation function on X we rst construct a certain three manifold M
X
with b oundary X which we call the connecting three manifold Technically the manifold M
X
can b e characterized as follows When X do es not have a b oundary then M X Z
X
where the group Z acts on X by and on the interval by the sign ip t t for
t Thus M consists of pairs x t with x a p oint on the double X and t in
X
mo dulo the identi cation x t x t For xed x the p oints of the form x t form a
segment the connecting interval joining the two pre images of a p oint in X When X has
a b oundary we obtain M from X Z by contracting the connecting intervals over
X
the b oundary to single p oints in such a way that M remains a smo oth manifold
X
It is readily checked that the b oundary of the connecting manifold M is indeed the double
X
X Moreover M connects the two pre images of a bulk p oint by an interval in such a manner
X
that the connecting intervals for distinct bulk p oints do not intersect Let us list a few examples
For a disk the connecting manifold is a solid three ball and the connecting intervals are all
p erp endicular to the equatorial plane Similarly when X is the annulus M is a solid torus
X
For X the cross cap the connecting manifold M is b est characterized by the fact that when
X
RP which coincides with the group glueing to its b oundary a solid ball we obtain S Z
manifold of the Lie group SO For closed orientable surfaces X the bundle M is just the
X
trivial bundle X e g when X is a sphere then M can b e visualized as consisting of
X
the p oints b etween two concentric spheres
The next step is to sp ecify a certain Wilson graph in M The prescription which is
X
illustrated in gure for the case of a disk with an arbitrary numb er of insertions in the bulk
and on the b oundary is as follows First for every bulk insertion j one joins the pre images
of the insertion p oint by a Wilson line running along the connecting interval Next one inserts
one circular Wilson line parallel to each comp onent of the b oundary and joins every b oundary
insertion k on the resp ective b oundary comp onent by a short Wilson line to the corresp onding
circular Wilson line Moreover the circular Wilson lines are required to run close to the
b oundary in the sense that none of the connecting intervals of the bulk elds passes b etween
the circular Wilson lines and the b oundary of X
So far we have only sp eci ed the geometric information for the conformal blo cks To pro ceed
we also must attach a primary lab el of the chiral conformal eld theory to each segment of the
Wilson graph For the bulk p oints this prescription is immediate as we are dealing with the
charge conjugation mo dular invariant Similarly we are naturally provided with the lab els k for
the short Wilson lines that connect the b oundary insertions with the circular Wilson lines In
addition the segments of the circular Wilson lines should enco de the b oundary conditions of the
corresp onding b oundary segments Recalling that those b oundary conditions which preserve all
bulk symmetries can b e lab elled by the primary elds of the chiral conformal eld theory
we attach such a primary lab el a to every segment of the circular Wilson lines Finally we must
consider the three valent junctions on the circular Wilson lines For each of them we cho ose
an element in the space of chiral couplings b etween the lab el k for the b oundary eld and ...
...
Figure Wilson graph for the disk correlators
the two adjacent b oundary conditions a b The dimension of this space of couplings is given
a
by the fusion rules N of the chiral theory Indeed it is known that b oundary op erators need
k b
an additional degeneracy lab el that takes its values in the space of chiral three p oint blo cks
As a matter of fact every segment of the Wilson graph should also b e equipp ed with a
framing in other words we should not just sp ecify a graph but a ribb on graph Moreover
the b oundary X of M must b e endowed with additional structure to o A careful discussion
X
of these issues will b e presented in As a side remark we mention that the circular Wilson
lines already come with a natural thickening to ribb ons which is obtained by connecting them
to the pre image of the b oundary of X in X In gure this is indicated by a shading Note
that in the case of symmetry breaking b oundary conditions the lab els of b oundary elds
and b oundary conditions can b e more general than in the bulk This can b e implemented in
our picture as the corresp onding part of the graph with the circular Wilson line is disconnected
from the rest of the Wilson graph
Using appropriate surgery on three manifolds we can prove that the correlation functions
obtained by our prescription p ossess the correct factorization or sewing prop erties and that
they are invariant under large di eomorphisms or in more technical terms under the relative
mo dular group For a detailed account of these issues we refer to Here we restrict
ourselves to the analysis of a few situations of particular interest we also show how to recover
known results for the structure constants from our formulas
In our approach the structure constants are obtained as the co e cients that app ear in the
expansion of the sp eci c element in the space of conformal blo cks that represents a correlation
function in a standard basis for the conformal blo cks For two p oints on the b oundary of a solid
three ball such a standard basis is given by a Wilson line with trivial framing connecting the
two p oints while for three p oints one takes a Mercedes star shap ed junction of three Wilson
lines Our general strategy for computing the co e cients is then to glue another three manifold
to the connecting manifold so as to obtain the partition function or in mathematical terms
the link invariant for a closed three manifold The values of such link invariants are available
in the literature see e g
Our rst example is the correlator of two bulk elds on S a closed and orientable surface
For the space of blo cks to b e non zero the two elds must b e conjugate i e carry lab els j and
j resp ectively According to our prescription the connecting manifold then consists of the
lling b etween two concentric two spheres and the Wilson graph consists of two disjoint lines
connecting the spheres b oth lab elled by j this is depicted in gure The space of conformal
blo cks for this situation is one dimensional its standard basis is displayed in gure Thus the
relevant three manifold is given by the disconnected sum of two balls each of which carries a
single Wilson line To b oth manifolds we glue two balls in which a Wilson line lab elled by j is
running In the case of the correlation function the resulting manifold is a three sphere with
an unknot lab elled by j for which the value of the link invariant is S S is the mo dular S
j
transformation matrix of the chiral conformal eld theory and the lab el refers to the vacuum
primary eld When applied to the manifold in gure the glueing pro cedure pro duces two
disjoint copies of S each with an unknot lab elled by j the corresp onding partition function is
S Comparing the two results we see that the two p oint function on the sphere is expressed
j
in terms of the standard basis as
C S j j S B S j j B S j j
j
In other words the normalization of the bulk elds j di ers from the more conventional pre
scription in which they are canonically normalized to one by a factor of S
j
Figure C S j j
Figure B S j j B S j j
Next we discuss an example featuring an orientable surface with b oundary we compute the
one p oint amplitude for a bulk eld j on a disk D with b oundary condition a Again the space
of blo cks is one dimensional Our task is then to compare the Wilson graph of gure with the
standard basis that is displayed in gure In the present context this particular conformal
blo ck is often called an Ishibashi state We now obtain the three sphere by glueing with a
single three ball When applied to the graph of gure we get the unknot with lab el j in S
for which the partition function is S In the case of gure we get a pair of linked Wilson
j
lines with lab els a and j in S the value of the link invariant for this graph is S Comparison
a j
thus shows that the correlation function is S S times the standard two p oint blo ck on the
a j j
sphere
C D j S S B S j j
a a j j
Taking into account the normalization of the bulk elds as obtained in formula we recover
the known result that the correlator for a canonically normalized bulk eld j on a disk with
p
b oundary condition a is S S times the standard two p oint blo ck on the sphere This
a j j
relation forms the basis of the so called b oundary state formalism
Figure C D j
a
Figure B S j j
As a third example we study again a one p oint correlator of a bulk eld j now on the
cross cap RP which do es not have a b oundary but is non orientable The latter prop erty
forces us to b e careful with the framing The structure constants are obtained by comparing
the correlator C RP j with the cross cap state This state is de ned in gure it is
j
similar to the basis element B S j j of the two p oint blo cks on S but now the Wilson line
in the three ball has a non trivial framing and accordingly in gure we have drawn a ribb on
instead of a line A priori we could twist the line either by thereby obtaining some state
i
j
or by and obtain another state with These two vectors di er by a factor of e
j j
the conformal weight of j Salomonically we de ne the cross cap state as
j
i i
j j
e e
j
j j
Again the comparison of the correlator C RP j with the standard basis is carried out by
j
glueing a three ball with a Wilson line to the ball of gure In contrast to the previous cases
however this line is given a non trivial framing cho osing the framing in such a way that the
twist of the cross cap state is undone glueing the ball to the cross cap state yields S with the
unknot with partition function Z S j S
j
As already mentioned glueing the three ball to the connecting manifold of the cross cap
yields SO It is also known that SO can b e obtained from S by a surgery on the unknot
with framing Following how the framed graph is mapp ed by the surgery one may visualize
the situation as in gure Taking all framings prop erly into account we obtain
X
S T S T P Z SO j T
k j k k j
j
k
i c
j
with T e for the invariant of this three manifold where in the second equality we
j
expressed the result through the matrix P T ST ST We have thereby recovered
the known formula
C RP j P S
j j j
for the one p oint correlator on the cross cap
j
i
j
Figure The state e
j j
j
k
Figure A visualization of C RP j
As a nal example consider three b oundary elds i j k on a disk The relevant Wilson
graph in the three ball is of the typ e shown in gure without any vertical Wilson lines
along connecting intervals it consists of a circular line with segments lab elled a b c with
three short Wilson lines lab elled i j k attached to it We must compare it to the standard
basis for three p oint blo cks on the sphere which is a Mercedes star shap ed junction This
comparison can b e made by p erforming a single fusing op eration followed by a contraction
of the lo op For b oundary elds it is natural to de ne the correlation functions as linear
forms on the degeneracy spaces for b oundary op erators Denoting a basis of the degeneracy
ac
by fe ica g normalized by the quantum trace condition space for the b oundary op erator
i
tr e ica e i ac we nd that
X
S
0 0
i c a
e k j i g f C D i j k e ica e j ab e k bc
a b c
b j k
S
k 0
i j k
denotes a fusing matrix or quantum j symb ol g where the symb ol f
l m n
One imp ortant conclusion we can draw from our results is that the construction of correlation
functions from conformal blo cks can b e p erformed in a completely mo del indep endent manner
All structure constants for any arbitrary conformal eld theory can b e expressed in terms
of purely chiral data such as conformal weights the mo dular S matrix fusing matrices and
the like All sp eci c prop erties of concrete mo dels already enter at the chiral level Physical
quantities such as the magnetization of an op en spin chain or op en string amplitudes in the
background of D branes can b e expressed in terms of the correlators studied in this letter
References
D H Friedan Z Qiu and S H Shenker Conformal invariance unitarity and critical exp onents
in two dimensions Phys Rev Lett
I A eck A current algebra approach to the Kondo e ect Nucl Phys B
A A Belavin A M Polyakov and A B Zamolo dchikov In nite conformal symmetry in two
dimensional quantum eld theory Nucl Phys B
M Oshikawa and I A eck Defect lines in the Ising mo del and b oundary states on orbifolds
Phys Rev Lett
J L Cardy Critical p ercolation in nite geometries J Phys A L
J Polchinski Dirichlet branes and Ramond Ramond charges Phys Rev Lett
V Alessandrini A general approach to dual multilo op diagrams Nuovo Cim A
J Fuchs and C Schweigert Branes from free elds to general backgrounds Nucl Phys B
E Witten Quantum eld theory and the Jones p olynomial Commun Math Phys
J Fr ohlich and C King Two dimensional conformal eld theory and three dimensional top ology
Int J Mo d Phys A
V G Turaev Quantum Invariants of Knots and Manifolds de Gruyter New York
G Felder J Fr ohlich J Fuchs and C Schweigert in preparation
J L Cardy Boundary conditions fusion rules and the Verlinde formula Nucl Phys B
J Fuchs and C Schweigert Symmetry breaking b oundaries I General theory preprint hep
th Nucl Phys B in press Symmetry breaking b oundaries I I More structure examples
preprint hep th
M Bianchi and A Sagnotti Op en strings and the relative mo dular group Phys Lett B
G Mo ore and N Seib erg Lectures on RCFT in Physics Geometry and Topology H C Lee
ed Plenum Press New York p
M Bianchi and A Sagnotti The partition function of the SO b osonic string Phys Lett
B