- Email: [email protected]
The semiclassical approximation for the Chern–Simons partition function
15 January 1998
Physics Letters B 417 Ž1998. 53–60
The semiclassical approximation for the Chern–Simons partition function David H. Adams
1
School of Mathematics, Trinity College, Dublin 2, Ireland Received 24 September 1997 Editor: P.V. Landshoff
Abstract The semiclassical approximation for the partition function in Chern–Simons gauge theory is derived using the invariant integration method. Volume and scale factors which were undetermined and had to be fixed by hand in previous derivations are automatically taken account of in this framework. Agreement with Witten’s exact expressions for the partition function in the weak coupling Žlarge k . limit is verified for gauge group SUŽ2. and spacetimes S 3 , S 2 = S 1 , S 1 = S 1 = S 1 and LŽ p,q .. q 1998 Elsevier Science B.V.
1. Introduction There has been much interest in the pure Chern– Simons gauge theory with spacetime a general 3-dimensional manifold since E. Witten in 1989 gave a prescription for obtaining exact expressions for the partition function and expectation values of Wilson loops w1x. This prescription, which is based on a correspondence with 2D conformal field theory, leads in the case of the partition function to a new topological invariant of the 3-manifold 2 , and in the case of Wilson loops it leads to the Jones knot polynomial Žand generalisations.. However, it is far from clear that Witten’s exact prescription is compatible with standard approaches to quantum field theory, in par-
1
Email: [email protected] This was subsequently shown by K. Walker w7x to coincide with the 3-manifold invariant constructed in a rigorous framework by Reshetikhin and Turaev w8x. 2
ticular with perturbation theory. It is of great theoretical interest to compare results obtained by Witten’s prescription with those obtained by standard approaches; this may lead to new insights into the scope or limitations of quantum field theory in general. A basic prediction of perturbation theory is that the partition function should coincide with its semiclassical approximation in the weak coupling limit, corresponding to the asymptotic limit of large k in the case of Chern–Simons gauge theory. This has been investigated in a program initiated by D. Freed and R. Gompf w2x, and followed up by other authors w3–5x. In these works the large k asymptotics of the expressions for the partition function obtained from Witten’s prescription was evaluated for various classes of spacetime 3-manifolds with gauge group SUŽ2. and compared with expressions for the semiclassical approximations. Agreement was obtained, but only after fixing by hand the values of certain undetermined quantities Žvolume and scale factors.
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 3 4 3 - 9
54
D. H. Adamsr Physics Letters B 417 (1998) 53–60
appearing in the expressions for the semiclassical approximation. Our aim in this paper is to derive a complete, self-contained expression for the semiclassical approximation for the Chern–Simons partition function in which undetermined quantities do not appear. We do this using the invariant integration method introduced by A. Schwarz in App. II of w6x. An important property of the resulting expression is that it is independent of the choice of invariant inner product in the Lie algebra of G which is required to evaluate it. We provide an explicit demonstration of this; it amounts to showing that the expression is independent of the scale parameter l determining the inner product ² a,b : s y l1 TrŽ ab . in the Lie algebra. The resulting expression for the semiclassical approximation is explicitly evaluated for gauge group SUŽ2. and spacetime 3-manifolds S 3 , S 2 = S 1 , S 1 = S 1 = S 1 and arbitrary lens space LŽ p,q .. The expression for the semiclassical approximation involves an integral over the moduli space of flat gauge fields, and the calculations involve cases where the moduli space is both discrete Ž S 3 , LŽ p,q .. and continuous Ž S 2 = S 1 , S 1 = S 1 = S 1 .. After including the standard geometric counterterm in the phase factor we find complete agreement with the exact expressions for the partition function in the large k limit. The techniques and mathematics involved in Witten’s exact prescription Žconformal blocksrrepresentation theory for Kac–Moody algebras, surgery techniques. are very different from those used to obtain the semiclassical approximation Žgauge theory, Hodge theory, analytic continuation of zeta- and eta-functions.. It is remarkable that not only the general features such as the asymptotic k-dependence, but also the precise numerical factors Žincluding factors of p ., are reproduced by the semiclassical approximation. Some details of our calculations, along with a more detailed description of the invariant integration method, will be provided in a forthcoming paper w9x. 2. The invariant integration method We briefly recall the invariant integration method of Schwarz App. II of w6x. Let M be a closed manifold, G a compact simple Lie group, g the Lie algebra of G , V q Ž M, g . the space of q-forms on M
with values in g , A s V 1 Ž M, g . the space of gauge fields Žfor simplicity we are assuming trivial bundle structure although this is not necessary., G the group of gauge transformations, i.e. the maps f : M ™ G acting on A by f P A s f A fy1 q f d fy1 . A choice of metric on M and G-invariant inner product in gdetermine a G-invariant inner product in each V q Ž M, g . and G-invariant metrics on A and G , G. Let which in turn determine a metric on ArG d qA : V q Ž M, g . ™ V qq 1 Ž M, g . denote the exterior derivative twisted by gauge field A Žthen yd 0A s y= A is the generator of infinitesimal gauge transformations of A.. Consider the partition function of a gauge theory with action functional SŽ A. , formally given by 1 1 ZŽ a . s D A exp y 2 S Ž A . Ž 2.1 . VŽ G. A a
ž
H
/
where a is the coupling parameter and V Ž G . is the formal volume of G . Rewrite: 1 ZŽ a . s VŽ G.
HArG D w A x V Ž w A x . exp
=
ž
1 y
a2
S Ž A.
/
Ž 2.2 . where w A x s G P A , the orbit of G through A , and V Žw A x. is its formal volume. Let C ; A denote the G subspace of absolute minima for S , and M s CrG its moduli space. For Au g C expand S Ž Au q B . s S Ž Au . q SAŽ2.u Ž B . q SAŽ3.u Ž B . q . . .
Ž 2.3 . where SAŽ up. Ž B . is of order p in B g V 1 Ž M, g .. Then the asymptotics of ZŽ a . in the limit a ™ 0 is given by 1 Z sc Ž a . s VŽ G.
HM D w A
=
u
=
HT˜
ž
x V Ž w Au x . exp y
ž
D w B x exp y
w Au x
1
a2
1
a2
SAŽ2.u Ž B .
S Ž Au .
/
/
Ž 2.4 .
G .rTw Au x M . Writing where T˜w Au x s Tw Au xŽ ArG ² : SwŽ2. Au x Ž B . s B , DAu B
Ž 2.5 .
D. H. Adamsr Physics Letters B 417 (1998) 53–60
where DAu is a uniquely determined self-adjoint operator on V 1 Ž M, g . with kerŽ DAu . s TAu C one gets
HT˜
ž
D w B x exp y
w Au x
s
HkerŽ D
s det
X
ž
Au
.H
1
a
S Ž2. B 2 Au Ž .
ž
D B exp y
1
a2
/
² B, DA B : u
/
y1 r2
1
pa 2
DAu
/
Ž 2.6 .
Let HAu ; G denote the isotropy subgroup of Au , i.e. the subgroup of gauge transformations which leave Au invariant. HAu consists of constant gauge transformations, corresponding to a subset of G which we also denote by HAu Žsee, e.g., p. 132 of w10x.. Using the one-to-one map (
Gr H A ™ G P Au s w Au x ,
3. The semiclassical approximation in Chern– Simons gauge theory In Chern–Simons gauge theory, with 3-dimensional M and gauge group G s SUŽ N . , the negative number y a12 in Ž2.1. is replaced by the purely imaginary number ik Ž k g Z .. It is therefore natural to take C to be the set of all critical points for the Chern–Simons action functional in this case. Then the elements Au of C are the flat gauge fields and M is the moduli space of flat gauge fields. Expanding the Chern–Simons action functional S Ž A. s
1
H Tr 4p M
u
s V Ž G . VG Ž HAu .
y1
SAŽ2.u Ž B . s
1
.
detX
Au 0
žŽd .
)
d 0Au
1r2
/
² a,b :g s y
where VG Ž HAu . is the volume of HAu considered as a subspace of G . Substituting Ž2.6. and Ž2.7. in Ž2.4. leads to Schwarz’s expression for the semiclassical approximation App. II, Eq. Ž9. of w6x:
AnAnA
3
/
Ž 3.1 .
Au 1
Ž 3.2 .
1
l
Tr Ž ab .
Ž 3.3 .
specified by the scale parameter l g Rq. Thus:
l SAŽ2.u Ž B . s ² B , DAu B : ,
DAu s y
4p
) d1Au
Ž 3.4 .
where ) is the Hodge operator. Using the regularisation procedure of w5x we get
Z sc Ž a .
HM D w A
u
=detX
2
HMTr Ž B n d B .
4p
Ž 2.7 .
s
A n dA q
To obtain DAu from this we need a metric on M and invariant inner product in g to determine the inner product in V 1 Ž M, g .. The G-invariant inner products in g are those of the form
one gets V Ž w Au x . s
ž
around a flat gauge field Au one finds
f ¨ f P Au
u
55
y1
ž ž
x VG Ž HAu . exp y Au 0
žŽd .
)
d 0Au
1r2
/
detX
1
a2 1
pa 2
S Ž Au .
/
detX
y1 r2
DAu
/
ž
DAu
p
s det
Ž 2.8 . In the cases of interest M is finite-dimensional Že.g. in the Yang–Mills theory it is an instanton moduli space., and the determinants in Ž2.8. can be zeta-regularised, leading to a finite expression for Z sc Ž a . Žmodulo any difficulties that may arise from M not being a smooth compact manifold..
y1 r2
yik
=
X
ž
/
ik l 4p kl
ž / 4p
2
y1 r2
) d1Au 2
/
sexp
ž
ip 4
h Ž Au .
y z Ž Au .r2
detX
Au 1
žŽd .
)
d1Au
/
y1 r4
/ Ž 3.5 .
where h Ž Au . and z Ž Au . are the analytic continuations to s s 0 of the eta function h Ž s;) d1Au . and
D. H. Adamsr Physics Letters B 417 (1998) 53–60
56
zeta function z Ž s; <) d1Au <. respectively. In Eq. Ž23. of w5x we showed that
z Ž Au . s dim H 0 Ž Au . y dim H 1 Ž Au .
Ž 3.6 .
where H q Ž Au . is the q’th cohomology space of d Au . Substituting in Ž2.8. gives the following expression for the semiclassical approximation for the Chern– Simons partition function: Z sc Ž k . s
HM D w A
u
x VG Ž HAu .
žž
p
=exp i
=
y1
4
h Ž Au . q kS Ž Au .
//
y z Ž Au .r2
kl
t X Ž Au .
ž / 4p 2
1r2
Ž 3.7 .
where t X Ž Au . is the Ray–Singer torsion of d Au w11x. Note that this expression does not involve any undetermined quantities; all its ingredients are determined by the choice of metric on M and scale parameter l in the invariant inner product for g. We will discuss below the metric dependence of Žand its removal from. this expression. But first we derive the following: Theorem. The semiclassical approximation for the Chern–Simons partition function given by Ž3.7. is independent of the scale parameter l. Proof. Since a scaling of the inner product in g is equivalent to a scaling of the metric on M in Ž3.7. the theorem can be obtained by the general metricindependence arguments of Schwarz in §5 of w12x. But let us give an explicit derivation. We will show that the l-dependence of V Ž HAu . and Dw Au x factors out as V Ž HAu . ; lyŽ dim H Au .r2
Ž 3.8 .
D w Au x ; lyŽ dim Tw Aux
Ž 3.9 .
M .r2
Then, since dim HAu s dim H 0 Ž Au . and Žin the generic case. dimTw Au x M s dim H 1 Ž Au . , we have D w Au x V Ž HAu .
y1
; lŽdim H
0
Ž Au .ydim H 1Ž Au ..r2
independent of l and it follows that Ž3.7. is l-independent as claimed. To derive Ž3.8. – Ž3.9. we begin with a general observation on the change in the volume element under a scaling of inner product in a vectorspace U of dimension d. Let u1 , . . . ,u d be an orthonormal basis for U , then the volume element vol g L d U ) is the dual of u1 n PPP n u d g L d U , i.e. volŽ u1 n PPP n u d . s 1. If we scale the inner product in U by ² P ,P : ™ ² P ,P :l s l1 ² P ,P : then an orthonormal basis for the new inner product is u1l, . . . ,u ld where u lj s 'l u j . The new volume element voll , given by vollŽ u1l n PPP n u dl . s 1 , is voll s lyd r2 vol. The relations Ž3.8. and Ž3.9. follow from this observation together with the fact that the metrics on HAu and M depend on l through a factor l1 due to Ž3.3.. This completes the proof. 3 Were it not for the metric-dependent phase factor exp Ž i4p h Ž Au . . the semiclassical approximation Ž3.7. would be independent of the metric on M by the general arguments in §5 of w12x. A related observation is that, as it stands, Ž3.7. cannot reproduce Witten’s exact formulae for the partition function at large k because the latter are not only metric-independent but also involve a choice of framing of M. In w1x Witten resolved both of these problems by putting in by hand in the semiclassical approximation a phase factor Ž‘‘geometric counterterm’’. depending both on the metric and on the framing of M. It cancels the metric-dependence of exp Ž i4p h Ž Au . . and transforms under a change of framing in the same way as the exact expression for the partition function. We will also put in this factor here. We will carry out the calculations in the canonical framing of Atiyah w14x; then the inclusion of the geometric counterterm in the phase amounts to replacing h Ž Au . ™ h Ž Au . y h Ž0. in Ž3.7. w2x. To explicitly evaluate Ž3.7. we use the fact that the moduli space M can be identified with HomŽp 1Ž M .,G .rG , the space of homomorphisms p 1Ž M . ™ G modulo the conjugation action of G.
.
Ž 3.10 . 3
This l-dependence cancels against the l-dependence of Ž k lr4p 2 .yz Ž Au .r2 in Ž3.7. due to Ž3.6.. It is easy to see that all the other ingredients in Ž3.7. are
In a previous preprint w13x we arrived at a l-dependent expression for the semiclassical approximation for M s S 3. This was due to an error in our calculation equivalent to assuming voll s l d r2 vol instead of lyd r2 vol in the argument above.
D. H. Adamsr Physics Letters B 417 (1998) 53–60
This leads, at least for the examples we consider below, to a one-to-one correspondence of the form (
M˜ ' Ž G1 = PPP = Gs . r W l M ,
u l w Au x
Ž 3.11 . where each Gi is a subspace of G , W is a finite group acting on the Gi ’s and s is the number of generators of p 1Ž M . which can be independently associated with elements of G to determine a homomorphism p 1Ž M . ™ G. The inner product Ž3.3. determines a measure Du on M˜ , and D w Au x s < J1 Ž u . < Du Ž 3.12 . where the Jacobi determinant < J1Ž u .< depends only on the metric on M and Du depends only on l. Similarly, VG Ž HA . s < J0 Ž u . < V Ž HA . Ž 3.13 . u
u
where V Ž HAu . is the volume of HAu as a subspace of G , depending only on l , and < J0 Ž u .< depends only on the metric on M. ŽExplicitly, < J 0 Ž u .< s V Ž M .Ždim H Au .r2 .. Putting all this into Ž3.7. gives Z sc Ž k . s
Du V Ž HAu .
HM˜
ž
=
kl
ž / 4p
2
4
Ž h Ž Au . y h Ž 0. . q kS Ž Au .
/
y z Ž Au .r2
t Ž Au .
1r2
Ž 3.14 .
where
t Ž Au .
1r2
we must choose a value for l ; the answers are independent of the choice due to the theorem in §3. A basis for g s suŽ2. is 1 0 i 1 0 y1 a1 s a2 s i 0 0 2 2 1 1 i 0 a3 s Ž 4.1 . 2 0 yi Since TrŽ a i a j . s y 12 d i j a convenient choice for l is
ž
/
ž
ž
s < J0 Ž u .
1r2
.
Ž 3.15 .
This quantity is the square root of the Ray–Singer torsion ‘‘as a function of the cohomology’’, introduced and shown to be metric-independent in §3 of w15x. Since h Ž Au . y h Ž0. is known to be metric-independent w16x we see that the resulting expression Ž3.14. for Z sc Ž k . is metric-independent as discussed above. 4. Explicit evaluations of the semiclassical approximation We evaluate Z sc Ž k . in the cases where G s SUŽ2. and M is S 3 , S 2 = S 1 , S 1 = S 1 = S 1 and LŽ p,q . , and compare with the expressions ZW Ž k . for the partition function obtained from Witten’s exact prescription in the large k limit. To do the calculations
/
/
l s 1r2 , Ž 4.2 . then a1 ,a 2 ,a 34 is an orthonormal basis for suŽ2. , determining a left invariant metric on SUŽ2.. The volume of SUŽ2. corresponding to this metric can be calculated to be V Ž SU Ž 2 . . s 16p 2 . Define UŽ1. ; SUŽ2. by
Ž 4.3 .
°
u
U Ž 1 . s~exp Ž a u . s 3
¢
y1
p
=exp i
57
¶• ß
ž /
exp i
0
2
u
ž /
0
exp yi
u g w 0,4p w
2
0
Ž 4.4 .
Since a 3 is a unit vector in su Ž 2 . , d < ds ss 0 exp Ž a 3 Ž u q s . . s a 3 is a unit tangent vector to SUŽ2. at exp Ž a3 u . and it follows that the volume of UŽ1. in SUŽ2. is V Ž U Ž 1. . s
4p
H0
d u s 4p .
Ž 4.5 .
M s S 3: p 1Ž S 3 . is trivial, so M consists of a single point corresponding to Au s 0. Then HAu s H0 s G s SUŽ2. , dim H 0 Ž0. s 3 , dim H 1 Ž0. s 0 , z Ž0. s 3 y 0 s 3. In w9x we calculate the torsion Ž3.15. to be
t Ž 0. s 1 Substituting in Ž3.14. we get Z sc Ž k . s
1 V Ž G. 1
s
16p 2
kl
y z Ž0 .r2
ž /
t Ž 0.
4p 2
k
1
2 4p 2
0
Ž 4.6 . 1r2
y3 r2
s '2 p ky3 r2 .
Ž 4.7 .
D. H. Adamsr Physics Letters B 417 (1998) 53–60
58
This coincides with the exact formula Eq. Ž2.26. of w1x in the large k limit: ZW Ž k . s
(
p
2 kq2
sin
ž
kq2
; '2 p ky3 r2 2
Ž4.13. that M˜ can be identified with w0,2 p x = w0,4p w=w0,4p w and
HM˜Du Ž PPP .
/
for k ™ ` .
Ž 4.8 .
1
MsS =S : p 1Ž S 2 = S 1 . ( Z , so by standard arguments M ( Hom Ž Z,SU Ž 2 . . rSUŽ2. ( U Ž 1 . r Z 2 ' M˜
Ž 4.9 . where UŽ1. is given by Ž4.4. and the action of Z 2 on UŽ1. is generated by exp Ž a3 u . ™ exp Ž ya3 u . . It follows that M˜ can be identified with w0,2 p x and HM˜ Du Ž PPP . s Hw0,2 p x du Ž PPP . where u is the parameter in Ž4.4.. The isotropy group HAu is the maximal subgroup of SUŽ2. which commutes with exp Ž a3 u . , so HAu s UŽ1. for u / 0. Hence V Ž HAu . s 4p , dim H 0 Ž Au . s dim H 1 Ž Au . s 1 , z Ž Au . s 1 y 1 s 0. In w9x we show that SŽ Au . s 0 , h Ž Au . s 0 for all Au , and calculate 2
t Ž Au . s Ž 2 y 2cos u . .
Ž 4.10 .
s
We have dim H 1 Ž Au . s dimTu M˜s 3 , HAu s UŽ1. Žexcept for u s Ž0,0,0. and several other isolated points., so dim H 0 Ž Au . s dimŽUŽ1.. s 1 and z Ž Au . s 1 y 3 s y2. In w9x we show that SŽ Au . s 0 , h Ž Au . s 0 for all Au , and calculate
t Ž Au . s 1 .
Ž 4.14 .
Putting all this into Ž3.14. we get Z sc Ž k . s
Hw0,2p x=w0,4p w=w0,4p w V Ž HA u .
s
kl
1
s
s
Hw0,2p xdu V Ž H Hw0,2p x
du
1 4p
Au
k
/ .ž 4p 2
1
2 4p 2
0
t Ž Au .
0
ž /
1
2 4p 2
t Ž Au .
4p 2
0
1r2
du 1 du 2 du 3
1
P 1sk
ZW Ž k . s 1
Ž 4.12 . 1
MsS =S =S : p 1Ž S 1 = S 1 = S 1 . s Z = Z = Z , so by standard arguments M ( Hom Ž Z = Z = Z ,SU Ž 2 . . rSUŽ2. ( Ž U Ž 1 . = U Ž 1 . = U Ž 1 . . r Z 2 ' M˜
Ž 4.13 .
Set u s Ž u 1 , u 2 , u 3 . where u 1 , u 2 , u 3 are 3 copies of the parameter for UŽ1. in Ž4.4.; it follows from
Ž 4.15 .
which coincides in the large k limit with the exact formula Eq. Ž4.32. of w1x: ZW Ž k . s k q 1 .
Ž 2 y 2cos u . s 1
which coincides with the exact formula Eq. Ž4.31. of w1x: 1
4p
1r2
Ž 4.11 .
1
1
k
du 1 du 2 du 3
y z Ž Au .r2
Hw0,2p x=w0,4p w=w0,4p w =
y z Ž Au .r2
kl
1
=
Putting all this into Ž3.14. we get Z sc Ž k .
du 1 du 2 du 3 Ž PPP .
Hw0,2p x=w0,4p w=w0,4p w
Ž 4.16 .
M s L Ž p,q . : In this case the quantities of interest have been calculated in w2,3x and we quote the results. LŽ p,q . s S 3r Z p for a certain free action of Z p on S 3 specified by p and q Žwhich must be relatively prime., so p 1Ž LŽ p,q .. s Z p and M ( Hom Ž Z p ,SU Ž 2 . . rSUŽ2. ( exp Ž a 3 Ž 4p nrp . . < 0 F n F pr2 4 ' M˜
Ž 4.17 . Žsee w2x.. Thus the moduli space is discrete, u ™ n , Au ™ A n , H M˜ Du Ž PPP . ™ Ý0 F nF p r2 Ž PPP . and dim H 1 Ž A n . s dim M s 0. The isotropy group HA n is the maximal subgroup of SUŽ2.whose elements commute with exp Ž a 3 Ž 4p nrp . . , so for 0 - n -
D. H. Adamsr Physics Letters B 417 (1998) 53–60
pr2 HA n s UŽ1. , V Ž HA n . s 4p , dim H 0 Ž A n . s dim HA n s 1 and z Ž A n . s 1 y 0 s 1. For n s 0 , and for n s pr2 if pr2 is integer, exp Ž a3 Ž 4p nrp . . s "1 , so in this case HA n s SUŽ2. , dim HA n s 3 and z Ž A n . s 3 y 0 s 3. By Ž3.14., the k-dependence of the summand is ; kyz Ž A n .r2 and it follows that the terms corresponding to n s 0 and n s pr2 Žif integer. do not contribute to the large k asymptotics of Z sc Ž k . , so we discard these in the following; i.e. restrict to 0 - n - pr2. In Eq. Ž5.3. and prop.5.2 of w3x it was shown that p exp i Ž h Ž A n . y h Ž 0 . . q kS Ž A n . 4 s iexp Ž 2p iq ) Ž k q 2 . n 2rp . Ž 4.18 . ) where q q s 1 Žmod p .. The torsion t Ž A n . is obtained from the calculations in w2x to be 16 2 2p n 2p q ) n t Ž An . s sin sin2 Ž 4.19 . p p p It follows that the large k asymptotics of Ž3.14. in this case is k™` 1 Z sc Ž k . , Ý VŽH . An 0-n-pr2 p =exp i Ž h Ž A n . y h Ž 0 . . q kS Ž A n . 4 k l y z Ž A n .r2 1r2 = t Ž An . 4p 2
ž
/
ž / ž
/
ž
/
ž /
w
py1
x
2
s
1
Ý ns1
k
=
=
s
4p
iexp Ž 2p iq ) Ž k q 2 . n 2rp .
0 'p
sin
Acknowledgements 2p n
ž / ž p
sin
2p q ) n p
/
'2 p Ý iexp Ž 2p iq ) Ž k q 2. n2rp . 'k ns1 =
1
'p
totics of the exact partition function ZW Ž k . derived in Eq. Ž5.7. of w3x. The calculation of t Ž Au . in the preceding examples is carried out in w9x using the equality between Ray–Singer torsion and the combinatorial R-torsion Žboth considered as functions of the cohomology. w17x. The R-torsion is calculated in each case using a suitable cell decomposition of M for which the combinatorial objects corresponding to the < Jq Ž u .<’s in Ž3.15. are equal to 1; then the combinatorial object corresponding to t X Ž Au . is calculated to give Ž4.6., Ž4.10., Ž4.14. and Ž4.19. respectively. For the cases where M is S 2 = S 1 and S 1 = S 1 = S 1 the result h Ž Au . s 0 is obtained in w9x by decomposing V 1 Ž M, g . into the direct sum of finite-dimensional subspaces invariant under ) d1Au , and showing that ) d1Au has symmetric spectrum on each of these subspaces. In future work we will be checking the agreement between the expression Ž3.14. for the semiclassical approximation and the exact formulae for the partition function in the large k limit for other more complicated situations, e.g. when M is a torus bundle w3,18x or a Seifert manifold w4x, and for gauge groups other than SUŽ2.. In certain cases, e.g. for certain Seifert manifolds, the k-dependence ; 4 k max u yz Ž Au . r2 predicted by the semiclassical approximation fails w4x. This is related to the moduli space M having certain singularities. It is an interesting problem to refine the derivation of the semiclassical approximation given here so that it also works for these cases.
y1 r2
1
2 4p 2 4
59
sin
2p n
ž / ž p
sin
2p q ) n p
/
.
Ž 4.20 . This is precisely the formula for the large k asymp-
I am grateful to Siddhartha Sen for many helpful discussions and encouragement during this work, which was supported by Forbairt and Hitachi Labs, Dublin.
References w1x E. Witten, Commun. Math. Phys. 121 Ž1989. 351. w2x D.S. Freed and R. Gompf, Phys. Rev. Lett. 66 Ž1991. 1255; Commun. Math. Phys. 141 Ž1991. 79.
60
D. H. Adamsr Physics Letters B 417 (1998) 53–60
w3x L. Jeffrey, Commun. Math. Phys. 147 Ž1992. 563. w4x L. Rozansky, Commun. Math. Phys. 171 Ž1995. 279; L. Rozansky, in: Knots and Applications, ed. L. Kauffman ŽWorld Scientific, 1995. w5x D.H. Adams and S. Sen, Phys. Lett. B 353 Ž1995. 495. w6x A.S. Schwarz, Commun. Math. Phys. 64 Ž1979. 233. w7x K. Walker, Preprint. w8x N. Reshetikhin and V. Turaev, Invent. Math. 103 Ž1991. 547. w9x D.H. Adams Žin preparation.. w10x S. Donaldson, The Geometry of Four-Manifolds ŽClarendon Press, Oxford, 1990.. w11x D. Ray and I. Singer, Adv. Math. 7 Ž1971. 145.
w12x w13x w14x w15x
A.S. Schwarz, Commun. Math. Phys. 67 Ž1979. 1. D.H. Adams and S. Sen, T.C.D. Preprint, hep-thr9503095. M. Atiyah, Topology 29 Ž1990. 1. D. Ray and I. Singer, in: Proceedings of Symposia in Pure Mathematics, Vol.23 ŽAmerican Mathematical Society, Providence, 1973. p. 167. w16x M. Atiyah, V. Patodi and I. Singer, Math. Proc. Cambridge Philos. Soc. 78 Ž1975. 405. w17x J. Cheeger, Ann. Math. 109 Ž1979. 259; W. Muller, Adv. ¨ Math. 28 Ž1978. 233. w18x J. Andersen, Aarhus Univ. Preprint ŽMath. dept., 1995 series, no. 13..
Physics Letters B 417 Ž1998. 53–60
The semiclassical approximation for the Chern–Simons partition function David H. Adams
1
School of Mathematics, Trinity College, Dublin 2, Ireland Received 24 September 1997 Editor: P.V. Landshoff
Abstract The semiclassical approximation for the partition function in Chern–Simons gauge theory is derived using the invariant integration method. Volume and scale factors which were undetermined and had to be fixed by hand in previous derivations are automatically taken account of in this framework. Agreement with Witten’s exact expressions for the partition function in the weak coupling Žlarge k . limit is verified for gauge group SUŽ2. and spacetimes S 3 , S 2 = S 1 , S 1 = S 1 = S 1 and LŽ p,q .. q 1998 Elsevier Science B.V.
1. Introduction There has been much interest in the pure Chern– Simons gauge theory with spacetime a general 3-dimensional manifold since E. Witten in 1989 gave a prescription for obtaining exact expressions for the partition function and expectation values of Wilson loops w1x. This prescription, which is based on a correspondence with 2D conformal field theory, leads in the case of the partition function to a new topological invariant of the 3-manifold 2 , and in the case of Wilson loops it leads to the Jones knot polynomial Žand generalisations.. However, it is far from clear that Witten’s exact prescription is compatible with standard approaches to quantum field theory, in par-
1
Email: [email protected] This was subsequently shown by K. Walker w7x to coincide with the 3-manifold invariant constructed in a rigorous framework by Reshetikhin and Turaev w8x. 2
ticular with perturbation theory. It is of great theoretical interest to compare results obtained by Witten’s prescription with those obtained by standard approaches; this may lead to new insights into the scope or limitations of quantum field theory in general. A basic prediction of perturbation theory is that the partition function should coincide with its semiclassical approximation in the weak coupling limit, corresponding to the asymptotic limit of large k in the case of Chern–Simons gauge theory. This has been investigated in a program initiated by D. Freed and R. Gompf w2x, and followed up by other authors w3–5x. In these works the large k asymptotics of the expressions for the partition function obtained from Witten’s prescription was evaluated for various classes of spacetime 3-manifolds with gauge group SUŽ2. and compared with expressions for the semiclassical approximations. Agreement was obtained, but only after fixing by hand the values of certain undetermined quantities Žvolume and scale factors.
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 3 4 3 - 9
54
D. H. Adamsr Physics Letters B 417 (1998) 53–60
appearing in the expressions for the semiclassical approximation. Our aim in this paper is to derive a complete, self-contained expression for the semiclassical approximation for the Chern–Simons partition function in which undetermined quantities do not appear. We do this using the invariant integration method introduced by A. Schwarz in App. II of w6x. An important property of the resulting expression is that it is independent of the choice of invariant inner product in the Lie algebra of G which is required to evaluate it. We provide an explicit demonstration of this; it amounts to showing that the expression is independent of the scale parameter l determining the inner product ² a,b : s y l1 TrŽ ab . in the Lie algebra. The resulting expression for the semiclassical approximation is explicitly evaluated for gauge group SUŽ2. and spacetime 3-manifolds S 3 , S 2 = S 1 , S 1 = S 1 = S 1 and arbitrary lens space LŽ p,q .. The expression for the semiclassical approximation involves an integral over the moduli space of flat gauge fields, and the calculations involve cases where the moduli space is both discrete Ž S 3 , LŽ p,q .. and continuous Ž S 2 = S 1 , S 1 = S 1 = S 1 .. After including the standard geometric counterterm in the phase factor we find complete agreement with the exact expressions for the partition function in the large k limit. The techniques and mathematics involved in Witten’s exact prescription Žconformal blocksrrepresentation theory for Kac–Moody algebras, surgery techniques. are very different from those used to obtain the semiclassical approximation Žgauge theory, Hodge theory, analytic continuation of zeta- and eta-functions.. It is remarkable that not only the general features such as the asymptotic k-dependence, but also the precise numerical factors Žincluding factors of p ., are reproduced by the semiclassical approximation. Some details of our calculations, along with a more detailed description of the invariant integration method, will be provided in a forthcoming paper w9x. 2. The invariant integration method We briefly recall the invariant integration method of Schwarz App. II of w6x. Let M be a closed manifold, G a compact simple Lie group, g the Lie algebra of G , V q Ž M, g . the space of q-forms on M
with values in g , A s V 1 Ž M, g . the space of gauge fields Žfor simplicity we are assuming trivial bundle structure although this is not necessary., G the group of gauge transformations, i.e. the maps f : M ™ G acting on A by f P A s f A fy1 q f d fy1 . A choice of metric on M and G-invariant inner product in gdetermine a G-invariant inner product in each V q Ž M, g . and G-invariant metrics on A and G , G. Let which in turn determine a metric on ArG d qA : V q Ž M, g . ™ V qq 1 Ž M, g . denote the exterior derivative twisted by gauge field A Žthen yd 0A s y= A is the generator of infinitesimal gauge transformations of A.. Consider the partition function of a gauge theory with action functional SŽ A. , formally given by 1 1 ZŽ a . s D A exp y 2 S Ž A . Ž 2.1 . VŽ G. A a
ž
H
/
where a is the coupling parameter and V Ž G . is the formal volume of G . Rewrite: 1 ZŽ a . s VŽ G.
HArG D w A x V Ž w A x . exp
=
ž
1 y
a2
S Ž A.
/
Ž 2.2 . where w A x s G P A , the orbit of G through A , and V Žw A x. is its formal volume. Let C ; A denote the G subspace of absolute minima for S , and M s CrG its moduli space. For Au g C expand S Ž Au q B . s S Ž Au . q SAŽ2.u Ž B . q SAŽ3.u Ž B . q . . .
Ž 2.3 . where SAŽ up. Ž B . is of order p in B g V 1 Ž M, g .. Then the asymptotics of ZŽ a . in the limit a ™ 0 is given by 1 Z sc Ž a . s VŽ G.
HM D w A
=
u
=
HT˜
ž
x V Ž w Au x . exp y
ž
D w B x exp y
w Au x
1
a2
1
a2
SAŽ2.u Ž B .
S Ž Au .
/
/
Ž 2.4 .
G .rTw Au x M . Writing where T˜w Au x s Tw Au xŽ ArG ² : SwŽ2. Au x Ž B . s B , DAu B
Ž 2.5 .
D. H. Adamsr Physics Letters B 417 (1998) 53–60
where DAu is a uniquely determined self-adjoint operator on V 1 Ž M, g . with kerŽ DAu . s TAu C one gets
HT˜
ž
D w B x exp y
w Au x
s
HkerŽ D
s det
X
ž
Au
.H
1
a
S Ž2. B 2 Au Ž .
ž
D B exp y
1
a2
/
² B, DA B : u
/
y1 r2
1
pa 2
DAu
/
Ž 2.6 .
Let HAu ; G denote the isotropy subgroup of Au , i.e. the subgroup of gauge transformations which leave Au invariant. HAu consists of constant gauge transformations, corresponding to a subset of G which we also denote by HAu Žsee, e.g., p. 132 of w10x.. Using the one-to-one map (
Gr H A ™ G P Au s w Au x ,
3. The semiclassical approximation in Chern– Simons gauge theory In Chern–Simons gauge theory, with 3-dimensional M and gauge group G s SUŽ N . , the negative number y a12 in Ž2.1. is replaced by the purely imaginary number ik Ž k g Z .. It is therefore natural to take C to be the set of all critical points for the Chern–Simons action functional in this case. Then the elements Au of C are the flat gauge fields and M is the moduli space of flat gauge fields. Expanding the Chern–Simons action functional S Ž A. s
1
H Tr 4p M
u
s V Ž G . VG Ž HAu .
y1
SAŽ2.u Ž B . s
1
.
detX
Au 0
žŽd .
)
d 0Au
1r2
/
² a,b :g s y
where VG Ž HAu . is the volume of HAu considered as a subspace of G . Substituting Ž2.6. and Ž2.7. in Ž2.4. leads to Schwarz’s expression for the semiclassical approximation App. II, Eq. Ž9. of w6x:
AnAnA
3
/
Ž 3.1 .
Au 1
Ž 3.2 .
1
l
Tr Ž ab .
Ž 3.3 .
specified by the scale parameter l g Rq. Thus:
l SAŽ2.u Ž B . s ² B , DAu B : ,
DAu s y
4p
) d1Au
Ž 3.4 .
where ) is the Hodge operator. Using the regularisation procedure of w5x we get
Z sc Ž a .
HM D w A
u
=detX
2
HMTr Ž B n d B .
4p
Ž 2.7 .
s
A n dA q
To obtain DAu from this we need a metric on M and invariant inner product in g to determine the inner product in V 1 Ž M, g .. The G-invariant inner products in g are those of the form
one gets V Ž w Au x . s
ž
around a flat gauge field Au one finds
f ¨ f P Au
u
55
y1
ž ž
x VG Ž HAu . exp y Au 0
žŽd .
)
d 0Au
1r2
/
detX
1
a2 1
pa 2
S Ž Au .
/
detX
y1 r2
DAu
/
ž
DAu
p
s det
Ž 2.8 . In the cases of interest M is finite-dimensional Že.g. in the Yang–Mills theory it is an instanton moduli space., and the determinants in Ž2.8. can be zeta-regularised, leading to a finite expression for Z sc Ž a . Žmodulo any difficulties that may arise from M not being a smooth compact manifold..
y1 r2
yik
=
X
ž
/
ik l 4p kl
ž / 4p
2
y1 r2
) d1Au 2
/
sexp
ž
ip 4
h Ž Au .
y z Ž Au .r2
detX
Au 1
žŽd .
)
d1Au
/
y1 r4
/ Ž 3.5 .
where h Ž Au . and z Ž Au . are the analytic continuations to s s 0 of the eta function h Ž s;) d1Au . and
D. H. Adamsr Physics Letters B 417 (1998) 53–60
56
zeta function z Ž s; <) d1Au <. respectively. In Eq. Ž23. of w5x we showed that
z Ž Au . s dim H 0 Ž Au . y dim H 1 Ž Au .
Ž 3.6 .
where H q Ž Au . is the q’th cohomology space of d Au . Substituting in Ž2.8. gives the following expression for the semiclassical approximation for the Chern– Simons partition function: Z sc Ž k . s
HM D w A
u
x VG Ž HAu .
žž
p
=exp i
=
y1
4
h Ž Au . q kS Ž Au .
//
y z Ž Au .r2
kl
t X Ž Au .
ž / 4p 2
1r2
Ž 3.7 .
where t X Ž Au . is the Ray–Singer torsion of d Au w11x. Note that this expression does not involve any undetermined quantities; all its ingredients are determined by the choice of metric on M and scale parameter l in the invariant inner product for g. We will discuss below the metric dependence of Žand its removal from. this expression. But first we derive the following: Theorem. The semiclassical approximation for the Chern–Simons partition function given by Ž3.7. is independent of the scale parameter l. Proof. Since a scaling of the inner product in g is equivalent to a scaling of the metric on M in Ž3.7. the theorem can be obtained by the general metricindependence arguments of Schwarz in §5 of w12x. But let us give an explicit derivation. We will show that the l-dependence of V Ž HAu . and Dw Au x factors out as V Ž HAu . ; lyŽ dim H Au .r2
Ž 3.8 .
D w Au x ; lyŽ dim Tw Aux
Ž 3.9 .
M .r2
Then, since dim HAu s dim H 0 Ž Au . and Žin the generic case. dimTw Au x M s dim H 1 Ž Au . , we have D w Au x V Ž HAu .
y1
; lŽdim H
0
Ž Au .ydim H 1Ž Au ..r2
independent of l and it follows that Ž3.7. is l-independent as claimed. To derive Ž3.8. – Ž3.9. we begin with a general observation on the change in the volume element under a scaling of inner product in a vectorspace U of dimension d. Let u1 , . . . ,u d be an orthonormal basis for U , then the volume element vol g L d U ) is the dual of u1 n PPP n u d g L d U , i.e. volŽ u1 n PPP n u d . s 1. If we scale the inner product in U by ² P ,P : ™ ² P ,P :l s l1 ² P ,P : then an orthonormal basis for the new inner product is u1l, . . . ,u ld where u lj s 'l u j . The new volume element voll , given by vollŽ u1l n PPP n u dl . s 1 , is voll s lyd r2 vol. The relations Ž3.8. and Ž3.9. follow from this observation together with the fact that the metrics on HAu and M depend on l through a factor l1 due to Ž3.3.. This completes the proof. 3 Were it not for the metric-dependent phase factor exp Ž i4p h Ž Au . . the semiclassical approximation Ž3.7. would be independent of the metric on M by the general arguments in §5 of w12x. A related observation is that, as it stands, Ž3.7. cannot reproduce Witten’s exact formulae for the partition function at large k because the latter are not only metric-independent but also involve a choice of framing of M. In w1x Witten resolved both of these problems by putting in by hand in the semiclassical approximation a phase factor Ž‘‘geometric counterterm’’. depending both on the metric and on the framing of M. It cancels the metric-dependence of exp Ž i4p h Ž Au . . and transforms under a change of framing in the same way as the exact expression for the partition function. We will also put in this factor here. We will carry out the calculations in the canonical framing of Atiyah w14x; then the inclusion of the geometric counterterm in the phase amounts to replacing h Ž Au . ™ h Ž Au . y h Ž0. in Ž3.7. w2x. To explicitly evaluate Ž3.7. we use the fact that the moduli space M can be identified with HomŽp 1Ž M .,G .rG , the space of homomorphisms p 1Ž M . ™ G modulo the conjugation action of G.
.
Ž 3.10 . 3
This l-dependence cancels against the l-dependence of Ž k lr4p 2 .yz Ž Au .r2 in Ž3.7. due to Ž3.6.. It is easy to see that all the other ingredients in Ž3.7. are
In a previous preprint w13x we arrived at a l-dependent expression for the semiclassical approximation for M s S 3. This was due to an error in our calculation equivalent to assuming voll s l d r2 vol instead of lyd r2 vol in the argument above.
D. H. Adamsr Physics Letters B 417 (1998) 53–60
This leads, at least for the examples we consider below, to a one-to-one correspondence of the form (
M˜ ' Ž G1 = PPP = Gs . r W l M ,
u l w Au x
Ž 3.11 . where each Gi is a subspace of G , W is a finite group acting on the Gi ’s and s is the number of generators of p 1Ž M . which can be independently associated with elements of G to determine a homomorphism p 1Ž M . ™ G. The inner product Ž3.3. determines a measure Du on M˜ , and D w Au x s < J1 Ž u . < Du Ž 3.12 . where the Jacobi determinant < J1Ž u .< depends only on the metric on M and Du depends only on l. Similarly, VG Ž HA . s < J0 Ž u . < V Ž HA . Ž 3.13 . u
u
where V Ž HAu . is the volume of HAu as a subspace of G , depending only on l , and < J0 Ž u .< depends only on the metric on M. ŽExplicitly, < J 0 Ž u .< s V Ž M .Ždim H Au .r2 .. Putting all this into Ž3.7. gives Z sc Ž k . s
Du V Ž HAu .
HM˜
ž
=
kl
ž / 4p
2
4
Ž h Ž Au . y h Ž 0. . q kS Ž Au .
/
y z Ž Au .r2
t Ž Au .
1r2
Ž 3.14 .
where
t Ž Au .
1r2
we must choose a value for l ; the answers are independent of the choice due to the theorem in §3. A basis for g s suŽ2. is 1 0 i 1 0 y1 a1 s a2 s i 0 0 2 2 1 1 i 0 a3 s Ž 4.1 . 2 0 yi Since TrŽ a i a j . s y 12 d i j a convenient choice for l is
ž
/
ž
ž
s < J0 Ž u .
1r2
.
Ž 3.15 .
This quantity is the square root of the Ray–Singer torsion ‘‘as a function of the cohomology’’, introduced and shown to be metric-independent in §3 of w15x. Since h Ž Au . y h Ž0. is known to be metric-independent w16x we see that the resulting expression Ž3.14. for Z sc Ž k . is metric-independent as discussed above. 4. Explicit evaluations of the semiclassical approximation We evaluate Z sc Ž k . in the cases where G s SUŽ2. and M is S 3 , S 2 = S 1 , S 1 = S 1 = S 1 and LŽ p,q . , and compare with the expressions ZW Ž k . for the partition function obtained from Witten’s exact prescription in the large k limit. To do the calculations
/
/
l s 1r2 , Ž 4.2 . then a1 ,a 2 ,a 34 is an orthonormal basis for suŽ2. , determining a left invariant metric on SUŽ2.. The volume of SUŽ2. corresponding to this metric can be calculated to be V Ž SU Ž 2 . . s 16p 2 . Define UŽ1. ; SUŽ2. by
Ž 4.3 .
°
u
U Ž 1 . s~exp Ž a u . s 3
¢
y1
p
=exp i
57
¶• ß
ž /
exp i
0
2
u
ž /
0
exp yi
u g w 0,4p w
2
0
Ž 4.4 .
Since a 3 is a unit vector in su Ž 2 . , d < ds ss 0 exp Ž a 3 Ž u q s . . s a 3 is a unit tangent vector to SUŽ2. at exp Ž a3 u . and it follows that the volume of UŽ1. in SUŽ2. is V Ž U Ž 1. . s
4p
H0
d u s 4p .
Ž 4.5 .
M s S 3: p 1Ž S 3 . is trivial, so M consists of a single point corresponding to Au s 0. Then HAu s H0 s G s SUŽ2. , dim H 0 Ž0. s 3 , dim H 1 Ž0. s 0 , z Ž0. s 3 y 0 s 3. In w9x we calculate the torsion Ž3.15. to be
t Ž 0. s 1 Substituting in Ž3.14. we get Z sc Ž k . s
1 V Ž G. 1
s
16p 2
kl
y z Ž0 .r2
ž /
t Ž 0.
4p 2
k
1
2 4p 2
0
Ž 4.6 . 1r2
y3 r2
s '2 p ky3 r2 .
Ž 4.7 .
D. H. Adamsr Physics Letters B 417 (1998) 53–60
58
This coincides with the exact formula Eq. Ž2.26. of w1x in the large k limit: ZW Ž k . s
(
p
2 kq2
sin
ž
kq2
; '2 p ky3 r2 2
Ž4.13. that M˜ can be identified with w0,2 p x = w0,4p w=w0,4p w and
HM˜Du Ž PPP .
/
for k ™ ` .
Ž 4.8 .
1
MsS =S : p 1Ž S 2 = S 1 . ( Z , so by standard arguments M ( Hom Ž Z,SU Ž 2 . . rSUŽ2. ( U Ž 1 . r Z 2 ' M˜
Ž 4.9 . where UŽ1. is given by Ž4.4. and the action of Z 2 on UŽ1. is generated by exp Ž a3 u . ™ exp Ž ya3 u . . It follows that M˜ can be identified with w0,2 p x and HM˜ Du Ž PPP . s Hw0,2 p x du Ž PPP . where u is the parameter in Ž4.4.. The isotropy group HAu is the maximal subgroup of SUŽ2. which commutes with exp Ž a3 u . , so HAu s UŽ1. for u / 0. Hence V Ž HAu . s 4p , dim H 0 Ž Au . s dim H 1 Ž Au . s 1 , z Ž Au . s 1 y 1 s 0. In w9x we show that SŽ Au . s 0 , h Ž Au . s 0 for all Au , and calculate 2
t Ž Au . s Ž 2 y 2cos u . .
Ž 4.10 .
s
We have dim H 1 Ž Au . s dimTu M˜s 3 , HAu s UŽ1. Žexcept for u s Ž0,0,0. and several other isolated points., so dim H 0 Ž Au . s dimŽUŽ1.. s 1 and z Ž Au . s 1 y 3 s y2. In w9x we show that SŽ Au . s 0 , h Ž Au . s 0 for all Au , and calculate
t Ž Au . s 1 .
Ž 4.14 .
Putting all this into Ž3.14. we get Z sc Ž k . s
Hw0,2p x=w0,4p w=w0,4p w V Ž HA u .
s
kl
1
s
s
Hw0,2p xdu V Ž H Hw0,2p x
du
1 4p
Au
k
/ .ž 4p 2
1
2 4p 2
0
t Ž Au .
0
ž /
1
2 4p 2
t Ž Au .
4p 2
0
1r2
du 1 du 2 du 3
1
P 1sk
ZW Ž k . s 1
Ž 4.12 . 1
MsS =S =S : p 1Ž S 1 = S 1 = S 1 . s Z = Z = Z , so by standard arguments M ( Hom Ž Z = Z = Z ,SU Ž 2 . . rSUŽ2. ( Ž U Ž 1 . = U Ž 1 . = U Ž 1 . . r Z 2 ' M˜
Ž 4.13 .
Set u s Ž u 1 , u 2 , u 3 . where u 1 , u 2 , u 3 are 3 copies of the parameter for UŽ1. in Ž4.4.; it follows from
Ž 4.15 .
which coincides in the large k limit with the exact formula Eq. Ž4.32. of w1x: ZW Ž k . s k q 1 .
Ž 2 y 2cos u . s 1
which coincides with the exact formula Eq. Ž4.31. of w1x: 1
4p
1r2
Ž 4.11 .
1
1
k
du 1 du 2 du 3
y z Ž Au .r2
Hw0,2p x=w0,4p w=w0,4p w =
y z Ž Au .r2
kl
1
=
Putting all this into Ž3.14. we get Z sc Ž k .
du 1 du 2 du 3 Ž PPP .
Hw0,2p x=w0,4p w=w0,4p w
Ž 4.16 .
M s L Ž p,q . : In this case the quantities of interest have been calculated in w2,3x and we quote the results. LŽ p,q . s S 3r Z p for a certain free action of Z p on S 3 specified by p and q Žwhich must be relatively prime., so p 1Ž LŽ p,q .. s Z p and M ( Hom Ž Z p ,SU Ž 2 . . rSUŽ2. ( exp Ž a 3 Ž 4p nrp . . < 0 F n F pr2 4 ' M˜
Ž 4.17 . Žsee w2x.. Thus the moduli space is discrete, u ™ n , Au ™ A n , H M˜ Du Ž PPP . ™ Ý0 F nF p r2 Ž PPP . and dim H 1 Ž A n . s dim M s 0. The isotropy group HA n is the maximal subgroup of SUŽ2.whose elements commute with exp Ž a 3 Ž 4p nrp . . , so for 0 - n -
D. H. Adamsr Physics Letters B 417 (1998) 53–60
pr2 HA n s UŽ1. , V Ž HA n . s 4p , dim H 0 Ž A n . s dim HA n s 1 and z Ž A n . s 1 y 0 s 1. For n s 0 , and for n s pr2 if pr2 is integer, exp Ž a3 Ž 4p nrp . . s "1 , so in this case HA n s SUŽ2. , dim HA n s 3 and z Ž A n . s 3 y 0 s 3. By Ž3.14., the k-dependence of the summand is ; kyz Ž A n .r2 and it follows that the terms corresponding to n s 0 and n s pr2 Žif integer. do not contribute to the large k asymptotics of Z sc Ž k . , so we discard these in the following; i.e. restrict to 0 - n - pr2. In Eq. Ž5.3. and prop.5.2 of w3x it was shown that p exp i Ž h Ž A n . y h Ž 0 . . q kS Ž A n . 4 s iexp Ž 2p iq ) Ž k q 2 . n 2rp . Ž 4.18 . ) where q q s 1 Žmod p .. The torsion t Ž A n . is obtained from the calculations in w2x to be 16 2 2p n 2p q ) n t Ž An . s sin sin2 Ž 4.19 . p p p It follows that the large k asymptotics of Ž3.14. in this case is k™` 1 Z sc Ž k . , Ý VŽH . An 0-n-pr2 p =exp i Ž h Ž A n . y h Ž 0 . . q kS Ž A n . 4 k l y z Ž A n .r2 1r2 = t Ž An . 4p 2
ž
/
ž / ž
/
ž
/
ž /
w
py1
x
2
s
1
Ý ns1
k
=
=
s
4p
iexp Ž 2p iq ) Ž k q 2 . n 2rp .
0 'p
sin
Acknowledgements 2p n
ž / ž p
sin
2p q ) n p
/
'2 p Ý iexp Ž 2p iq ) Ž k q 2. n2rp . 'k ns1 =
1
'p
totics of the exact partition function ZW Ž k . derived in Eq. Ž5.7. of w3x. The calculation of t Ž Au . in the preceding examples is carried out in w9x using the equality between Ray–Singer torsion and the combinatorial R-torsion Žboth considered as functions of the cohomology. w17x. The R-torsion is calculated in each case using a suitable cell decomposition of M for which the combinatorial objects corresponding to the < Jq Ž u .<’s in Ž3.15. are equal to 1; then the combinatorial object corresponding to t X Ž Au . is calculated to give Ž4.6., Ž4.10., Ž4.14. and Ž4.19. respectively. For the cases where M is S 2 = S 1 and S 1 = S 1 = S 1 the result h Ž Au . s 0 is obtained in w9x by decomposing V 1 Ž M, g . into the direct sum of finite-dimensional subspaces invariant under ) d1Au , and showing that ) d1Au has symmetric spectrum on each of these subspaces. In future work we will be checking the agreement between the expression Ž3.14. for the semiclassical approximation and the exact formulae for the partition function in the large k limit for other more complicated situations, e.g. when M is a torus bundle w3,18x or a Seifert manifold w4x, and for gauge groups other than SUŽ2.. In certain cases, e.g. for certain Seifert manifolds, the k-dependence ; 4 k max u yz Ž Au . r2 predicted by the semiclassical approximation fails w4x. This is related to the moduli space M having certain singularities. It is an interesting problem to refine the derivation of the semiclassical approximation given here so that it also works for these cases.
y1 r2
1
2 4p 2 4
59
sin
2p n
ž / ž p
sin
2p q ) n p
/
.
Ž 4.20 . This is precisely the formula for the large k asymp-
I am grateful to Siddhartha Sen for many helpful discussions and encouragement during this work, which was supported by Forbairt and Hitachi Labs, Dublin.
References w1x E. Witten, Commun. Math. Phys. 121 Ž1989. 351. w2x D.S. Freed and R. Gompf, Phys. Rev. Lett. 66 Ž1991. 1255; Commun. Math. Phys. 141 Ž1991. 79.
60
D. H. Adamsr Physics Letters B 417 (1998) 53–60
w3x L. Jeffrey, Commun. Math. Phys. 147 Ž1992. 563. w4x L. Rozansky, Commun. Math. Phys. 171 Ž1995. 279; L. Rozansky, in: Knots and Applications, ed. L. Kauffman ŽWorld Scientific, 1995. w5x D.H. Adams and S. Sen, Phys. Lett. B 353 Ž1995. 495. w6x A.S. Schwarz, Commun. Math. Phys. 64 Ž1979. 233. w7x K. Walker, Preprint. w8x N. Reshetikhin and V. Turaev, Invent. Math. 103 Ž1991. 547. w9x D.H. Adams Žin preparation.. w10x S. Donaldson, The Geometry of Four-Manifolds ŽClarendon Press, Oxford, 1990.. w11x D. Ray and I. Singer, Adv. Math. 7 Ž1971. 145.
w12x w13x w14x w15x
A.S. Schwarz, Commun. Math. Phys. 67 Ž1979. 1. D.H. Adams and S. Sen, T.C.D. Preprint, hep-thr9503095. M. Atiyah, Topology 29 Ž1990. 1. D. Ray and I. Singer, in: Proceedings of Symposia in Pure Mathematics, Vol.23 ŽAmerican Mathematical Society, Providence, 1973. p. 167. w16x M. Atiyah, V. Patodi and I. Singer, Math. Proc. Cambridge Philos. Soc. 78 Ž1975. 405. w17x J. Cheeger, Ann. Math. 109 Ž1979. 259; W. Muller, Adv. ¨ Math. 28 Ž1978. 233. w18x J. Andersen, Aarhus Univ. Preprint ŽMath. dept., 1995 series, no. 13..