We examine the electric-magnetic duality for a U(1) gauge theory on a general
4-manifold. The partition function for such a theory transforms as a modular
form of specific weight. However, in the canonical approach, we show that
S-duality, like T-duality, is generated by a canonical transformation leading
to a modular invariant partition function.Comment: latex, 10 page

Kehagias, A. A.

English

arXiv

We examine the electric-magnetic duality for a U(1) gauge theory on a general 4-manifold. The partition function for such a theory transforms as a modular form of specific weight. However, in the canonical approach, we show that S-duality, like T-duality, is generated by a canonical transformation leading to a modular invariant partition function

Kehagias, A A

CERN Document Server

hep-th/9508159
NTUA-95/51
August 1995
A CANONICAL APPROACH TO S-DUALITY IN
ABELIAN GAUGE THEORY
Alexandros A. Kehagias 1
Physics Department
National Technical University
15780 Zografou Campus Athens, Greece
E-Mail: kehagias@sci.kun.nl
Abstract
We examine the electric-magnetic duality for a U(1) gauge theory on a general four
manifold which generates the SL(2;Z) group. The partition functions for such a theory
transforms as a modular form of specic weight. However, in the canonical approach, we
show that S-duality for the abelian theory, like T-duality, is generated by a canonical trans-
formation leading to a modular invariant partition function.
1Address after September 1995: Phys. Dept., Inst. fu¨r Theor. Phys., Technische Univ., Mu¨nchen,
D-85748 Garching, Germany
According to the S-duality assertion, certain four-dimensional gauge theories with a -
term are invariant under SL(2;Z) modular transformations on the complex coupling
 =

2
+ i
4
g2
; (1)
where , g are the theta angle and the coupling constant, respectively [1]. This assertion
has been tested for N=4 supersymmetric Yang-Mills theory by calculating the corresponding
partition function [2] and the result was in favour of S-duality if the gauge group is replaced
by its dual. For an abelian U(1) gauge theory, S-duality extends the electric-magnetic duality
[3],[4] of electromagnetism on R4 to the full action of SL(2;Z) for a general four-manifold.
The corresponding partition function fails to be modular invariant but rather it transforms
as a modular form of specic weight [5]{[7]. This can be seen by compacting M over a
torus. The reduced two-dimensional theory looks like a linear sigma-model where the internal
components of the photon are scalar elds. S-duality is then reduced to T-duality [8] and
the modular anomaly appears because there is no dilaton eld to compensate it [9],[10].
However, when the four manifold is of the form RX and a time direction can be chosen,
by employing the canonical approach we will see that S-duality, like T-duality [11],[12], is
just a canonical transformation leading to a modular invariant partition function.
Let us consider a U(1) gauge eld A, i.e., a connection on a line bundle L over a four
dimensional manifold M with corresponding eld strength F = @A − @A. This theory
is described by the action
S[A] =
1
e2
Z
M
F ^ F − i

82
Z
M
F ^ F ; (2)
where F = 1
2
Fdx
 ^ dx and F = 1
4
F dx
 ^ dx. We assume that M is Euclidean
so that, the saddle points of (2) turns out to be the Hodge-de Rham equations
dF = 0 ; d F = 0 ; (3)
for harmonic two-forms. In general, the number of harmonic n-forms on M is bn where bn is
the nth Betti number, i.e., the dimension of the nth de Rham cohomology group H2(M;R)
while the Euler number of M is given by the algebraic sum  =
P
n(−1)
nbn.
The partition function of the U(1) gauge theory dened on the four-manifold M is given
by
Z = C
1
jGj
Z
M
DAe−S[A] ; (4)
where C is a regularization constant, the integration is over all the U(1) gauge elds on
M and we divide by the volume jGj of the gauge group as usual. Moreover, a sum over
1
isomorphic classes of the line bundle L over M is understood. By employing the Faddeev-
Popov procedure to factor out this volume, the partition function may easily be evaluated
to be
Z = C
det0e−1FP
(det0e−2A)1=2
Zcl ; (5)
where
Zcl =
X
saddle points
e−S[Acl ] ; (6)
is the sum of all saddle-point congurations of the action (2). det0e−1FP ; det0e−2A denote
the Faddeev-Popov determinant and the determinant of the kinetic term for the gauge eld
(without counting the zero modes). The former is the determinant of the scalar Laplacian
0 while the latter of the Laplacian 1 for one-forms. One may easily verify that
det0e−2A = (
1
e2
)A(0)det0A ; (7)
where (s) is the generalised zeta function for the operator . Since (0) is just the
dimension of  (innite if not properly regularized) without counting the zero modes, we
may write
A(0) = dim1 − b1 ; (8)
Similarly, we have
det0e−1FP = (
1
e2
)
1
2
FP (0)det0FP ; (9)
with
FP (0) = dim0 − b0 : (10)
Thus, by choosing the constant C in eq.(4) to be
C = (Im )
1
2
(dim1−dim 0)(4)
1
2
(dim0−dim1+b1−b0) ; (11)
the partition function is properly regularized and may be written as
Z = (Im )
1
2
(b1−b0)
det0FP
(det0A)1=2
Zcl ; (12)
where numerical factors have been omitted. Note that we may express the action (2) in
terms of self-dual (F+ = F + F ), anti-self-dual (F− = F − F ) elds and the complex
coupling  dened in eq.(1) as
S =
i
16
Z
M
(F− ^ F− − F+ ^ F+): (13)
2
Let us suppose that in M there exist non-trivial two-cycles, that is closed surfaces which
do not bound any 3-dimensional sub-manifold of M . One may then consider the flux of
the Maxwell eld strength F, through these cycles which satises the Dirac quantization
condition Z
SI
F = 2nI ; (14)
where nI 2 Z and I = 1; :::; b2. We may dene a basis I of harmonic two-forms (dI =
dI = 0) normalised as
R
SI
J = IJ which generates the harmonic representatives in
H2(M;R) and allows us to express the eld strength F as
F = 2
X
I
nII : (15)
We may also dene a basis in the space of self-dual/anti-self-dual harmonic two-forms by
I = I  I : (16)
By employing eqs.(15,16), we may express the action (13) as
S[Acl] = i

4
nIH+IJn
J − i

4
nIH−IJn
J ; (17)
where
HIJ =
Z
M4
 ^  ; (18)
is the intersection form for harmonic self-dual and anti-self-dual two-forms. The partition
function in eq.(12) may then be written as
Z = (Im )
1
2
(b1−b0)
det0FP
(det0A)1=2
X
nI
ei

4
nIH−IJn
J−i
4
nIH+IJn
J
: (19)
To examine S-duality transformations which are generated by T :  !  + 1 and S :
 ! −1= , let us recall one of the basic invariants of a 4-manifold M , the intersection form
!. If M has a smooth structure, it is dened by using the de Rham cohomology H(M) as
!(; ) =
Z
M
 ^  ; (20)
for ;  2 H2(M;R) [13]. The dimensionality of ! is b2 and the number of its positive
(negative) eigenvalues is b+2 (b
−
2 ). The signature  of ! is dened as the number of positive
eigenvalues minus the negative ones, i.e.,  = b+2 − b
−
2 and it is a topological invariant. For
a simply-connected spin manifold, !(; ) is an even integer, otherwise is odd. Thus, for
3
a spin manifold by taking  = F=2 one may verify that the action (2) is invariant under
 !  + 1 while for a non-spin manifold (2) is invariant under  !  + 2 [2].
To examine the transformation properties of the partition function under  ! −1= , let
us consider the dual theory which can be found from the rst-order action
~S =
1
g2
Z
G ^ G− i

82
Z
G ^G+
i
2
dG ^B ; (21)
where G = 1
2
Gdx
^dx and B = Bdx are a 2-form and a 1-form, respectively. It can be
constructed by requiring invariance under gauge transformations of the \third kind":A !
A+B [14]. The partition function for this theory is given by
Z = C
Z
M
DB
j ~Gj
DGe−
~S ; (22)
where we have divide by the volume j ~Gj of the gauge group which transforms B according
to B ! B + d. By integrating out the Lagrange multiplier eld B on a topologically
trivial manifold M , we get dG = 0 which implies that G = d ~A and the original U(1) gauge
theory described by (2) is recovered. However, in a non-trivial manifold M , in order G to
be the curvature (eld strength) of a U(1) connection (gauge eld), the Dirac quantization
condition must be taken into account, i.e., eq.(22) must be implemented by the sumX
nI2Z


nI −
Z
SI
G
2

=
X
mI2Z
e
−imI
R
SI
G
; (23)
where the sum in the righthand side in the above equation is over the dual lattice. We may
also integrate the G-eld and for this it is more convinient to express (21) as
~S =
i
16
Z
M

G− ^G− − G+ ^G+

−
i
8
Z
M

G+ ^W+ +G− ^W−

; (24)
where W = dB and G = GG, W = W W are the self-dual and anti-self-dual parts
of G and W . Since self-dual and anti-self-dual forms are orthogonal, we may integrate them
separately. The result is, ignoring numerical factors,
Z( ) = C−
1
2
B−2 
1
2
B+2
Z
DB
j ~Gj
e−
~S ; (25)
where
~S =
i
16
Z
M

(−
1

)W− ^W− − (−
1

)W+ ^W+

; (26)
and B2 is the number of self-dual, anti-self-dual 2-forms on M in a lattice regulatization.
Note that (26) does describe a U(1) gauge theory since by integrating out G we also getZ
SI
W
2
= mI 2 Z ;
4
for the fluxes of the dual theory. Thus, eq.(25) is indeed the partition function for a U(1)
gauge theory. As follows now from eqs. (4,11),
Z(−
1

) = C( )−
1
2
(B1−B0)
Z
DA
jGj
e−
i
16
R
M
((− 1 )F−^F−−(−
1

)F+^F+) ; (27)
where B1 = dim1; B0 = dim0. By comparing eqs.(25,27) we get
Z(−
1

) = 
1
2
(B0−B1+B
+
2 )
1
2
(B0−B1+B
−
2 )Z( ) (28)
= 
1
4
(−)
1
4
(+)Z( ) ; (29)
and therefore Z( ) transforms as a modular form of weight (1
4
(− ); 1
4
(+ )).
Let us now examine S-duality in the Hamiltonian approach. We will assume that the
four-manifold M is Lorentzian and by separating space and time, M turns out to be of the
form RX where X is a three-manifold. We will also assume that M is endowed with the
product metric
ds2 = −dt2 + gijdx
idxj ; (30)
where gij is the metric on X. In this case the action (2) turns out to be
S =
Z
dtd3x
p
g
 
−
1
2e2
FF
 +

322
FF
!
; (31)
which is of course real. The canonical momenta are easily found to be
i =
2
e2
F0jg
ij +

82
ijkFij ; (32)
and by performing a Legendre transform, the Hamiltonian is given by
H =
e2
4
i
i + i@iA0 −
e2
162
i
ijkFjk + (
1
2e2
+
e22
1284
)FijF
ij ; (33)
while the symplectic structure is provided by the equal-time Poisson bracket
fAi(x); 
j(y)g = ji 
(3)(x− y) : (34)
The partition function is given by
Z = N
Z
DAD
1
jGj
ei
R
dtd3x
p
g(i _Ai−H) ; (35)
where N is a regularization constant and we have divide as usual by the volume of the U(1)
gauge group. It contains a sum over the isomorphic classes of the line bundle L over X
5
and in each class we have to integrate over the momenta i and A. Every i integration
produces a factor (1=e2)1=2  Im 1=2 while every Ai one produces a factor (
1
2e2
+ e
22
1284
)−1=2 
(Im= )1=2 (A0 integrations do not produce any factors but simply a delta function). By
using a lattice regularization, the number of Ai integrations is B1 − B0 where B1; B0 is the
number of 1-forms and 0-forms on X, respectively. Since the number of i integrations
equals the number of Ai ones, the cut-o dependent term coming from the integration is
Im
1
2
(B1−B0)Im
1
2
(B1−B0)( )−
1
2
(B1−B0) : (36)
Thus, a properly regularized, cut-o independent partition function is given by
Z( ) = ImB0−B1(  )
1
2
(B1−B0)
Z
DAD
1
jGj
ei
R
dtd3x
p
g(i _Ai−H) : (37)
As usual, A0 has no kinetic term and it is just a Lagrange multiplier leading to the constraint
ri
i = 0 ; (38)
where r is the covariant derivative on M . Let us now perform a canonical transformation
generated by
G =
1
4
Z
X
d3x
p
g( ~Ai
ijkFjk +Ai
ijk ~Fjk) ; (39)
so that
i = G
Ai
=
1
4
ijk ~Fjk; (40)
~i = − G
 ~Ai
= −
1
4
ijkFjk : (41)
It follows from (41) that the dual momenta ~i satisfy the constraint
ri~
i = 0 ; (42)
which can be incorporated in the dual Hamiltonian by means of a Langrange multiplier ~A0.
Therefore, the dual Hamiltonian is
~H = 162(
1
2e2
+
e22
1282
)~i~
i +
e2
162
~iijk ~Fjk
+
e2
322
~Fij ~F
ij +
1
4
ijk ~Fjk@iA0 + ~A0ri~
i : (43)
The integration of A0 gives the constraint
ijkri ~Fijk = 0 ; (44)
6
which is just the Bianchi identity and allows us to express ~Fij locally as
~Fij = @i ~Aj − @j ~Ai : (45)
However, such expression will fail in general to hold globally in view of possible non-zero
magnetic fluxes Z
SI
ijk ~FijdSk = 4n
I ; (46)
through non-trivial two-cycles of X according to the Dirac condition. By performing an
inverse Legendre transform in (43), the dual Lagrangian is found to be
~L = −
1
2~e2
~F ~F
 +
~
322
 ~F ~F ; (47)
where
~e2 = e2(

42
+
162
e4
) ;
~ = −(

42
+
162
e4
)−1 : (48)
Thus, the classical dual theory is related to the original one by the transformation  ! −1= .
Since the partition function is invariant under canonical transformations we get
Z( ) = ImB0−B1(  )
1
2
(B1−B0)
Z
D ~AD~
1
V olG
ei
R
dtd3x
p
g(~i
~_Ai− ~H) : (49)
By employing the transformation  !−1= in eq. (37) we get the righthand side of (49) so
that
Z(−1=) = Z( ) ; (50)
and therefore the partition function for the U(1)-gauge theory in the canonical approach is
modular invariant.
Let us also note that there are no quantum corrections to the generating functional since
it is linear to Ai and ~Ai [15]. Moreover, physical states k[Ai] and  k[ ~Ai] in the original and
in the dual theory are related through
 k[ ~Ai] =
Z
DAeiG(A;
~A)[Ai] : (51)
These states are invariant under gauge transformations, i.e. k[Ai + @i] = k[Ai] so that
G(A− @; ~A)−G(A; ~A) = 2n ; n 2 Z : (52)
7
This condition is satised as long as the Dirac quantization condition (46) is satised.
As we have seen above, the partition function in the canonical approach is modular
invariant. On the other hand, in T-duality the modular anomaly is compensated by a change
of the dilaton eld leading to a modular invariant theory. Similarly here, it seems that the
anomaly is compensated by a change of the generalised momenta i. In this respect, i
integrations imitates metric integrations in the string case. Finally, one should expect that
the above discussion would also be carried out for the non-abelian case as well. However, in
this case, one cannot non-trivially satisfy the Gauss’ law constraint Dii = 0 and the dual
theory is identical to the original one. The non-abelian case will be discussed elsewhere.
While the present work was being proof-reading, we became aware of Ref[16] where
S-duality is also confronted as a canonical transformation.
8
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9
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10


A canonical approach to S-duality in abelian gauge theory

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A canonical approach to S-duality in Abelian gauge theory

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