Anomalies involving the space of couplings and the Zamolodchikov metric
HJE
Anomalies involving the space of couplings and the Zamolodchikov metric
CFT Correspondence 0 1
0 Kashiwa , Chiba 2778583 , Japan
1 Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo
The anomaly polynomial of a theory can involve not only curvature twoforms of the avor symmetry background but also twoforms on the space of coupling constants. As an example, we point out that there is a mixed anomaly between the Rsymmetry and the topology of the space of exactly marginal couplings of class S theories. Using supersymmetry, we translate this anomaly to the Kahler class of the Zamolodchikov metric. We compare the result against a holographic computation in the large N limit.
Anomalies in Field and String Theories; Supersymmetry and Duality; AdS

Zamolodchikov
1 Introduction and summary 2 3
the space of couplings [1{3].1
One example can be constructed as follows. Take an arbitrary target space M and
a U(N ) gauge eld A on it. Consider a Ddimensional free theory of a dynamical chiral
fermion
in the fundamental of U(N ) coupled to a background scalar
which is a map
to M in the following manner: the fermion
is minimally coupled to the pullback under
of the U(N ) eld A. Of course the theory has the anomaly polynomial
AD+2 = A^(T X) tr ei (F)
(1.1)
where X is the worldvolume of the theory and F is the curvature twoform on M. When
the scalar
is considered dynamical, this is known under the name of the model anomaly,
and makes the theory illde ned when not canceled. When
is considered as a background
eld, this anomaly polynomial involves di erential forms on the space M of couplings, and
serves as one of the characteristic properties of the theory, just as the ordinary 't Hooft
anomalies do.
To readers who found the example above rather arti cial, let us provide a more
meaningful case. In [5] Gaiotto introduced a large class of 4d N =2 theories obtained by
compactifying a 6d N =(2; 0) theory on a Riemann surface C, possibly decorated with punctures.
Here let us consider a simple case where C is of genus g without any puncture. This
construction gives rise to a family of 4d N =2 superconformal eld theories (SCFTs), now
1Strictly speaking in these papers the scalars were considered dynamical. But before 't Hooft [
4
] the
gauge
elds in the anomalies were also mostly considered dynamical. In this sense the anomaly involving
the space of couplings is known from the early days.
{ 1 {
known as class S theories, whose space of exactly marginal couplings is the moduli space
Mg of the genusg Riemann surfaces. We show below that this theory has a
hithertounappreciated term in the anomaly polynomial of the form
A6
c2 (R) h ! i
2
(1.2)
where c2(R) is the second Chern class of the background SU(2)R gauge eld and [!] is
a degree2 cohomology class on Mg, which can be determined from the known anomaly
polynomial of the 6d N =(2; 0) theory [6{8].
Now, a mixed anomaly between the Weyl transformation and the Kahler
transformation of the space of exactly marginal couplings of general 4d N =2 SCFTs was described
in [9{11].2 When one reads their derivation carefully, one nds that their analysis already
implies an anomaly of the form (1.2) above, with an added bonus that [!] is proportional
to the cohomology class [!Z] of the Kahler form !Z of the Zamolodchikov metric. Turning
the logic around, this means that we can easily x [!Z] in terms of the anomaly polynomial
of the 6d N =(2; 0) theory.
Finally, we note that the Zamolodchikov metric and therefore !Z is computable in
the large N limit by means of the AdS/CFT correspondence [13] using the holographic
dual of the class S theories [14, 15]. This is known to be proportional to the standard
WeilPetersson metric on Mg,3 but the precise proportionality coe cient has not been
computed to the author's knowledge. We will show below that the class of the Kahler form
computed holographically is compatible with the computation from the anomaly as above,
using a classic mathematical result by Wolpert [16, 17].
The rest of the note is devoted to implement the computations outlined above: in
section 2 we compute A6 of the class S theory from the anomaly of the 6d N =(2; 0)
theory and then use it to compute the Kahler class [!Z] of the Zamolodchikov metric.
Then in section 3 we compute the Zamolodchikov metric using holography, determine the
proportionality coe cient with respect to the standard WeilPetersson metric, and compare
it against the result in section 2.
2
2.1
Field theoretical computations
The 4d anomaly from the 6d anomaly
We start from the anomaly polynomial [6{8] of the 6d N =(2; 0) theory of type G =
An 1; Dn; E6;7;8:
h_GdG
24
rG
48
A8 =
p2(N Y ) +
p2(N Y )
p2(T Y ) + (p1(N Y )
p1(T Y ))2 :
(2.1)
1
4
2There they conjectured that the Kahler potential would be globally well de ned but this was answered
3The author does not know who originally noticed this; he forgot from whom he rst learned the fact.
This is surely a common knowledge among those who study class S theories using AdS/CFT.
{ 2 {
Here Y is the worldvolume of the theory, T Y is its tangent bundle, N Y is the SO(5)
Rsymmetry bundle, and p1, p2 are the Pontryagin classes; h_G, dG and rG are the dual Coxeter
number, the dimension and the rank of the Lie algebra of type G. We use the convention
that the anomaly polynomial gives the (D + 2)dimensional phase exp(RYD+2 2 iAD+2).
The N =2 class S theory of our interest is obtained by compactifying the 6d theory
on a Riemann surface C of genus g without any punctures so that we introduce a nonzero
curvature to the subgroup SO(2)
SO(5) of the Rsymmery which cancels the curvature
of C. This means that the Chern roots of N Y is
2
and
t where
are the Chern
roots of the SU(2)R background eld and t is the c1 of the tangent bundle of C. We note
that c2(R) =
2
.
Using p1 = P
i i2 and p2 = Pi<j i2 j2 when the Chern roots are
i, we easily get
HJEP12(07)4
A6
h_GdG +
6
rG
12
c2 (R)
Z
C
t2:
In the last factor, t is considered as the c1 of the relative tangent bundle of C over Mg
(i.e. the tangent bundle of the universal bundle U where C ,
! U
Mg minus the pull
back of the tangent bundle of Mg). Then t2 is a 4form on U , and we obtain a 2form on
Mg by integrating over the ber C.
2.2
Finding the Kahler potential
Suppose now that a given 4d N =2 SCFT has a space of exactly marginal couplings
parameterized by M with local complex coordinates I
marginal operators so that they enter in the deformation of the Lagrangian as
. We normalize the corresponding exactly
following the convention of [9{11]. We then de ne the Zamolodchikov metric gIZJ by the
formula
to be accompanied by a shift of the contact terms
In [11] the authors identi ed that the Kahler transformation K 7! K + F + F needs
S 7! S +
1
where we used the N =2 supergravity super elds as used in [11]. Using the component
expansion given in (5.5) of [18], we see that this shift contains the terms of the form
that the second Chern class is given by
Z
hOI (x)OJ (0)i = gIJ :
x8
c2(R) =
d4x
16 2 R(V )abijR~(V )abij
{ 3 {
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
we see that K 7! K + F + F is accompanied by
The Kahler form of the Zamolodchikov metric is
where
A =
Therefore, K 7! K + F + F does
HJEP12(07)4
A 7! A +
This means that the shift of the contact term (2.8) is exactly the gauge variation one
obtains from the anomaly polynomial
1
2
A6
c2 (R) !Z
24
2
Choosing the type to be AN 1 and taking the large N limit, we nd that our results (2.2)
and (2.11) imply
as cohomology classes.
3
Holographic computations
!Z
2
4N 3
Z
C
t
2
holds as di erential forms, not just as cohomology classes.
!WP
2 2 =
Z
C
t
2
{ 4 {
WeilPetersson metric. First let us recall the WeilPetersson metric on the moduli
space Mg of the genusg Riemann surface. Given Beltrami di erentials I on a Riemann
surface C, we de ne the Hermitean structure on them by
It is a classic mathematical result by Wolpert [16, 17] that the relation
hIWJP =
Z
C
I J dA
!WP = gIWJPd I d J =
12 hIWJPd I d J :
where dA is the area form of curvature
form of the WeilPetersson metric is given by
1, so that RC dA = 4 (g
1). Then the Kahler
MaldacenaNun~ez solution.
The holographic dual of the class S theory of type AN 1
on a genusg Riemann surface is given in [14, 15]. The 11d metric is of the form
where ds2AdS5 is the AdS5 metric of unit radius, ds2H2= is the metric on the Riemann surface
C represented as a quotient of the Poincare disk of curvature radius 1 by a discrete group
, the coordinates , ,
and
parameterize the internal space which is topologically of
the form S4, and W := 1 + cos2 . Our convention is that the 11d action is
space of genusg Riemann surfaces. The space Mg appears as the target space of the
massless scalars in ve dimensional supergravity, which in turn can be identi ed with the
space of exactly marginal couplings of the dual SCFT.
Reduction to
ve dimensions. Let us proceed with our computation. Integrating over
the foursphere part, we have
where the indices are appropriately placed. This means that upon reduction to 5d one nds
S5
4 (g
16 G(N7)
{ 5 {
with the 7d action
where
which means that
We now reduce it further to ve dimensions, including the deformation of the Riemann
surface. In general, under a small deformation g 7! g + h, we have
Now note that the Beltrami di erential := I zzI deforms the internal metric as
jdzj2 7! jdz + dzj2
g 7! g + h;
h := 2( + )
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
Translation to the dual SCFT.
At this point, we can use the formula [19] for the
central charge a
c to compute
which reproduces the standard result.
We are more interested in the Zamolodchikov metric on Mg, for which we use the
formula for the twopoint function under AdS/CFT given in [20]. The formula says that
given the action
HJEP12(07)4
for a real scalar in ve dimensions, the corresponding operator has the twopoint function
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
with the caveat that the deformation is introduced via the coupling
without an additional factor of 2 in the denominator as in (2.3).
Carefully collecting all the factors, one nds
meaning that
gIZJ =
16 G(N7) RAdS5 2 2hIWJP( 2)2 = 4N 3 gIWJP
1 3 24
!Z
2
4N 3 !WP
2 2 = 4N 3
Z
C
t
2
c =
;
in the large N limit. This is consistent with the result (2.12) we obtained from the
consideration of the anomaly in the last section.
Acknowledgments
The author thanks D. R. Morrison for discussions, and T. Max eld for pointing out a minor
error in (2.2) in an older version of the manuscript. The author is partially supported in part
byJSPS KAKENHI GrantinAid (WakateA), No.17H04837 and JSPS KAKENHI
GrantinAid (KibanS), No.16H06335, and also supported in part by WPI Initiative, MEXT,
Japan at IPMU, the University of Tokyo.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
{ 6 {
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