Anomalies, conformal manifolds, and spheres
HJE
Anomalies, conformal manifolds, and spheres
Jaume Gomis 0 1 3 6 7 8 9
PoShen Hsin 0 1 3 4 7 8 9
Zohar Komargodski 0 1 3 5 7 8 9
Adam Schwimmer 0 1 3 5 7 8 9
Nathan Seiberg 0 1 2 3 7 8 9
Stefan Theisen 0 1 3 7 8 9
0 Rehovot 76100 , Israel
1 Princeton , NJ 08544 , U.S.A
2 School of Natural Sciences, Institute for Advanced Study
3 Waterloo , Ontario, N2L 2Y5 , Canada
4 Department of Physics, Princeton University
5 Weizmann Institute of Science
6 Perimeter Institute for Theoretical Physics
7 14476 Golm , Germany
8 Princeton , NJ 08540 , U.S.A
9 is modi ed by simply replacing @
The twopoint function of exactly marginal operators leads to a universal contribution to the trace anomaly in even dimensions. We study aspects of this trace anomaly, emphasizing its interpretation as a sigma model, whose target space M is the space of conformal eld theories (a.k.a. the conformal manifold). When the underlying quantum theory is supersymmetric, this sigma model has to be appropriately supersymmetrized. As examples, we consider in some detail N = (2; 2) and N = (0; 2) supersymmetric theories in d = 2 and N = 2 supersymmetric theories in d = 4. This reasoning leads to new information about the conformal manifolds of these theories, for example, we show that the manifold is KahlerHodge and we further argue that it has vanishing Kahler class. For N = (2; 2) theories in d = 2 and N = 2 theories in d = 4 we also show that the relation between the sphere partition function and the Kahler potential of M follows immediately from the appropriate sigma models that we construct. Along the way we nd several examples of potential trace anomalies that obey the WessZumino consistency conditions, but can be ruled out by a more detailed analysis.
Supersymmetric gauge theory; Anomalies in Field and String Theories

eld
1 Introduction 2 3 5
The anomaly associated with the metric on M
N = (2; 2) supersymmetry in d = 2
4 (0,2) supersymmetric theories
N = 2 supersymmetry in d = 4
A Normalization of the anomaly
B The FTPR operator and its properties
C Review (and conventions) of twodimensional supersymmetry
C.1 (2; 2)
C.2 (0; 2)
D (2; 2) and (0; 2) supersymmetric backgrounds in superconformal gauge
means that when we add them to the action with coupling constants I
Some ddimensional conformal eld theories have exactly marginal operators fOI g. This
S =
1
d=2
X Z
I
ddx I
OI (x)
hOI (x)OJ (y)i =
gIJ ( K )
(x
y)2d
the theory remains conformal. The coe cients I parameterize the space of conformal eld
theories, a.k.a. the conformal manifold M. The twopoint functions
de ne a metric, known as the Zamolodchikov metric [1]. It is the metric on the conformal
manifold [2]. It carries nontrivial information that cannot be removed by rede nitions
of the coupling constants
I [3]. For example, the Ricci scalar associated to gIJ ( K ) is
invariant under all such rede nitions.
The purpose of this note is to explore the geometry and the topology of M.1 Our
main tool will be the conformal anomaly
rst discussed in [5]. By allowing
I to be
1In worldsheet string theory, M can be interpreted as the space of classical vacua of the theory. In the
AdSd+1/CFTd correspondence, the conformal manifold of the CFTd is interpreted as the space of vacua in
AdSd+1 (see e.g. [4]). These correspondences allow to connect our results to various other topics.
{ 1 {
spacetime dependent background elds, [5] derived a contribution to the trace of the
energymomentum tensor, which depends on the Zamolodchikov metric gIJ .
In supersymmetric theories the anomaly above must be supersymmetrized. This
introduces a few new elements into the analysis. First, it leads to restrictions on the local
form of the metric gIJ and it also leads to global restrictions. Second, the anomaly forces
us to introduce some contact terms. We will study both aspects in detail.
In section 2 we review the analysis of these conformal anomalies (without
supersymmetry). Here we spell out the conditions they have to satisfy and show how a careful analysis
leads to new constraints beyond the WessZumino consistency conditions [6].
In the remaining sections we will study N = (2; 2) and N = (0; 2) theories in two
dimensions and N = 2 theories in four dimensions.
Our discussion of twodimensional N
= (2; 2) theories in section 3 leads to a new
proof2 that M factorizes into a space Mc depending on chiral couplings ( , ) and a space
Mtc depending on twisted chiral couplings (e, e). Their Kahler potentials are Kc( ; ) and
Ktc(e; e). We will show that M must be Hodge3 and will further argue that its Kahler class
should be trivial. This, in particular, shows that M cannot be a smooth compact manifold.
We will also study the sphere partition function. Without supersymmetry, there are
counterterms that render it ambiguous. With N = (2; 2) supersymmetry there are two
ways, denoted A and V , to place the theory on the sphere and the partition function has
universal content. It is given by
ZA =
e Kc( ; ) ;
ZV =
e Ktc(e;e) :
(1.3)
r
r0
c
3
r
r0
c
3
Here r is the radius of the sphere and r0 a (scheme dependent) scale. The dependence
on r re ects the ordinary conformal anomaly. As we will show, the appearance of Kc or
Ktc re ects another contribution to the conformal anomaly depending on exactly marginal
couplings. The identi cations (1.3) were conjectured in [8] and proven in [9] as well as
in [10] based on the work of [11{13].
In section 4 we will discuss N
restrictions on the metric on M and shows that M is Hodge and suggests that its Kahler
class is trivial. But we will argue that the twosphere partition function is not universal.
Section 5 is devoted to N = 2 theories in d = 4. Again, the sphere partition function
has universal content and computes the Kahler potential on M
= (0; 2) theories in d = 2. Our analysis leads to
Z =
r
r0
4a
eK=12 :
(1.4)
This relation was proven in [10] as well as in [14] and was further used in [15{17]. N = 2
supersymmetry xes an additional contribution to the conformal anomaly depending on
a fourtensor in M in terms of the Riemann tensor of the Zamolodchikov metric. As in
2In this paper we assume that the coupling constants can be promoted to N = (2; 2) chiral and twisted
chiral super elds. This assumption is nontrivial as it can fail in some cases [7].
3A KahlerHodge manifold is a Kahler manifold for which the ux of the Kahler twoform through any
twocycle is an integer.
{ 2 {
is trivial.4
two dimensions, our analysis shows that M is Hodge and suggests that its Kahler class
Our discussion is reminiscent of that of [19, 20]. In both cases nontrivial contact
terms are identi ed. They cannot be absorbed by supersymmetric local counterterms and
therefore correspond to anomalies. They re ect short distance physics and can be analyzed
in the at space theory. Then, these contact terms have interesting consequences when the
theory is placed on the sphere.
For a related supersymmetric analysis of conformal anomalies in N = 1 theories in
d = 4 and N = 2 theories in d = 3 see [21, 22].
Four appendices contain technical results. Appendix A concerns with the
normalizaPaneitzRiegert (FTPR) operator, which appears in the anomaly in d = 4. Appendix C
reviews (2; 2) and (0; 2) supersymmetry in two dimensions and their linearized supergravities.
Appendix D considers (2; 2) Poincare supergravity in superconformal gauge (which always
exists locally). We classify the allowed rigid supersymmetric backgrounds in this gauge.
2
The anomaly associated with the metric on
M
In momentum space the twopoint functions (1.2) take the following form
hOI (p)OJ ( p)i
gIJ
p
d
p2n log
2
p2
d = 2n + 1
d = 2n :
The explicit scale (or cuto )
in the logarithm does not violate scale invariance. The
reason is that rescaling
changes the answer by a polynomial in p2, which is a contact
term. The correlation function at separated points (1.2) therefore remains intact under
rescaling . Such logarithms appear abundantly in conformal eld theories (CFTs). Even
though they do not violate the conformal Ward identities, they lead to anomalies (i.e.
the nonconservation of the dilatation charge in the presence of nonvanishing background
elds). One way to detect it is to make the couplings, I , xdependent. Then, the trace
anomaly in even dimensions includes a term, roughly of the form [5, 23]:5
T
gIJ I d2 J :
(2.1)
(2.2)
The precise action of the Laplacian could be to the left and to the right. We will specify
this later for d = 2 and d = 4.
We study CFTs with spacetime metric
and spacetime dependent coupling
constants
4We do not make this claim for N = 1 theories in four dimensions. In fact, [18] suggested a construction
5This term is in addition to the ordinary conformal anomalies, which depend only on the spacetime
{ 3 {
the partition function Z[
formations
; I ] and its variation
log Z under in nitesimal Weyl
trans
of compact support. Naively, conformal invariance means that the
variation vanishes. But because of the anomaly, it does not. This variation satis es a
number of important properties:
; I ] is a nonlocal functional of its arguments. However, its variation
2.
log Z must be coordinate invariant in spacetime.
3. It must be coordinate invariant in M. Below we will argue that it should also be
globally well de ned on M.
4. It must obey the WessZumino consistency condition [6]
1 2 log Z
2 1 log Z = 0 :
(2.4)
5. A term in
log Z that is obtained by a Weyl variation of a local term is considered
trivial. An anomaly is a \cohomologically nontrivial" term. It cannot be removed
by changing a counterterm. Equivalently, it cannot be removed by changing the
renormalization scheme [24]. Therefore, even though the anomaly arises due to a
short distance regulator, it is universal  it does not depend on the regularization.
Let us start in d = 2. The in nitesimal Weyl variation of log Z responsible for the
trace anomaly (2.2) and the ordinary trace anomaly is given by
I
J :
(2.5)
Here R is the Ricci scalar and the rst term is the universal contribution due to the central
charge c. The normalization of the second term is worked out in appendix A.
The anomaly functional (2.5) includes a sigma model with target space M. It
manifestly obeys the WessZumino consistency condition because it is Weyl invariant. There is
no local counterterm, whose Weyl variation yields (2.5). Therefore, (2.5) is cohomologically
nontrivial. In the language of [25] one could refer to the rst term in (2.5) as a typeA and
to the second term as a typeB anomaly.6
An important part of our discussion will be the analysis of the allowed local
counterterms (related to item 5 in the list above). In two dimensions, an important counterterm is
Z
d2x p
RF ( I ) :
(2.6)
6A typeA anomaly vanishes for xindependent
when the background
elds have trivial topology.
I
with an antisymmetric twoform BIJ on the conformal manifold M. Also, we have two
HJEP03(216)
Its Weyl variation is typeA anomalies and
Z
d2x p
RF ( I ) =
2
Z
d2x p
F ( I ) :
log Z
log Z
Z
Z
p
p
VI
VeI
I
I
;
:
(2.7)
(2.8)
(2.9)
(2.10)
They are characterized by oneforms VI and VeI on M. We note that (2.10) is invariant
under the gauge transformation VeI ! VeI + @I fe. On the other hand, (2.9) transforms under
VI ! VI + @I f , but the change is cohomologically trivial. It can be absorbed in the Weyl
variation of the local counterterm (2.6) with F
VeI as connections on the conformal manifold M.
f ( I ). This allows us to identify VI and
with
operators
=
We will now show that even though (2.8), (2.9), (2.10) obey the WessZumino
consistency conditions, a more detailed analysis leads to further restrictions, ruling out these
anomalies. This demonstrates that constraints that go beyond the standard cohomological
analysis can further restrict anomalies.
First, a simple argument excludes all typeB anomalies that are beyond that in (2.5).
We recall that a typeB anomaly is associated to a logarithm appearing in a correlation
function. Without loss of generality we can study the theory in
at Euclidean spacetime
. Consider the momentum space correlation function of the exactly marginal
hOI1 (p1)OI2 (p2)
OIn (pn)i = log
X pr AI1I2 In +
;
(2.11)
where the ellipses on the righthand side represent terms independent of the UV cuto
. Since the operators OI are exactly marginal, the coe cient of the logarithm must be
ultralocal, i.e. a polynomial in momentum (otherwise, there would be a beta function for
the couplings I ). Scale invariance constrains AI1I2 In to be quadratic polynomials in the
momenta pr. Therefore, we can determine AI1I2 In by picking speci c simple combinations
of momenta. For example, for p2 =
p1 and p3 = p4 =
= 0 it is clear that AI1I2 In
@In gI1I2 . Similar other speci c cases show that AI1I2 In is determined entirely
by derivatives of the Zamolodchikov metric. This means that the additional parity odd
anomaly (2.8) controlled by a twoform cannot be present. More precisely, the argument
above shows that HIJK = @[I BJK] = 0 and hence BIJ is locally given by BIJ = @[I BJ] for
some oneform BJ . By integration by parts we nd that this anomaly is now identical to
the typeA anomaly (2.10). We will discuss it below.
{ 5 {
Next, we argue that VI and VeI in the typeA anomalies (2.9), (2.10) must satisfy
@[I VJ] = @[I VeJ] = 0, i.e. these connections are at. These anomalies can be extracted from
the following energymomentum correlator
h
T (z)OI (x)OJ (y)i :
(2.12)
Using the conformal Ward identity at separated points, the correlator must be proportional
to gIJ . There could also be contributions with support at x = y 6= z, or z = x 6= y, or
z = y 6= x. In the rst case the only contact term allowed by dimensional analysis contains
T , which has zero separatedpoints correlation functions. In the second and third case,
we can have the contact term T
(x)OI (0)
(2)(x)MI OK (0) with some matrix MIK .
K
This would lead to a logarithmic term in the threepoint function (2.12), of the type already
analyzed above, and hence, it is proportional to the Zamolodchikov metric and does not
contribute to the anomalies VI and VeI .
Therefore, all the terms in (2.12) that are associated to separated points physics are
proportional to gIJ . They cannot lead to nonzero \ eld strengths" @[I VJ] or @[I VeJ], which
are antisymmetric in I and J . Thus, at least locally, the connections VI and VeI are pure
gauge and the associated anomalies vanish.
Summarizing, we have seen that even though (2.5), (2.9), (2.10), are a priori allowed
anomalies that obey the WessZumino consistency conditions, they can be all excluded.
This will have important consequences in what follows.
Thus far we limited ourselves to deformations by exactly marginal operators with
coe cients I as in (1.1). If the CFT also has conserved currents ja, then it is natural
to couple them to classical background elds Aa and examine the anomaly as a function
of these elds. The anomaly sigma models now depend on the spacetime metric
couplings I , and the gauge elds Aa . The operators OI are taken to carry charges
, the
qIa and
the coupling constants I thus carry charges qIa. Away from
I = 0 some of the symmetries
generated by ja may be thus explicitly broken. Related expressions appear in [23]. In
addition to the previous requirements of conformal invariance, coordinate invariance in
spacetime and on M, we should now also demand gauge invariance. The equation (2.5)
I ! r
I
iqJa Aa J . In addition, one could
encounter new anomalies that contain the eld strength F a . There could also be 't Hooft
anomalies under gauge transformations.
In d = 4 the local functional that reproduces the logarithm in the twopoint
function (2.1) is a fourderivative local term. One can construct it by starting with the ansatz
log Z
R d4x
p gIJ
I
J +
and covariantize this expression both in spacetime
After some work7 one nds the expression8
and in M. One also requires that it satis es the WessZumino consistency condition (2.4).
1
3
I
R
R
J
: (2.13)
log Z
1
is proportional to the Zamolodchikov metric
at the
xed point plays a very important role in perturbative proofs of the strong version of the
atheorem [5, 26{29]. The situation in d = 6 unfortunately appears to be more complicated [30, 31].
For a review see [32].
{ 6 {
Zamolodchikov metric. sions
Z
d4x p
Above b I =
K , where IJK is the usual Christo el connection on M.
The ordinary Laplacian
is enriched to b so that the anomaly is coordinate invariant
on M, as we demand in general. At this juncture
I
JK could be an arbitrary connection,
not necessarily the LeviCivita one. However, demanding that (2.13) satis es the
WessZumino consistency condition forces IJK to be the LeviCivita connection. Note that (2.13)
coincides with expressions that appeared in [28] and [31] in related contexts. The
combination (2.13) can be viewed as an interesting variant of the FradkinTseytlinPaneitzRiegert
operator [33{36], which we discuss further in appendix B.
While we do not present an exhaustive classi cation of anomalies in four dimensions,
there is an additional conformal anomaly that depends on a fourtensor on M with
components cIJKL that we would like to mention:
log Z
Z
d4x
I
J
(2.14)
The fourtensor cIJKL may be either an independent rankfour tensor on the manifold
M, or it may be xed by the Zamolodchikov metric, e.g. cIJKL
RIKJL + RJKIL, where RIKJL is the Riemann tensor on M.9 The WessZumino
consistency condition (2.4) does not imply a relation between cIJKL and the Zamolodchikov
gIJ gKL or cIJKL
metric. However, in section 5 we will show that in N = 2 supersymmetric theories, such
a relation must exist, and cIJKL is proportional to the Riemann curvature tensor of the
For future references, let us also list some of the allowed counterterms in four
dimenR2 F1( I ) + R
2 F2( I ) + R
2
F3( I ) +
:
(2.15)
We will be particularly interested in the case where the underlying theory is
supersymmetric. Then, the exactly marginal couplings I reside in various super elds [38]. If the
superconformal eld theory (SCFT) can be regularized in a supersymmetric manner, then
we must further require that the local anomaly functionals above be supersymmetrized.
We will study some of the consequences of supersymmetrizing (2.5) and (2.13). We will
show that the remaining ambiguity (2.6) in the renormalization scheme in two dimensions
and (2.15) in four dimensions is restricted to have holomorphic dependence on the coupling
constants. This fact has several important consequences. In particular, it makes the sphere
partition function meaningful (up to a Kahler transformation generated by a holomorphic
= (2; 2) supersymmetry in d = 2
{ 7 {
function [10, 14]).
3
N
in N
Our goal in this section is to determine the conformal anomaly and analyze its consequences
= (2; 2) supersymmetric theories in two dimensions.
Here the exactly marginal
parameters belong either to background chiral multiplets or twisted chiral multiplets, which
we denote by I and eA, respectively.
9We thank Y. Nakayama for a discussion on the topic and for stressing the potential relevance of the
anomaly (2.14) to the question of locality in AdSd+1. See for instance [37].
First, we should supersymmetrize the anomaly (2.5) and the counterterm (2.6). For
that we need to place the theory not only in curved space but in curved superspace [39].
N = (2; 2) supergravity was discussed in [40{44] and, in particular, the possibilities for
rigid supersymmetry in curved space were analyzed in [44]. (We repeat this analysis in the
superconformal gauge in appendix D.)
We should discuss two distinct supergravity formulations known as U(1)V and U(1)A
supergravities [40]. These are labeled by whether the U(1) symmetry preserved in the
Poincare supergravity theory is vector or axial.10
In terms of the (2; 2) SCFT this distinction is the following. The (2; 2) SCFT has
a U(1)V
U(1)A Rsymmetry. We can couple either U(1)V or U(1)A to a background
gauge eld, but an anomaly prevents us from coupling both of them to background elds.
Correspondingly, the coincident points divergences and the associated contact terms can
preserve either U(1)V or U(1)A Rsymmetry but not both. These contact terms are
described by the corresponding supergravity. Equivalently, we assume that the theory can
be regularized while preserving di eomorphism invariance and supersymmetry as well as
either U(1)V or U(1)A. In particular, we assume that there are no gravitational anomalies
so that cL = cR.
We nd it convenient to use a simpli cation speci c to two dimensions. Since locally
every twodimensional metric is conformally
at, we can describe the metric using the
conformal factor
 the Liouville eld. This statement is easily supersymmetrized. Every
supergravity background can be described locally by a superconformal factor in a super eld.
In U(1)A supergravity it is in a chiral super eld
and in U(1)V supergravity it is in a
twisted chiral super eld e (see appendix C). The corresponding superconformal variations,
whose anomalies we are interested in, are
and
e respectively. In what follows we will
concentrate mainly on U(1)A. It is straightforward to repeat it for U(1)V .
The supersymmetrization of the conformal anomaly (2.5) is then straightforward. In
the regularization preserving U(1)A, the anomaly is given by
where K is a complex function of the exactly marginal couplings.11 Clearly, these
expressions obey the WessZumino consistency condition.
One might try to integrate (3.1) to
nd the
dependence of log ZA. Although this
can be done as a local expression in terms of , the answer is nonlocal. The point is that
it is valid and local in the superconformal gauge, but it is nonlocal in other gauges. This
property makes it particularly interesting, as it cannot be absorbed in local counterterms.
10To follow the discussion below (in our analysis of twodimensional theories) no familiarity with
supergravity is necessary.
11In the full supergravity without using the conformal gauge the anomaly takes the form
that contains the Ricci scalar in its 2 component. The rst term represents the ordinary anomaly.
where E is the chiral superspace measure, E is the Berezinian super eld, and here R is a chiral super eld
{ 8 {
In order to proceed we must nd the most general supersymmetric expression, which is
local in any gauge, and can serve as a local counterterm. This is the supersymmetrization
of (2.6). In U(1)A the local counterterm is [10]
SA =
2
erm (3.3) depends only on the chiral parameters
Under a superWeyl transformation
is the chiral curvature super eld in superconformal gauge. The
countertand the dependence is holomorphic.12
SA =
Further restrictions on K can be found by expanding (3.1) in components and requiring
that the forbidden twodimensional anomalies (2.8), (2.9), (2.10) are absent. Note that this
goes beyond the WessZumino consistency conditions. After some algebra, the conclusion
is that K is real and
K = Kc( ; )
Ktc(e; e) ;
and therefore the metric on M is a product metric of two Kahler manifolds M = Mc
The Kahler potential on Mc is Kc and it depends only on the chiral parameters and the
Kahler potential on Mtc is Ktc and it depends only on the twisted chiral parameters. This
splitting between the chiral and the twisted chiral parameters is well known and is natural
in the context of type II string theory, where (2; 2) worldsheet theories lead to N = 2
supersymmetry in spacetime. The hypermultiplet and the vector multiplet metrics are
factorized as a consequence. Here we see that it follows from properties of anomalies on
is trivially invariant under
In addition, under the Kahler transformation
the anomaly shifts by the superWeyl variation (3.5) of the supersymmetric local
counterterm (3.3).
12In the full supergravity without using the conformal gauge the counterterm is
K ! K + G(e) + G(e) :
K ! K + F ( ) + F ( )
4
SA = 1 Z d2x d2 E R F ( ) + c.c. :
{ 9 {
(3.5)
(3.6)
ways with interesting consequences. First, we can simply state that the Kahler
transformation should be accompanied by a change in a local counterterm. Second, we can assign
a transformation law to
!
+
F ( )
c
6
c
and use the rst term in (3.7) to achieve full Kahler invariance of the anomaly (here we
assume c 6= 0). This perspective means that e 6 is a holomorphic section of a line bundle,
whose rst Chern class is the cohomology class of the Kahler form on M and therefore
M must be Hodge. This result, which we have now derived using the anomaly, is known
for sigma models with CalabiYau target spaces and for general (2; 2) theories. It is also
natural in the context of string compacti cation as a property of the fourdimensional
supergravity theory [45] (see a re nement of this statement in [46{49]). In that context
the action depends on
c
6
(
+
) + K where
equivalently, it is the spacetime conformal compensator. This is similar to our , which
is the twodimensional conformal factor. Indeed, integrating the anomaly (3.7) we obtain
the anomalous piece of the e ective action in superconformal gauge
is the spacetime dilaton super eld, or
(3.10)
Finally, we can try to use this analysis to suggest a stronger result. It is well known,
and we have used it extensively, that the anomaly variation is a well de ned local term.
The lack of strict Kahler invariance means that our anomaly is not quite well de ned. If
the Kahler class of M is trivial, there is no immediate problem since we are not forced
to perform the Kahler transformations (3.9) and we thus have a global description of the
theory. Di erent presentations of the theory might be related by Kahler transformations,
but this can be absorbed in a local counterterm or in a rede nition of . However, when
the Kahler class of M is nontrivial, there is a di culty. In that case we must cover M with
patches and transition functions that involve Kahler transformations and correspondingly
a change in the counterterm (2.6).
Now, consider the couplings changing in spacetime in such a way that we must use the
transition functions (e.g. spacetime wraps a nontrivial cycle in M). Here di erent parts
of spacetime have coupling constants in di erent patches in M and since the transition
functions between them need a counterterm, e.g. (2.6), there is no single Lagrangian in all
of spacetime! We suggest that such a situation is inconsistent. This would mean that M
and the various elds on it are such that no such transition functions are needed. Therefore,
we arrive at the conclusion that M must have vanishing Kahler class. This argument is
analogous to that of [50], with the di erence being that we are considering the properties
of the space of theories rather than the usual target space of a speci c sigma model.
More generally, we should always require that the scale variation of the partition
function is a local, globallyde ned functional of the background
elds. In our context,
the anomaly functional contains K explicitly and is therefore not invariant under Kahler
transformations. The anomaly functional is well de ned only if the Kahler class vanishes.
Let us now extract some useful physical information from our anomalies (3.7). It
su ces for our purposes to evaluate the anomaly keeping only the bottom components of
the multiplets of the exactly marginal parameters I and e
A and of
where a parameterizes the U(1)V transformation. We nd
B
=
+ i a ;
+
+
c
6
A
=
Ae =
i
2
i
2
A :
Here we have integrated by parts and used the metrics gIJ and geAB on Mc and Mtc as
well as the pullback of the Kahler oneforms A
and Ae , whose exterior derivatives give
the Kahler twoforms of Kc and Ktc. We note that only the Kahler potential Kc for the
chiral multiplets appears in the term proportional to
in (3.13). This is due to the fact
that
is a chiral multiplet. This point will be important below.
Another way of stating our equations uses the supercurrent multiplet. As we review
in appendix C, the relevant axial supercurrent multiplet consists of real (in Lorentzian
signature) J
and a chiral W satisfying
D J
=
D
W :
In a conformal theory W = 0. Our anomaly is
W =
c
24
R +
1
4
D2(Kc( ; )
Ktc(e; e)) ;
where R = D
2
is the chiral curvature super eld. As above, it is invariant under Kahler
transformations of Ktc, but not under Kahler transformations of Kc.
We can absorb
Kahler transformations of Kc by improvements of the energymomentum tensor multiplet.
Alternatively, we can make it invariant by also shifting
. From the rst point of view
it thus follows that if the Kahler class of M is nonvanishing, upon letting the coupling
constants wrap some twocycle in M, we would not be able to de ne a single
energymomentum tensor throughout our twodimensional space. This again suggests that the
Kahler class of Kc vanishes.
In the absence of supersymmetry, the last term
Kc in the rst line of (3.13)
would be cohomologically trivial and could be tuned away by an appropriate choice of
1
2
+ a a
Kc
!
;
HJEP03(216)
(3.12)
(3.13)
(3.14)
(3.15)
regularization scheme. Indeed, this term is proportional to the variation of the local
term,
R d2xp RKc.
However, since we are, by assumption, de ning the partition
function using a supersymmetric regulator, cohomologically trivial terms must arise from
the Weyl variation of U(1)A supergravity invariants.
The most general such term is
R d2xp
R(F ( ) + F ( )) with holomorphic F ( ) (see (3.3)). Therefore, modulo Kahler
transformations, the anomaly in (3.13) is a genuine new contribution to the trace anomaly
in N = (2; 2) SCFTs.
Even in the absence of supersymmetry, the terms in (3.13)
1 Z
2
d2x
B
(3.16)
are cohomology nontrivial since they cannot be generated by the Weyl variation of any
local term. These are precisely the terms we discussed in (2.5). This part of the anomaly
is captured by a nonlocal term in the e ective action, whose Weyl variation reproduces
the anomaly (3.16). Supersymmetry relates this nonlocal universal term to local terms
that upon a Weyl transformation give rise to the term
Kc in (3.13). We can thus
reconstruct these terms in the e ective action by integrating the anomaly sigma model.
In the evaluation of the partition function for constant sources I and eA and vanishing
a, it su ces to focus on the term
Kc. First we covariantize it
Using
p R =
2p
we learn that the partition function contains
(3.17)
(3.18)
(3.19)
We repeat that this is not a supersymmetric local term. It is related by supersymmetry to
some nonlocal terms that generate the anomaly (3.16). This is the reason the coe cient
of (3.18) is physical. The superWeyl invariant terms in ZA vanish identically since all
twodimensional supergravity backgrounds are superconformally
at.
Upon evaluating the partition function on the twosphere S2 and for constant sources
we obtain that the S2 partition function of a N = (2; 2) SCFT regularized preserving
U(1)A is
ZA[S2] =
e 81 RS2 d2xp RKc =
r
r0
c
3
e Kc ;
where we exhibit the radius of the sphere r, which arises from the ordinary central charge
anomaly. Note that in agreement with the picture above, we either say that ZA is not
Kahler invariant, or we accompany Kahler transformations with r ! re 3c (F ( )+F ( )). We
remark that (3.19) is correct for any compact manifold with the topology of the twosphere,
the prefactor being reexpressed in terms of the area of the manifold. This is consistent
with [9], who argued that the S2 partition function is independent of squashing.
The analysis extends almost verbatim in U(1)V supergravity.
In this case the
anomaly (3.7) becomes
e
log ZV
1 Z
4
d2x d4 ( e +
e) Ktc( ; )
Kc(e; e) :
(3.20)
And using e =
e
log ZV =
+ ia
1 Z
2
d2x
B
e
+ a a
e e
Ktc
!
:
(3.21)
(3.22)
r
r0
c
3
ZV [S2] =
e 81 RS2 d2xp RKtc =
r
r0
c
3
e Ktc :
Now the Kahler potential for the twisted chiral multiplets Ktc appears explicitly in the
anomaly since e is a twisted chiral multiplet. Integrating the anomaly, as above, we
arrive at
HJEP03(216)
In summary, we have rederived the result that supersymmetric S2 partition functions
of N = (2; 2) SCFTs are expressed in terms of the Kahler potential on the appropriate
moduli space of theories [8, 9] (see also [10]). Our derivation shows that this phenomenon
follows directly from a new trace anomaly in supersymmetric eld theories. The new trace
anomaly is tied by supersymmetry to the anomaly associated with the Zamolodchikov
metric. Our methods also led to the suggestion that the Kahler class of M vanishes.
It is important that the anomalies we discussed re ect UV physics. They are
independent of the background spacetime and can be explored locally in
at space. The sphere
partition function was used as a tool to extract this anomaly. We note that we simply
substituted
of a sphere.
In appendix D we discuss a classi cation of supersymmetric backgrounds using our
superconformal gauge formalism. Speci cally, we consider
=
+ ia + 2w with various
, a, and w. The round sphere discussed above corresponds to a = 0 but with
and w
nonvanishing. We would like now to make a few comments on the other possible
supersymmetric backgrounds, and in particular, about the topologically twisted background on
the twosphere a = i (such that
= R = 0). Due to the anomaly c=6 R d2x a a in (3.13),
the partition function needs to transform with a nonzero phase under U(1)V
transformations. Hence, the S2 partition function on the twisted sphere vanishes (see [51{53]). If one
introduces the parameter w, then this phase can be absorbed by including w c=3 in the
partition function. The partition function can be argued to be holomorphic as a function
of the coupling constants. But since the holomorphic counterterm (3.3) does not vanish
(the antiholomorphic one vanishes), only the singular part of the dependence on coupling
constants is physical. It would be interesting to understand what these singular pieces
mean. They were recently computed in [54, 55]. Another interesting open question
concerns Calabi's diastasis, which is a nice Kahler invariant observable in (2; 2) SCFTs. It has
an elegant interpretation in terms of conformal interfaces [56], and it would be interesting
to see if our methods shed light on it.
(0,2) supersymmetric theories
Here we consider (0; 2) SCFTs and study their trace anomaly. Our conventions are such
that the supersymmetry is rightmoving. The exactly marginal operators are necessarily
in Fermi multiplets [57] (see [58] for background on (0; 2) models). The corresponding
couplings are in chiral super elds I
. We will determine their contribution to the
conformal anomaly.
We plan to place the theory in a nontrivial supergravity background. This is simpler
when the supergravity theory is anomaly free. First, to avoid gravitational anomalies we
must relate the leftmoving and the rightmoving central charges cL = cR. Similarly, cR
determines the anomaly in the rightmoving U(1) current and we assume that there is also
a leftmoving U(1) current with the same anomaly. Then we can gauge an anomaly free
linear combination of these two currents. Note that even if we do not have such an anomaly
free setup, we can imagine adding decoupled elds to achieve it.
Under these assumptions, the supergravity transformations (which include gauge
transformations for a U(1) gauge eld) are nonanomalous. One can then naturally couple the
theory to the corresponding supergravity and study it in nontrivial supergravity
backgrounds.
We will refer to the gauged U(1) symmetry as axial (as in our discussion of
U(1)A (2; 2) supergravity above) and then the vector U(1) symmetry is a global symmetry,
su ering from an anomaly.
Before delving into a technical discussion, let us summarize what we nd. The trace
anomaly in (0; 2) models contains a term depending on the Kahler potential,
supersymmetrizing the ordinary bosonic anomaly (2.5). But there is an additional term in the trace
anomaly that depends on a new function of the couplings, H. This function is not xed
by the (0; 2) theory. It depends on precisely how we couple the theory to the background
elds. It is instructive to consider a (2; 2) theory viewed as a (0; 2) theory. Then this
function H is given as
H
Kc
Ktc ;
(4.1)
i.e. in this case the function is physical and unambiguous. But in general (0; 2) models
it is ambiguous. The sphere partition function depends on H and therefore the sphere
partition function in such theories is not universal. However, the arguments leading to
the conclusion that the Kahler class vanishes do hold in (0; 2) models and in particular,
the moduli space of SCFTs cannot be compact. This is in accord with intuition from the
heterotic string, where (0; 2) models are used to construct N = 1 supergravity theories in
spacetime. In such cases it is known that the vacuum manifold is KahlerHodge [45] (see
also [46{49]). For a related stringy discussion see [57, 59].
As in the (2; 2) theory, we nd it convenient to use the superconformal gauge. But
unlike the (2; 2) theory there are two natural \conformal gauges." The di erence between
them is in the gauge condition imposed on the U(1) gauge eld.
One possibility is to use the gauge A
= 0, where A
couples to the rightmoving
U(1) current j++ in the superconformal algebra. In this case the remaining degrees of
freedom are in a real super eld V =
+ 2i +
+ + 2i +
+ + + +
A++. In the linearized
+ ia + i +
+
i + +
These two multiplets are almost identical. Given
supergravity approximation which we review in appendix C (see also [60]) this corresponds
to H++ = H
= 0 with V = 12 H.
Alternatively, we impose Lorentz gauge on that gauge eld @ A
= 0, which is solved
locally by A
=
@ a. Then the remaining degrees of freedom are in a chiral multiplet
i D+D+V is nonlocal. The lack of locality a ects
but not in V . Conversely, a constant mode of A++
we can write V = 12
(
+
). But
given V , the chiral super eld
=
only a. Its zero mode is present in
correspond to a linearly growing a.
these elds
(4.2)
(4.3)
The gauge invariant chiral curvature super eld can be expressed using either of
HJEP03(216)
R
D+ :
SuperWeyl transformations are associated with a chiral
and simply shift . Their
action on V is 2 V =
+
. The shift of a by a constant represents the action of the
global vector U(1) symmetry that is not gauged. This global symmetry shifts
by an
imaginary constant and does not act on V .
Then, the most general expression for the anomaly action is
I
+ c.c. :
So far, AI ; BI are arbitrary functions of the couplings.
in our two slightly di erent versions of conformal gauge:
The rst term is the ordinary central charge anomaly. It can also be written as follows
c Z
6
(
+
) = i
(
V :
(4.4)
The functions AI ; BI represent the anomalies that arise in the presence of exactly marginal
coupling constants. Already from this we can infer that the metric on M is Hermitian.
The components expansion of (4.3) leads to a term proportional to
J + c.c.. However as we discussed in section 2, this is consistent only if
(locally) AI = @I K. Furthermore, K has to be real in order to eliminate typeA anomalies
that are present upon expanding the rst term. Similar considerations show that BI = @I H
with some real function H.
Therefore, the anomaly must be of the form
V + H
2c
3
+ (
+
I
!
:
(4.5)
The expression (4.3) satis es the usual consistency conditions including the
WessZumino conditions. We therefore see that our additional considerations in section 2
concerning which anomalies are allowed show that the metric on M must be Kahler (in accord
with intuition from heterotic compacti cations, which lead to N = 1 supergravities). Below
we will also nd some global restrictions on M.
As in (3.14){(3.15), we can express the anomaly as an operator statement. The theories
we study have a supercurrent multiplet with real R
appendix C)
and T
satisfying [60] (see
D+ (T
R
= 0 ;
When the theory is conformal we also have D+R
that we assumed exists in the CFT. Our anomaly modi es D+R
= 0 to
is the leftmoving current
c
12
D+R
= i
R
+
I
I
H) :
Next we should identify the ambiguity (i.e. the cohomologically trivial terms that
arise from variations of (0; 2)supersymmetric local counterterms). This will allow us to
determine the actual anomaly. For that we should supersymmetrize (2.6). We can either
use the full nonlinear supergravity (see e.g. [61]), or simply use linearized supergravity as
in [60] and appendix C to show that the local counterterm is
i
4
i Z
8
d2xd +h( I )R ;
where h( I ) is holomorphic. This counterterm allows us to absorb some holomorphic
transformations on H and K but most of the information in (4.5) is cohomologically nontrivial.
Even though the anomaly associated with H seems like a nontrivial anomaly, which
cannot be absorbed in a local counterterm, in fact, it is not physical in (0; 2) theories.
It can be absorbed in a rede nition of V .13 Physically, this means that we rede ne the
metric and its superpartners by some function of the coupling constants I ; I . In other
words, when we allow the couplings to be general functions, we can add new terms to the
Lagrangian that vanish upon setting the couplings to constants. Such a freedom exists
in (0; 2) theories and it leads to the anomaly H. There is no a priori principle that xes
H, unless the theory is a (2; 2) theory in which case this freedom does not exist and H
becomes physical (4.1).
After removing H we conclude that the anomaly can be written as
log Z = i
c
12
Z
c.c. :
(4.9)
13Note that we cannot absorb H in
+
by a local transformation. However, this rede nition is indeed
a truly local transformation in (0; 2) supergravity. It can be understood in linearized supergravity before
picking any gauge. There we simply shift H by the function H( I ; I
theory to curved space by additional terms in the Lagrangian, which depend on the coupling constants.
For conformal theories, this modi cation only depends on derivatives of the coupling constants. Such
). This modi es the couplings of the
ambiguities do not play a role in (2; 2) theories.
(4.6)
HJEP03(216)
(4.7)
(4.8)
Expanding it in components with the only nonzero background elds
=
A
c
6
(
1
4
I
K +
I
1
4
I
(4.10)
HJEP03(216)
The rst term is the ordinary anomaly. The second term is the anomaly (2.5) associated
with the metric on M.
c
The situation with Kahler transformations is as in (2; 2) theories. Kahler
transformations can be absorbed in a shift of . As there, e 6
is a holomorphic section of a line
bundle over M and therefore M is not only Kahler, but it is also Hodge. Also, as in (2; 2)
theories, we suggest that the Kahler class of M is in fact trivial.
These results are consistent with the expectation from the string application of these
models. When the (0; 2) theory is used as the worldsheet of a compacti ed heterotic string
it leads to N = 1 supersymmetry in four dimensions and M is the target space of some
of its chiral super elds. In this case it is known that M should be Kahler [57, 59] and
Hodge [45]. We extend these conclusions to all (0; 2) SCFTs and argue that the Kahler
class of M should be trivial.
While the anomaly functional (4.10) contains the term
K, the partition function depends on the choice of H (which we have set to zero for simplicity) and therefore it is not universal.
5
N
= 2 supersymmetry in d = 4
We now proceed to the supersymmetric generalization of the conformal anomaly (2.13).
For N = 2 supersymmetry the appropriate superspace expression is
log Z
1
+
)K( I ; I ) :
The superWeyl parameters and are chiral and antichiral super elds, respectively.
They can be viewed as a conformal compensator in N = 2 supergravity [62].
are chiral and antichiral super elds with Weyl weight zero, whose lowest components are
the exactly marginal couplings, which we also denote as I and
. K( ; ) is the Kahler
I
potential on the conformal manifold M.
In addition to the anomaly that contains the moduli, we also have the usual Weyl
anomaly, which depends only on the supergravity multiplet. Its superspace expression is
(5.1)
I and
I
where
A and
A are N = 2 chiral and antichiral multiplets with Weyl weight w = 0,
respectively. For our anomaly (5.2) we will then specify to
K( A
; A) =
1
(
+
) K( I ; I ) :
For calculating the component expansion of (5.3) we follow [64]. We start with the
special case K = A B
, where A and B are chiral and antichiral multiplets respectively.
Keeping only the bottom components Aj = A, Bj = B, and the metric background (i.e.
dropping the bosonic auxiliary elds in the supergravity multiplet), we get
S =
Z
d4xp
r2A r2B
2 r A
R
1
3
R
r B
:
expressed as a sum P
term in the sum. Doing this we arrive at
In order to nd the answer for a generic K( A
; A) we expand around a reference point
and then use the fact that product of chiral multiplets with Weyl weight w = 0 is a chiral
multiplet with w = 0 and similarly for antichiral multiplets. Then K( A
; A) can be
i AiBi with chiral Ai and antichiral Bi and we can use (5.5) for each
S =
Z
d4xp
KABCDr
+ KABC r
2 KABr
A
A
r
r
A R
B
B
r
C
r
D + KABC r
A
r
B
C
C + KAB
1
3
R
r
A
B
;
B
S =
1 Z
4
; A) ;
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
an integral over chiral N = 2 superspace (see e.g. [63] and appendix B for details)
log Z
+ (c
a) W
W
and
are chiral super elds. W
is the Weyl super eld, while
is constructed
from curvature super elds that appear in the commutators of supercovariant derivatives
in curved superspace.
expansion of an action of the general form
To work out the component eld expansion of (5.1), we need to know the component
where A = ( ; I ) ( A = ( ; I ). Using (5.4) and the following de nitions for the metric,
connection and curvature on a Kahler manifold with Kahler potential K
Z
d4xp
log Z
+
+
1
2
i
2
+ i R
gIJ
K
2
b
2 gIJ r
I
+
b
1
6
I
r
1
3
R
J
J
(
r r
I
r
J
RIKJLr
1
3
R
I
r
r
+ i gIJ rb rb
rI K r
I
rI K r
K r R r
+ K
R
we arrive, after several integrations by parts, at
J
I
1
3
I
r
r
r
I
J
R
J
r
K
r
L
r r
rb rb
)
a :
rbI rbJ K r
rbI rbJ K r
I
r
J + rI K b
I
rI K b
J
r
I
r
I
a
a
(5.8)
HJEP03(216)
As in (2.13), the hats denote covariant derivatives with respect to target space di
eomorphisms acting on the I
. Note that this action is completely covariant under target space
di eomorphisms and we nicely identify the term (2.13) in the second line of (5.8). We also
identify the new anomaly (2.14) in the rst line. It appears with the Riemann tensor of
M. To take into account the complete anomaly we have to add (5.2) to (5.8).
Using the expressions in appendix B one realizes that the terms in the third line of (5.8)
can be written as a variation of a local term. Speci cally,
p
where c is an arbitrary function of the moduli and
4 the FTPR operator (see appendix B).
For c = 0 this is precisely the combination that appears in the N = 2 supersymmetric
version of the GaussBonnet invariant (see e.g. [65]). The supersymmetric GaussBonnet
term may include as a prefactor an arbitrary holomorphic function of the moduli.
Therefore, (5.8) is cohomologically trivial if K = F + F is a sum of a holomorphic and
antiholomorphic function of the moduli, in which case it reduces to
(5.9)
(5.10)
(5.11)
(5.12)
Indeed, consider the following local superspace counterterm
The combination
W
W
contains the Euler combination E4
R and its Weyl
variation is the supersymmetrization of (5.10).
To arrive at the S4 partition function we simply need to integrate the combination
2
3
appearing in (5.9) on S4. Using
d4x p (F
4 + c.c.) :
W
W
+ c:c:
1
192 2
Z
Z
Z
S4
p
E4
2
3
R + c C2
= 64 2
Z[S4] =
r
r0
4a
eK=12 ;
as claimed in [10] and [14]. We note that (5.13) is true for any superconformally
at
compact fourmanifold if we express the prefactor in terms of its volume.
The Kahler ambiguity K ! K + F + F of the partition function is taken care of by the
ambiguous local counterterm (5.11). As in d = 2, we will now see that the trace anomaly
is invariant under a correlated Kahler shift and Weyl transformation.
To nd the change of the anomaly polynomial (5.2) under an in nitesimal Weyl
transformation
e, we use (cf. [63])
where , the chiral projection operator, is the N = 2 supersymmetric generalization of the
FTPR operator (see also appendix B). The Weyl super eld W
transforms homogeneously
with weight one, while the chiral superspace density E transforms with weight
2 (the full
superspace density E is invariant). We then nd
On the other hand, under a Kahler shift K ! K + F + F it transforms as
F
Therefore, choosing
Z
e
1
(5.13)
(5.14)
(5.15)
(5.17)
d4x d8 E(
F +
F ) =
F + c.c. (5.16)
e =
1
24a
F ;
the anomaly polynomial is invariant under an in nitesimal joint KahlerWeyl
transformation and therefore also under a nite transformation. The invariance can be explicitly seen
to hold for the partition function (5.13).
As in two dimensions, this means that M is not only Kahler, but it is also Hodge. In
addition, using background
I that vary in spacetime and wrap a nontrivial cycle in M,
we argue that the Kahler class of M
must be trivial. (For certain cases with an N = 4
AdS5 dual, it has been argued in [66] that M is specialKahler. It would be interesting to
understand when this happens in general.)
Acknowledgments
We would like to thank C. Bachas, C. Closset, S. Cremonesi, L. Di Pietro, N. Ishtiaque,
S. Kuzenko, D. Morrison, Y. Nakayama, V. Niarchos, H. Ooguri, H. Osborn, K.
Papadodimas, K. Skenderis, E. Witten, and J. Zhou for useful discussions. Z.K. would like to thank
the Perimeter Institute for its very kind hospitality during the course of this project. The
work of NS was supported in part by DOE grant DESC0009988. Z.K. is supported by
14Since the S4 background is superconformally at, the superWeyl invariant terms in Z vanish.
whose anomalous Weyl variation is
1
jxj4 =
1
32
(anom) 1
2 log2(x2 2) ;
jxj4 =
2
(2)(x) :
This is the contact term that is reproduced by the anomaly functional (2.5).
In d = 4 we write
whose anomalous Weyl variation is
the ERC STG grant 335182, by the Israel Science Foundation under grant 884/11, by the
United StatesIsrael Bi national Science Foundation (BSF) under grant 2010/629 as well
as by the Israel Science Foundation center for excellence grant (grant no. 1989/14). A.S.
and Z.K. are supported by the ICORE Program of the Planning and Budgeting
Committee. P.H. is supported by Physics Department of Princeton University. A.S. and S.T.
acknowledge support from GIF  the GermanIsraeli Foundation for Scienti c Research
and Development. This research was supported in part by Perimeter Institute for
Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada
through Industry Canada and by the Province of Ontario through the Ministry of Research
and Innovation. J.G. also acknowledges further support from an NSERC Discovery Grant
and from an ERA grant by the Province of Ontario. Any opinions, ndings, and
conclusions or recommendations expressed in this material are those of the authors and do not
necessarily re ect the views of the funding agencies.
A
Normalization of the anomaly
The normalization of the anomaly (2.5) and (2.13) is xed as follows. We compute the
change of the contact term in d = 2 under constant rescaling of the coordinates x ! e x
by writing the twopoint function as
(A.1)
(A.2)
(A.3)
(A.4)
(B.1)
1
jxj8 =
1
768
3 log(x2 2)
x2
;
(anom) 1
jxj8 =
2
96
2 (4)(x) :
Z
d4x p
4 ;
This is matched by the anomaly (2.13).
B
The FTPR operator and its properties
We collect some properties of the FradkinTseytlinPaneitzRiegert operator [33{36]. It
arises in our context in the case that there is only one exactly marginal modulus and
one integrates (2.13) by parts. One obtains, up to cohomologically trivial terms, that the
anomaly is
where
+ 2 R
r r
R
is the FTPRoperator with the de ning property that under Weyl rescaling of the metric,
when it acts on a scalar. Another property of
4 which is used in section 5 is
This can be derived using
E4
R
=
4
R
+ 4 4
:
R
6
;
and the expression for the Euler density which we normalize to
such that
E4 = C
C
R
2 2
R
3
Z
S4
d4x p
E4 = 64 2
:
Here C is the Weyl tensor.
In two dimensions instead of
E2 =
R =
2
R
d = 2, N = (2; 2) this is the chiral projection operator r
which transforms as r
! e
under superWeyl transformations, which are parameterized by a chiral super eld
2
2
The conformally covariant operators have generalizations in chiral superspace. In
a similar transformation for the antichiral projector r2).
For N = 2 in d = 4 the analog of the FTPR operator is the chiral projection operator
with the in nitesimal transformation
= 2
under a superWeyl transformation
parameterized by a chiral scalar super eld
. Its precise de nition in terms of
supercovariant derivatives and curvature super elds is reviewed in [63], where one also
nds
references to the original literature.
C
C.1
(2; 2)
Review (and conventions) of twodimensional supersymmetry
We will use the notation x
for the coordinates, which makes it easier to compare with
spinors. In Euclidean signature x++ is the complex conjugate of x
and in Lorentzian
signature they are two real independent coordinates. The Ricci scalar is given by Rp
=
2
1 log in the conformal gauge, where
= det
.
The supercovariant derivatives, which can be obtained from the fourdimensional ones
of Wess and Bagger by dimensional reduction, are
D
=
i
;
D
=
+ i
;
The algebra is
Chiral super elds
and twisted chiral super elds e are de ned by
= 0
D+e = D e = 0 :
HJEP03(216)
There are two interesting energymomentum supermultiplets in a (2; 2) theory with
an Rsymmetry [60]. They are related by mirror symmetry. First, there is the U(1)V
supermultiplet
D
R
=
;
D+
= D
= 0 ;
D+
= D
+ :
It immediately follows that the bottom component, R
current
V
j++ = 0 :
Often, there exists a twisted chiral operator Te such that
j
V
is a conserved vector
Often, there is a chiral W such that
and then (C.8) is replaced by
A
j++ = 0 :
= D
W
D
W :
(C.1)
(C.2)
(C.3)
(C.4)
(C.5)
(C.6)
(C.7)
(C.8)
(C.9)
(C.10)
(C.11)
and then the last two equations in (C.4) are automatically satis ed and the rst becomes
+ = D+Te ;
=
D T
e
D+R
=
D Te ;
D
R++ =
D+Te :
The U(1)A supermultiplet is obtained formally by acting with a mirror symmetry
transformation on (C.4)
D+Y
+ D Y+ = 0 :
In this case jA
J
is a conserved axial current
D J
=
Y ;
D Y
= 0 ;
D Y
= 0 ;
H
then takes the form
Now we discuss linearized coupling to supergravity (see also the analysis of [44].) We
start from the case of U(1)A supergravity. The supergravity multiplet is (H
; ), where
is real (in Lorentzian signature) and
is chiral. The linearized coupling to matter
L =
Z
d
4
X
H
J
+
Z
d
2
W + c.c. :
This is invariant under the linearized transformations
H
= D L
D2(D+L
D L ;
D L+) ;
Above is chiral and is antichiral. U is a general multiplet.
H++ ! H++ +
H
! H
++ +
++ ;
;
reality of H
relations (C.4).
where L
are arbitrary super elds. Note that the rst line of (C.13) is consistent with the
and that the action (C.12) is invariant under (C.13) by using the de ning
In the superconformal gauge H
= 0 and the only degree of freedom is
=
+ia+: : :.
is the conformal factor in the metric and a represents the U(1)A gauge eld in
Lorentz gauge A
=
@ a. Under the superWeyl transformations
is shifted by a chiral
super eld
. The shift of a by a constant is a U(1)V global symmetry transformation.
The situation in U(1)V supergravity is analogous. In that case the only mode in the
superconformal gauge is the twisted chiral super eld e.
C.2
(0; 2)
three super elds, R
; T
For (0; 2) theories with a conserved Rsymmetry the supercurrent multiplet consists of
, which are real in Lorentzian signature. They satisfy [60]
R
= 0 ;
j++ = 0. In components we have
R++ = j++
T
R
= j
= T
i +S+++
i +S+
i +
i +
S+++
S+
+
+
T++++ ;
T++
;
+ +
We couple this theory to linearized supergravity in a standard fashion. We introduce
three real super elds, H++; H; H
, and the linearized coupling takes the form
L =
Z
d +d +
T
H++ + R
H + R++H
) :
The complete superdi eomorphism group is generated by the following transformations
(C.12)
(C.13)
(C.14)
(C.15)
(C.16)
(C.17)
The curvature is in the invariant chiral super eld
R
H
4
i +(Rp
2iF ) :
(C.18)
are components of the \gravitino" and originate from H and H
,
respectively. R and F are the Ricci scalar and the eld strength of the U(1) Rgauge eld,
D
(2; 2) and (0; 2) supersymmetric backgrounds in superconformal gauge
In this appendix we repeat the classi cation of supersymmetric backgrounds of [44]. These
authors used linearized supergravity to nd the equations for supersymmetric backgrounds
and then covariantized them. Instead, we will use the superconformal gauge. This way we
will not have to rely on linearized or the full nonlinear supergravity. The point is that every
(2; 2) background in two dimensions can be brought locally to a superconformal gauge and
then all the information is contained in a chiral super eld
. (More precisely, we will be
using U(1)A supergravity where the conformal factor is in a chiral multiplet. It is trivial
to repeat the analysis in U(1)V .)
As we said, the advantage of using this presentation is that there is no need to use
supergravity. We simply use at space ordinary superconformal symmetry.
We will set the fermionic components of
to zero; i.e.
=
+ia+ 2w,
=
ia+ 2w.
We will assume that , which is the conformal factor, is real. But we will allow nonunitary
backgrounds in which a can be complex and w is not the complex conjugate of w.
Most of our analysis will be local. The global considerations are easily implemented
later. We will denote the Killing spinors for the supersymmetry variation as
;
with
, and will view them as four independent complex variables (no particular reality).
The conditions for supersymmetry are
ia +
i
2
i
2
e +ia
e
ia
i
2
i
2
we +ia + = 0
we
ia + = 0 :
(D.1)
The rst equation is the standard restriction due to at space superconformal symmetry.
The remaining equations state that the fermionic components of e and e
are invariant.
Note that + and
have the same U(1)A Rcharge, and e +ia and w are neutral under it.
The N = (2; 2) supersymmetric background
can be classi ed as follows according
to the preserved supercharges given by the Killing spinors
; .
The rst class of backgrounds preserves one supercharge of a given U(1)A charge.
Without loss of generality we can take the Killing spinors as nonzero ( +;
) and
+ =
again (D.2) with
which is related to (D.2) by
constrains
and a
where + =
+(x++) and
=
) are arbitrary functions of x++ and x
respectively. (In Euclidean space they are holomorphic and antiholomorphic functions.) Recall
that our analysis is local; global considerations restrict these functions.
charge, denoted as ( 1+; 1 ) and ( 2+; 2 ). Imposing invariance under ( 1+; 1 ) we
The second class of backgrounds preserves two supercharges with opposite U(1)A
Rnd
! 1. Imposing invariance under ( 2+; 2 ) we nd another expression,
a. Consistency of the two solutions
HJEP03(216)
+ +
+ ia + log 2+(x++) 1 (x
and a are invariant under the vector v
a superconformal transformation by log 2+(x++) 1 (x
)).
+ +
topological twist and
deformation respectively.
generality take 1+ = 0 with nonzero 1 . The solution (D.2) gives
There are two such cases depending on whether v = 0 or v 6= 0. They lead to the
For v = 0 it su ces to assume that one component of v vanishes. Without loss of
( x++; 1) and ( 2+; 2 ) = (1;
The corresponding isometry vector is v =
are arbitrary function of the invariant combination x++x
satis es 1 2 + 2 1 = iLv 6= 0. And the background is
i.e. a = i , w = 0 with arbitrary
and w. This is the topological twist. The antitopological
twist corresponds to a =
i .
In the second case v 6= 0 and
and a must have an isometry given by the vector
v. Up to a conformal transformation we can take the Killing spinors to be ( 1+; 1 ) =
x
). Here
is a constant describing the
deformation.
x
). Therefore,
and a
. The supersymmetry algebra
= 0. We nd for every
and a
=
+ ia
2i 2
+
2 +
ia + 2i
+ ia + log
ia + log +
(D.2)
(D.3)
(up to
(D.4)
(D.5)
(D.6)
= 2w;
= 2 ;
2
x
= ( + ia)
= (
ia) + 2i
2i 2 1
( + ia) ;
ia + log x++ :
Two limiting cases are interesting. For
! 0, the solution (D.5) becomes the
topological background (D.4). For a = i
with nonzero , equation (D.5) reproduces the
twodimensional background in [44]
= 0;
= 2 + 2i
) :
The background (D.5) preserves maximally four supercharges, if and only if the
spacetime metric is maximally symmetric and the U(1)A gauge
eld has zero curvature
a = 0. For a sphere the maximally supersymmetric background is given by
log 1 + x2 + 2 2i=
log 1 + x2 +
1 + x2
2 2i
1 + x2
;
where x2
. The sphere background (D.7) is not unitary since
result agrees with the supersymmetric sphere background of [
11, 12
].
Similarly we can classify the N = (0; 2) supersymmetric backgrounds. They are
determined by imposing the vanishing rightmoving supersymmetry variation on the chiral
super eld e = e +ia
+ = 0 ;
ia +
= 0 :
There is only one class of N
referred to as (anti) topological halftwist in [58].
= (0; 2) smooth supersymmetry backgrounds. They are
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