#### Surveying 4d SCFTs twisted on Riemann surfaces

Received: April
Surveying 4d SCFTs twisted on Riemann surfaces
Antonio Amariti 0 1 2 3
Luca Cassia 0 1 2
Silvia Penati 0 1 2
Open Access 0 1 2
c The Authors. 0 1 2
Supersymmetry, Conformal Field Theory
0 Piazza della Scienza 3 , 20161, Milano , Italy
1 Sidlerstrasse 5 , Bern, ch-3012 , Switzerland
2 Institute for Theoretical Physics, University of Bern
3 Albert Einstein Center for Fundamental Physics
Within the framework of four dimensional conformal supergravity we consider N = 1; 2; 3; 4 supersymmetric theories generally twisted along the abelian subgroups of the R-symmetry and possibly other global symmetry groups. Upon compacti cation on constant curvature Riemann surfaces with arbitrary genus we provide an extensive classi cation of the resulting two dimensional theories according to the amount of supersymmetry that is preserved. Exploiting the c-extremization prescription introduced in arXiv:1211.4030 we develop a general procedure to obtain the central charge for 2d N = (0; 2) theories and the expression of the corresponding R-current in terms of the original 4d one and its mixing with the other abelian global currents.
Supergravity Models; Anomalies in Field and String Theories; Extended
1 Introduction 2 3 4
Twisted reduction of N = 1 SCFTs
Twisting with avors
Classi cation of the solutions
Twisted reduction of N = 2 SCFTs
Twisting with avors
Classi cation of the solutions
Twisted reduction of N = 3 SCFTs
Classi cation of the solutions
Twisted reduction of N = 4 SCFTs
Classi cation of the solutions
Further directions
A The anomaly polynomial
B Supersymmetry variations of the auxiliary
elds in N = 3; 4 SCFTs
Introduction
Two dimensional (super) conformal eld theories ((S)CFTs) play a central role in the
worldsheet description of string theory and in the formulation of the AdS3/CFT2
correspondence. Moreover, being the conformal group in nite dimensional, many exact results
can be extracted from their algebraic structure. Classifying 2d CFTs is anyway a di cult
nding new examples of conformal theories is not straightforward.
A powerful laboratory to build in nite families of 2d CFTs is supersymmetry. SCFTs
in 2d can be obtained by compactifying 4d SCFTs on curved compact 2d manifolds. In
this process some of the original supersymmetry charges survive whenever Killing spinor
equations arising from requiring fermion variations to vanish, admit non-trivial solutions.
In general, this does not happen since on curved manifolds there are no covariantly
constant Killing spinors. However, as suggested in [1] (see also [2, 3]), this problem can be
circumvented by performing a (partial) topological twist, i.e. by turning on background
gauge elds for (a subgroup of) the R-symmetry group along the internal manifold in such
a way that its contribution to the Killing spinor equations compensates the contribution
from the spin connection. More generally, one can also turn on properly quantized
backuxes for other non-R
avor symmetries. In this case preserving supersymmetry
also requires to set to zero the associated gaugino variations.
Although this procedure does not allow to extract the matter content of the 2d theory,
useful information on its IR behavior is provided by the 2d global anomalies that can be
obtained in terms of the 4d ones and of the background
Focusing on 2d theories with N
= (0; 2) (or equivalently, N
= (2; 0))
supersymmetry, the corresponding central charge cL (cR) is proportional to the anomaly of
the abelian R-symmetry current inherited from the exact 4d R-current JR, obtained by
a-maximization [5]. However, under dimensional reduction 4d abelian global currents can
mix with the exact 4d J , hence the exact 2d R-current has to be determined by
extremizing the 2d central charge cL (cR) as a function of such a mixing. The program of
such c-extremization principle was derived in [4].
An interesting phenomenon regarding the mixing of global currents with J
Riemann surfaces. There, it was observed that even though there is a global (baryonic)
symmetry, that does not mix with J
mixing with J
richer structure of baryonic symmetries.
xed point [18, 19], it has a non-trivial
xed point. This phenomenon is generalizable to cases with a
Motivated by the former discussion, in this paper we engineer the partial
topological twist in the natural setup of conformal supergravity and study systematically the
twisted compacti cation on constant curvature Riemann surfaces of 4d SCFTs with
different amount of supersymmetry. In this uni ed framework we investigate the cases of
is a genus g Riemann surface.
We study the conditions to preserve di erent
amounts of supersymmetry in 2d by solving the Killing spinor equations arising from
setting to zero the variations of the gravitino and of the auxiliary fermions in the Weyl
mulwe also turn on vector multiplets associated to global non-R symmetries. In this case an
additional constraining equation for Killing spinors arises from setting to zero the variation
of the corresponding gaugino.
= 3; 4 theories, where
avor symmetries are absent, we also discuss the
possibility of twisting in two steps. This consists in a rst twist along an abelian subgroup
of SU(3)R
U(1)R or SU(4)R, reducing the R-symmetry and leaving some vector multiplets
associated to non-R global symmetries. A further twist along such symmetries corresponds
vide the (formal) expression for the anomaly coe cients and the central charge in terms
of the 4d anomalies and the
uxes, as obtained by c-extremization. Concurrently, the
explicit expression for the exact 2d R-current is given as a linear combination of the 4d
R-current and global non-R symmetries. In section 7 we conclude by commenting on some
possible future lines of research. In appendix A few necessary details on the anomaly
polynomial are collected. In appendix B we provide further details on the vanishing of the
Twisted reduction of N
= 1 SCFTs
manifold M = R1;1
is a Riemann surface of genus g and constant scalar
curvature. Twisted compacti cation of this class of theories has been already discussed
x the general scheme that we will use in the N -extended cases.
R1;1 coordinates are labelled (x0; x1), while the ones on
connection ! on
satis es the relation
are (x2; x3). The spin
where R2(!) is precisely a representative 2-form for the rst Chern class of the tangent bundle
class is usually denoted as c1(T ) 2 H2( ; Z). The curvature for a Riemann surface can
be written in terms of the volume form d Vol and the Gaussian curvature K as
For later convenience we de ne the normalized scalar curvature
sgn(K), the normalized
volume form
R(!) = 2
R(!) = K d Vol
8> jKjd Vol for K 6= 0
for K = 0
and the total normalized volume
so that R(!)
= 2
In general, compacti cation on
breaks supersymmetry completely, since on
arbitrarily curved manifolds there are no covariantly constant Killing spinors. Along the lines
of [21], in order to put a 4d theory on a curved manifold and preserve some supersymmetry
we couple the theory to a conformal supergravity background that reproduces the desired
spacetime geometry. The whole superconformal group is gauged and the corresponding
gauge elds are organized into the Weyl multiplet as follows (we use notations and
conare indicated in table 1. Here Pa,Ka are vector generators of translations and special
conformal transformations, Mab and
are generators of Lorentz rotations and dilatations, Q
and S are the spinorial supercharges. The U(1)R R-symmetry generator TR assigns charge
1 to the positive chirality supercharges Q
and charge +1 to their conjugates
we will often write TR =
5 with 5 = i 0123.
The supersymmetry transformation laws of the independent gauge elds read
ea =
b =
= D "
where "; are the Majorana spinors associated to Q and S transformations, respectively.
The covariant derivative is de ned as D "
iA TR)".
Since we are only interested in theories on curved manifolds with rigid supersymmetry,
we x the Weyl multiplet to be a collection of background
elds describing the geometry
of spacetime. In order to preserve Lorentz invariance on R1;1 we set all the spinor elds to
zero and assign possibly non-vanishing components to bosonic forms only in the (x2; x3)
directions. As follows from (2.8), in general this choice breaks superconformal invariance.
However, some Q-supersymmetry survives if the geometry admits non-trivial covariantly
may have non-trivial solutions if we turn on a non-zero background also for the R-symmetry
gauge connection A [1] such that the two contributions coming from A and !ab in the
= 0).1 This equation
covariant derivative cancel each other.
More precisely, focusing on constant solutions, we rst apply the exterior derivative to
] =
iR (A) 5 " = 0
where R (!23) and R (A) are the curvatures of the connections !23 and A , respectively.
Given the particular form of the curvature R
(!) =
, we choose A such that its
curvature is also proportional to the normalized volume form
R (A) =
where the parameter a is constrained by the Dirac quantization condition
Substituting (2.10) in (2.9), we then obtain
R(A) =
a 5 " = 0
We postpone the search and classi cation of non-vanishing solutions to section 2.2.2.
1To begin with one could solve the equation
= 0 for non-vanishing , by setting = 14D=" [23]. The
We now consider the case in which the original 4d theory also admits a global abelian
nonR symmetry that can be either avor or baryonic symmetry. With an abuse of notation,
we call it U(1) avor.
This symmetry can be weakly gauged by turning on a background connection.2
However, in order to preserve the original superconformal symmetry one has to turn on a whole
gauge vector potential B , the gaugino
and the auxiliary scalar Y , all in the adjoint
representation of the avor symmetry. The corresponding supersymmetry transformations are
where R (B) is the curvature 2-form of the gauge connection B and the covariant
derivative on spinors is de ned as in eq. (2.8).
connection with constant curvature
Similarly to the case of the R-symmetry background in (2.10), we can choose a U(1) avor
Y = i" 5
R (B) = b
Y i 5 " = 0
together with vanishing background gaugino. In order to preserve some supersymmetry we
have to require
where jej = e22e33
e23e32 is the vielbein determinant on .
23. Therefore, setting Y =
bjej 23 we nally obtain the condition
01) " = 0
We then see that in principle, turning on a background for an abelian non-R global
symmetry, introduces additional constraints on the supersymmetry generators.
More generally, we can consider 4d theories with rank{n avor symmetry group, i.e.
with n generators Ti in the Cartan subalgebra. In this case we can gauge one vector
multiplet (Bi ; i; Y i) for each Cartan generator. If the corresponding auxiliary scalars are
bijej 23) we are led to the same
constraints (2.16).
2Similar discussions appeared in [6, 8, 24].
=2 = 0
a + =2 = 0
are automatically
irreducible representations of the R-symmetry group corresponding to charge 1.
Classi cation of the solutions
We are now ready to discuss the most general solutions of the two supersymmetry
preserving conditions
a 5 " = 0 ;
01) " = 0
as Spin(3; 1) ! Spin(1; 1)
also splits as
where the constant a signals the presence of a non-trivial U(1)R background, eq. (2.10),
while bi are associated to Bi connections for U(1) avor symmetries, eq. (2.14). We note
that the second equation is nothing but a 2d (anti)chirality condition.
In order to nd solutions to these equations, we write the Majorana spinor " in terms of
its Weyl components, " = (
to the positive chiral spinor
_ ), and with no loss of generality we restrict the discussion
transforming in the 2 of SL(2; C).
On the product manifold R1;1
the original Lorentz group of 4d Minkowski is reduced
Spin(2) , and consequently the spinorial representation of
Here the representations on the right hand side are labelled by the eigenvalues of the
hermitian generators 01 and i 23 of Spin(1; 1) and Spin(2) , respectively. The generator
01 corresponds also to the chirality operator on R1;1, hence we refer to 11;1 and 1 1; 1 as
the 2d positive (left) and negative (right) chirality representations respectively, and denote
the corresponding spinors as + and
As summarized in table 2, for
to + for a = 2 and
for a =
. The second equation in (2.17) does not restrict the
Killing spinors any further, since we can always choose bi such that (2.16) projects on the
same chirality as that of the Killing spinor. Therefore, independently of the presence of
for a =
. These solutions are compatible with the quantization condition a 2 Z, being
an even number.
In the special case of compacti cation on a torus,
= 0, when no avor symmetry
to zero, the Killing
spinor equation reduces to @
R-symmetry U(1)left
U(1)right generated by the two combinations T
= 12 TR
M23, where
M23 is the Lorentz generator on
. Supersymmetry can be reduced by gauging some avor
symmetry. In this case, in fact, the second equation in (2.17) constrains the supercharges
b = 0
b 6= 0
a = 2
N = (2; 0)
N = (2; 0)
a =
N = (0; 2)
N = (0; 2)
b = 0
b 6= 0
N = (2; 2)
N = (2; 0) or (0; 2)
1; 0 in terms of the surviving amount of supersymmetry in 2d. We include the
possibility of a twist along the avor symmetries, with ux b.
N = (0; 2) for Yi =
in table 3, where the resulting 2d theories are classi ed in terms of the surviving amount
of supersymmetry.
Twisted reduction of N
= 2 SCFTs
We now consider a N
U(1)R. The Lie
Pauli matrices.
The four-dimensional chiral supercharges Q I are in the (2; 2) 1 representation of the
group Spin(3; 1)
form in the (2; 2)+1 representation. In particular, the U(1)R generator TR acts on the
supercharges as
group U(1)RN =1 generated by the combination
TRN =1 =
manifold M = R1;1
Here we give a systematic derivation within the superconformal gravity setup.
and then gauge xing the background Weyl multiplet as to reproduce the desired geometry
with possibly non-trivial uxes turned on in order to preserve some supersymmetry.
group ea ; f a; b ; !ab, the superconnections
I associated to supersymmetries QI and
SI , the connections A
and V A for the R-symmetry groups U(1)R and SU(2)R and the
auxiliary elds Tab; D (bosonic) and
I (fermionic), needed to close the algebra o -shell.
Under supersymmetry transformations the fermionic
elds of the gravity multiplet
transform as
I = @ + 1 b + 1 !ab
I =
4 D= Tab"IJ "J
V A(i A)IJ "J
Rab(A)i 5"
In order to preserve Lorentz invariance on R1;1 the background fermions must be set to
zero. This choice automatically sets to zero the Q-supersymmetry variation of all bosonic
elds, which can then be chosen such that the Q-variation of the fermions vanish as well.
From (3.2) and (3.3) we deduce that we can safely set the background elds b and
Tab to zero and simplify these expressions to
I =
I =
V A(i A)IJ "J
1 ab Rab(A)i 5"I + Rab(V A)(i A)IJ "J
2 Z, and
The remaining background connections A
topological twist as we now describe.
Turning on a background
ux for V A breaks explicitly the SU(2)R invariance of the
theory down to a U(1) subgroup of it. Without loss of generality we choose this subgroup
to be the one generated by i 3. Namely, we parametrize the R-symmetry gauging as follows
and V A can then be used to perform partial
R (A) =
R (V A=1;2) = 0;
R (V 3) =
is the normalized volume form of . This choice is actually equivalent to gauging the
1-parameter subgroup of SU(2)R
U(1)R generated by a1TR + a2 3.
Looking for constant spinor solutions of (3.4) and (3.5) we can apply the exterior
covariant derivative to
thus turning the Killing spinor equation into an equation for
the curvatures. Substituting the background (3.6) we nd
I] =
= i
I =
where (3.8) is obtained by substituting (3.7) in (3.5) and therefore it is only valid on the
components of I that are actual solutions of the Killing spinor equation.
are then left with a single de ning equation for Killing spinors.
Twisting with
Before solving the Killing spinor equation (3.7) we generalize the discussion to the case
of 4d SCFTs admitting some global abelian non-R symmetry U(1) avor. Weakly gauging
(B ; X; I ; Y A). Such a multiplet contains one gauge
with curvature R
one complex scalar X, two gaugini I forming a SU(2) doublet, and one auxiliary eld
the supersymmetry variations of the bosonic components of the multiplet are identically
vanishing, and they can be chosen to satisfy
I =
Rab(B) ab JI + Y A(i A)IJ "
Gauging the global symmetry along
with R (B) = b
, and setting for instance
Y 1;2 = 0, Y 3 =
2b for the positive chirality component of "J we obtain
J = 0
( 01 + 1) 1 = 0
1) 2 = 0
where we have used i 23 = 01 5 and 5
J = J
The previous condition is equivalent to requiring that the two components of the I
Another possibility to perform the avor twist would be via a two-step procedure. We
section 2. Observe that we could engineer such a reduction also in the absence of avor
symmetries. In that case we should rst perform a R-symmetry twist that preserves four
supercharges. This twist would break R-symmetry and leave an unbroken U(1) that could
be treated as avor symmetry useful for further twisting.
Classi cation of the solutions
chirality components I in the (2; 2) representation as
In order to
nd solutions to eq. (3.7) we observe that the selected background breaks
SU(2)R ! Spin(1; 1)
U(1) 3 , and correspondingly the positive
where on the r.h.s.
indices denote the 2d chirality of the reduced spinors
I =
I =
We can nd solutions to (3.7) by appropriately choosing the values of the twisting
parameters ai as summarized in table 4. A further constraint comes from eq. (3.10) when a global
non-R symmetry is also gauged.
= 0, separately.
amount of supersymmetry preserved in 2d. All the other choices are related by a trivial
change of basis of the symmetries or a di erent choice of sign for the auxiliary elds.
For a1 =
2 which form a
SU(2)R doublet. The 4d R-symmetry is left unbroken and the 2d theory is a chiral
=2 = 0
a1 + a2 + =2 = 0
=2 = 0
a2 + =2 = 0
are preserved when the twisting parameters ai satisfy the corresponding equations in the column
on the right.
imply that only one of the two components of the doublet can be preserved
according to the particular choice of the auxiliary
For a1 = 0 and a2 =
broken to U(1)2 with generators T
2 the preserved supersymmetries are 1
M23 and the preserved supersymmetry
eld Y A in the vector multiplet, hence
M23 + 12 3 becomes a avor symmetry in two dimensions since, by de
nition, the preserved supercharges transform trivially under it. In this case, gauging
a global non-R symmetry with the corresponding connection B
together with the
choice of auxiliary Y 3 =
2b , does not constrain the Killing spinors any further (see
For a1 + a2 =
(0; 2) with U(1) R-symmetry. In this case there are two new abelian avor symmetries
that were not present in the original 4d theory, generated by the two combinations
2 TR + M23
Turning on a avor ux B does not constrain this solution any further.
case where there is no twist, since the dimensional reduction on at space preserves
all supersymmetry. The compacti ed theory
ows to N = (4; 4) in 2d with global
symmetry SU(2)
U(1)2 where the two abelian groups are generated by the
combi
M23. Both sectors (4; 0) and (0; 4) provide a four dimensional
real representation of the SU(2) R-symmetry group.
Another possible choice of supersymmetry preserving background on the torus
corre1 that transform trivially with respect to the background symmetry
b = 0
b 6= 0
N = (4; 4)
N = (2; 2)
N = (2; 2)
N = (0; 2) or (2; 0)
6= 0
b = 0
b 6= 0
a1 =
2 , a2 = 0
a1 = 0, a2 =
a1 + a2 =
N = (0; 4)
N = (0; 2)
N = (2; 2)
N = (2; 2)
N = (0; 2)
N = (0; 2)
1; 0 in terms of the surviving amount of supersymmetry in 2d. We include the
possibility of a twist along the avor symmetries, with ux b.
with the auxiliary Y 3 =
2b further breaks supersymmetry to 1 , as can be seen
avor symmetries which correspond precisely to the T background (3.14) and the left
R-symmetry T+ (under which the right sector is invariant). Alternatively, choosing
The results of this section are summarized in the table 5.
V A, A = 1;
transformations, respectively.
Twisted reduction of N
= 3 SCFTs
can exist at strong coupling. These theories have SU(3)R
U(1)R R-symmetry and their
non-R global symmetries.
, a partial topological twist
can be performed on
using an abelian subgroup of the R-symmetry group. In this section
we study all possible solutions of the Killing spinor equations for such a twist, classifying
all di erent con gurations of preserved supercharges in two dimensions in terms of the
di erent choices of the uxes for the R-symmetry group.
a curved manifold is N
the corresponding non-linear supersymmetry transformations have been recently derived
; 8 are the gauge elds associated to the R-symmetry U(1)R and SU(3)R
# of real d.o.f.
1; : : : ; 8. We choose a basis in which the SU(3) can be embedded into the top left 3
of SU(4), so that the rst 8 generators of SU(4) reduce straightforwardly to the generators of
SU(3). The U(1)R group is obtained by mixing the U(1) from the decomposition of SU(4)R
into SU(3)R
U(1) and the chiral U(1) that enhances the superalgebra from PSU(2; 2j4)
to SU(2; 2j4) [31, 35]. We observe that these two U(1) groups act proportionally to each
N = 3 Weyl multiplet.
As in the previous cases, we are interested in preserving supersymmetry while coupling
the SCFT to a curved background describing the geometry of the manifold M . We choose
a background Weyl multiplet where, together with the fermions, all the bosonic elds are
set to zero except for ea , A , V A and DJI . Consequently, the conditions for the fermion
variations to vanish read [34]
IJ =
I =
= 0
I = @ +
1 DKI "K
V A(i A)IJ "J = 0
abRab(V A)(i A)JK)"L = 0
1 abRab(A)i 5"I = 0
These provide the set of constraints that select the surviving Killing spinors in two
dimensions. In order to nd non-trivial solutions, we choose the R-symmetry V A and A
background elds such that
R (V 8) =
R (V A) = 0 for
A 6= 3; 8
R (V 3) =
R (A) =
I] =
= i
and subject to appropriate quantization conditions (see the remark at the end of section 5).
The non-trivial Killing spinor equations then reduce to
R (V 3)(i 3)IJ + R (V 8)(i 8)IJ "
together with the two auxiliary conditions (4.2), (4.3).
a1 + a2 + a3
a1 + a2 + a3
=2 = 0
=2 = 0
=2 = 0
Classi cation of the solutions
In order to
nd non-trivial solutions to equation (4.7) we restrict the discussion to the
positive chirality components of the "I spinors.
We observe that under the breaking
U(1)R ! Spin(1; 1)
U(1)R realized
still indicate the 2d chirality as de ned in (3.12). The spinors are charged
under U(1) 3
U(1)R according to 1
p23 ; 1). Supersymmetry preserving equations are then given in table 7. Once the
need to be satis ed. In appendix B we prove that solutions to
exist if we appropriately choose the value of the components of the auxiliary eld DJI .
IJ = 0 and
I = 0 always
In table 8 we list all possible solutions to the conditions in table 7 together with the
corresponding preserved supersymmetries and the remaining 2d R-symmetry. We focus on
the cases with mostly right supersymmetry and for each possibility we pick up just a choice
of uxes. All the other possibilities can be obtained through a change of basis for the SU(3)R
it associated to the diagonal generator ( 2 i 23
case, one extra U(1) symmetry emerges from the topological twist, which is generated by
T itself (or any linear combination of T with the
avor symmetry generator). Although
under T the supercharges are charged, this symmetry cannot be a R-symmetry of the low
energy SCFT. It might be that this symmetry is not a symmetry of the 2d theory, or
appears as an outer automorphism of the 2d supersymmetry algebra.3 However, in order
to get more insight on it one should know the actual SCFT algebra that emerges from the
twisted reduction and the relation of T with the rest of the superalgebra generators.
From table 8 we note that, while for
to further reduce supersymmetry.
3We are grateful to Nikolay Bobev for raising this interesting point.
N = (6; 6)
N = (4; 4)
N = (2; 2)
6= 0
N = (2; 4)
N = (0; 6)
N = (2; 2)
N = (0; 4)
N = (0; 2)
a1 = 0, a2 = 0, a3 = 0
a1 = 0, a2 + a3 = 0
a1 + a2 + a3 = 0
a1 = 0, a2 =
3 , a3 =
a1 = 0, a2 = 0, a3 =
a1 =
2 , a2 + a3 = 0
a1 = 0, a2 + a3 =
a1 + a2 + a3 =
surfaces in terms of the surviving amount of supersymmetry in 2d. In the last column we indicate
the subgroup of 4d R-symmetry that is compatible with the twisted compacti cation.
However, also in the
without avor symmetries. This works as follows. First we perform a R-symmetry twist
that preserves either four or eight supercharges. This twist breaks R-symmetry as well,
leaving some avor symmetries with the associated vector multiplets. The second step of
multiplet that preserves only half of the supercharges. For example, if we use this procedure
the original SU(3)R
U(1)R survives as avor symmetry. In the second step we can gauge
an abelian subgroup of this avor symmetry. The corresponding gaugino background then
Twisted reduction of N
= 4 SCFTs
This case has been extensively discussed in the literature [2{4, 6, 36]. For completeness,
here we brie y review the main results in the language of conformal supergravity.
The supercharges are in the antifundamental representation of the SU(4)R R-symmetry
in which the Cartan subalgebra is spanned by
3 = 6
8 = p
15 = p
# of real d.o.f.
and an auxiliary
parameters "I satisfying
the theory on the curved manifold4 M
= R1;1
I , the bosonic auxiliary elds C, EIJ , TaIbJ DKIJL and the fermionic auxiliaries
I , IKJ . In table 9 we list the corresponding SU(4)R representations. For a complete
, by freezing the Weyl multiplet to
contain as only non-vanishing components the vielbein, a R-symmetry background V A
eld DKIJL. Supersymmetry is (partially) preserved if there exist spinor
I = @ "I +
IKJ =
1 DKIJL"L
V A(i A)IJ "J = 0
1 ab K[I Rab(V A)(i A)JL]"L = 0
I is identically zero in the selected background. In order to nd non-trivial solutions
we choose the R-symmetry gauge eld such that
R (V 3) =
R (V A) = 0 for
R (V 8) =
A 6= 3; 8; 15
R (V 15) =
subject to appropriate quantization conditions (see the remark at the end of this section).
Equations (5.2) and (5.3) then reduce to
I] =
= i
Classi cation of the solutions
The selected background induces the breaking Spin(3; 1) SU(4)R ! Spin(1; 1) Spin(2)
U(1) 15 under which the chiral supersymmetry parameters split as
where, once again, the
indices indicate chirality as de ned in (3.12). The spinors are
charged under U(1) 3
U(1) 15 according to 1
Therefore, equation (5.6) translates into the set of supersymmetry preserving equations
listed in table 10. For any set of ai parameters satisfying one of the conditions in the
= 0
N = (8; 8)
N = (4; 4)
N = (2; 2)
6= 0
N = (4; 4)
N = (0; 6)
N = (2; 2)
N = (0; 4)
N = (0; 2)
a1 = 0, a2 = 0, a3 = 0
a1 = 0, a2 + a3 = 0
a1 + a2 + a3 = 0
a1 = 0, a2 =
3 , a3 =
a1 = 0, a2 = 0, a3 =
a1 =
2 , a2 + a3 = 0
a1 = 0, a2 + a3 =
a1 + a2 + a3 =
a1 + a2 + a3
a1 + a2 + a3
=2 = 0
=2 = 0
=2 = 0
=2 = 0
1; 0 in terms of the surviving amount of supersymmetry in 2d. In the last column
we indicate the subgroup of 4d R-symmetry that is compatible with the twisted compacti cation.
previous table, equation (5.3) can be satis ed by a suitable choice of the background
auxiliary elds DKIJL without further constraining the I parameters.
In table 11 we list explicit solutions for the ai parameters and the corresponding
2d surviving supersymmetry with its R-symmetry group.
We focus on the cases with
mostly right-handed supersymmetry and for each possibility we pick up just one particular
solution with
emerges from the topological twist. Although T acts non-trivially on the supercharges, this
cannot be a R-symmetry of the low energy SCFT, but it could be identi ed as an outer
automorphism of the 2d superconformal algebra.
there are no
avor symmetries, we can further reduce supersymmetry by performing a
two step reduction. The rst step consists of turning on an R-symmetry twist, breaking
half of the supercharges are preserved.
avor symmetry, such that only
if we look at the N = (4; 4),
6= 0 case in table 11 the solutions a2 =
=3 and a3 =
symmetry bundle would be ill-de ned. However, in this case the quantization condition
that one has to actually impose is that the combination T
background symmetry that has been gauged by the twist) assigns integer charges to every
eld/representation of the theory. Substituting the explicit values of a2 and a3 we can see
that the background symmetry T corresponds precisely to the U(1) R-symmetry of the
a2p3 8 + a3p6 15 (i.e., the
N = (4; 4) theory
T =
( 3 8) + ( 6 15) =
The quantization condition then becomes 2
2 Z, which is satis ed for any choice of genus
g. A similar analysis applies to the other cases, leading to the same conclusion.
by twisted compacti cation of N -extended supersymmetric theories in four dimensions, as
described in the previous sections. In particular, we determine a general expression for the
central charge and the other 2d global anomalies.
4d anomaly polynomial I6 for the U(1) global symmetries, including the abelian symmetry
coupled to the twisting supergravity background, and integrate it along the
The resulting expression is a 4-form that can be identi ed with the anomaly polynomial
I4 of the 2d theory. From this expression we can then infer the 2d anomalies as functions
of the 4d anomalies and of the background uxes.
In this procedure we have to take into account that, even if the R-symmetry we start
with is the exact R-symmetry in 4d, along the dimensional ow the U(1)R can mix with
other abelian
avor symmetries. The exact 2d central charge is then reconstructed by
extremizing a trial central charge as a function of the mixing coe cients [4]. Because of this
potential mixing, in the reduction procedure we can start with any trial U(1) R-symmetry
TR in four dimensions, as di erent choices will simply shift the mixing parameters of the
2d theory without a ecting the nal result of the extremization procedure.
We consider a generic SCFT in four dimensions with di erent amount of
supersymmetries will have di erent matrix forms, but they can be identi ed up to a mixing with
the abelian avor symmetries
where ti are the generators of the abelian avor symmetries U(1)i in the 2d representation,
while i are the mixing coe cients. The relation (6.1) represents the most general trial 2d
R-current, involving abelian currents that do not necessarily mix with the R-current in the
4d SCFT, as the baryonic symmetries in toric quiver gauge theories [18, 19].
Our discussion can be applied also to the case of extended supersymmetry. In that
be treated as avor symmetries that can potentially mix with the 2d R-symmetry. In the
In order to compute the anomaly polynomial I6, which encodes all the global and
gravitational anomalies of the twisted theory,5 we rst couple each global symmetry to a
background connection on R1;1, which being topologically trivial can be compacti ed into
a torus T = S1
S1. The topological twist introduces additional background components
for U(1)R and U(1)i also along the
Following the notations of appendix A, if we denote fR the rst Chern class of the
R-symmetry bundle and fi the class associated to the gauging of the abelian U(1)i avor
symmetries, then we can write
where, the components in the direction of
are de ned by (2.10) and (2.14) as
fR = f
fi = fiT + fi
R =
so that the total Chern class of the global symmetry bundle E (see appendix A for the
de nition) restricted to the Riemann surface
where TR and Ti are the 4d generators and the trace means summing over positive
(negative) chirality fermions with plus (minus) sign. Here the twisting parameter a is xed by
the Killing spinor equation (2.12) to the value
. We can then interpret the combination
2 TR + Pi biTi to be the abelian symmetry which generates the topological twist on .
5The gauge theory is assumed to be free of local gauge anomalies, i.e., anomalies for symmetries coupled
to dynamical gauge vectors.
According to formula (A.14), the anomaly polynomial is given by the six-form
I6 = ch3(E)
Tr[TR3 ]fR3 +
1 X Tr[TR2 Tj ]fR2 fi
1 X Tr[TRTiTj ]fRfifj +
1 X Tr[TiTj Tk]fifj fk
where Tr[TA1
Having compacti ed the theory on
it is natural to identify the anomaly polynomial
of the corresponding two-dimensional theory with the expression obtained by integrating
I6 on the Riemann surface. The result of the integration is
kA1:::Al are the l-degree 't Hooft anomaly coe cients of the
I6 =
Tr[TR2 T ] fR2 + X Tr[TRTiT ]fRfi + X Tr[TiTj T ]
which can be compared to the general formula for the anomaly polynomial in 2d
I4 = ch2(E)
kRR fR2 + X kRifRfi + X kij fifj
kRR =
kRi =
kij =
Tr[TRTiT ]
Tr[TiTj T ]
k =
leading to the following identities
is de ned in (2.4). We note that (6.8) relates 4d 't Hooft anomaly coe cients on
the right hand side with 2d anomaly coe cients, kAB
Tr[tAtB], on the left hand side.
As already mentioned, when we ow to two dimensions the generator TR corresponding
to a trial four dimensional R-symmetry can mix with the other global U(1)'s to give rise
to the exact two dimensional R-symmetry. Therefore, reinterpreting equation (6.8) in a
two-dimensional language, requires substituting the generator TR with (6.1). Explicitly,
kRtrRial =
kRtriial =
kij =
k =
The mixing parameters i are now determined by extremizing the trial central charge
3kRtrRial (a sign appears, due to our choice of 2d chirality matrix 01, see table 2)
which implies
0 =
= 0
Equation (6.14) can be solved by inverting the matrix kij , provided that it has
nonvanishing determinant. The expression for the extremized central charge is nally given by
cr = 3 2
in terms of the anomaly coe cients of the original four dimensional SCFT.
We note that eq. (6.14) determines the coe cients i at the (possible) 2d
superconformal xed point, giving raise to the exact 2d R-current once plugged in (6.1). These
coe cients may di er from the ones appearing in the 4d exact R-current.
There are
abelian currents that do not mix in 4d but their mixing in 2d is in general non-vanishing.
This is for example the case of the baryonic symmetries in the Y pq models discussed in [13].
Further directions
We conclude our analysis by discussing some open questions and future lines of research. A
rst generalization of the program of constructing 2d SCFTs from four dimensions consists
of decorating the Riemann surfaces discussed here with punctures. A possible way to study
such a problem consists of exploiting the doubling trick discussed in [38, 39]. In this case
one can gain information on the e ective number of 2d chiral fermions by gluing a Riemann
surface with a copy of itself (with the opposite orientation), thus obtaining a closed surface.
One can apply our results to classes of 4d SCFTs with a gravitational dual. For example
one can consider theories associated to D3 branes probing the tip of three dimensional
Calabi-Yau cones. The analysis of such models was initiated in [13], for the in nite Y pq
family of [40]. Such theories are characterized by the presence of a SU(2)
U(1) mesonic
avor symmetry and a U(1) baryonic symmetry. The baryonic symmetry does not mix with
the 4d R-current, but it has been observed that this mixing is non-trivial once the theory is
reduced to 2d. For more general quivers the gauge group is a product of U(N )i factors. In
the IR the U(1)i
U(N )i are free and decouple. The non anomalous combinations of these
U(1)s are the baryonic symmetries. While in the Y pq case there is just a single baryonic
symmetry, in other cases one can have a richer structure. The formalism developed in
section 6 is necessary for extending the analysis to such families.
One can also study the problem from the AdS dual setup along the lines of [13],
reconstructing the central charge from the gravitational perspective. The solution in this
case should correspond to D1 branes probing a type IIB warped AdS3
where M7 represents (locally) a U(1) bundle over a 6d Kahler manifold. It should be
! M7 geometry,
possible to formulate the central charge and its extremization in terms of the volumes of
M7, in the spirit of [41].
It should be also possible to study models arising from the compacti cation of 6d
theories, such as class S theories [42] or theories with lower supersymmetry, as the Sk
problem, especially because the central charges a and c can be computed along the lines
of [29]. The analysis of the gravitational dual mechanism of the topological twist in this
case can be performed by studying the consistent truncation of [35] in gauged supergravity.
uxes are turned on. In such a case it might be possible to compare the eld theory and
the supergravity results.
Acknowledgments
S(M ) = S
the bundle S
operator D= =
Mazzucchelli for useful discussions. The work of A.A. is supported by the Swiss National
Science Foundation (snf) under grant number pp00p2 157571/1.
This work has been
supported in part by Italian Ministero dell'Istruzione, Universita e Ricerca (MIUR), Istituto
Nazionale di Fisica Nucleare (INFN) through the \Gauge Theories, Strings, Supergravity"
(GSS) research project and MPNS-COST Action MP1210 \The String Theory Universe".
The anomaly polynomial
In this section we brie y review the general formalism of the anomaly polynomial that has
been used in section 6. We refer the reader to the original paper [45] for further details
(see also [46] for a review).
We begin by recalling the Atiyah-Singer index theorem for the Dirac operator on a
compact manifold. Let M be a compact closed manifold of even dimension 2l and E a
smooth complex vector bundle over it, associated to some representation of a Lie group
G. If M is a spin manifold, we can de ne fermionic elds as sections of the spinor bundle
are the two chiral irreducible spinor representations of the
spin group of M . A chiral fermion eld charged under G is then described by a section of
The gamma matrices a act on spinors S
exchanging their chirality, hence the Dirac
can be represented by the o -diagonal 2-by-2 matrix
where the operators D= : S
Singer index theorem then states that
E are the adjoints of each others. The
Atiyahindex(D= +)
dim ker D= +
dim ker D=
A^(M ) ch(E)
D= =
= + D0
where A^(M ) is the so called A-roof genus of the tangent bundle of M
A^(M ) = 1
4p2(M )] + : : :
expressed in terms of the Pontryagin classes pi(M ) 2 H4i(M; Z), while ch(E) is the Chern
character of the bundle E.
dimensional and the bundle E can be decomposed as a Whitney sum of line bundles
E = L
(r) is de ned, up to isomorphisms, by its rst Chern class
(n), one for each representation (particle species). It follows that each
c1(L(r)) =
where R(A(r)) is the cohomology class of the curvature of the associated abelian
connection A(r). In this case the Chern character is de ned additively as
ch(E) =
X chk(E) =
Since each line bundle L
(r) is associated to a unitary one-dimensional representation of
integer charge q(r), we can equivalently describe the bundle E as follows. If we de ne fG to
be the rst Chern class of the line bundle of unit charge,6 then for each L
(r) we can write
Qim=1 U(1)i, we have to consider the bundle
E =
(r) = L1
where each L
(r) is a tensor product representation for the group G, labelled by the set of
charges (q1(r); : : : ; qm(r)). If, as before, we de ne fi to be the rst Chern class of the line
6Note that a principal U(1) bundle and the associated line bundle of charge 1 have the same rst
from which we obtain
where we assembled all the charges q(r) into the diagonal matrix TG, which now
represents the Lie algebra part of the connection on E. Using this rede nitions, the Chern
character (A.5) is
c1(L(r)) = q(r)fG
c1(E) = Tr[TG]fG
ch(E) =
bundle of unit charge for the U(1)i symmetry, we can write
chk(E) =
k! r=1 i=1
i1 ik r=1
e ective action of a massless chiral fermion
where we used the property of the rst Chern class, c1(Li(r)
L(jr)) = c1(Li(r))+c1(L(jr)). The
diagonal matrices Ti can be taken to be the hermitian generators of the U(1)i symmetries,
written in the representation associated to E.
In fact, under a chiral rotation the fermionic path integral picks up a non-zero phase
eiW =
R iD= +
W =
2 index(D= +)
proportional to the index of the Dirac operator.
Since the anomaly for a negative chirality fermion is minus that of a positive chirality
index(D= )
dim ker D=
dim ker D= + =
index(D= +)
in a theory with many fermions of both chiralities, the total chiral anomaly is given by the
sum of the anomalies of the positive-chirality fermions minus the sum of the anomalies of
the negative-chirality ones.
Finally, in [45] it was shown that one can construct a Dirac operator in 2l+2 dimensions
in such a way that its index reproduces the gauge anomaly for a charged chiral fermion in
2l dimensions. The corresponding index density is a (2l + 2)-form
A^(M ) ch(E)
which is called the anomaly polynomial.
Supersymmetry variations of the auxiliary
= 3; 4 SCFTs
In this section we show that it is always possible to satisfy the conditions
IJ = 0,
I = 0
in (4.2), (4.3) and
auxiliary eld DJI and DKIJL, respectively.
as usual to the positive chirality transformation, the IIJ variation reads
that can be trivially solved by setting the corresponding DKIJL components to zero.
1 abRab(V )JJ J = 0
After twisted compacti cation the J spinors decompose as i 23 eigenvectors and we
1 according to the 2d chirality of the spinor.
where we chose DIILJ to be diagonal in the J; L indices.
write i 23 J = sJ J with eigenvalue sJ =
Using equation (5.6)
we eventually nd
J = R (V )JJ J
R23(!23) =
"J = 0
Note to the reader: in this section we do not assume Einstein summation notation for
repeated R-symmetry indices.
along a subgroup of the Cartan of SU(4), the curvature R(V )IJ
R(V A)(i A)IJ is diagonal
in the adjoint (I; J ) indices. As a consequence, the Killing spinor equations (5.6) split into
to the preserved supersymmetries, whereas the rest of the components are set to zero.
Having this in mind, we now discuss the condition
IJ = 0, where the variation is
generated by the preserved supercharges. Three possible cases can arise.
If K 6= I; J from (5.3) we immediately nd
IKJ =
1 X DKIJL"L = 0
I vanishing.
always possible to choose a non-vanishing DNM background that makes the supersymmetry
6= 0, from table 11 it turns out
that for each xed J solution only one chirality is present and (B.4) can be always satis ed
by an appropriate choice of the DIIJJ components.
Finally, if "J is a Killing spinor but "I is not, the
IJ variations do not vanish in
general. However it is possible to show with a case by case analysis that these components
always decouple from the representation of the supersymmetry algebra and they are not
relevant for the counting of the supersymmetries.
Open Access.
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