#### One-loop tests of supersymmetric gauge theories on spheres

HJE
One-loop tests of supersymmetric gauge theories on spheres
Joseph A. Minahan 0 1 3
Usman Naseer 0 1 2
0 Cambridge , MA 02139 , U.S.A
1 Box 516 , SE-751 20 Uppsala , Sweden
2 Center for Theoretical Physics, Massachusetts Institute of Technology , USA
3 Department of Physics and Astronomy, Uppsala University
We show that a recently conjectured form for perturbative supersymmetric partition functions on spheres of general dimension d is consistent with the at space limit of 6-dimensional N = 1 super Yang-Mills. We also show that the partition functions for N = 1 8- and 9-dimensional theories are consistent with their known at space limits.
Field Theories in Higher Dimensions; Supersymmetric Gauge Theory
1 Introduction 2 3
Review of localization and its analytic continuation
One-loop divergences from partition functions
3.1
3.2
3.3
3.4
Eight supersymmetries in 4d
Eight supersymmetries in 6d
Sixteen supersymmetries in 4d and 6d
Sixteen supersymmetries in 8d and 9d
4
Summary and discussions
1
Introduction
on Euclidean spaces [
1
]. For instance, a single vector multiplet in a 4d N = 1 gauge theory
has a Majorana fermion as a superpartner to the gauge eld in Minkowski space. In rotating
to Euclidean space it is impossible to maintain the reality condition on the fermion, so the
supercharges are naturally complexi ed. As shown by Zumino [
1
], requiring the spinors to
be real leads to extra supersymmetry with additional elds in the supermultiplet, namely
two extra fermion degrees of freedom as well as two scalars, where one scalar has the wrong
sign kinetic term. In hindsight we can easily understand these extra elds as arising from
a dimensional reduction of a Minkowskian six-dimensional vector multiplet, where one of
the reduced directions is the time direction [2, 3].
A similar issue occurs when we consider a vector multiplet in six dimensions. In
Minkowski space an N = 1 vector multiplet has a real Weyl spinor. Analytically continuing
to Euclidean space complexi es the spinor. In order to have real spinors one must increase
the amount of supersymmetry to N = 2 with 16 supersymmetries. Now there will be four
scalars where one of them has the wrong sign kinetic term. Note that these issues do not
mean that we cannot have minimal supersymmetry on Euclidean spaces, it just means we
cannot have it with only real elds.
However, the standard localization procedure often starts with ten-dimensional super
Yang-Mills and dimensionally reduces to super Yang-Mills on lower dimensional spheres [4{
7]. As such, the elds are real. Nevertheless, a simple argument shows that it should be
possible to put a theory with N
vector multiplet on R4, with no chiral multiplets charged under the U(1). This theory
is free, hence it is conformal. Therefore one can make a conformal transformation to
S4 and preserve the supersymmetry, albeit with complex elds. In any case, one can
= 1 supersymmetry on S4. Suppose one has a U(1)
{ 1 {
use the methods of Festuccia and Seiberg, starting with o -shell supergravity to put any
N = 1 theory on AdS4 and analytically continue to S4 [8]. However, the full partition
function for a generic interacting N = 1 theory has been shown to be scheme dependent [9],
questioning whether one could obtain any scheme independent results from a localized
partition function. Nonetheless, certain theories might have further symmetries, opening
up the possibility of localizing the theory. A prominent example of this type is the N = 1
theory where it was shown that one can obtain unambiguous and scheme independent
quantities constructed from the partition function [10]. However, there is no known way
to localize an N = 1 theory on S4. The di culty arises in trying to construct a positive
de nite localization term. It is possible to construct a positive de nite Q-exact term from
a supersymmetry generator, however the known generators that give such terms do not
themselves close to a symmetry of the Lagrangian [11].1 This issue is also discussed in [12].
For six-dimensional N = 1 super Yang-Mills the situation is worse. In the four
dimensional minimal supersymmetric case one expects to be able to put the theory on S4
because of the existence of an appropriate superalgebra, namely OSp(1j4), which has four
supercharges and a bosonic SO(5) subalgebra corresponding to the isometry of S4. For six
dimensions we would want a superalgebra with a bosonic SO(7) subalgebra and 8
supercharges transforming in a spinor representation of SO(7), but no such superalgebra exists.
The F (4) supergroup has an SO(7)
SU(1; 1) bosonic subalgebra and 16 supercharges,
hence this is appropriate for N = 2 supersymmetry.
We encounter a similar problem when considering super Yang-Mills in eight and nine
dimensions. In this case there are sixteen supersymmetries, and again, one would like
to be able to put these theories on spheres. But once more there is no corresponding
superalgebra with the appropriate bosonic subalgebras, namely SO(9) and SO(10) for S8
and S9 respectively.2 Even if one were able to de ne such theories it might be the case that
their partition functions su er from ambiguities just like N = 1 theories in four dimensions.
In the next section we will review explicitly the obstacles in putting these supersymmetric
theories on the spheres.
Nevertheless, there are some indications that there are theories akin to N = 1 six,
eight and nine dimensional supersymmetric theories on spheres. In particular, evidence was
presented in [14] showing that the perturbative partition functions for super Yang-Mills
with 8 supersymmetries on S3, S4 and S5 have a natural analytic continuation, such that
one can continue up to six dimensions. Likewise theories with 16 supersymmetries on Sd
with d = 3; 4; 5; 6; 7 also have a natural analytic continuation which can then be continued
up to d = 8; 9. Although, we do not have an explicit construction of Lagrangians for these
theories, it is reasonable to assume that in the decompacti cation limit, they reduce to
usual gauge theories in at space. The main objective of this paper is to demonstrate that
the partition functions are consistent with this picture. These partition functions include a
1We thank Guido Festuccia for several discussions on this point.
2There are superalgebras with SO(9) and SO(10) bosonic subalgebras, for example, OSp(9j2) and
OSp(10j2). However, these algebras have supercharges that transform in the vector representation of SO(9)
one-loop determinants produce the well known physics of the at space theories.
The paper is organized as follows: in section 2 we brie y review the localization
procedure in [5] and the analytic continuation in [14]. In section 3 we compute one-loop
divergences from the analytically continued expressions for one-loop determinants and compare
them with well known results in the literature. In section 4 we summarize our results and
discuss some further issues.
2
Review of localization and its analytic continuation
In this section we review the localization procedure for gauge theories with eight
supersymmetries on spheres. We use the conventions in [5] which generalizes the procedure on S4 [4],
where one starts with N = 1 super Yang-Mills in ten dimensions and dimensionally reduces,
including along a time-like direction so that the unreduced dimensions are Euclidean. We
then review analytic continuation of the localized result to other dimensions.
In ten dimensions the Lagrangian in at space is given by [15]
HJEP07(21)4
L =
1
g10
aMb and ~M ab are real and symmetric. The
Lagrangian is invariant under the supersymmetry transformations
symmetry transformation to
where the constants I are given by
= 12 MN FMN +
I I r
;
I
2
FIJ = [ I ; J ] :
where is any constant real spinor. We then dimensionally reduce to d dimensions so that
the gauge elds are A ,
= 1; : : : d, while the remaining bosonic
elds are scalars with
I
AI , I = 0; d + 1; : : : 9. The scalar 0 will have a wrong-sign kinetic term in the action
because it came from the dimensional reduction of the time direction. This also leads to a
noncompact R-symmetry, SO(1; 9
d). The dimensionally reduced eld-strengths become
Putting the d-dimensional Euclidean space on the sphere modi es the fermion
super(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
The index I in (2.4) is summed over. The supersymmetry parameters
are special cases
of the superconformal Killing spinors on Sd that satisfy
r
= ~ ~;
r ~ =
2
:
= 21r , which has 32 independent components. We reduce this to 16 components
r
, ~
=
~
= 1,
;
T =
for consistency with (2.6). A
=
0 ~8 9, showing that this construction works up to d = 7. If we
while at the same time modifying the extra terms in the Lagrangian to
5 by imposing the extra condition
= +
, where
=
6789. The fermions then
split into the components of a vector multiplet , with
= +
, and a hypermultiplet
with
=
. The gauge elds A and scalars 0
, d+1 : : : 5 are the bosonic elds of the
vector multiplet, while
6; : : : 9 are the bosonic elds for the hypermultiplet. With fewer
supersymmetries we can add mass terms to the Lagrangian for the hypermultiplets. This
also modi es the I for the hypermultiplet scalars to
then the solutions to (2.7) are given by
1
where s is a constant spinor. The supersymmetric Lagrangian is given by
{ 4 {
s ;
r2
I = 6 : : : 9
I = 6; 7
I = 8; 9 ;
2(d
4)
r
and the quadratic terms for the hypermultiplet to
where
L
L
2
d
=
=
1
1
2
gYM
2
gYM
( imTr
) ;
d2 r2I Tr I I
d(d
4
;
2)
1 loop are vector- and hyper-multiplet one-loop determinants. It was
observed in [14] that the one-loop determinants of all known examples with eight
superd
5, can be written in the more general form
Zvec
1 loop ( ) Y
h ; i2 =
Y
n=1 h n2 + h ; i
Y
2
(n + d
2)2 + h ; i
(n+d 2)
5, but also agrees with the one-loop determinants on S6 and S7 for sixteen supercharges.
It now seems quite natural to analytically continue the dimension past d = 5 in (2.16)
and (2.17) up to d = 6, even though there is no appropriate
and
that can accommodate
the above construction of vector and hypermultiplets on S6. At this point we will assume
that our presumed theory has 8 supersymmetries and reduces to standard N = 1 super
Yang-Mills in the at space limit. One possibility is that the SO(7) symmetry is broken to
SO(6) with the spinors having opposite chiralities on the north and south poles.
Likewise, it seems natural to analytically continue the dimension past d = 7 in (2.18),
even though there is no appropriate
for the construction. In the d = 8 and d = 9 cases
we will again assume that our presumed theory reduces to standard N = 1 SYM in the at
space limit. On the spheres we assume that the rotational symmetry is broken to SO(8).
{ 5 {
In this section we will use the analytically continued expressions for one-loop
determinants to compute e ective couplings for theories with eight and sixteen supersymmetries
in diverse dimensions. The ultraviolet divergences of the gauge coupling at one-loop can
then be compared with the counter terms for supersymmetric theories at one-loop. In four
dimensions, upon taking the decompacti cation limit one can compute the beta function
of the theory. We show that results obtained from the analytically continued one-loop
determinants are in agreement with explicit one-loop computations in these theories.
3.1
Eight supersymmetries in 4d
Recall that for a gauge theory in four dimensions with Nf Dirac fermions in representation
Rf and Ns complex scalars in representation Rs of a semi-simple gauge group, the one
loop beta function is given by
(g) =
1
C2 (R) is the quadratic Casimir in the representation R of the gauge group. For N =
2 theory with Nh hypermultiplets in the representation R of the gauge group the beta
function becomes
(g) =
g
3
The contribution from the vector multiplet was previously found in [4] by taking the
hypermultiplet mass to in nity in the N = 2 theory. We want to reproduce (3.2) by using
the analytically continued one-loop determinant for the vector and hyper multiplets given
in equations (2.16) and (2.17).
To do so we need to determine O
To proceed we replace
by t in the expressions for the one-loop determinants. The
parameter t keeps track of the order of . Focusing only on the vector multiplet, one can
2 terms appearing in the one-loop determinants.
easily nd that
d log Z1vecloop
dt2
where
For d = 4
we nd
+ X 1
>0
t
2 =
X
>0
F (x; y; z)
=
1
X
n=0
i
2z
(n + x)
1
(n + 1) (x) (n + y)2 + z2
1
y + iz 2F1 (x; y + iz; y + iz + 1; 1)
c:c: :
, we expand the r.h.s. in powers of t and . Keeping only the leading terms,
h ; i2 (F (d
2; 0; t h ; i) + F (d
2; d
2; t h ; i)) ;
d log Z1vecloop
dt2
=
2
{ 6 {
(3.2)
(3.3)
(3.4)
(3.5)
HJEP07(21)4
2
For a gauge multiplet and Nh hypermultiplets, the contribution to the O
the one-loop determinants can be combined with the O
as given in equation (2.15) to get
2 term in the xed point action
2 term from
From this we can easily obtain
where g ( ) is the running coupling constant at the renormalization scale
is the bare coupling. From the above equation one can easily obtain the beta function,
Since the explicit expression for one loop determinants for eight supersymmetries in 4d
are known in terms of in nite products, the above results can be reproduced by
regularizing those expressions by introducing a
nite cut o
parameter
r and then taking the
decompacti cation limit r ! 1. As explained earlier, it is not known how to localize a
six dimensional theory with eight supersymmetries. In this case the expression (2.16) is a
genuine ansatz. In this subsection we will perform a non trivial check on that ansatz by
computing the e ective coupling. It is well known that the six dimensional theory with
eight supersymmetries has a quadratic divergence at one-loop [18, 19]. We will compute the
e ective coupling using the one loop determinant (2.16) and show that it has a quadratic
divergence in the decompacti cation limit.
Since dimensional regularization is only sensitive to logarithmic divergences we will
use a hard cuto to isolate the quadratic divergence. At leading order in the divergence
this is expected to be consistent with supersymmetry. However, there could be issues with
sub-leading divergences, if for example imposing the cuto leaves o the super-partners of
modes at or near the cuto . However, assuming that the proposed dimensional
regularization respects the supersymmetry we can show that the logarithmic divergences coming
from the hard cuto are consistent with the result coming from dimensional regularization,
even if the log divergence is sub-leading.
{ 7 {
We use d = 6 in (2.16) and truncate the in nite product at nmax =
nd
quadratic dependence on the energy cuto
. It is straightforward to nd that the divergent
contribution to the 2 term from the vector one-loop determinant is
By combining this with the xed point action, we nd the e ective coupling given by
In the r ! 1 limit only the leading terms in r survive and one obtains
1
g2
=
1
g
2
0
2
96 3 C2 (Adj) :
We see that the e ective coupling diverges quadratically with the scale .
It is also known that the six dimensional theory can be made
nite at one loop by
adding a suitable hypermultiplet. This would be the case if the hypermultiplet and the
vector multiplet contribute to the quadratic divergence with opposite sign. This is also
consistent with the one-loop determinant (2.17). For a hypermultiplet in representation
R, the contribution to O
2 term is given by
log Z1hy ploop =
+
log ( r) C2 (R) 2 +
:
So that the e ective coupling with Nh hypermultiplets in the representation R is given by:
2r2 + 5 r
6
2
1
3
1
g2
=
1
g
2
0
96 3 (C2 (Adj)
NhC2 (R)) :
In particular, for a single hypermultiplet in the adjoint representation the quadratic
divergence vanishes as expected [18].
3.3
Sixteen supersymmetries in 4d and 6d
In four and six dimensions, explicit expressions for the one-loop determinants for sixteen
supersymmetries are known. We will compute the e ective coupling at one-loop using both
of these expressions and show that it is consistent.
For four dimensions, we compute e ective coupling from (2.18) using d = 4. We will
truncate the in nite product at nmax =
r. By doing so one nds that the contribution
to O
2 term vanishes.
log Zvec
1 loopZ1hy ploop
= 0 +
:
Hence the coupling is not a ected by one-loop e ects and is independent of the cuto
scale . Now we can easily check that this result is consistent with analytically continued
(3.12)
(3.13)
(3.14)
(3.15)
{ 8 {
expression, i.e., expanding (2.18) in powers of for d = 4
. Replacing
by t one can
easily obtain
d
Which is consistent with the result obtained from the explicit expression.
Similarly, in six dimensions the contribution to the O
2 term from (2.18) takes
1
2
+
1
2
+
the form
given by:
log Zvec
1 loopZ1hy ploop
= 3 log ( r)
xed point action, we obtain the e ective coupling which is
1
We see that the coupling has a logarithmic dependence on the scale . It is easy to see
that Logarithmic dependence is produced by using dimensional regularization for d = 6
in the analytically continued expression. Doing so we nd that:
log Zvec
1 loopZ1hy ploop
=
3
Combining this with the contribution from the xed point action and noting that in 6
dimensions gr22 has mass dimension
we get:
1
where a -independent in nite piece is absorbed in g12 . This gives the same logarithmic
0
dependence on the energy scale . Note that in the decompacti cation limit this
logarithmic divergence vanishes, consistent with the fact that the six dimensional theory with 16
supersymmetries is nite at one-loop.
3.4
Sixteen supersymmetries in 8d and 9d
For d = 8; 9, it is not known how to localize. Here we show that the analytically continued
expression for the one-loop determinant is consistent with known results. It is known that
for d = 8; 9, none of the terms present in the tree level Lagrangian need a counter term at
one-loop [19, 20]. Hence, the e ective coupling determined from the analytically continued
expressions for one-loop determinants should not have any divergences. This can be easily
demonstrated by using the methods of this section. A short calculation shows that the
contribution to O
2 term from the one-loop determinant (2.18) for d = 8 is
log Zvec
1 loopZ1hy ploop
=
5
6
( r)2
4
+
3 r
2
!
+ 5 log ( r) C2 (Adj) 2
:
(3.21)
{ 9 {
(3.18)
(3.19)
(3.20)
This leads to the e ective ccoupling
1
Here we see that in the decompacti cation limit the dependence on the energy scale
vanishes. A similar computation for d = 9 yields
log Zvec
which leads to following expression for e ective coupling:
1
in the decompacti cation limit. The same calculation
can be repeated for d = 10 and it can be shown that the one-loop determinants do not
contribute any divergences to the gauge coupling in the decompacti cation limit.
4
Summary and discussions
In this note we have presented some nontrivial evidence in favor of the analytically
continued expressions for one-loop determinants of supersymmetric gauge theories on Sd. We
have computed the one-loop contribution to the gauge coupling as predicted from these
determinants and showed that it is consistent with the well known one-loop behavior of
minimally supersymmetric gauge theories in diverse dimensions. Unfortunately, at this
point it is not clear what are the actual Lagrangians for these gauge theories when they
are on the sphere. Attempts to construct these theories by starting from ten dimensional
SYM have not been successful so far.
Another possible way to nd supersymmetric theories on spheres is to consider o shell supergravity coupled to vector multiplets and then take the rigid limit, as in [8, 21].
An o -shell formalism is known for N = 1 supergravity in six dimensions, [22{24]. This
possibility was analyzed partially in [25] and it was shown that the o -shell theory with
R-symmetry gauging does not admit S6 as a supersymmetric background. It would be
interesting to complete this analysis by determining whether or not the o -shell theory
without R-symmetry gauging admits S6 as a supersymmetric background, although in this
case the supersymmetry might be enhanced to N = 2. For eight and nine dimensions we
are unaware of an o -shell supergravity formulation.
It is reasonable to believe that the theories, assuming they exist, will have Lagrangians
that have explicit terms that break the bosonic symmetries down to SO(6) ' SU(4) in the
six dimensional case, leaving an invariance under the SU(4j1) superalgebra. Likewise, we
expect the eight and nine dimensional theories to only have an SO(8) symmetry and be
invariant under an OSp(8j2) superalgebra. From the supergravity perspective, it would
correspond to turning on non trivial background elds in addition to metric.
(3.23)
(3.24)
HJEP07(21)4
Acknowledgments
We thank Guido Festuccia and Luigi Tizzano for discussions. J.A.M also thanks the CTP at
MIT for hospitality during the course of this work. The research of J.A.M. is supported in
part by Vetenskapsradet under grants #2012-03269 and #2016-03503 and by the Knut and
Alice Wallenberg Foundation under grant Dnr KAW 2015.0083. Work of U.N is supported
by the U.S. Department of Energy under grant Contract Number de-sc0012567.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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