#### The character of the supersymmetric Casimir energy

HJE
The character of the supersymmetric Casimir energy
Dario Martelli 1 3 4
James Sparks 1 2 4
The Strand 1 4
London 1 4
Andrew Wiles Building 1 4
Radcli e Observatory Quarter 1 4
0 to the anomaly polynomial. As
1 Woodstock Road , Oxford, OX2 6GG , U.K
2 Mathematical Institute, University of Oxford
3 Department of Mathematics, King's College London
4 complex coordinate. Since @
We study the supersymmetric Casimir energy Esusy of N = 1 eld theories with formalism. These backgrounds admit an ambi-Hermitian geometry, and we show that the net contributions to Esusy arise from certain twisted holomorphic modes on R respect to both complex structures. The supersymmetric Casimir energy may then be identi ed as a limit of an index-character that counts these modes. In particular this explains a recent observation relating Esusy on S1 further applications we compute Esusy for certain secondary Hopf surfaces, and discuss how the index-character may also be used to compute generalized supersymmetric indices.
Supersymmetric gauge theory; Di erential and Algebraic Geometry
1 Introduction 2
Supersymmetric backgrounds
2.1
2.2
Background geometry
Hopf surfaces
2.2.1
2.2.2
Primary Hopf surfaces
Secondary Hopf surfaces
2.3
Flat connections
3
Supersymmetric Casimir energy
Path integral formulation
Hamiltonian formulation
Twisted variables
Unpaired modes on R
M3
4
Primary Hopf surfaces
3.1
3.2
3.3
3.4
4.1
4.2
4.3
4.4
5.1
5.2
5.3
Solving for the unpaired modes
The character
Zeta function versus heat kernel regularization
Rewriting as a Dirac character
5
Secondary Hopf surfaces and generalizations
Lens spaces
Fixed point formula
More general M3
5.3.1
5.3.2
Poincare Hopf surface
Homogeneous hypersurface singularities
5.4
Full supersymmetric Casimir energy
6
Discussion
A Supersymmertic index from the character
A.1 Primary Hopf surfaces
A.2 Secondary Hopf surfaces
{ 1 {
Introduction
In recent years the technique of localization [1] has provided access to a host of exact results
in supersymmetric eld theories de ned on certain curved backgrounds. This method can
be used to compute a number of observables in strongly coupled
eld theories. These
in general depend on the background geometry, leading to a richer structure than in
at
space. In this paper we will consider the supersymmetric Casimir energy, introduced in [2]
and further studied in [3{5]. We will focus on four-dimensional N = 1 theories with an
Rsymmetry, de ned on manifolds S1
M3, with M3 a compact three-manifold. These arise
as rigid supersymmetric backgrounds admitting two supercharges of opposite R-charge,
which are ambi-Hermitian, with integrable complex stuctures I [6, 7].
Moreover, the
backgrounds are equipped with a complex Killing vector eld K of Hodge type (0; 1) for
both complex structures. Denoting this as K = 12
(
2 [0; ) parametrizes
S1 = S1, is a nowhere zero vector on M3 (the Reeb vector eld), generating a transversely
holomorphic foliation. When all orbits of
close, this means that M3 is a Seifert
bred
three-manifold, with
generating the bration.
On such a background, one can consider the partition function of an N = 1 theory
with supersymmetric boundary conditions for the fermions. As is familiar from
nite
temperature eld theory, this computes
where the Hamiltonian Hsusy generates time-translations along @ . Supposing this has a
spectrum of energies fEigi2I , with Hsusy jii = Ei jii, then the minimum energy is E0
Esusy where evidently
ZS1 M3 = Tr e
Hsusy ;
Esusy =
lim
!1 d
d
ZS1 M3 :
(1.1)
(1.2)
the vacuum state.
Unlike the usual Casimir energy on S1
Thus the supersymmetric Casimir energy is given by Esusy = h 0jHsusyj0i, where j0i is
M3 (proportional to the
integral of the energy-momentum tensor T
over M3), this has been argued to be a
wellde ned observable of the theory, i.e. it is scheme-independent, in any supersymmetric
regularization [4].
We will be interested in computing hHsusyi = Esusy via canonical quantization. This
approach was initiated in [3] for the conformally at S1
S3 background, and further
elaborated on in [4]. One can dimensionally reduce the one-loop operators on M3 to obtain a
supersymmetric quantum mechanics on R , where the
! 1 limit e ectively
decompacties the circle S1. Most of the modes of the one-loop operators are paired by supersymmetry,
and these combine into long multiplets that do not contribute to hHsusyi in the
supersymmetric quantum mechanics [4]. In this paper we will show that the unpaired modes are
certain (twisted) holomorphic functions on R
M3, where there is one set of modes for
each of the two complex structures I . More precisely, here we will restrict attention to the
contribution of the chiral multiplet. We expect that the vector multiplet contributions will
{ 2 {
also arrange into short multiplets, and will similarly be related to (twisted) holomorphic
functions. However, we will not perform this analysis in this paper.
When R
M3 = X n fog is the complement of an isolated singularity o in a Gorenstein
canonical singularity X, one can elegantly solve for these unpaired modes that contribute
to the supersymmetric Casimir energy. These include of course M3 = S3, as well as
singularity, previously studied in the literature; but this construction also includes many
other interesting three-manifolds. A large class may be constructed from homogeneous
hypersurface singularities. Here X comes equipped with a C action, which is generated
by the complex vector eld K, and X n fog
bres over a compact orbifold Riemann surface
= X are isomorphic as complex varieties, but the relative sign of the
complex structures on
bre and base are opposite in the two complex structures I . We
will show that the modes that contribute to the supersymmetric Casimir energy in a chiral
matter multiplet take the form
=
P
are the globally de ned nowhere zero holomorphic (2; 0)-forms
of de nite Reeb weight under , that exist because X+ = X
is Gorenstein. Furthermore,
k
denote the R-charges of the relevant elds; in particular, k+ = r
2, k
= r, where
r 2 R is the R-charge of the top component of a chiral multiplet. These correspond
to fermionic ( +) and bosonic (
) modes, respectively. The essential point in (1.3) is
that F
are simply holomorphic functions on X . More precisely, in general the path
integral (1.1) splits into di erent topological sectors, labelled by
at gauge connections,
and for the trivial at connection F
are holomorphic functions; more generally they are
holomorphic sections of the associated at holomorphic bundles. For example, for quotients
of M3 = S3, such as the Lens spaces L(p; 1) = S3=Zp, the relevant holomorphic modes may
be obtained as a projection of the holomorphic functions on the covering space.
The supersymmetric Casimir energy is computed by \counting" these holomorphic
functions according to their charge under the Reeb vector . As such, Esusy is closely
related to the index-character of [8]. In this reference, it was shown that the volume of
a Sasakian manifold Y can be obtained from a certain limit of the equivariant index of
the @ operator on the associated Kahler cone singularity X = C(Y ). In a similar vein,
here we will show that the supersymmetric Casimir energy is obtained from a limit of an
index-character counting holomorphic functions on R
M3. In the case of M3 = S3, this
explains a conjecture/observation made in [5], where it was proposed that Esusy may be
computed using the equivariant anomaly polynomial.
The rest of the paper is organized as follows. In section 2 we review and expand on
the relevant background geometry, emphasizing the role of the ambi-Hermitian structure.
In section 3, after recalling how the supersymmetric Casimir energy arises, we formulate
the conditions for (un-)pairing of modes on R
M3. In section 4 we discuss the
indexcharacter counting holomorphic functions, and make the connection with [5] in the case
{ 3 {
We are interested in studying four-dimensional N = 1 theories with an R-symmetry on
M3, where M3 is a compact three-manifold. In Euclidean signature, the relevant
supersymmetry conditions are the two independent rst-order di erential equations
(r
iA )
+ iV
+ iV (
)
= 0 ;
are spinors of opposite chirality. Here we use the spinor conventions1 of [7],
are two-component spinors with corresponding Cli ord algebra generated by
(
)a = ( ~ ; i1 2), where a = 1; : : : ; 4 is an orthonormal frame index and ~ = ( 1; 2; 3
)
are the Pauli matrices. In particular the generators of SU(2)
Spin(4) = SU(2)+
The eld V is assumed to be a globally de ned one-form obeying r V = 0, and will not
play a role in this paper. The eld A is associated to local R-symmetry transformations,
with all matter elds being charged under this via appropriate covariant derivatives.
The Killing spinors
equip M4 with two commuting integrable complex structures2
of primary Hopf surfaces. Extensions to secondary Hopf surfaces, and more general M3
realized as links of homogeneous hypersurface singularities, are discussed in section 5. We
conclude in section 6. We have included an appendix A, where we discuss the relation of
the index-character to the supersymmetric index [9] and its generalizations.
Supersymmetric backgrounds
Background geometry
(2.1)
(2.2)
(2.3)
(2.4)
(
)ab =
1
4
a b
b a
(I )
2i
j j
2
y (
)
:
:
The metric gM4 is Hermitian with respect to both I , but where the induced orientations
are opposite, which means the geometry is by de nition ambi-Hermitian. This structure
also equips M4 with a complex Killing vector eld
K
+ +
:
This has Hodge type (0; 1) for both complex structures, and satis es K K
= 0. We
assume that K commutes with its complex conjugate K , [K; K ] = 0.3 It then follows
1Di erently from previous literature, we denote the Killing spinors and associated complex structures
with
subscripts.
This emphasizes the fact that the two spinors and complex structures are on an
equal footing.
2We adopt the same sign conventions as [10, 11] for the complex structures. Our main motivation for
3If [K; K ] 6= 0 the metric is locally isometric to R
S3 with the standard round metric on S3 [7].
{ 4 {
that we may write K = 12
(
nowhere zero vector eld on M3.
Following [2], we assume the metric on M4 = S1
M3 to take the form
where the local form of the metric on M3 may be written as
gM4 =
2 d 2 + gM3 ;
gM3 = (d
+ a)2 + c2dzdz :
(2.5)
(2.6)
HJEP08(216)7
Here
= @ generates a transversely holomorphic foliation of M3, with z a local transverse
are both Killing vectors the positive conformal
factor is
=
(z; z), while c = c(z; z) is a locally de ned non-negative function and
a = az(z; z)dz + az(z; z)dz is a local real one-form. Notice that any Riemmanian
threemanifold admitting a unit length Killing vector
may be put into the local form (2.6).
Notice also that this geometry is precisely the rigid three-dimensional supersymmetric
geometry of [12, 13], for which there are two three-dimensional supercharges of opposite
R-charge.
We shall refer to
induces splits into three types: regular, quasi-regular and irregular. In the rst two cases
all the leaves are closed, and hence
generates a U(1) isometry of M3. If this U(1) action
is free, the foliation is said to be regular. In this case M3 is the total space of a circle
bundle over a compact Riemann surface
2, which can have arbitrary genus g
local metric c2dzdz then pushes down to a (arbitrary) Riemannian metric on
0. The
the one-form a is a connection for the circle bundle over
2. More generally, in the
quasiregular case since
is nowhere zero the U(1) action on M3 is necessarily locally free, and
the base
2
M3=U(1) is an orbifold Riemann surface. Topologically this is a Riemann
surface of genus g, with some number M of orbifold points which are locally modelled on
C=Zki , ki 2 N, i = 1; : : : ; M. The induced metric on
2 then has a conical de cit around
each orbifold point, with total angle 2 =ki. The three-manifold M3 is the total space of
a circle orbibundle over 2. Such three-manifolds are called Seifert bred three-manifolds,
and they are classi ed.
In the irregular case
has at least one open orbit. Since the isometry group of a
compact manifold is compact, this means that M3 must have at least U(1)
U(1) isometry,
with
being an irrational linear combination of the two generating vector elds. Notice
that M3 is still a Seifert manifold, by taking a rational linear combination, and that the
corresponding base 2 inherits a U(1) isometry. There are then two cases: either this U(1)
action is Hamiltonian, meaning there is an associated moment map, or else 1( 2) is
nontrivial. In the rst case 2 = WCP[2p;q] is necessarily a weighted projective space [14], while
in the second case instead
2 = T 2. In particular in the rst case M3 is either S1
S2, or
it has nite fundamental group with simply-connected covering space S3.
In addition to the local complex coordinate z, we may also introduce
w
i + P (z; z) ;
(2.7)
{ 5 {
where P (z; z) is a local complex function. Taking this to solve
where recall that a is the local one-form appearing in the metric (2.6), and de ning
h = h(z; z)
the metric (2.5) may be rewritten as
In these complex coordinates we have the complex vector elds
gM4 =
2 (dw + hdz)(dw + hdz) + c2dzdz :
Y =
Here s is a complex-valued function which appears in the Killing spinors
, where the
vector Y , like K in (2.4), is de ned as a spinor bilinear via
HJEP08(216)7
s
y
Y
+ :
1
sc
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
1+;0.
(2.15)
Following [10], we also de ne
Y
1
2
j j
2
K
1
which again have natural expressions as bilinears. The dual one-forms to K and Y are
K[ =
2(dw + hdz) ;
Y [ = sc dz :
These both have Hodge type (1; 0) with respect to I+, showing that z and w are local
holomorphic coordinates for this complex structure. In fact K[; Y [ form a basis for
On the other hand K[; (Y [) form a basis for
1;0. It follows that K generates a complex
transversely holomorphic foliation of M4, where the transverse complex structure has
opposite sign for I , while the complex structure of the leaves is the same for both I . In
other words, z is a transverse holomorphic coordinate for I+, but it is z that is a transverse
holomorphic coordinate for I . In the quasi-regular and regular cases, this means that
the induced complex structure on the (orbifold) Riemann surface
2 = M3=U(1) has the
opposite sign for I .
Finally, let us introduce the complex two-form bilinears
P
1
2
(
)
dx ^ dx :
These are nowhere zero sections of 2;0
L2 , where L
= ( 2;0) 1=2 are spinc line bundles
for the Killing spinors
. We shall consider a class of geometries in which the background
Abelian gauge eld A that couples to the R-symmetry is real. In this case we may write
2
s
P+ = (det gM4 )1=4s (dw + hdz) ^ dz =
3c e i! dw ^ dz ;
P
= (det gM4 )1=4
(dw + hdz) ^ dz =
3c ei! (dw + hdz) ^ dz ;
(2.16)
{ 6 {
where (det gM4 )1=4 =
2c and s =
e i!, with ! real [2]. Notice that the latter implies
Y = Y in (2.13), where the star denotes complex conjugation. By de nition
where Q
where dc
and thus K
structures).
2.2
Hopf surfaces
dP
=
iQ
^ P ;
are the associated Chern connections. We calculate
Q
A =
= dc log( 3c)
d! ;
1
2 Q+ =
1
for the I complex structure. Notice that dA in fact has Hodge type (1; 1) for both I ,
2;0 is the canonical bundle
are both holomorphic line bundles (with respect to their relevant complex
In most of the paper we will focus on backgrounds M4 = S1
M3, where the
threemanifold M3 has nite fundamental group. This means that the universal covering space
of M3 is a three-sphere S3, and moreover M3 = S3= , where
SO(4).4
These
socalled spherical three-manifolds are classi ed:
is either cyclic, or is a central extension
of a dihedral, tetrahedral, octahedral, or icosahedral group. The cyclic case corresponds
to Lens spaces L(p; q), with fundamental group
= Zp. Another particularly interesting
case is when
is the binary icosahedral group: here M3 is the famous Poincare homology
sphere. Being a homology sphere means that
is a perfect group (equal to its commutator
subgroup), and hence has trivial Abelianization. In fact 1(M3) =
has order 120, while
H1(M3; Z) is trivial. Of course our three-manifold M3 also comes equipped with extra
structure, and M4 = S1
see, one can realise such supersymmetric S1
for
SU(2)
U(2)
SO(4).
2.2.1
Primary Hopf surfaces
M3 must be ambi-Hermitian with respect to I . As we shall
M3 backgrounds as Hopf surfaces, at least
Let us rst describe this structure in the case when M3 = S3. Here M4 is by de nition a
primary Hopf surface | a compact complex surface obtained as a quotient of C2 n f0g by
a free Z action. These were studied in detail in [2], and in what follows we shall review
and extend the analysis in this reference.
In the I+ complex structure global complex coordinates (z1+; z2+) on the covering space
C2 nf0g are expressed in terms of the local complex coordinates z, w de ned in the previous
subsection via
4This is Thurston's elliptization conjecture, now a theorem.
z
z
1+ = ejb1j(iw z) ;
2+ = ejb2j(iw+z) :
{ 7 {
(2.17)
(2.18)
(2.19)
(2.20)
S3 is the quotient of C2 n f0g by the Z action generated by
(z1+; z2+) ! (p+z1+; q+z2+) ;
(2.21)
p
p+1, q
q+1.
%; ; 1
; 2 via
where the complex structure parameters are p+
e jb1j, q+
e jb2j.5 Notice that we may
equivalently reverse the sign of the generator in (2.21), with (z1+; z2 ) ! (p z1+; q z2+) and
+
We may further express these complex coordinates in terms of four real coordinates
w =
1
2jb1j
1 +
1
2jb2j
2
i
iQ(%) ;
z = u(%)
i
1
2jb1j
1
1
2jb2j
equations that may be found in [2], although their precise form won't be relevant in what
follows.6 We then have
(2.23)
(2.24)
(2.25)
(2.26)
and the quotient by (2.21) simply sets
a general class of metrics on M3 = S3 was studied, with U(1)
+ , with
a coordinate on S1 = S1. In [2]
U(1) isometry. The latter
The complex structure I
also equips M4 = S1
S3 with the structure of a Hopf
surface. Global complex coordinates on the covering space C2 n f0g are now
z
z
1 = e jb1j[i(w+2iQ)+z) = e jb1j ejb1j(Q u)e i 1 ;
2 = e jb2j[i(w+2iQ) z) = e jb2j ejb2j(Q+u)e i 2 :
In particular notice in these coordinates the complex structure parameters are p
e jb1j = p+1, q
e jb2j = q+1. Notice also that w + 2iQ and z are local complex
coordinates for I , the former following from dw + 2idQ = (dw + hdz) + 2i@zQdz, both
of which have Hodge type (1; 0) with respect to I . The fact that (z1 ; z2 ) cover C2 n f0g
follows from an analysis similar to that in [2] for the I+ complex structure.
Another fact that we need from [2], that will be particularly important when we come
to solve globally for the modes in section 4, is that
! =
1
2 :
5For a general primary Hopf surface these parameters may be complex.
6Compared to reference [2] we have de ned i = sgn(bi)'i, i = 1; 2, and recall from footnote 2 that we
have also reversed the overall sign of the two complex structures I compared with that reference, meaning
that zi jhere= zi jthere.
z
z
1+ = ejb1j ejb1j(Q u)ei 1 ;
2+ = ejb2j ejb2j(Q+u)ei 2 ;
Recall here that s =
e i!, which for example enters the Chern connections (2.18), and
hence the background R-symmetry gauge eld (2.19). This choice of phase in s is xed
uniquely by requiring that A is a global one-form on M3 = S3. The Killing spinors
are then globally de ned as sections of trivial rank 2 bundles over M4 = S1
S3. Gauge
transformations A ! A + d of course shift ! ! !
SU(2), with SU(2) acting on C2 in the standard
twodimensional representation 2. The generator of this U(1) subgroup of the isometry group
U(1)
U(1) is the Killing vector
follows that M4 = S1
M3 is isomorphic to the secondary Hopf surface (C2 n f0g)=(Z
Zp),
in both complex structures. The three-manifold M3 is the Lens space L(p; 1) in this case.
Notice that
commutes with the Reeb vector eld , and hence jb1j; jb2j (which determine
the complex structure parameters p , q ) can be arbitrary.
We may also realise supersymmetric backgrounds with non-Abelian fundamental
groups. Here we may take
SU(2) to act on C2 in the representation 2. In order for
to
act isometrically we assume the isometry group to be enlarged to U(2) = U(1) Z2 SU(2),
with the Reeb vector embedded along U(1). This means jb1j = jb2j. The metric on M3 = S3
is then that of a Berger sphere
1
2
(2.27)
(2.28)
(2.29)
ds2M3 = d 2 + sin2 d'2 + v2(d& + cos d')2 ;
where v > 0 is a squashing parameter, and & =
1 +
2, ' =
2. This special case
of a Hopf surface background was studied in appendix C of [2], and has b1 =
b2 = 1=2v,
and I+ complex coordinates
z
1+ = p
2 e 2v cos ei 1 ;
2
z
2
+ =
p
2 e 2v sin ei 2 :
In particular jz1+j2+jz2+j2 = 2e =v is invariant under SU(2). The I complex coordinates are
(z1 ; z2 ) = e v (z1+; z2+) ;
meaning that the SU(2) group acts in the complex conjugate representation 2 in the I
complex structure. As is well known, 2 = 2, and thus again M4 = S1
M3 is isomorphic
to the secondary Hopf surface (C2 n f0g)=(Z
) in both complex structures. Of course
nite subgroups
SU(2) have an ADE classi cation, where the A series are precisely
the Abelian
= Zp quotients of primary Hopf surfaces described at the beginning of this
subsection, while the D and E groups are the dihedral series and tetrehedral E6, octahedral
E7 and icosahedral E8 groups, respectively.
{ 9 {
We may also describe the complex geometry of the associated Hopf surfaces
algebraically. Consider the polynomials
fAp 1 = Z1p + Z22 + Z32 ;
fE6 = Z13 + Z24 + Z32 ;
fDp+1 = Z1p + Z1Z22 + Z32 ;
on C3 with coordinates (Z1; Z2; Z3). The zero sets
have an isolated singularity at the origin o of C3.
These are all weighted
homogeneous hypersurface singularities, meaning they inherit a C
action from the C action
(Z1; Z2; Z3) ! (qw1 Z1; qw2 Z2; qw3 Z3) on C3, where wi 2 N are the weights, i = 1; 2; 3, and
q 2 C . For example, fAp 1 has degree d = 2p under the weights (w1; w2; w3) = (2; p; p),
while fE8 has degree d = 30 under the weights (w1; w2; w3) = (10; 6; 15). The smooth
locus X n fog = R
M3, where M3 = S3= ADE , while the quotients (X n fog)=Z are
precisely the ADE secondary Hopf surfaces described above. Here Z
C is embedded as
n ! qn for some xed q > 1. The Reeb vector eld action is quasi-regular, generated by
C . The quotient 2 = M3=U(1) is in general an orbifold Riemann surface of
2.3
Flat connections
The path integral of any four-dimensional N = 1 theory with an R-symmetry on one of
the supersymmetric backgrounds S1
M3 of section 2.1 localizes. In particular, the
supercharges generated by
localize the vector multiplet onto instantons and anti-instantons,
respectively [2], which intersect on the
at connections. In the Hamiltonian formalism for
computing the supersymmetric Casimir energy, we will then need to study at connections
M3. The two spaces S1
M3 and R
M3 have respectively
Recall that at connections on M4 with gauge group G are in one-to-one
corresponon the covering space R
periodic with period , and
2 R.
dence with
Hom( 1(M4) ! G)=conjugation :
(2.32)
so that 1(M4) =
connection on M3.
In particular a at G-connection is determined by its holonomies, which de ne a
homomorphism % : 1(M4) ! G, while gauge transformations act by conjugation. In the path
integral on M4 = S1
M3 we have 1(M4) = Z
1(M3), with 1(S1) = Z. A
at
connection is then the sum of pull-backs of at connections on S1 and M3, and we denote the
former by A0. On the other hand in the Hamiltonian formalism instead M4 = R
M3,
1(M3), and a at connection on M4 is simply the pull-back of a at
When 1(M3) is nite, which is the case for the primary and secondary Hopf surfaces
in section 2.2, the number of inequivalent at connections on M3 is also
nite. The path
integral on S1
M3 correspondingly splits into a nite sum over these topological sectors,
together with a matrix integral over the holonomy of A0. In the Hamiltonian formalism on
R
M3, instead for each at connection on M3 we will obtain a di erent supersymmetric
quantum mechanics on R.
A matter multiplet will be in some representation R of the gauge group G. In the
presence of a non-trivial at connection on M4 = R
M3, this matter multiplet will be
a section of the associated
at vector bundle, tensored with K+
R-charge k. The latter follows since recall that the background R-symmetry gauge eld A
k=2 if the matter eld has
is a connection on K+
1=2 = K
+1=2. For the Hopf surface cases of interest this will always
be a trivial bundle, albeit with a generically non- at connection, and we hence suppress
this in the following discussion. Concretely then, composing % : 1(M4) ! G with the
representation R of G determines a corresponding at connection in the representation R,
and the scalar eld in the matter multiplet is a section of the vector bundle
is the fundamental group, V = CM is the vector space associated to R, and the action
of 1 on V is determined by the at R-connection described above. The scalar eld in
the matter multiplet is then a section of the bundle (2.33), which is a CM vector bundle
S3 for Hopf surfaces), 1 =
over M4.
To illustrate, let us focus on the simplest non-trivial example, namely the Lens space
M3 = S3=Zp = L(p; 1). For a G = U(1) gauge theory the at connections on R
M3 may
be labelled by an integer 0
m < p, which determines the holonomy
exp i
Z
A
= e2 im=p :
%(!p) = diag(!pm1 ; : : : ; !pmN ) 2 U(N ).
Here A is the dynamical U(1) gauge eld, while the circle
generates 1(R
M3) = Zp.
The associated homomorphism % : Zp ! U(1) is generated by %(!p) = !pm, where !p
e2 i=p is a primitive pth root of unity. For a U(N ) gauge theory the at connections are
similarly labelled by 0
mi < p, where i = 1; : : : ; N runs over the generators of the
Cartan U(1)N subgroup of U(N ). These are permuted by the Weyl group, so without
loss of generality one may choose to order m1
m2
mN , and label the
at
U(N ) connection by a vector m = (m1; : : : ; mN ). Now % : Zp ! U(N ) is generated by
An irreducible representation of U(1) is labelled by the charge
2 Z, so R = R . In the
presence of the at connection (2.34), a matter eld in this representation becomes a section
of the line bundle L over R
given by c1(L)
m mod p. Equivalently, on the universal covering space Mf4 = R
S3 =
L(p; 1) with rst Chern class c1(L) 2 H2(R
L(p; 1); Z) = Zp
C2 n f0g, the relevant sections of Vmatter may be identi ed with functions on Mf3 which pick
up a phase e2 ic1(L)=p under the generator of the Zp action. More generally, for a U(N )
gauge group we may decompose the representation R =
V into weight spaces, with
weights . This then essentially reduces to the line bundle case above, with the part of
the matter eld in V now being a section of L with c1(L)
(m) mod p. For example,
the fundamental representation of U(N ) has weights i(m) = mi, i = 1; : : : ; N , the adjoint
representation has weights ij = mi
mj , etc.
In this section we review the two approaches to de ne the supersymmetric Casimir energy
Esusy, involving the path integral formulation on a compact manifold S1
M3, and the
Hamiltonian formalism on its covering space R
M3, respectively.
We also present a
geometric interpretation of the shortening conditions previously discussed in [4, 10].
On general grounds [11], the localized path integral of a four-dimensional N = 1 theory with
an R-symmetry on M4 = S1
M3 is expected to depend on the background geometry only
via the complex structure(s) of M4. For example, for the primary Hopf surfaces described
in section 2.2.1 the complex structure parameters are p
= e jb1j; q
= e
may equivalently be thought of as speci ed by the choice of Reeb vector eld
jb2j, which
in (2.24)
(together with
). For a secondary Hopf surface S1
M3, the localized partition function
also carries information about the
nite fundamental group
=
1(M3). Of course the
partition function will also depend on the choice of N = 1 theory, through the choice of
gauge group, matter representation, and in particular on the R-charges of the matter elds.
In analogy with the usual zero point energy of a
eld theory, the supersymmetric
Casimir energy was de ned in [2] as a limit of the supersymmetric partition function
ZSsu1syM3 , namely the path integral with periodic boundary conditions for the fermions
along S1. More precisely,
Esusy
lim
d
log ZSsu1syM3 :
(3.1)
This may be computed using localization. As already mentioned in section 2.3, the vector
multiplet localizes onto at connections for the gauge group G, while at least for primary
Hopf surfaces the matter multiplet localizes to zero. The localized partition function
comprises the contributions of one-loop determinants for the vector and chiral multiplets of
the theory, evaluated around each such BPS locus, and one then integrates/sums over the
space of at connections. For primary Hopf surfaces (M3 = S3), the only non-trivial gauge
eld holonomy is for the at connection A0 along S1 [2]. On the other hand, if 1(M3) is
non-trivial one should also sum or integrate over at connections on M3, in the cases that
1(M3) is nite, or in nite, respectively [
15
].
For primary Hopf surfaces the partition function factorises ZSsu1syS3 = e
Esusy(jb1j;jb2j)I,
where I is a matrix integral over the gauge eld holonomies on S1, known as the
supersymmetric index [9]. The latter does not contribute to the limit (3.1), and thus in order to
compute Esusy one can e ectively set the gauge eld A0 = 0 in the one-loop determinants.
The regularization of these determinants is rather delicate and it was proved in [4] that
regularizations respecting supersymmetry give rise to a partition function with large and
small
limits consistent with general principles [16]. See appendix C of [4].
For secondary Hopf surfaces the partition function is a sum of contributions over
sectors with a xed at connection on M3. Let us label these sectors as
that in the special case that M3 = L(p; 1) = S3=Zp is a Lens space and G = U(N ) we
2 M at. Recall
Casimir energy is given by
may identify M at with the space of vectors m = (m1; : : : ; mN ), where 0
mN . Then from the de nition (3.1) it is clear that the supersymmetric
where for each
we have de ned a \supersymmetric Casimir energy in the sector
" as
2M at fEsusy; g ;
min
lim
d
log Z :
(3.2)
(3.3)
HJEP08(216)7
In the Lens space case M3 = L(p; 1) = S3=Zp the partition functions Z , which include
the Casimir contributions Esusy; , have been computed in [
15
].
Because the geometries of interest are of the form M4 = S1
vector generating translations on S1, we can consider the theories on the covering space
M4 = R
M3, employing the Hamiltonian formalism.7 These two approaches have been
shown to yield equivalent results for both the supersymmetric Casimir energy, as well as
the index I, for primary Hopf surfaces, M3 = S3. It was argued in [9] that the
supersymmetric index cannot depend on continuous couplings of the theory or the RG scale,
and therefore may be computed in the free limit (assuming this exists). We return to
discussing the supersymmetric index in appendix A. The supersymmetric Casimir energy can
also be obtained as the vacuum expectation value of the supersymmetric (Weyl ordered)
Hamiltonian Hsusy, and again it can be reliably computed in a free theory [4]. This can be
further Kaluza-Klein reduced on M3 to give a supersymmetric quantum mechanics on R,
with an in nite number of elds, organised into multiplets of one-dimensional
supersymmetry. Then Esusy = hHsusyi, where Hsusy is the total Hamiltonian for this supersymmetric
quantum mechanics. If supersymmetric regularizations are employed, then this de nition
has been shown to agree with (3.1) in the primary Hopf surface case M3 = S3 [4].
This formalism can also be utilised when
1(M3) is non-trivial (and
nite), as we will
see in more detail later in the paper. In this case there is a supersymmetric quantum
mechanics for each
at connection on M3. This leads to a de nition of \supersymmetric
Casimir energy in the sector
" that will depend on the at connection
2 M at, thus
Esusy;
= hHsusy; i :
(3.4)
We will see that this quantum-mechanical de nition of Esusy;
coincides with the path
integral de nition given previously, in any sector
= m, for Lens space secondary Hopf
surfaces with M3 = L(p; 1) = S3=Zp. Of course, the actual supersymmetric Casimir energy
of the theory will be given by the minimum Esusy; among all at connections.
7On M4 = R
M3 one usually works in Lorentzian signature. In this paper, however, we will always
remain in Euclidean signature. One can then take the point of view that the Wick rotation (t = i ) to pass
from Euclidean to Lorenztian signature can be done after the reduction to one dimension. In practice, we
will never need to perform this last step.
In the simplest case, where M3 = Sr3ound, the Hamiltonian formalism can be used to
obtain explicitly all of the modes and their eigenvalues [3, 4, 17]. Only a subset of unpaired
modes contribute to Esusy [3]. These modes where shown in [4] to correspond to short
1d supersymmetry multiplets (chiral and Fermi multiplets). This feature extends to more
general geometries, where the unpaired modes obey shortening conditions taking the form
of linear rst order di erential equations [10].
In what follows we will focus attention on a chiral multiplet. Using a set of \twisted
variables" [10], the fermion of a chiral multiplet can be replaced by a pair of anticommuting
elds B and C. Thus such a multiplet comprises the four scalar
elds
( ; B; C; F ), with R-charges (r; r
2; r; r
2), respectively. There is also a set of tilded
elds ( ~; B~; C~; F~) with opposite sign R-charges, that are eventually simply related to the
untilded elds by complex conjugation. The localizing deformation8 in these variables takes
the simple form
Lloc = 4 ~ bos + 2 ~
where ~ = (B~; C~),
= (B; C)T , and we have de ned the operators
bos
(L^K L^K + L^Y L^Y ) ;
with the rst order operators
^
ikA
iA ) :
Here U is one of the four complex vector elds K; K; Y; Y , de ned in section 2.1, k is the
R-charge of the
eld on which the operator is acting, and A
denotes the localized
at
gauge connection, acting on the eld in the appropriate representation R. As discussed in
section 2.3, such matter elds may equivalently be identi ed with functions on the covering
space that transform appropriately under the action of 1 =
1(M3) determined by the
at connection A . This action commutes with K and K, as was necessary to preserve
supersymmetry. We note the following relations
[L^K ; L^K ] = 0 ;
[L^K ; L^Y ] = 0 ;
[L^K ; L^Y ] = 0 :
(3.8)
These were proven in [10] in a xed (local) R-symmetry gauge where s = s(z; z) (and
without the
at connection), although it is obvious that they are valid in any gauge.
In particular they are valid in the unique global non-singular gauge (2.26), relevant for
Hopf surfaces.
The unpaired modes were shown in [10] to satisfy the shortening conditions
L^Y B = 0 ;
^
LY
= 0 ;
iL^K B =
iL^K
=
BB ;
;
8This coincides with the standard chiral multiplet Lagrangian for a particular choice of the parameter .
fer
fer
F~F ;
i
^
LK
^
LY
LY L^K
^
!
;
(3.5)
(3.6)
(3.7)
(3.9)
2
i n +
where we have denoted the modes B, , to distinguish them from the closely related
modes to be introduced momentarily. It is worth emphasizing that these equations are
valid both on S1
the two cases. In particular, on S1
M3; however, the eigenvalues
M3 one expands all elds in Kaluza-Klein modes over
(xi) e in , where n 2 Z and xi, i = 1; 2; 3 are coordinates on M3.
, where
2 R is the Reeb charge of the modes
On the other hand, using the equations (3.9) in the context of the Hamiltonian formalism
M3 [4], one has e ectively to set n = 0, and therefore in this case
=
.
In order to compute Esusy in principle one should consider the Hamiltonian
canonically conjugate to (3.5), insert all modes obeying their (free) equations of motion, and then
reduce the problem to one dimension [3]. Alternatively, one can focus on the unpaired
modes, giving rise to short 1d multiplets, and determine their -charge, for example by
analysing the reduced supersymmetry transformations [4]. Here
is the Hermitian
operator appearing in the one-dimensional supersymmetry algebra
L^Y B = 0 = L^K B ;
fQ; Qyg = 2(Hsusy
) ;
Q2 = 0 ;
[Hsusy; Q] = [ ; Q] = 0 :
Then Esusy is determined using the fact that for every multiplet hHsusyi = h i [4].
3.4
Unpaired modes on R
M3
In the path integral formalism, localization reduces the problem to computing the
oneloop determinant associated to (3.5). Correspondingly, in the Hamiltonian formulation, we
consider modes obeying the equations of motion following from (3.5), namely
fermionic zero mode,
so
bos
It is simple to show that modes satisfying the equations in (3.12) are paired by
supersymmetry. Indeed, if
is a bosonic zero mode,
bos
= 0, one can check using (3.8) that
= (L^Y ;
^
LK )T is a fermionic zero mode, so
fer
= 0. Conversely, if (B; C)T is a
fer(B; C)T = 0, one can check that
C is a bosonic zero mode,
= 0. Modes that are paired this way form long multiplets that do not contribute
to the supersymmetric Casimir energy. Notice that a fermionic zero mode satis es
bos
= 0 ;
fer
= 0 :
(3.12)
L^Y C =
L^K B ;
L^K C = L^Y B :
The net contribution to Esusy comes from unpaired modes. These are bosonic/fermionic
zero modes for which the putative fermionic/bosonic partner is identically zero. Thus these
are fermionic (B; 0) modes satisfying (using (3.13))
(3.11)
(3.13)
(3.14)
and bosonic
modes satisfying
^
LY
= 0 = LK
^
:
(3.15)
Recalling the de nition (3.7) and using the preliminaries in section 2.1, one recognises
denotes the (0; 1)+ part of d
ikA
the two operators L^Y and L^K as the components of the twisted @A+; A
iA, where the twisting is determined by the
Rdi erential. This
symmetry connection A in (2.19) and at connection A. In particular, the unpaired B
modes in (3.14) obey
and are therefore (twisted) holomorphic in the I+ complex structure. Similarly, one can
show the unpaired
modes in (3.15) satisfy
sections of Vmatter
matter vector bundle (2.33).
in the I
denotes the (0; 1) part of d ikA iA, and are therefore (twisted) holomorphic
complex structure. Notice that more precisely the unpaired B and
modes are
K+
(r 2)=2 and Vmatter
K
r=2, respectively, where Vmatter is the at
writing a mode as
It is simple to see that the above holomorphic modes may be decomposed into modes
on M3 which have de nite charge under the (twisted) Reeb vector eld iL^1 . In particular,
2
( ; xi) = e 2
(xi) ;
and using iL^K = L 1 @ + iL^1 , one sees that
2 2
these become
motion on R
on S1
relation [7]
^
LK ( ; xi) = 0
()
This shows that the modes on R
M3 de ned by (3.9) were indeed independent of , and
therefore de ned on M3, as already remarked below equation (3.9). Thus we can think of
the modes (3.18) as the \lifting to the cone" of the modes in the previous section. In fact
setting r = e
one sees that the metric on R
M3 is conformally related to the metric on
the cone C(M3): gC(M3) = dr2 + r2gM3 . Notice also that upon the Wick rotation t = i ,
(t; xi) = e2i t (xi), as expected for modes solving the free equations of
M3 in Lorentzian signature [3]. These have to be contrasted with the modes
M3 discussed earlier, namely
( ; xi) = e in
(xi).
Recall that the supersymmetry algebra acting on elds contains the anti commutation
f +
;
g = 2iL^K = 2 L 1 @ + iL^1
2 2
;
(3.20)
where
denote supersymmetry variations with respect to the
Killing spinors,
respectively. Comparing this with the anti-commutator in (3.11), one can identify the
eigenvalues of the quantum mechanical operators Hsusy and
with those of the operators9
9After performing the Wick rotation t = i to go to Lorentzian signature.
(3.16)
HJEP08(216)7
(3.17)
(3.18)
(3.19)
LK
and
iL^1 , acting on the classical modes, respectively. Therefore, the condition
2
= 0 obeyed by the holomorphic modes (on the cone) may be interpreted as showing
that the Hamiltonian eigenvalues are equal to their Reeb charge, and is the counterpart of
hHsusyi = h i in the supersymmetric quantum mechanics.
To summarise, the supersymmetric Casimir energy is computed by summing the Reeb
charges of (twisted) holomorphic modes on R
M3, with fermionic and bosonic modes
corresponding to each complex structure I , respectively.
4
Primary Hopf surfaces
In this section we re-examine the supersymmetric Casimir energy for the primary Hopf
surfaces S1
S3 in the above formalism. This was rst de ned and computed in the path
integral approach in [2]. Since M3 = S3 there are no at connections on M3.
Solving for the unpaired modes
Recall that the unpaired B and
modes, that contribute to the supersymmetric Casimir
energy, are zero modes on C2 n f0g of the twisted holomorphic di erentials @A+, @A ,
respectively, where the background R-symmetry gauge
eld A is given by (2.19) and the
operators are understood to act on elds of R-charge k. The curvature of A has Hodge
type (1; 1) with respect to both I , and thus both di erentials are nilpotent.
Using the global complex coordinates de ned in section 2.2.1, it is straightforward to
solve explicitly for these zero modes. In what follows we assume that we are working in a
weight space decomposition of the matter representation R, so that for a xed weight
we
have B = B is a single scalar eld. For the unpaired B modes we rst note from (2.19)
that the (0; 1)+ part of A is
In particular notice that we have used
A(0;1)+ =
2
2
(z1+z2+)k=2 = jz1 z2 j
+ + k=2 e i(k=2)! ;
where ! =
1
2
. Recall that
is globally a nowhere zero function, while near the
c
j j
complex axes (i.e. z1+ = 0 and z
2+ = 0) the real function c behaves to leading order as
jz1+j, jcj
jz2+j, respectively. This is required for regularity of the metric [2]. It
follows that the factor in front of B inside the square bracket in (4.2) is a real nowhere
zero function on C2 n f0g. A basis of regular solutions is hence
where n1; n2 2 Z 0.
B = Bn1;n2
k=2
(z1+)n1 (z2+)n2 ;
3
c
+ +
(4.1)
(4.2)
(4.3)
(4.4)
HJEP08(216)7
and one obtains a basis of regular solutions given by
i
2
3
c
1
2
=
n1;n2
k=2
(z1 )n1 (z2 )n2 :
The prefactors in front of the holomorphic monomials in the modes (4.4), (4.6) also
have a simple geometric interpretation. Recall that the Hermitian structure (gM4 ; I+)
equips C2 n f0g with the (2; 0)+-form
P+
1
2 +( +)
+ dx ^ dx
=
3c e i!dw ^ dz :
On the other hand, C2 has the global holomorphic (2; 0)+-form10
+
1
2jb1jjb2j
dz1+ ^ dz2+ = iz1+z2+ dw ^ dz ;
where we have used (2.20). Then
One may similarly solve for the unpaired
zero modes. Since from (2.19) we now have
is simply the modulus of the ratio of these two canonically de ned (2; 0)+-forms. A similar
computation shows that
3
c
+ +
=
P+ ;
+
3
c
=
P
;
iL^1 =
2
i
2 L +
k
2
;
L 1 ! =
2
1
2 (jb1j + jb2j) :
scalars
where
1
2jb1jjb2j
dz1 ^ dz2
= iz1 z2 (dw + hdz) ^ dz :
To summarize: the unpaired B modes are jP+= +jk=2 times a holomorphic function on
C2 with respect to the I+ complex structure, while the unpaired
times a holomorphic function on C2 with respect to the I complex structure. Here k = r 2
modes are jP =
for B, while k = r for , where r is the R-charge of the matter multiplet.
As discussed in section 3.4, the contributions of these modes to the supersymmetric
Casimir energy is determined by their eigenvalues under iL^1 , where recall that acting on
2
10This is not to be confused with the conformal factor , especially in the following formulae.
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
j
k=2
(4.12)
(4.13)
The eigenvalues are then easily computed:
where we have used that the Reeb vector is given by (2.24), and hence
L zi =
ijbijzi ;
i = 1; 2 :
We may now further reinterpret the eigenvalues B,
, using our earlier description of
HJEP08(216)7
the holomorphic volume forms
. Recall that A is a connection on K+
on sections of K+
holomorphic section
k=2. In the case at hand K+ =
2+;0 is a trivial bundle over C2 nf0g, but the
+ of K+ leads to a canonical lifting of the U(1)
U(1) action, with
generators (q1; q2) 2 U(1)
multiplication by q1q2 on
U(1) acting on C
2 = C
+. With this understanding, the eigenvalue
of the ordinary Lie derivative iL 1 acting on holomorphic sections of K+
2
on the bre contributes precisely
k
2
(
) = k2
to iL 1 , since iL 1
2 2
+ =
+.
+ satis es L@ i + = i +, i = 1; 2, the
C as (z1+; z2 ) ! (q1z1+; q2z2+) act as
+
B is the eigenvalue
k=2. Here the action
A similar reasoning applies to the
modes. Here
(q1; q2) 2 U(1)
k=2. Again the canonical bundle is trivial, but the action of
U(1) above on the
bre is now (q1q2) 1. This follows from the relative
minus signs in the phases in (2.23), (2.25). The action on the bre then again contributes
precisely k2
to iL 1 , since now iL 1
2 2
=
.
Notice that with these de nitions K+ = (K ) 1 as equivariant holomorphic line
bundles under U(1)
U(1).
4.2
The character
The supersymmetric Casimir energy is (formally, before regularization)
(4.14)
(4.15)
Esmuasytter =
X
B
n1;n2 +
X
where the eigenvalues are those on the right hand side of (4.14). Here we have introduced
the superscript \matter" to emphasize that in what follows we focus on the contribution
of a single weight
in a weight space decomposition of the chiral matter representation
R. We have seen that the eigenvalues in (4.16) are precisely Reeb charges, under iL 1 , of
2
holomorphic sections of K+
k=2 and K
k=2, respectively, where k = r 2 for the B modes and
k = r for the
modes. Thus it is natural at this point to introduce the index-character
of [8] that counts such holomorphic sections according to their U(1)
U(1) charges. We
take the U(1)
U(1) generators to be (q1; q2), which act as
(z1 ; z2 ) ! (q1 1z1 ; q2 1z2 ) :
(4.17)
For the B modes we have the associated index-character
K+k=2 ; (q1; q2)) =
X
The left hand side is de ned as the trace of the action of (q1; q2) on the zero modes of the
by analytically continuing to jq1j; jq2j < 1 the series converges to give
K+k=2 . The right hand side of (4.18) is a divergent series for jq1j = jq2j = 1, but
This then e ectively regularizes the eigenvalue sum. Indeed, setting q1 = etjb1j, q2 = etjb2j
and formally expanding (4.18) in a Taylor series around t = 0, the coe cient of
precisely 2 nB1;n2 =
n1jb1j
n2jb2j + k . Recalling that the B modes have k = r
we hence see that according to this \character regularization" their contribution to the
t is
2,
supersymmetric Casimir energy is
(4.18)
(4.19)
where the second equality is by a simple direct computation. This is indeed the correct
contribution of the unpaired B modes to the supersymmetric Casimir energy!
The
modes work similarly. The relevant character is now
X
Summing the series for jq1j; jq2j > 1 we obtain
(1
(q1q2) k=2
(q1q2) (k 2)=2
q1)(1
q2)
:
Recalling that
has R-charge r, we see that their contribution is also precisely the right
hand side of the rst line of (4.20). Thus they contribute equally to the supersymmetric
Casimir energy, as expected, Esusy = EsBusy.
4.3
Zeta function versus heat kernel regularization
At rst sight the result just obtained is somewhat remarkable, because we regularized
the eigenvalue sum (4.16) using the index-character (via analytic continuation to a simple
geometric series), while in previous work the sum in (4.16) is regularized using the Barnes
double zeta function. The two regularization schemes lead to the same result.
This may be explained as follows. In order to regularize each sum in (4.16) in a
supersymmetric fashion one should replace11
X
n
n !
X
n
n f ( n; t) ;
transform.
zeta function, de ned as
with f (x; t) a function chosen so that the sum converges. Requiring that f (x; 0) = 1, the
value of the regularized sum is given by the nite part in the limit that the parameter t ! 0.
Indeed, supersymmetric counterterms exist that may be added to remove divergences
appearing as poles in t 2 and t 1. However, the fact that nite supersymmetric counterterms
do not exist [18] implies that the
nite part is unambiguous, and therefore independent
of the details of the regularization. There are two natural choices. Picking f ( n; t) =
leads to the spectral zeta function regularization, while the choice f ( n; t) = e t n leads
to the heat kernel regularization, which as we shall see is the \character regularization" we
have used above. It is well known that these two are related to each other via the Mellin
n
t
In the case of interest the sums in (4.16) were regularized in [4] using the Barnes double
X
where x = r for the physical case of interest. Here we have focused on the
modes. The
sum in (4.24) converges for Re t > 1 and one analytically continues to t =
1 obtaining [19]
where we have de ned u = (r
1) = x
. Note that
Esmuasytter =
u
3
6jb1jjb2j
so that the contributions to Esmuasytter of the modes
and B are indeed identical.
Alternatively, in the heat kernel regularization we are led to consider
S(t; jb1j; jb2j; x)
X
and we extract Esmuasytter from the coe cient of t in a series around t = 0. This is precisely
the character regularization we introduced above. Concretely,
(4.23)
(4.24)
(4.25)
(4.26)
(4.27)
S(t; jb1j; jb2j; x) =
=
(1
(1
e tx
e tjb1j)(1
(q1q2) r=2
(4.28)
where recall that x = r , and in the second line we have precisely the character (4.22) for
the
modes.
11Below n denotes a multi-index.
4.4
In the above discussion we saw that both the B and
unpaired modes lead to the same
contribution to the supersymmetric Casimir energy. However, the discussion is not quite
symmetric because B has R-charge k = r
has R-charge k = r. One can put
these on the same footing, with overall R-charge r
1, by e ectively further twisting the
@ operators, thus viewing them as (part of) a Dirac operator.
Let us begin with the
zero modes. The relevant operator is
where we have de ned
HJEP08(216)7
L
;
L
K
(r 1)=2 :
K+
(r 1)=2 :
1
jb1jjb2jt2
u
jb1jjb2jt
+
u
3
u
4
3!jb1jjb2j
4!jb1jjb2j
u
2
t
+
q2)
1=2.
;
:
:
b21 + b22
24jb1jjb2j
(4.29)
(4.30)
(4.31)
(4.32)
(4.33)
(4.34)
(4.35)
(4.36)
5760jb1jjb2j
7(b21 + b2)
2
4b21b22 t2 + O(t3) :
Let us denote the weight on L
as
= (q1q2) (r 1)=2. Then the relevant character is
(q1q2) 1=2
=
(1
(q1q2)1=2
q1)(1
where the (q1q2) 1=2 in the numerator comes from the twisting by K
Similarly for the B zero modes the operator is
where
Notice that the weight on L+ is also
relevant character is
= (q1q2) (r 1)=2, and indeed L+ = L . Thus the
K+
1=2
q1)(1
q2)
This makes manifest that the two modes have the same character. In both cases the
L , which may be viewed as part of a Dirac-type operator
twisted by L . From this point of view, the explicit (q1q2)1=2 factors come from the fact
that the modes transform as spinors under the U(1)
U(1) action.
We may thus de ne
C(Dirac; (q1; q2; ))
(1
(q1q2)1=2
q1)(1
q2)
Setting q1 = etjb1j, q2 = etjb2j,
= e tu, we may expand in a Laurent series around t = 0
=
n
Y
i=1
xi
2 sinh xi=2
;
while the denominator is the Euler class
(x1; : : : ; xn) =
n
Y xi :
i=1
In the usual index theorem the xi would be the rst Chern classes of the line bundles that
arise on application of the splitting principle. In the equivariant setting these are replaced
by xi + i i, where the group action on the complex line
bre is multiplication by ei i .
The Euler class cancels against the numerator of (4.38), which leads to the rst equality
in (4.37). The A-roof class may be expanded as
A^ = 1
1
24 p1 +
1
5760
(7p12
4p2) +
;
where the Pontryagin classes pI are the Ith elementary symmetric functions in the xi2.
Thus in particular for complex dimension n = 2 we have p1 = x21 + x22, p2 = x2x2. These
1 2
comments of course explain the structure of the right hand side of (4.36). Analytically
continuing q1 = etjb1j; q2 = etjb2j amounts to sending i i ! tjbij above. Then (4.36) may be
rewritten as
C(Dirac; (q1; q2; )) =
e tu
1
4 sinh(tjb1j=2) sinh(tjb2j=2)
=
1
jb1jjb2jt2
b21 + b22 t2 +
24
7(b21 + b2)2
2
5760
4b21b22 t4 +
We immediately see that the divergent \index", which is given by setting t = 0, arises as
a second order pole, while the coe cient of the linear term in
t precisely reproduces the
regularized supersymmetric Casimir energy (setting u to its physical value of u = (r
Of course this is simply equivalent to the computation in (4.20), although now the equal
contribution of the B and
modes is manifest.
The appearance of the A-roof class in the expansion (4.36) is explained by the following
identity:
A^(i 1; i 2)
(i 1; i 2)
=
1
q1 1=2)(q21=2
2
q 1=2)
=
(1
(q1q2)1=2
q1)(1
q2)
;
where q1 = ei 1 , q2 = ei 2 . Here the numerator on the left hand side is the A-roof class,
which in general is de ned as
The middle term in brackets is the contribution from the A-roof class. This of course
explains the observation in [5] that the supersymmetric Casimir energy on the primary
Hopf surface is obtained (formally) by an equivariant integral on R
4 associated to the
Dirac operator. This arises naturally in the way we have formulated the problem. Here
the supersymmetric Casimir energy is the coe cient of
t in an expansion of the
indexcharacter of the Dirac operator, where the latter is regularized by analytically continuing
a divergent geometric series into its domain of convergence. Mathematically, this arises as
a heat kernel regularization, as opposed to a (Barnes) zeta function regularization.
(4.37)
(4.38)
(4.39)
(4.40)
(4.41)
e tu :
5.1
Lens spaces
The simplest way to generalize the primary Hopf surfaces studied in the previous section is
to take a
= Zp quotient. These secondary Hopf surfaces were described at the beginning
of section 2.2.2. With respect to either complex structure I the Zp action is generated by
spinors
(z1; z2) ! (e2 i=pz1; e 2 i=pz2), where zi = zi , i = 1; 2. This action preserves the Killing
The quotient M3 = S3=Zp = L(p; 1) is then a Lens space.
Since
1(M3) = Zp, the space R
M3 now supports non-trivial at connections. As
discussed in section 3, the localized partition function on S1
M3 splits into associated
topological sectors, which are summed over. In the Hamiltonian approach, each such
sector leads to a distinct supersymmetric quantum mechanics on R. Following the end
of section 2.3, here we consider a U(N ) gauge theory with matter in a representation R
in a weight space decomposition. The modes B = B ,
=
then become sections of
K+
L and K
Chern class c1(L)
of K+
L and K
sector m, we thus want to compute a twisted character, which counts holomorphic sections
L, respectively, where the line bundle L over R
L(p; 1) has rst
(m) mod p. In the Hamiltonian approach, and for xed topological
L according to their U(1)
U(1) charges (where as usual k = r
2
Recall that holomorphic functions on C2 are counted by
(1
1
q1)(1
q2)
:
The Dirac index-character (4.35) is constructed from this by multiplying by (q1q2)1=2 ,
which takes account of the lifting of the U(1)
holomorphic sections of L over C2=Zp, where c1(L)
U(1) action to K
k=2. More generally,
mod p, are counted by the twisted
character
q1 (1
(q1q2)p
(1
(1
p
q )
2
:
Here
is understood to lie in the range 0
< p, and as usual one expands the
denominator in a geometric series, for jq1j; jq2j < 1. Perhaps the simplest way to derive (5.2) is
via an appropriate projection of (5.1).
(z1; z2) ! (!pz1; !p 1z2), where !p
Recall that the Zp action on C2 is generated by
e2 i=p. The twisted character is then
p 1
j=0 (1
!p j
!pjq1)(1
!p j q2)
:
One easily veri es that this may be simpli ed to give (5.2). For zero twist, meaning
= 0,
we are simply counting holomorphic functions on C2=Zp, and (5.2) reads
C(@; (q1; q2); C2=Zp) =
(1
(q1q2)p)
=
(1
(q1q2))(1
1 + q1q2 + (q1q2)2 +
q1p)(1
p
q )
2
+ (q1q2)p 1
(1
q1p)(1
p
q )
2
:
This is the index-character of an Ap 1 = C2=Zp singularity.
(5.1)
(5.2)
(5.3)
(5.4)
Thus the contribution of a matter eld, for weight
and xed at connection m, leads
to a supersymmetric Casimir energy (in the sector
= m 2 M at) given by the character
(q1q2)1=2 hq1 (1
(q1q2)p
(1
(q1q2))(1
(1
q )
2
i
As in section 4, the Casimir energy is obtained by substituting q1 = etjb1j, q2 = etjb2j,
= e tu, and extracting the coe cient of
t in a Laurent series around t = 0. This is
easily done, and we nd
Esmuasytt;emr =
1
24jb1jjb2jp
4u3
(b12 + b22
2jb1jjb2j(p2
+2jb1jjb2j(jb1j jb2j) (
6 p + 6 2
p)(2
p) ;
1))u
where
= [(m). Here the hat indicates that is understood to lie in the range 0
and thus (m) 2 Z should be reduced mod p to also lie in this range. Recall that we xed
the convention that 0
u = (r
1) , where
mi < p, and ordered m1
mN . As usual we should also put
= (jb1j + jb2j)=2. This is the contribution from the weight ; one
should of course then sum over weights to get the total contribution of the matter eld, in
The partition function on S1
L(p; 1) has been computed in [
15
], and xing the sector
(z1; z2) ! (q1u(1j) qu(2j) z1; q1v1(j) qv2(j) z2) :
2 2
(5.5)
(5.6)
< p,
(5.7)
(5.8)
(5.9)
the sector m.
m one can check that indeed
for each topological sector.
5.2
Fixed point formula
Esmuasytt;emr =
log Zmmatter :
lim
!1 d
d
See equations (5.32){(5.34) of [
15
]. Thus the Hamiltonian approach does indeed correctly
reproduce the supersymmetric Casimir energy, de ned in terms of the partition function,
In [8] it was explained that the index-character may be computed for a general isolated
singularity by rst resolving the singularity, and using a xed point formula. In the case
at hand C2=Zp = Ap 1 is well-known to admit a crepant resolution, meaning that the
holomorphic (2; 0)-form extends smoothly to the resolved space, by blowing up p
1
twospheres. The action of U(1)
U(1) on C2=Zp extends to the resolution, which is hence
toric, with p isolated xed points. Each such
xed point is of course locally modelled by
C2, and the general formula in [8] expresses the index-character of C2=Zp = Ap 1 in terms
of a sum of the index-characters for C2, for each xed point. Labelling the xed points by
j = 0; : : : ; p
1, explicitly we have
C(@; (q1; q2); C2=Zp) =
p 1
X
u(j) = (u(1j); u
(2j)); v(j) = (v1(j); v2(j)) 2 Z2 as
U(1) on each xed origin of C2 is speci ed by the two vectors
One nds (for example using toric geometry methods) that
u(j) = (p
j; j) ;
v(j) = ( p + j + 1; j + 1) ;
and (5.8) reads
X
which one can verify agrees with (5.4).
Let us de ne the matter contribution to the supersymmetric Casimir energy for S3 as
4u3
24b1b2
:
Then (5.8) leads to the following xed point formula for the Casimir for S1
L(p; 1) (with
trivial at connection):
Esmuasytter[L(p; 1); b1; b2] =
X Esmuasytter[S3; b(1j); b(2j)]
=
p 1
j=0
4u3
24jb1jjb2jp
[(jb1j + jb2j)2
2jb1jjb2jp2]u
:
Here we have de ned
1
pjb1j
j(jb1j + jb2j) ;
2
pjb1j + (j + 1)(jb1j + jb2j) :
In fact (b(1j); b(2j)), j = 0; : : : ; p
1, are precisely the Reeb weights at the p xed points. In
this precise sense, we may write the supersymmetric Casimir energy for the secondary Hopf
surface (S1
S3)=Zp as the sum of p Casimir energies for primary Hopf surfaces S1
S3,
where each xed point contribution has a di erent complex structure, determined by (5.14).
This data is in turn determined by the equivariant geometry of the resolved space.
5.3
More general M3
In section 2.2.2 we discussed more general classes of secondary Hopf surfaces, realised as
ADE
SU(2) quotients of primary Hopf surfaces. The A series is precisely the
Lens space case discussed in the previous subsection, while the D and E series result in
non-Abelian fundamental groups. The formalism we have described gives a prescription
for computing the supersymmetric Casimir energy Esusy (or at least the matter
contribution Esmuasytter) for such backgrounds. One rst needs to classify the inequivalent at
Gconnections on M3 = S3= , via their corresponding homomorphisms % :
! G. A given
matter representation R of G then gives a corresponding
at R-connection, from which
one constructs the matter bundle (2.33). For each such
at connection one then needs to
compute the index-character of this bundle, namely one counts holomorphic sections via
their Reeb charges. The supersymmetric Casimir energy, in this topological sector, is then
obtained as a limit of this index-character.
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
g 2
In practice, one thus rst needs to understand the representation theory of the relevant
nonAbelian groups, before one can compute the associated index-characters. An interesting but
simple example is provided by the exceptional group
=
E8 : this is the binary icosahedral
group, which has order 120. The quotient M3 = S3= is the famous Poincare sphere, which
has the homology groups of S3, despite the very large fundamental group. This follows
since
E8 is equal to its commutator subgroup, and hence its Abelianization (which equals
H1(M3; Z)) is trivial. Related to this fact is that consequently any homomorphism into
an Abelian group is necessarily trivial.
This is easy to see: since any group element
may be written as g = hvh 1
v 1, then for any homomorphism % :
have %(g) = %(h)%(v)%(h) 1%(v) 1 = identity, where in the last step we used that G is
Abelian. This shows that, for example, any
at U(1) connection over the Poincare sphere
is necessarily trivial. Because of this, to compute the supersymmetric Casimir energy we
need only the index-character of C2= . But this is easily computed by realizing the latter
as a homogeneous hypersurface singularity
! G we
HJEP08(216)7
4u3 + 539b2u
720b2
:
The Reeb vector eld acting on (z1 = jz1jei 1 ; z2 = jz2jei 2 ) is
b
2
while for a matter multiplet of R-charge r we have u = (r
1)jbj=2.
5.3.2
Homogeneous hypersurface singularities
For an Abelian gauge theory on the Poincare Hopf surface just discussed, any at
connection over S3= E8 is trivial, and thus the index-character that counts holomorphic functions
on C2= E8 is su cient to compute the supersymmetric Casimir energy. However, more
C2= E8 = ffE8
Z13 + Z25 + Z32 = 0
C3 :
Here the polynomial fE8 has degree d = 30 under the weighted C action on C
weights (w1; w2; w3) = (10; 6; 15). From the general formula in [20] we thus compute the
1
q30
q10)(1
q15)
= 1 + q6 + q10 + q12 + q15 + : : : :(5.16)
Here q 2 C acts diagonally on C2= E8 as (z1; z2) ! (q1=2z1; q1=2z2). Notice that the centre
of E8 is Z2, which acts as multiplication on (z1; z2) by
1. The holomorphic (2; 0)-form
thus has weight q under the C
action, and the supersymmetric Casimir energy for an
Abelian gauge theory on the \Poincare Hopf surface" is
Esmuasytter =
q15) q = etjbj; = e tu coe cient of t
(5.15)
3 with
(5.17)
(5.18)
generally we may easily extend the above discussion to compute Esusy for Z quotients of
homogeneous hypersurface singularities in the sector with trivial at connection. These are
compact complex surfaces of the form M4 = S1
M3, where M3 is the link of the singularity.
Consider a general weighted homogeneous hypersurface singularity in C3. Here the
weighted C action on C3 is (Z1; Z2; Z3) ! (qw1 Z1; qw2 Z2; qw3 Z3), where wi 2 N are the
weights, i = 1; 2; 3, and q 2 C . The hypersurface is the zero set X
weighted homogeneous polynomial f = f (Z1; Z2; Z3), where
f
f = 0
C3 of a
f (qZ1; qZ2; qZ3) = qdf (Z1; Z2; Z3) ;
which de nes the degree d 2 N. We assume that f is such that X n fog = R
smooth, where o is the origin Z1 = Z2 = Z3 = 0. The associated compact complex surface
is obtained as a free Z quotient of X n f0g, where Z
C is embedded as n ! qn for some
xed q > 1. The Reeb vector eld action is quasi-regular, generated by q 2 U(1)
C , and
the quotient 2 = M3=U(1) is in general an orbifold Riemann surface. This construction of
course includes all the spherical three-manifolds in section 2.2.2, for which M3 = S3= ADE
and
2 has genus g = 0, but it also includes many other Seifert three-manifolds. For
example, taking weights (w1; w2; w3) = (1; 1; 1) and f to have degree d, then M3 is the
total space of a circle bundle over a Riemann surface
2 of genus g = (d
1)(d
2)=2.
Such homogeneous hypersurface singularities are Gorenstein canonical singularities,
meaning they admit a global holomorphic (2; 0)-form
0, de ned on the complement of
the isolated singularity at Z1 = Z2 = Z3 = 0. With respect to the I+ complex structure,
so that we identify
0 =
+, we may then write
The
modes work similarly, with respect to the second complex structure I . This may
be de ned globally in this setting as follows. The singularity X may be viewed as a complex
where z and w are the local coordinates de ned by supersymmetry on R
M3, de ned in
section 2.1, and
= (z; w) is a local holomorphic function. The argument in section 4.1
then generalizes to give that the unpaired B modes that contribute to the supersymmetric
Casimir energy are
0 =
dz ^ dw ;
B =
P+
+
k=2
F ;
where jP+= +j =
3c=j j is a real, globally de ned, nowhere zero function on X n fog , and
F is a holomorphic function on X. This follows since P+ and
+ are both globally de ned,
and being both (2; 0)-forms are necessarily proportional. The holomorphic functions F on
X are spanned by monomials Zn1 Zn2 Z3n3 , where ni 2 Z 0, modulo the ideal generated by
1 2
the de ning polynomial f . The index-character that counts such holomorphic functions
according to their weights under q 2 C is
1
q
d
(1
M3 is
(5.20)
(5.21)
(5.22)
cone over the orbifold Riemann surface
2 = M3=U(1). Here R
M3 may be identi ed
bration over 2, with the isolated singularity arising by contracting
the whole space to a point. In terms of the coordinates de ned by supersymmetry, the
C action is generated by the complex vector eld K. The I
complex structure is then
obtained by reversing the sign of the complex structure on the base
2, while keeping that
of the C
bre. This leads to the same complex manifold, although of course the map
between the two copies is not holomorphic. As for the primary Hopf surfaces in section 4,
the unpaired
modes then give an identical contribution to the B modes above.
It follows that the relevant character is
HJEP08(216)7
(5.23)
1=2),
(5.24)
C(q; ; X)
q( d+Pi3=1 wi)=2
where C(@; q; X) is the index-character (5.22). Here the power of q is precisely 12 the
charge of the holomorphic (2; 0)-form (arising as usual since A is a connection on K+
and q 2 C is the generator of the C action. The supersymmetric Casimir energy in this
case is obtained as usual by setting q = etjbj,
= e tu, and extracting the coe cient of
t in a Laurent series about t = 0. A simple calculation shows that this leads to the
supersymmetric Casimir energy
Esmuasytter =
4du3
(w12 + w22 + w2
3
24b2w1w2w3
d2)db2u
:
Here u = (r
1) for a matter multiplet of R-charge r, where now 1=2 the Reeb charge of
the holomorphic (2; 0) form is
= ( d + Pi3=1 wi)jbj=2. For example, the Lens space case
L(p; 1) in sections 5.1, 5.2 is w1 = 2, w2 = w3 = p, d = 2p (with jb1j = jb2j = jbj), while
the Poincare Hopf surface in section 5.3.1 is w1 = 10, w2 = 6, w3 = 15, d = 30. We stress
again that (5.24) gives the matter contribution to the supersymmetric Casimir energy in
the topological sector with trivial at gauge connection. For non-trivial at connections
one would instead need to compute the index-character of the relevant ( at) matter bundle.
5.4
Full supersymmetric Casimir energy
As in much of the previous literature, in this paper we have focused attention on the
contribution of a matter multiplet to the supersymmetric Casimir energy. However, we
expect that the vector multiplet contribution will also arrange into short multiplets, and
will similarly be related to (twisted) holomorphic functions. At least for primary Hopf
surfaces, and secondary Hopf surfaces with M3 = L(p; 1), previous results in the literature
imply that the contribution of a vector multiplet to the supersymmetric Casimir energy is
(formally) obtained from the contribution of a matter multiplet by (i) setting the R-charge
r = 0 (since the dynamical gauge eld has zero R-charge), (ii) replacing weights
by roots
of the gauge group G, and
nally (iii) reversing the overall sign. In this subsection we
will simply conjecture this is true more generally, at least in the sector with trivial at
connection on which we focus.
Given this conjecture, it is straightforward to combine the matter multiplet
result (5.24) for a general homogeneous hypersurface singularity with the vector multiplet
result, and sum over relevant weights/roots. Remarkably, we nd the following simple
formula for the total supersymmetric Casimir energy
27 w1w2w3
b
dc1
3 w1w2w3
c) :
c1
3
d + X wi ;
i=1
d2 + X wi2 ;
3
i=1
which depend on the weights (w1; w2; w3) and degree d of the hypersurface singularity,
HJEP08(216)7
while a and c denote the usual trace anomaly coe cients,
a =
c =
3
32
1
32
(3Tr R3
(9Tr R3
Tr R) =
5Tr R) =
3 h
32
1 h
32
2jGj + X
3(r
4jGj + X
9(r
1)3
(r
1)3
5(r
1) jR j ;
i
1) jR j ;
i
with R being the R-symmetry charge, and the trace running over all fermions.
By setting (w1; w2; w3) = (2; p; p), d = 2p, which correspond to Ap 1 singularities with
corresponding secondary Hopf surfaces S1
L(p; 1), one sees that (5.25) reduces to
Esusy =
16jbj (3c
27p
2a) + 4jbjp (a
3
c) :
This agrees with the
! 1 limit of the partition function in [
15
], and reproduces the
original primary Hopf surface result of [2] when p = 1.
One can make a number of interesting observations about the general formula (5.25).
Firstly, it depends on the choice of supersymmetric gauge theory only via a and c. Secondly,
the coe cient of the term (3a
2c) is related to the Sasakian volume of M3 via
vol(M3) =
d
1
w1w2w3 jbj2
vol(S3) :
Here vol(S3) = 2 2 is the volume of the standard round metric on the unit sphere, and the
Reeb vector is normalized as
= jbj , where
generates the canonical U(1)
C action on
the hypersurface singularity. M3 is the link of this singularity, and any compatible Sasakian
metric on M3 has volume given by (5.29), as follows from the general formula in [20]. The
metric on M3 is not in general Sasakian, but the point is that M3 is equipped in general with
an (almost) contact one-form
= d
+ a. The corresponding contact volume 12 RM3
^ d
brie y comment further on this in the discussion section.
We also note that in (5.25)
c1 =
of the orbifold Riemann surface 2 = M3=U(1) (more precisely, global sections of K o1rb are
d + Pi3=1 wi is the rst Chern class (number) of the (orbifold) anti-canonical bundle
given by weighted homogeneous polynomials of degree c1). Thirdly, we have suggestively
denoted c2 =
d2 + Pi3=1 w2. Of course this is not supposed to suggest the second Chern
i
class/number of a line bundle, which is zero, but rather is a quadratic invariant of the
singularity that takes a similar form to c1. It would be interesting to understand the
geometric interpretation of the second term, proportional to (a
c), in (5.25).
(5.25)
(5.26)
(5.27)
(5.28)
(5.29)
In this paper we have shown that the supersymmetric Casimir energy Esusy of
fourdimensional N
= 1
eld theories de ned on S1
M3 is computed by a limit of the
index-character counting holomorphic functions on (or more generally holomorphic
sections over) the space R
M3. In particular, the latter is equipped with an ambi-Hermitian
structure, and the short multiplets contributing to the supersymmetric Casimir energy are
in one-to-one correspondence with (twisted) holomorphic functions, with respect to either
complex structure. As examples of Seifert three-manifolds M3 we considered S3, as well
as the links S3= ADE of ADE hypersurface singularities in C3. For M3 = S3 our analysis
explains the relation of the supersymmetric Casimir energy to the anomaly polynomial,
pointed out in [5]. In the case of M3 = L(p; 1) we obtained formulas that may
independently be derived using the path integral results of [
15
]; while, to our knowledge, the
formulas for the D and E singularities have not appeared before. We have also presented
a formula (5.25) for the supersymmetric Casimir energy when M3 is the link of a general
homogeneous hypersurface singularity, in the trivial at connection sector, and assuming
a conjecture for the vector multiplet contribution.
Our analysis can be extended in various directions. The localization results of [
2, 15
]
strongly suggest that in the supersymmetric quantum mechanics the contributions of the
vector multiplet will also also arrange into short multiplets. One should show explicitly
that these are indeed related to (twisted) holomorphic functions, and therefore ultimately
to the index-character we have studied (and in particular hence prove (5.25)). In this paper
we have explained how to incorporate the contributions of discrete at connections on M3,
considering M3 = L(p; 1) as concrete example. It may be interesting to work out more
examples. Moreover, here we have not addressed the role of continuous at connections
arising when 1(M3) is in nite. Ultimately, the complete supersymmetric Casimir energy of
a theory should be obtained by appropriately minimizing over the set of all at connections,
and it would be nice to see whether this quantity may be used as a new test of dualities
between di erent eld theories and/or geometries.
Using the formulas presented in appendix A one can also easily obtain new
supersymmetric indices for theories de ned on S1
M3, where M3 is the Seifert link of the D and
E type hypersurface singularities. It would be interesting to explore their properties, as
they involve a generalization of the elliptic gamma function appearing for M3 = S3 [2] and
M3 = L(p; 1) [
15, 23, 24
].
We close our discussion by recalling that it is not clear how to reproduce the
supersymmetric Casimir energy with a holographic computation in a supergravity solution, even
for M3 = Sr3ound. See for example [28, 29] for some attempts and further discussion. Let
us point out that the formula (5.25) shows that in the large N limit the
supersymmetric Casimir energy (in the trivial at connection sector) is proportional to N 2 vol(M3).
We expect that it should be possible to reproduce this result from a dual holographic
computation, and indeed we will report on this in [30].
HJEP08(216)7
Acknowledgments
D. M. is supported by the ERC Starting Grant N. 304806, \The Gauge/Gravity Duality
and Geometry in String Theory". J. F. S. was supported by the Royal Society in the early
stages of this work. We thank Benjamin Assel for useful comments.
A
Supersymmertic index from the character
In this appendix we return to the supersymmetric index I [9], clarifying its relation to the
index-character, that is the main subject of this paper.
HJEP08(216)7
A.1
Primary Hopf surfaces
We begin with the case M3 = S3 and consider the modi cations needed for the extension
to more general M3 in the next subsection. Following [9], we can work on M4 = R
Sr3ound, with the complex structure parameters of the Hopf surfaces emerging as fugacities
associated to two commuting global symmetries [2, 11]. The supersymmetric index may
be de ned quite generally for any theory that admits the superalgebra (3.11), in terms of
a trace over states in the Hilbert space, as
I(x) = Tr( 1)F x ;
where F is the fermion number. A standard argument then shows that the net contribution
to the trace arises from states obeying
Hsusy
= 0. As this quantity does not depend
on continuous parameters it can be computed in the free theory, where it takes the form
of a plethystic exponential
I(x) = Pexp (f (x))
exp
X1 1
k=1
k
f (xk)
Physically, this is the grand-canonical partition function written in terms of the single
particle partition function f (x), counting single particle states (annihilated by
) of the
free theory. In practice, the operator
appearing in the superalgebra is given by
=
(2J3L + R), where R is the R-symmetry and J3L is the angular momentum associated to
SU(2)R. One can introduce a second fugacity
y conjugated to the angular momentum J R associated to rotations in U(1)
3
SU(2)L
SU(2)R. After changing variables,12 setting p1 = xy and p2 = x=y, the single
SU(2)R
particle index for a chiral multiplet is given by [21]
f matter(p1; p2) =
(p1p2) 2
r
(p1p2) 2
2 r
(1
p1)(1
and the contribution of a chiral multiplet to the supersymmetric index then reads
I
matter(p1; p2) =
1
Y
=
((p1p2)r=2; p1; p2) ;
(A.4)
where (z; p1; p2) is the elliptic gamma function.
12In this section we will denote p1; p2 the variables in which the index is written naturally in terms of
elliptic gamma functions. We will later make contact with the variables q1; q2 used in the previous sections.
(A.1)
(A.2)
(A.3)
It was noticed in [2, 22] that the supersymmetric Casimir energy can be extracted
from the single particle index by setting p1 = etjb1j, p2 = etjb2j, and taking the nite part
of the limit
Esusy(jb1j; jb2j) =
1
lim
2 t!0 dt
d
f (p1; p2) :
(A.5)
Below we will clarify the reason why this limit reproduces the supersymmetric Casimir
energy by relating f matter(p1; p2) to the index-character counting holomorphic functions.
For the computation of f matter(p1; p2) we can use the ingredients worked out in [3, 17].
In particular, the expressions for the operators Hsusy; R; J3L; J3R can be found in these
references,13 written in terms of bosonic and fermionic oscillators. For example, writing
bos + fer, we have
bos =
2
2
X
`=0 m;n= 2`
and
with
a
`m a`mna`ymn + a`ymna`mn
b
`m b`mnb`ymn + b`ymnb`mn
fer =
2
2
X
`
2
X
`=0 n= 2` m= 2` 1
1 X1
2
`
2
X
2` 1
X
`=1 n= 2` m= 2`
a
`m = ` + 2 + 2m ;
c
`m =
(` + 2 + 2m) ;
c
`m c`mnc`ymn
c`ymnc`mn
d
`m d`mnd`ymn
d`ymnd`mn ;
(A.7)
b
`m = ` + 2m ;
d
`m =
` + 2m ;
and jd`;m;ni = d`ymnj0i. However, the only zero-modes of
are
states in the Fock space, namely ja`;m;ni = a`ymnj0i, jb`;m;ni = b`ymnj0i, jc`;m;ni = c`ymnj0i,
jb`; 2` ;ni ;
jc`; 2` 1;ni ;
while there are no zero-modes of the a-type and d-type states. These have m =
m =
2
1, respectively, which are precisely the shortening conditions obeyed by the
and B modes, in the special case of the round three-sphere [4]. These two sets of modes are
contributing non-trivially to (A.3). Let us now show this explicitly. From the de nition
f matter(x; y) = tr( 1)F x y2J3R = fbos(x; y)
ffer(x; y) ;
13We use the notation of [3]. For simplicity, and to make contact with [21], we are setting the parameters
, in [3] to
= 1, = 1.
(A.8)
2` and
(A.9)
where here the trace is over the single particle states in (A.8), and we have
fbos(x; y) =
X xr+` X
y2n =
ffer(x; y) =
1
`=0
1
X x` r+2
`=0
2
n= 2`
2
X
n= 2`
(1
x
r
x
2 r
x ;
x :
(A.11)
(A.12)
HJEP08(216)7
To derive these we used14
and
jb`; 2` ;ni = (r + `)jb`; 2` ;ni ;
jc`; 2` 1;ni =
2
`)jc`; 2` 1;ni ;
J3R jb`; 2` ;ni = njb`; 2` ;ni ;
J3R jc`; 2` 1;ni = njc`; 2` 1;ni :
modes jc`; 2` 1;ni is
the fermionic anti-particles [21].
Notice that the R-charge of the bosonic modes jb`; 2` ;ni is r, while that of the fermionic
2). Thus f matter(x; y) is counting the bosonic particles minus
In order to make contact with the main part of the paper, one can see that upon
making the identi cations15 p1 = q1 1, p2 = q2 1, the rst term in (A.3) is precisely the
character C(@Kr=2 ; (q1; q2)) in (4.22), counting
character counting B~ modes. Notice that
K+
modes. On the other hand, the second
(r 2)=2 ; (q1; q2)), namely it can be identi ed with the
(A.13)
On taking the limit (A.5), the opposite signs in front of the fermionic part and in its
exponent cancel each other, e ectively giving the same result as the limit of the character,
or Dirac character, that we considered before.
A.2
Secondary Hopf surfaces
Let us now discuss secondary Hopf surfaces M4 = S1
M3, starting with the case that
the fundamental group of M3 is
= Zp.
Thus M3 = L(p; 1) is a Lens space.
The
supersymmetric index in this case was studied in [
15, 23, 24
]. We can work on the space
with a round metric on S1
S3=Zp and obtain the modes by projecting from those on
the covering space S1
S3. In the absence of a
at connection the modes on L(p; 1) are
precisely the Zp-invariant modes on S3. For example, for the scalar eld , these are given
by the S3 hyperspherical harmonics Y`mn satisfying 2n
0 mod p. More generally, in the
14Here the operators are normal ordered [21].
15The need for this change of variables originates from our de nition of the complex structures. See
footnote 2. This is of course just a convention.
where P = fn 2 f 2` ; : : : ; 2` g : 2n
mod pg. The sums are then computed exactly as
Expressing this in terms of the variables p1 = xy and p2 = x=y, we obtain
fbos(x; y) = xr (xy) (1
x2(p )) + ( xy )p (1
x2 )
(1
x2)(1 (xy)p)(1
( xy )p)
:
r
fbpo;s(p1; p2) = (p1p2) 2 C(@L; (p1; p2); C2=Zp) :
For the fermions in the complex conjugate multiplet, the projection condition has to
be modi ed as [24]
2n
mod p :
This e ectively swaps n1 and n2, or equivalently, p1 and p2. Therefore, the index counting
antifermions is given by
ffpe;r (p1; p2) = (p1p2) 2 2 r C(@L; (p2; p1); C2=Zp) ;
Again, it can be checked explicitly that ffpe;r (p1; p2) = fbpo;s(p1 1; p2 1), showing the
character contributing to the fermions is counting anti -holomorphic sections, as opposed to
the bosonic character, which counts holomorphic sections. Of course, the result of the
limit (A.5) reproduces precisely the supersymmetric Casimir energy in (5.6).
In order to compute the supersymmetric index using the plethystic exponential, it is
convenient to write the twisted Lens space character as
presence of a at connection with rst Chern class c1(L), the modes that descend to the
Lens space from S3 obey the condition [23, 25]
2n
c1(L) mod p :
Since the at connection can be removed locally by a gauge transformation, the
eigenvalues of the operators Hsusy; R; J3L; J3R are unchanged. One can then compute the
generating function by restricting the sums in (A.9) to the single particle states annihilated by
of
the previous subsection, and further obeying the projection (A.14), with c1(L) = (m) = .
Accordingly, the bosonic part is then given by
n2P
C(@L; (p1; p2); C2=Zp) =
(1
p
1
p +
p
p
2
(1
p1p2)(1
p :
Using this, it is immediate to obtain the index in the factorised form [24], namely
I mp;atter(p1; p2) =
((p1p2) r2 p2p ; p2p; p1p2) ((p1p2) 2 p1; p1p; p1p2) ;
r
where notice that this does not contain any Casimir energy contribution.
(A.14)
(A.15)
(A.16)
(A.17)
(A.18)
(A.19)
(A.20)
(A.21)
The reasoning that led to the expression of the single particle index above should be
valid more generally for a theory de ned on M4 = R
M3 (where 1(M3) is nite), with
a xed at connection in a sector
given by
2 M at. In particular, we expect that this is always
r
f matter(p1; p2) = (p1p2) 2 C(@ ; (p1; p2); M4)
(p1p2) r2 C(@ ; (p1 1; p2 1); M4) :(A.22)
However, we will not pursue this direction further here. To illustrate our prescription, below
we will derive expressions for the (chiral multiplet contribution to the) supersymmetric
index in the class of homogeneous hypersurface singularities, in the sector without at
connection.
the theory on R
not invariant under
As before, to evaluate the bosonic single letter partition function, we can start from
S3, and evaluate the sums as in (A.15) by projecting out the modes
SU(2). This is equivalent to counting holomorphic functions on
C2 that are invariant under . For
= Zp this is of course the case of the Lens space,
yielding (A.20). Let us then discuss the remaining D and E singularities. Implementing
the projection on the modes, we nd
fbDoEs (x) = x
r
1
x2d
(1
x2w1 )(1
x2w2 )(1
x2w3 )
Changing variable setting x = q1=2, we indeed nd that
where the weights and degrees of the singularities can be read o from the de ning equations
given in (2.30). For example, for the E8 singularity, corresponding to the Poincare Hopf
surface, the (minimal) set of weights is (w1; w2; w3) = (10; 6; 15), with degree d = 30. For
the Dp+1 series the weights are (w1; w2; w3) = (2; p
1; p) and the degree is d = 2p. Notice
that in all cases the series expansion of (A.23) does not contain odd powers of x. This is
because for
E ,
Z2, where this acts as Z2 : (z1; z2) !
(z1; z2).
fbDoEs (q) = qr=2
1
q
d
(1
qw3 )
= qr=2C(@; q; C2= ) :
we compute
w1 + w2 + w3
d = 1 ;
ffDerE (q) = fbDoEs (q 1) = q(2 r)=2C(@; q; C2= ) :
Thus the single particle index for the chiral multiplet reads
fDmEatter(q) =
qw1 )(1
qw2 )(1
and taking the plethystic exponential it results in the following triple in nite products
IDE
matter(q) =
Q1
Q1
n1;n2;n3 0 1
n1;n2;n3 0 1
q1 r=2qn1w1+n2w2+n3w3
1
qr=2+dqn1w1+n2w2+n3w3
qr=2qn1w1+n2w2+n3w3
1
q1 r=2+dqn1w1+n2w2+n3w3
(A.23)
(A.24)
(A.25)
(A.26)
(A.27)
Notice that this cannot be expressed in term of the ordinary elliptic gamma functions.
However, interestingly, using the condition (A.25), valid for the D and E singularities, we
nd that this can be written as
where
IDE
n1;n2;n3 0
(qr=2+d; qw1 ; qw2 ; qw3 )
(qr=2; qw1 ; qw2 ; qw3 )
(z; q1; q2; q3) =
1
zq1n1 q2n2 q3n3 )
(A.29)
is a generalization of the elliptic gamma function [26, 27].
Open Access.
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