The character of the supersymmetric Casimir energy

Journal of High Energy Physics, Aug 2016

We study the supersymmetric Casimir energy E susy of \( \mathcal{N}=1 \) field theories with an R-symmetry, defined on rigid supersymmetric backgrounds S 1 ×M 3, using a Hamiltonian formalism. These backgrounds admit an ambi-Hermitian geometry, and we show that the net contributions to E susy arise from certain twisted holomorphic modes on ℝ × M 3, with respect to both complex structures. The supersymmetric Casimir energy may then be identified as a limit of an index-character that counts these modes. In particular this explains a recent observation relating E susy on S 1 × S 3 to the anomaly polynomial. As further applications we compute E susy for certain secondary Hopf surfaces, and discuss how the index-character may also be used to compute generalized supersymmetric indices.

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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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Dario Martelli, James Sparks. The character of the supersymmetric Casimir energy, Journal of High Energy Physics, 2016, 117, DOI: 10.1007/JHEP08(2016)117