Witten indices of abelian M5 brane on $\mathbb{R}\times S^5$

Witten indices and partition functions are computed for abelian 6d tensor and hypermultiplets on $\mathbb{R}\times S^5$ in Lorentzian signature in an R gauge field background which preserves some supersymmetry. We consider a generic supersymmetric squashing that also admits squashing of the Hopf fiber. Wick rotation to Euclidean M5 brane amounts to Wick rotation of squashing parameters and the hypermultiplet mass parameter. We compute Casimir energies for tensor and hypermultiplets separately for general squashing, and match these with the corresponding gravitational anomaly polynomials. We extract Witten indices on $\mathbb{R}\times \mathbb{CP}^2$ and find that this is zero, again matching with the vanishing anomaly polynomial on an odd dimensional space.


Introduction
It has been conjectured that 6d (2, 0) theory [4] compactified on a circle, is equivalent to the dimensionally reduced 5d SYM theory [9,8]. Confirming checks of this conjecture have been done for the abelian case in [10,15,20]. We understand the non-abelian 6d theory in the large N limit by the AdS supergravity dual, so we can test the duality in the large N limit. In [19] there was such an attempt. This paper took the large N limit of the partition function of a round S 5 that was obtained in [11,13] for 5d SYM on round S 5 [12], and extracted the free energy. The partition function of squashed S 5 was obtained in [17,18]. The corresponding 6d object is the superconformal index [7,14,16,18,20,26]. For the round sphere case, an agreement with the corresponding supergravity computation in Euclidean AdS 7 , could be found if the hypermultiplet mass parameter in the 5d SYM theory was Wick rotated as we go from Lorentzian M5 to Euclidean M5 theory [19]. But no justification for this Wick rotation from the 6d perspective was given. This result was our original motivation to better understand the 6d superconformal index in both Lorentzian and Euclidean signatures, as a continuation of [22].

Radial quantization
The abelian M5 brane superconformal index was first obtained in [7], and has been generalized in [25] 1 . The computation was done in Euclidean signature using radial quantization by summing up the contributions from all BPS letters. The bosonic part of the superconformal group SO(1, 7) ⊃ SO(1, 1) × SO (6) has Cartan generators ∆ of SO(1, 1) which is the scaling dimension, and three Cartans j i of the rotation group SO(6) of S 5 . We have two Cartans R 1 and R 2 of SO(5) R symmetry. The BPS equation is ∆ − j 1 − j 2 − j 3 + 2 (R 1 + R 2 ) = 0 (1.1) We use the following index notation Q R 1 R 2 j 1 j 2 j 3 for spinors such that j i = ± refers to j i = ± 1 2 . The bosonic fields are a selfdual two-form gauge potential B M N and five scalar fields φ A . The fermionic fields are four real chiral fermions. It turns out that the selfdual field strength H M N P does not saturate the BPS bound.
We define the singlet single-particle index as I singlet L (β, m, a i ) := tr(−1) F e −β(∆+ 1−3a with a := (a 1 + a 2 + a 3 )/3. The BPS letters associated to the singlet supercharge Q −− −−− are summarized in the Table 1 below, We can apply any number of derivatives on any field 1 We would like to thank S. Benvenuti for bringing this reference to our attention.
(which leads to a geometric sum), and in doing so we have to subtract the letter index corresponding to the fermionic equation of motion (since this particular combination of derivatives acting on the fermion is zero just by that fermionic equation of motion, so it shall not be counted as a BPS state). We get the single particle index (1 − e −β(1+a) ) 3 and this simplifies further at m = ±( 1 2 + a 2 ), where we get We define the triplet single-particle index as I triplet,I L (β, m, a i ) := tr(−1) F e −β(∆+ 1+a with a := a 1 −a 2 +a 3 . This preserves the triplet supercharge Q −− +−+ and the corresponding BPS table is presented in Table 2. From this table, we read off the full triplet letter index Table 2: Triplet case, with a = a 1 − a 2 + a 3 letter ∆ (j 1 , j 2 , j 3 ) (R 1 , R 2 ) (−1) F e −β(∆+ 1 Translation along the Hopf fiber of S 5 is generated by j 1 + j 2 + j 3 and corresponds to taking a 1 = a 2 = a 3 = a. We may also consider another Hopf fibration of S 5 . We may assign the generator j 1 − j 2 − j 3 as the generator for translations along this new Hopf fiber. This new generator is obtained from the old generator by flipping the signs of j 2 and j 3 . After this sign flip we get the supercharge Q −− ++− . To preserve it, we also need to modify the definition for a to read a := a 1 + a 2 − a 3 . The index with respect to the new Hopf fibration reads There is a third Hopf fibration which corresponds to taking its generator as −j 1 −j 2 +j 3 and which preserves the supercharge Q −− −++ . For this case we need to define a := −a 1 + a 2 + a 3 . The index with respect to this third Hopf fibration is given by and it will be of the same form as I triplet,II (β, m, a 1 , a 2 , a 3 ). Let us now evaluate these three indices for squashing along their respective Hopf fibers, at a 1 = a 2 = a 3 = a. We find We also notice the following relation between these indices, I singlet (β, m, a, a, a) = I triplet,I (β, m, a, a, a) I singlet (β, m, −a, −a, a) = I triplet,II (β, m, a, a, a) We have two points at m = ±( 1 2 + a 2 ) where we have the simplification By a direct computation, we will reproduce these indices. However, there are a couple of surprises.
First, we obtain these indices with real-valued chemical potentials m and a i only in Lorentzian signature where β = iT with T a real time interval.
Second, when we compute the indices with j := j 1 + j 2 + j 3 , and H = ∆ + 1 2 (R 1 + R 2 ) + m (R 1 − R 2 ) (to be specified below), then we should expect to get results corresponding to I singlet and I triplet,I above. This is because we pick the same generator j = j 1 + j 2 + j 3 along the same Hopf fiber for the computation of both these indices. What explicit computations shows, is that while we do reproduce I singlet , instead of getting I triplet,I , we get I triplet,II that from the viewpoint of radial quantization as presented above appears to correspond to a different Hopf fiber! This result is hard to understand from radial quantization alone. It shows limitations of radial quantization when applied to the M5 brane. We attribute these limitations of radial quantization to the fact that for the M5 brane there are no real-valued fields and no real-valued Lagrangian in Euclidean signature and this motivates us to do a direct computation in Lorentzian signature.

Wick rotation
In Lorentzian signature we have a typical chemical potential of the form e −iT a 1 . Wick rotation is done by taking β real and positive. Let us assume that T is also positive. Then T = β where left-hand side is the quantity in Lorentzian signature and the right-hand side is the same quantity in Euclidean signature. Then we have e −βa E 1 = e −T a E 1 in Euclidean signature. We want this to be unchanged by Wick rotation. This amounts to taking a E and I triplet,II is the complex conjugate of this. Although E M 5 is complex in Euclidean signature, it is real in Lorentzian signature and is given by equation (5.1). We thus see that we have got a familiar modular form on a two-torus. This two torus is spanned by the time axis and the Hopf circle, and a is the translation along the Hopf circle that together with the Euclidean time axis makes up a slanted Euclidean torus. This modular form for the special case a = 0 (that is, a torus with no slanting) was first obtained in [14] for the singlet case. What is new here is that we generalize this to a slanted two-torus and also include the triplet case.
Let us also note that the index for 6d (0, 2) theory is the complex conjugate of the index for the 6d (2, 0) theory; changing sign of a corresponds to changing 6d chirality, as we will see in more detail later on.

Hamiltonian computation
In this paper we will not make use of the superconformal symmetry 2 . That is why we refer to the corresponding quantity as the Witten index, which we define as where tr is over single-particle states, and where we have the M5 brane Hamiltonian on S 5 that is evolving in Lorentzian time by an interval T > 0. The space S 5 is compact and the spectra of the Laplace operators and of the Dirac operator on S 5 are discrete. The spectrum of the Hamiltonian is therefore also discrete and has a mass gap. From the commutator of two supercharges one may deduce a BPS equation. Since spectrum is discrete and we have supersymmetric pairing of non-BPS states, the Witten index picks up contributions only from those BPS states.
If we consider round S 5 , then in our R-gauge field background (2.1) we find that our Hamiltonian can be expressed in the form where H 0 is the nonsupersymmetric Hamiltonian we compute with R gauge field turned off (and which would correspond to ∆ in radial quantization). Here by R 12 and R 34 we denote the two Cartan generators of SO(5) which is the R-symmetry of the M5 brane (and which we denoted as R 1 and R 2 above). However, a generic mass parameter m breaks this symmetry down to SU (2) R × SU (2) F where SU (2) R is the R symmetry and SU (2) F is another global symmetry. The 8 real supercharges (N = (1, 0) supersymmetry) that we preserve do not commute with H 0 . The purpose with turning on some R gauge field background is to preserve some supersymmetry. Thus supercharges that we preserve commute with H, and this is true for any value of the mass parameter m, so in particular they commute with R 12 − R 34 . The points at ±m = 1 2 have enhanced supersymmetry with 16 real supercharges. On a squashed S 5 where the squashing is along the Hopf fiber, it is plausible that this critical value gets changed to the value ±m = 1 2 + a 2 where we saw the simplification of the index was happening above. But to show this we would need to derive the Killing spinor solution on this squashed S 5 .
We can consider a general squashing on S 5 by turning on three chemical potentials a i = (a, b, c) for the three Cartan generators of the isometry group SO(6), without imposing any restrictions on these chemical potentials other than they shall be real-valued. If we use a parametrization z i = r i e iφ i of S 5 with r i lying on S 2 , say r i = r (sin θ sin φ, sin θ cos φ, cos θ), we get the induced metric on S 5 . In Lorentzian signature squashing by turning on these chemical potentials amounts to changing the metric on R × S 5 into [18] ds 2 = −dt 2 + dr 2 1 + r 2 1 (dφ 1 + adt) 2 + dr 2 2 + r 2 2 (dφ 2 + bdt) 2 + dr 2 3 + r 2 3 (dφ 3 + cdt) 2 Wick rotation of t alone would lead to a complex metric, so we propose that we should Wick rotate the a i 's as well so that the Euclidean metric becomes ds 2 = dt 2 + dr 2 1 + r 2 1 (dφ 1 + adt) 2 + dr 2 2 + r 2 2 (dφ 2 + bdt) 2 + dr 2 3 + r 2 3 (dφ 3 + cdt) 2 We have not been able to rigorously compute the Witten index with generic squashing parameters a i since the representation theory of SO(6) is complicated. Instead we will compute the Witten index in Lorentzian signature with a = b = c by compensating this with a certain R rotation that also depends on the parameter a, such that the whole rotation generator including the R rotation, commutes with some amount of supersymmety. By combining this result with the result we obtained above using radial quantization for general squashing, it easy to make a guess the result with general squashing.

The M5 brane Lagrangian
The Abelian M5 brane on R × S 5 where r is the radius of S 5 , may be described by the following supersymmetric Lagrangian 3 We use 11d gamma matrices where M = (0, m) and m = 1, ..., 5 is vector index on S 5 and A = 1, ..., 5 is a vector index of the SO(5) R-symmetry. The Dirac conjugate isψ := ψ † Γ 0 . We define the covariant derivative as is the R-gauge field, M AB are SO(5) generators 4 , and we have the (2, 0) supersymmetry where m = 1, ..., 5 denote tangent space indices of S 5 . We break the supersymmetry down to (1, 0) by the Weyl projections The (2, 0) tensor multiplet separates into one (1, 0) tensor multiplet and one (1, 0) hypermultiplet In this Lagrangian we have included a decoupled wrong chirality part to the selfdual tensor gauge field as a spectator field. 4 We normalize these generators so that M AB CD = 2iδ AB CD in the vector representation. Then we have M AB = i 2 Γ AB in the spinor representation.
To go further, we need to pick an R gauge field and fix a gamma matrix convention. We pick the nonzero R gauge field components as and fix the gamma matrices in terms of Pauli sigma matrices (σ 1 , σ 2 , σ 3 ) as Finally we choose With this setup, we get We will now compute the contributions to the Witten index or the partition function for each field separately.

The tensor multiplet scalar field contribution
The Lagrangian for the tensor multiplet scalar field φ := φ 5 is given by where g mn denotes the metric tensor on S 5 with radius r. We expand the scalar field in spherical scalar harmonics Y n,m 1 ,m 2 ,m 3 , where we define ensuring that φ is a real-valued field. Here m i denote the j i charges that are the Cartans of SO(6) isometry group of S 5 . When we do the separation into x n,m 1 ,m 2 ,m 3 and (x n,m 1 ,m 2 ,m 3 ) * we must cut the sum over m = (m 1 , m 2 , m 3 ) by half to avoid double counting of modes. Effectively this means that the degeneracy b n = 1 12 (n + 1)(n + 2) 2 (n + 3) (for how to get this dimension, see (A.1) in the appendix) of Y n,m 1 ,m 2 ,m 3 is cut by half, to bn 2 . Now this number b n turns out to not always be an integer, for example b 4 = 105. Nevertheless it works, which we will explain in a moment.
By noting the eigenvalues of the scalar Laplacian 5 as given by eq (A.2) in the appendix, we find the Hamiltonian is composed of a zero mode part and an oscillator mode part Let us isolate one typical oscillator mode for which we have the Hamiltonian on the form For the zero mode part there is just one set of creation and annihilation operators and N denotes their corresponding number operator. We define the single particle index as where tr is a sum over all states with N n = 1 for one-particle excitations, and with ∆H = H(N n ) − H(0) is the Hamiltonian of the single-particle excitation minus the Hamiltonian of the vacuum. This in particular means that the normal ordering constant does not enter in the single-particle index. We then compute the full index by taking the plethystic exponent, and multiplying this by an exponent of the Casimir energy as a prefactor. This prefactor contains the contributions of the zero point energies.
The single particle index is given by where the factor bn 2 has argued for above. Here we also see explicitly that real modes can be viewed as 1/2 of a complex mode. Namely the real zero mode, naturally gets incorporated in complex oscillator mode sum as we can write the sum in the following neat form treating the zero mode and the complex oscillator modes and real oscillator modes on the same footing all entering in the same summation over n here. This is the explanation why we can divide b n by 2 even if b n is an odd integer. The reason is that when b n is odd, we have that b n,m=0 is odd where m = m 1 + m 2 + m 3 (for example b n = 105 and b n,0 = 27 and this pattern is easily seen to be general since b n = b n,0 + 2 m>0 b n,m . Thus if b n is odd, then so is b n,0 , for more details, see appendix A). But for those modes we can always find a rearrangement such that these modes are all real modes of degeneracy b n,0 (rather than complex modes of degeneracy b n,0 2 , which would be problematic if this degeneracy is not an integer number).

The tensor multiplet fermion contribution
The Lagrangian is We have the Majorana condition We can use that to eliminate ψ − and replace it as follows, if we assume that +− = 1. We then define a single component complex spinor In terms of this single complex spinor, the Lagrangian becomes We expand in spinor harmonics and get the Lagrangian for the modes as Here µ n denotes the eigenvalue of the Dirac operator acting on a spinor harmonic. This is given by [6,14] µ n = ± 1 r n + 5 2 each sign comes with the degeneracy f n = 1 6 (n + 1)(n + 2)(n + 3)(n + 4). For more details on the spinor harmonics, we refer to the appendix and eq (A.1) there. We define the conjugate momentum to each mode as the left derivative The Hamiltonian is If we were to decompose this as then we would read off However, we would not be sure how to understand the zero point contribution to the R-charges using such an approach. The contribution to the single particle index is where the exponents are corresponding to µ n + 1 2r . Again we are not including the zero point energies in the single-particle index. The negative energy modes −n − 2 are treated in the standard fashion by filling up the Dirac sea and then they become positive energy modes for the antiparticles.
Using that f −1 = 0, we may write the sum in a more neat form as

The selfdual tensor field contribution
The simplest way to see how to quantize the oscillator modes for the tensor field is from the path integral. After BRST quantization, we get the following partition function of a non-selfdual two-form gauge potential, We now note that the non-harmonic spectrum of p separates into exact and coexact parts. To see this, we use Hodge decomposition of a non-harmonic p-form, We then find that Thus we have Using this, we can write the partition function in the form Now we assume the six-manifold is on the form R × S 5 where time direction is along R.
Then we can decompose a coexact p-form as Namely, had Y p−1 been exact (or harmonic), then also Y p−1 ∧ dt would be exact (or harmonic) as well. Thus we have Using this, we find that many factors cancel and we get This is of the form that we would get from a Hamiltonian that is given by B coex 2 B where B is a coexact two-form. We can expand coexact two-forms in a basis of of twoform spherical harmonics that form a reducible representation of SO (6). This reducible representation separates into one chiral and one antichiral representation. To get the contribution from the chiral two-form we expand in modes from the chiral representation only. These come with the degeneracies b + n = 1 4 n(n + 1)(n + 3)(n + 4) as given in (A.1), and with corresponding eigenvalues n(n + 4) + 4 = (n + 2) 2 . Then this will contribute with the term to the Witten index.

The tensor multiplet Witten index
The Witten index is the sum of the individual contributions weighted with a minus sign for the fermions, On the other hand, for the particle partition function there is no minus sign,

The hypermultiplet scalars contribution
We group the four hypermultiplet scalars in two pairs. We denote the scalars in one pair as φ a where a = 1, 2. We have their Lagrangian as We will refer to the parameter m as the hypermultiplet mass. We define one complex scalar and the Lagrangian is Conjugate momenta are defined as We expand the complex scalar field in scalar harmonics with complex modes z n , The scalar harmonics have the properties where n represents the multi-index (n, m, m , m ) that labels the scalar harmonics. We define L = S 5 L and get This type of Lagrangian has been quantized in [3]. Conjugate momenta are q n = ∂L ∂ż n =ż n + iµz n q n = ∂L ∂ż n =ż n − iµz n and by consistent matching of the two ways of computing the conjugate momenta, we get and we define We define oscillators as and get We quantize by choosing some ordering prescription and by imposing the canonical commutation relations when acting on the state where we have chosen the Weyl ordering for the number operators in the Hamiltonian. More explicitly, the contribution from φ a for a = 1, 2 to the Hamiltonian is given by We note that the zero point energy is independent of the chemical potentials when we use the Weyl ordering prescription. By also including the contribution from the remaining two scalar field φ 3 and φ 4 in the hypermultiplet for which we shall flip the sign of m, we find that the total Hamiltonian can be expressed in the following form

The hypermultiplet fermion contribution
The Lagrangian is where the natural index position is ψ s for the hyper fermions, where s = 1, 2. That means that (σ 3 ) s t is the third Pauli matrix, and +− = 1 is the charge conjugation matrix on the space of s, t, ...-indices. The Majorana condition is We use this to eliminate ψ α− and express the Lagrangian in terms of a single complex valued Dirac spinor ψ α := ψ α+ as We now expand the spinor field in a basis of commuting complex spinor harmonics Inserting this expansion into the Lagrangian, we get The conjugate momentum is defined as As before for the tensor multiplet fermion, also for the hypermultiplet fermion, on S 5 , we have the spectrum µ n = ± n + 5 2 where the plus sign comes with the degeneracy f n = 1 6 (n+1)(n+2)(n+3)(n+4) and the minus sign also comes with the same degeneracy f n .

The hypermultiplet Witten index
For the bosonic part we sum over all the states ⊗ n N n,α , N n,β , N n,α , N n,β , subject to the single-particle constraint that the sum of all number operators is one, We notice that the last two factors are given by We define the bosonic contribution to the single particle index as I bosons = b n e −iE(Nn,α,N n,β , Nn,α, N n,β )T where the sum runs over all single particle states, whose corresponding energy levels are given by respectively.
Since the hyper fermions have R 12 = −R 34 , we find a dependence on the hypermultiplet mass. We may express the Hamiltonian as where the factor of 2 comes from R 12 − R 34 = 2R 12 . However, by simply looking at our explicit expression for the Hamiltonian H in eq (3.4), we deduce that the fermionic contribution to the single particle index is Here E − n comes from the negative eigenvalue −n − 5 2 of the Dirac operator on a unit five-sphere, in which case we shall swap the interpretation of the vacuum which brings in an overall sign change of E − n that has the effect of changing the sign of m. The index is given by The partition function is given by Explicitly we find that and

Squashing the Hopf fiber
Following [16], we now compute the refined indices where j = j 1 + j 2 + j 3 is the U (1) Hopf generator that translates along the Hopf fiber of S 5 . For notational simplicity, we put α = −βa. Inserting this chemical potential amounts to a squashing along the Hopf fiber that leads to a reduction of isometry group as SO(6) → SU (3) × U (1) Hopf . As we show in appendix C the supercharges that are preserved by these indices carry charge j either 3 2 or j = − 1 2 . These correspond to SO(6) isometry spin labels (− − −) and {(+ + −), (+ − +), (− + +)} of singlet and triplet supercharges respectively. Since the supercharges carry charge − 1 2 under R 12 , we see that these indices respect the singlet and triplet supersymmetries respectively.
Upon reducing along time direction, this will amount to a field theory living on a squashed five-sphere for all fields that are neutral under R 12 . So for example the tensor multiplet tensor field and the scalar field will live on a squashed five sphere. For the fermion, which is charged under R 12 there will be an additional mass term, but otherwise this will again live on the same squashed five sphere. Thus the fermion mass is correlated with the squashing parameter of the geometry. Here we chose to insert R 12 instead of the symmetric combination 1 2 (R 12 + R 34 ). However, since R 12 = 1 2 (R 12 + R 34 ) + 1 2 (R 12 − R 34 ), we see that the indices above are related to the symmetric case by a shift of our mass parameter m; for the triplet case we shift m into m − a 2 , and for the singlet case we shift m into m + 3a 2 to go to the symmetric case. We label SU (3) representations by their Dynkin labels (p, q). Using Weyl's dimension formula, we have the dimension of such a representation as dim(p, q) = 1 2 (p + 1)(q + 1)(p + q + 2) We then refine the degeneracies by inserting a chemical potential α for the U (1) Hopf charge, for the boson harmonics and for the fermion harmonics f + n (α) = n p=0 dim(p, n − p)e α(2p−n− 3 2 ) + dim(p, n − p + 1)e α(2p−n+ 1 2 ) We also have which does not enter our problem. Let us anyway notice that with the obvious definition for b − n (α). We will give the physics interpretation of these relations at the end of the next subsection.

Tensor
For the tensor multiplet, the bosonic fields carry no R charges. However, the fermionic fields do carry some R charge. For the tensor multiplet we have that R 12 − R 34 = 0 since there is no dependence on the hypermass m. We can also see this from the Weyl projection which means that Γ 12 ψ = Γ 34 ψ for the vector fermion. However, for the hyper fermion we have Γ 12 ψ = − Γ 34 ψ so this will pick up dependence on m when we compute the usual standard index. From R 12 = R 34 = 1 2 N n − 1 2 we conclude that for a single particle excitation N n = 1 for some n, we have R 12 = R 34 = 1 2 . (We use the short-hand notation R 12 for the single-particle contribution R 12 (N n = 1) − R 12 (N n = 0)). Thus we shall shift j by the amount j − 3R 12 = j − 3 2 for the fermions. But we also should keep in mind that we swap the sign of the number operator itself in the second term. Thus we have the refined indices

Hyper
For the hyper we have R 12 = −R 34 and from the form of the Hamiltonian for the hyper fermion we deduce that for a one-fermi particle excitation we have R 12 = − 1 2 . But now also the scalar fields carry R-charges.
The index is given by I = I B − I F where for the singlet case One may note that by shifting m as we described above, we can get to the symmetric case. By adding the tensor and hyper contributions, it is then easy to see that our indices agree with the indices I singlet L (β, a) and I triplet,II L (β, a) that we obtained in the Introduction from radial quantization.
We note that the result we get using Hamiltonian quantization is a result from a straightforward computation, up to one subtle point. Namely the choice of chiralities for the spinor harmonics and for the tensor harmonics. We get different answers depending on how we choose these chiralities. We pick one chirality of the spinor. Then we adjust the chirality of the tensor gauge field harmonics so that we get an index like quantity. If we had chosen the opposite chirality two-form harmonics, we would not have seen the huge cancelation between bosons and fermions. There is now only one choice left, and this is the choice of chirality of the spinor harmonics. If we pick the other chirality, this just amounts to flipping the sign of the chemical potential α. So in conclusion, our computation gives a non-ambiguous result up to a sign flip of the chemical potential α. The only difference between the singlet and the triplet case is the change from 3α to −α when relating the expressions in (4.1) and (4.2) for the tensor multiplet (and we have the same type of relation for the hypermultiplet case). This is a very simple change to make in a Mathematica file. By doing this simple change, we map the singlet index into the triplet index. Since we matched our singlet index with the result from radial quantization, we are confident about the correctness of our result for the triplet case as well, despite this does not match with the expected result from radial quantization. We get I triplet,II L instead of the expected result I triplet,I L . In Hamiltonian quantization we consider R × S 5 with Lorentzian time along R. To relate with radial quantization, we need to Wick rotate time. For the tensor multiplet, we make a change associated with the fermions which carry R-charges when relating singlet and triplet cases. Fermions are sensitive to the signature in the sense that they can be real only in Lorentzian signature. Since we need the Wick rotation that makes fermions complex, there is no direct way to relate Hamiltonian quantization with radial quantization.

The supersymmetric Casimir energy
The supersymmetric Casimir energy is defined as the sum of zero point energies 1 2 E n for the bosons plus the zero point point energies − 1 2 E n for the fermions. 7 However this sum 7 We are dealing with a free theory and hence we have harmonic oscillators. Here E n denote the frequencies of these harmonic oscillators. The corresponding zero point energies when using the Weyl ordering prescription, are then given by ± 1 2 E n for bosonic and fermionic harmonic oscillators respectively.
is divergent. We regulate can regulate the sum in a supersymmetric way as

Our Hamiltonian is given by
In [23] it was argued that the supersymmetric Casimir energy is unambiguous. However, as was noted in [20], here it appears to ambiguous since we could pick another regulator. For example, it appears we could use a different mass parameter m in the regulator Hamiltonian. If we pick m = m, then after subtracting divergent terms and taking to zero, we obtain the following supersymmetric Casimir energies for the tensor multiplet and the hypermultiplet We then notice that at m = 1 2 we get E hyper = 1 240 and so we get in total E tensor + E hyper = − 1 24 . The full index then takes the form of the inverse of the Dedekind eta function. This was first observed in [14]. Here we see that this result is valid for the Lorentzian M5 brane. At m = 1 2 we should take m = m in order to keep all supersymmetry. If we pick m different from m at that point, then we break half the supersymmetry. We will pick m = m also for a generic values of m. This choice of m can be justified by matching the resulting full superconformal index (including the Casimir energy factor) with the 5d partition function [18].

Squashed five-sphere
We consider the singlet case indices with generic squashing parameters, Instead of Weyl ordering one could use some other prescription. However, the total ground state energy can not change due to supersymmetry which implies a lower bound on the ground state energy. We therefore believe that any other ordering prescription will give the same result for any physical quantity we compute in a supersymmetric theory. and where a = (a 1 + a 2 + a 3 )/3. At a = 0 we reproduce the total Casimir energy that was obtained in [18] with δ := 1 4 − m 2 . This is the sum of a contribution from the tensor, and from the hyper, For a generic value of a, we find which is the sum of a contribution from the tensor, E tensor = N tensor (a 1 , a 2 , a 3 ) 720(1 + a 1 )(1 + a 2 )(1 + a 3 ) and from the hyper, Now let us take a 1 = a 2 = a 3 = a. Then one finds which is the sum of a contribution from the tensor, and from the hyper, Finally we further take m = 1 2 (1 + a). Then which is the sum of a contribution from the tensor, and from the hyper,

Anomaly polynomials and Casimir energies
The anomaly polynomial for a single M5 brane was first obtained in [4]. Here we also like to obtain the separation of this anomaly polynomial into its tensor multiplet and hypermultiplet contributions. This was recently obtained in [21]. Let us here summarize the result. Anomaly polynomials are conventionally denoted by I. We hope this does not cause any confusion with our indices that we also denote by I. The M5 brane embedded in 11 dimensions has a normal bundle N with structure group SO(5), and a tangent bundle T with structure group SO(6) if we assume Euclidean signature of the M5 brane. The anomaly polynomial of a real Dirac fermion is and ch(S(N )) = 4 + Multiplying the factors together and picking out the 8-form, we get Adding the anomaly of the selfdual tensor gauge field we get the M5 brane anomaly polynomial If we denote the Chern roots of the tangent bundle as ω i , then the first and second Pontryagin classes are [1] Similarly, if we let σ 1 and σ 2 be the Chern roots of the normal bundle, then We can now express the M5 brane anomaly polynomial in terms of these Chern roots as We will now extract the anomaly polynomial for the hyper and the tensor multiplets separately, following [21]. The anomaly polynomial of a Dirac fermion is given by The structure group SO(5) of the normal bundle N is reduced to SO(4) = SU (2) L × SU (2) R when we reduce from (2, 0) to (1, 0) supersymmetry. For the normal bundles L and R with structure groups SU (2) L and SU (2) R we have the relations where e(N ) = σ 1 σ 2 is the Euler class of N . We then separate the contribution from a Dirac fermion into its normal bundle Weyl components as To see that I D = I hyper,F + I tensor,F , we notice that To the tensor multiplet anomaly we also have an additional contribution coming from the tensor gauge field, which is given by

Dictionary
If we use the BPS equation (1.1) to eliminate ∆, then the Witten index, for the singlet case, can be expressed as [24] where we define The dictionary of [24] amounts to replacing the Chern roots of the normal bundle in the anomaly polynomial with σ 1 and σ 2 , and the Chern roots of the tangent bundle with ω i .
If one does this in the anomaly polynomial, and then divides it with the tangent bundle Euler class, which after this substitution becomes e(T ) = ω 1 ω 2 ω 3 , then it turns out that the result agrees with the Casimir energy, up to a minus sign. That one shall divide with the Euler class was motivated in [24] as a result of having applied the Berline-Vergne fixed point formula 8 , where p labels all fixed points, and the evaluation in the right-hand side is done by picking the zero-form component out of I and e(T ) at the fixed point p. Note that we integrate over the M5 brane world volume R 6 , and before making I an equivariant characteristic class, this was an 8-form. After making I equivariant, this contains all degree forms, and it is the 6-form piece that we are integrating over R 6 . For a simple derivation of this fixed point formula, see for instance section 2.6 in [5]. In our case the only fixed point is the origin which is the fixed point of the U (1) 3 action generated by the Cartan generators j i . Thus the idea is to promote the anomaly polynomial, which is a characteristic class that is d-closed, into an equivariant characteristic class that is no longer d-closed, but (equivariantly) d V -closed, where d V = d + ι V . Here ι V refers to the contraction associated with the vector field V = 3 i=1 ω i ∂ ϕ i where ϕ i denote the angular coordinates that are corresponding to the three Cartan rotation generators j i in R 6 . We shall then integrate this equivariantly closed version of the anomaly polynomial over R 6 using the fixed point formula. In this fixed point formula the equivariant Euler class corresponds to the Jacobian det(∂ M V N ) = ω 1 ω 2 ω 3 . That is, if we write V = ω 1 (x 1 ∂ 2 − x 2 ∂ 1 ) + ... (dots representing similar terms corresponding to ω 2 and ω 3 ), then ∂ 1 V 2 = ω 1 and so on. Now if we identify this Jacobian as an equivariant Euler class of the tangent bundle, it is natural to associate ω i with Chern roots of the tangent bundle. Since σ 1 and σ 2 enter the index in much the same way and these are associated with rotations in the normal bundle to the M5 brane, it gets natural to identify these with Chern roots of the normal bundle. We will now confirm that this dictionary works for all cases that we have checked.
First, this dictionary gives us In this formula we are ignorant about sign factors. One sign could come from relating the Pfaffian with the square root determinant that determines the Euler characteristic, another sign factor comes from an overall factor of (−2π) 3 = −(2π) 3 in 6d. Determining the sign factor is an interesting problem.
Let us begin with the case with a 1 = a 2 = a 3 = a. We then get the following anomaly polynomials, I hyper,F = 17(1 + a) 4 − 120m 2 (1 + a) 2 + 80m 4 1920 Summing the the tensor fermion and the tensor gauge field contributions, we get I tensor = I tensor,F + I tensor,A = 11(1 + a) 4 240 whereas for the hyper, we have just the contribution from the fermions, I hyper = I hyper,F . We now see that indeed these anomaly polynomials match with corresponding Casimir energies, after we divide by the equivariant Euler class e(T ) = (1 + a) 3 , Using Mathematica we have confirmed that these kind of relations hold for generic squashing parameters a 1 , a 2 , a 3 and hypermultiplet mass parameter m, where we allow for a generic value of a = 1 3 (a 1 + a 2 + a 3 ). In [24] this relation was shown to hold, but only for the sum I M 5 = I tensor + I hyper and only at the point a = 0. Here we did a generalization of their result, and found that the relation between anomaly polynomial and Casimir energy still holds. to pick out modes with U (1) Hopf charges which are integer multiples of N . We can do that by computing

Some projected indices
We note that since the chemical potential a couples to the operator j 1 + j 2 + j 3 − 3R 12 rather than just j 1 + j 2 + j 3 , this means that the fields which are neutral under R 12 live on Lens space without any further modification. Fields which carry R charges will therefore also live on Lens space, but for those fields there will be additional terms in the Lagrangian which under dimensional reduction along time will become additional mass terms on Lens space. By using Mathematica, we get for the first few values on N the following results We can also compute the index at N = ∞, which amounts to projecting to zero charge, where the charge is what multiplies the chemical potential α. For the scalars, picking the zero charge sector out of b n (α), we get nonvanishing contributions only from even n = 2k where we get b 2k = dim(k, k) = (k + 1) 3 For the tensor gauge field the zero charge sector picks out For the spinor the zero charge sector is a bit more involved as the spinor is charged under R 12 . For the triplet case, we have the contribution e α(2p−n−2) dim(p, n − p) + e α(2p−n) dim(p, n − p + 1) Now we pick the zero charge sector of this. We immediately see that only even n can give zero charge, so we let n = 2k. Then the zero charge sector gives the contribution For the singlet case, we have the contribution and picking the zero charge contribution from this, for n = 2k, we get Then for the triplet case and for the singlet case We then get The power 2 in the exponent reflects that we compute an index on one dimension lower, on R × CP 2 , and yet we keep the same amount of supersymmetry. The dimension in the exponent becomes clear when we compute the partition function instead. We get It is now clear that the power 4 instead of 5 now reflects the dimension of CP 2 as opposed to S 5 . For the hyper, let us again project on zero U (1) Hopf charge for the singlet case. We find that the contributions from bosons and the fermions perfectly agree with each other, and is given by 2 ) e βm so the single-particle Witten index is zero also for the hypermultiplet in the singlet case.

Outlook
It would be interesting to see whether the anomaly polynomial on Lens spaces also matches with corresponding Casimir energies on Lens spaces. Here we leave this as an open problem. However, we can immediately check one special case, namely R × (S 5 /Z ∞ ) = R × CP 2 . Here since we found that the single-particle Witten index for the singlet case is zero, the Casimir energy is also zero. This is consistent with the fact that the anomaly polynomial is zero on odd dimensional spaces, and on R × CP 2 in particular.

A Representation theory
In this appendix we use notations and conventions from the Lie algebra book [2].

A.3.1 Scalar harmonics
Scalar harmonics are functions on S 5 of the form where C i 1 ···in are symmetric and traceless. Symmetric is obvious since it contracts x i 's that are commuting coordinates in R 6 . Traceless is because of the constraint x i x i = r 2 that we have on the surface of S 5 . These functions form the irreducible representation (0, n, 0) of SU (4). Under SU (4) → SU (3), we have the branching rule as can be easily seen by expanding out the symmetric traceless tensor in a complex basis replacing six real coordinates x i with three complex coordinates z a and their complex conjugates z a . The U (1) Hopf charges for these SU (3) representations are then given by 2p − n. We will introduce the notation b n = dim R (0,n,0) We will also need the refinement b n (α) = n p=0 dim(p, n − p)e α(2p−n) We will also need to introduce the degeneracy at a fixed given U (1) Hopf charge m = 2p−n, which is given by [14] b n,m = dim R ( n+m Here m = −n, −n + 2, · · · , n − 2, n. The Laplace operator acting on the scalar harmonics has the eigenvalue as Up to an overall constant, this eigenvalue is equal to the value of the Casimir operator in the SU (4) representation (0, n, 0) for any n = 0, 1, 2, ....

A.4 Consistency checks
As a consistency check of our charge assignments, let us consider We decompose m = (µ, y) where µ is on base and y is fiber coordinate, and write γ µ = γ µ + rV µ γ γ y = rγ where tilde is used for tensors on the base, V µ is the graviphoton for the fibration.
That is, we get a covariantly constant and electrically charged Killing spinor on CP 2 with electric charge e = 3 2 . If we study F instead, then we get ∂ y F = −i(s 1 + s 2 − 2s 1 s 2 )F and we consider now the quantity which vanishes for s 1 = s 2 = − 1 2 . We then get The 6d solution is now Let us now write these in terms of R . First we havē =¯ R g and then Now we notice the invariance relation and we get This motivates us to define new rotated fields In terms of these new fields, we get and we have the relation We use the invariance of gamma matrices and we get Thus we see that the only change is to replace derivatives by gauge covariant ones.