Supersymmetric Localization on dS: Sum over topologies

We find the exact quantum gravity partition function on the static patch of 3d de Sitter spacetime. We have worked in the Chern Simons Formulation of 3d Gravity. To obtain a non-perturbative result, we supersymmetrized the Chern Simons action and used the technique of supersymmetric localization. We have obtained an exact non-perturbative result for the spin-2 gravity case. We comment on the divergences present in the theory. We also comment on higher spin gravity theories and analyse the nature of divergences present in such theories.


Introduction
Quantum theory of gravity in 3 space-time dimensions does not cease to surprise us, owing to the richness of physical and mathematical structures that are being continually revealed for more than 3 decades starting from [1]. It is interesting that, gravity in 3 dimensions is devoid of local degrees of freedom. One of the main causes of non-triviality in 3D gravity is the BTZ black hole solution [2] for negative cosmological constant. The most interesting sector of solutions for the case of negative cosmological constant is asymptotically AdS. A huge body of work has stemmed from the seminal work of Brown and Henneaux [3], which showed that the asymptotic symmetries of asymptotically AdS space-time form two copies of Virasoro algebra; thereby hinting to a plausible conformal field theory (CFT) at the two dimensional asymptotic boundary. As an example of low dimensional holography, this generated a great deal of physical and mathematical curiosities; motivated just from the question of calculating partition function for quantum gravity and arriving at black hole entropy from it. Interested readers may refer Refs. [4] - [5] in recent times.
Analogous progress in the case for zero cosmological constant is being pursued recently, specially in the works of Refs. [6] - [7] . In this sector, one attempts at quantum gravity for asymptotically flat space-time, now equipped with the BMS 3 algebra. [8] contains a relatively extensive discussion of quantum gravity in 3 dimensions from the perspective of asymptotic symmetries for asymptotically non-AdS space-time, even including higher spin degrees of freedom.
Whereas these aspects of quantum gravity are under focus of intensive studies in recent times, one might be curious for the case of positive cosmological constant. Vacuum solution to the corresponding Einstein equation is the dS 3 space-time. However unlike Minkowski spacetime, here exists a horizon at thermal equilibrium. As argued in [9], correlation function of any quantum degree of freedom with respect to a time-like observer is a thermal correlator. The corresponding vacuum state, as discussed in [10] and named as the Hartle Hawking state, is the Euclidean partition function.
The choice of Hartle Hawking state as a candidate for vacuum state circumvents an otherwise conceptually difficult problem in the following manner. Standard wisdom says that isometries of a maximally symmetric space-time like de Sitter should fix the vacuum state. But if one wishes to incorporate effects from quantum gravity, one has to incorporate all possible quantum fluctuations on the de Sitter background, from a perturbative viewpoint. Hartle Hawking is however defined as the Euclidean path integral considering all possible geometries with some fixed boundary data. Now in de Sitter space, a time-like observer is in causal contact with what is known as the static patch, defined in Euclidean time as: Euclideanizing is done by setting t = −iτ and it makes the static patch geometry identical to that of It would therefore be natural to consider fluctuations over round S 3 background geometry to construct the Hartle Hawking state. However, as nicely pointed out in [9], there is an infinite class of topologically distinct manifolds which allow smooth local geometry as Eq. (1.1). These are of the form S 3 /Γ, where Γ is a discrete subgroup of the isometry group of S 3 . In terms of the coordinates in (1.1), these quotient spaces with smooth local dS geometry are understood by the following identifications: Here q, p are coprime positive integers with p always being the greater of the pair. That this identification indeed results into the topological quotient space S 3 /Z p can be easily understood by first defining Then the Z p action on it is: Finally defining (z 1 , z 2 ) = cos r e iτ , sin r e iφ makes the identification (1.2) clear. The resultant manifold is named as a Lens space L(p, q), now equipped with the smooth geometry (1.1). All of these manifolds are therefore valid classical smooth saddles of Einstein equation.
Since S 3 as well as all the quotients L(p, q) are closed, Hartle Hawking state, considering all quantum gravity effects, would simply be given by: where S E is the Euclidean action for the theory of gravity. Interestingly as reported in [9], the functional integral, when summed over all Lens space topologies diverges as a harmonic series in the integer p: , which cannot be regularized. This was computation was performed in a perturbative one-loop calculation in metric variables and cross-checked with results from a non-perturbative computation in first order formulation of gravity (Chern Simons (CS) theory) [11]. However, the divergence seems to be tamed, when including further degrees of freedom, like topological massive modes [12]. This was later established [13] using a twisted first order theory of gravity (again CS formulation) and a dimensionless parameter, which can be tuned to get rid of the divergent piece. Interestingly, using results from SU(N) topological invariants [14] in 3-manifolds one can repeat the calculations for higher spin cases. For this, one introduces a consistently truncated tower of higher spins over gravitational degrees of freedom, the sum over all Lens spaces become finite, for spins ≥ 4 [15].
One further motivation towards a definition of Hartle Hawking state in 3D quantum de Sitter comes from an analogous question in AdS 3 . Euclidean AdS has a topology of solid torus. The two dimensional toric boundary serves as the asymptotics. Using the fact that asymptotic symmetry in AdS 3 is given by 2D conformal algebra, one may come up with speculations [16] regarding a candidate 2D CFT at the boundary. An exact non-perturbative calculation for the bulk partition function (corresponding to fixed boundary modular parameter) can lead one a long way towards a definite answer regarding the dual field theory. A series of recent remarkable results in AdS, Refs. [17] - [18] have taken the approach of supersymmetrizing the gravity theory (CS formulation) and exploiting the elegant methods of supersymmetric localization [19]. Since the additional fermionic fields introduced for supersymmetrization were non-dynamical, it is believed that the non-perturbative result after localization would constrain completely the CFT dual to the original bosonic gravity theory. For further progress in localization in low-dimensional AdS space times, the interested reader is referred to Refs.
[20] - [21]. These references focus on the program of localization on non-compact manifolds.
In our present perspective we don't aim at the holographic point of view. But rather take cue from the above analysis as far as exact partition function is concerned. We use the first order CS formulation here, and supersymmetrize it write down the exact partition function. The main aim here was to investigate whether fermionic degrees freedom, even if non-dynamical, can bring in the analytic property of the partition function summed over Lens spaces.
The paper is organized as follows. In section 2, we introduce the CS formulation of 3d bosonic gravity. In subsection 2.1, we obtain the supersymmetric extension of bosonic CS theory. In subsection 3.1, we discuss the technique of supersymmetric localization of our theory. In subsection 3.2, we explicitly evaluate the partition function, obtained as a matrix model, for our case of spin-2 gravity. We also explicitly identify the divergent pieces in the partition function. In the following subsection 3.3 , we evaluate the same for higher spin cases and comment on the divergences observed. In section 4 , we comment on some future directions that may be explored. Section 5 carries a note on our definitions and conventions.
2 Chern Simons formulation for 3d gravity and its supersymmetrization 3D gravity is long known to be equivalent to a pure CS theory [1]. Let us first briefly take a detour through this equivalence, particularly for the case of positive cosmological constant in Euclidean setting. One can start off with a CS functional on a 3-manifold M out of a su(2) ⊕ su(2) valued 1-form (gauge field). Also the Lie algebra is equipped with an Ad invariant symmetric bilinear quadratic form Tr ≡ ·, · valued to be diag(k, k, k) and diag(−k, −k, −k) respectively on the first and the second su (2). The CS functional then can be written as difference of two su(2) CS functional, Tr now evaluating diag(1, 1, 1): One then sets A ± = ω ± e and k = 1 4G , keeping the positive cosmological constant = 1 and G the Newton's constant in 3 dimensions. e and ω respectively are the su(2) triad and connection respectively. It is easy to see that (2.1) is actually the action for first order gravity: If M , is closed (for example the manifolds we will be dealing with in this article, ie the static patch of Euclidean dS 3 ∼ S 3 or S 3 /Γ), ie ∂M = ∅, the variational principle holds for the action (2.1) without any concern for boundary terms. Equations of motion are flatness conditions of the CS connections, ie F ± = dA ± which translate into torsionless condition de I + IJK e J ∧ ω K = 0 and (2.3a) for gravity variables. Interestingly, the following actioñ with independent levels k ± also gives the same equations of motion (2.3) for gravity variables.
For sake of convenience we introduce a parameter γ such that, k ± = a(1/γ±1)

4G
and (2.4) gives back (2.1) at the limit γ → ∞ [13]. The equations of motion are independent of γ. This applies to the space solutions as well. On the other hand, other aspects of the dynamics of the theory, ie. canonical structures are parametrized by γ. For example, the pre-symplectic structure on the space of solutions (2.3):

Supersymmetrization
To evaluate the partition function given by Eqn. (1.5) exactly, we would use the recently developed supersymmetric Localization techniques of Pestun et al [19], adapted to our purpose. Towards this, we start by supersymmetrizing a CS gauge field A valued in some semi-simple Lie algebra. Later we will specialize to mainly su (2), the case of relevance to 3D gravity. We construct the 3d N = 2 vector multiplet, defined, as always as V = (A, σ, D, λ,λ).
The supersymmetric CS Theory action is written as Note that in the 3d N = 2 vector multiplet, the additional fields (σ, D, λ,λ) are not dynamical and give no kinetic terms contributions to the action.

Localization of the 3d Supersymmetric Chern Simons Theory on Lens Spaces
With the connection between 3d Euclidean gravity and the Supersymmetric CS Theory made explicit in Eqns. (2.1) and (2.2), we will now evaluate the partition function of the 3d Supersymmetric CS Theory via supersymmetric Localization techniques. Since, we are interested in evaluating gravity partition function on Lens Spaces, we would we would try localizing the CS Theory on Lens Spaces L(p, q).

Principle of Localization
Suppose we have a theory on a compact manifold M, defined by an action S[Φ] 1 , which has a Grassmann-odd symmetry 2 δ. Let us further assume that there exists an operator V which is invariant under the transformation δ 2 , i.e. δ 2 V = 0. Once we have established the existence of such a special V, let us now consider not the original partition function, but rather a perturbed one, viz.
Note that this function is independent of t as 3 This means that the original unperturbed partition function maybe evaluated by evaluating the perturbed partition function Z(t) for any value of t (that is dictated by convenience) and especially, for t → ∞. This is immediately useful. If the perturbing operator has a positive definite bosonic part, the integral localizes to a sub-space, often even a finite dimensional one, With this motivation, we will try and evaluate the partition function of Supersymmetric CS theory on L(p, q). Now, to have some supersymmetric actions on some curved 3-manifold, we need to find some background, off-shell supergravity theories that preserve some rigid supersymmetry. These theories can then be made to couple to some supersymmetric field theory. This is done via the stress tensor multiplet. For our specific case of 3d N = 2 theory, this supergravity theory was called the "new minimal supergravity" which has the following field content We define the (dualized) field strengths To ensure that we have rigid supersymmetry, we need to find Killing spinors (ζ,ζ) which satisfy the Killing spinor equations, given in terms of these fields, as In terms of these Killing spinors, the general Supersymmetric variations of the fields in the gauge multiplet for the 3d N = 2 theory are given by 4 We also recall that the 3d N = 2 super Yang-Mills (SYM) action on S 3 , given by 5,6 can also be written as The action given by Eqn. with M = L(p, q) and in the limit t → ∞ where the partition function localises to a finite dimensional integral and the evaluation is exact. The bosonic part of Eqn. (3.7), being expressed as the sum of squares, immediately gives us the BPS configurations. They are viz. , Here, solving the equations in (3.10), we face non trivialities due to difference in global topology of L(p, q) when compared to S 3 . It is evident that we need the classical saddles corresponding to (3.9) on L(p, q) on which the localized partition function will be supported. Non-triviality of this statement arises from the fact that the flat connections on a manifold are characterized by holonomies around noncontractible loops on the base manifold, modulo a homogeneous adjoint group action at the base point of the loop. These loops form the first fundamental group of the base manifold. Hence the moduli space of space of flat connections modulo gauge transformation is given by For the present case, L(p, q) is a free Z p quotient of the simply connected manifold S 3 . Therefore we have the first homotopy group as π 1 (L(p, q))= Z p . This implies that the CS saddles ie, the flat connections are labelled by g ∈ G, with g p = 1. If we take g to lie in the 5 Recall V = Aµ,λ, λ, σ, D is the field content of the 3d N = 2 theory. They are respectively a vector, two complex fermions, a scalar and an auxiliary scalar respectively. 6 Actually this is the action given not just on S 3 but also on quotient spaces of the kind S 3 /Zp. This is understood as such spaces are locally equivalent to 3-spheres and transformations generated by supercharges are local.
maximal torus (this can be always be done for simply connected lie groups by the Ad action), we have p m , m ∈ Λ/(pΛ) (3.12) where, Λ is the co-weight lattice of the group G and m is N dimensional vector, where N is the rank of group G. Note that Eqn. (3.12) would then imply that m j ∈ Z p . For example, for G=SU(N), we have We will takeσ 0 to lie in the Cartan sub-algebra h of the Lie Algebra g of the group G. Note that, the second equation of Eqn. (3.10) motivates why we can expand m in the same Cartan basis.
Classical Contribution : The classical (tree level) contribution to the action is obtained by plugging in the BPS configurations in S SCS . There will be two such contributions, one coming from the scalars, σ and D, which have been shown to be constant in Eqn.(3.14) and the flat gauge fields. The contribution from the scalars is The contribution from the flat gauge fields is The total classical contribution is then with q * is defined as q * q = 1 mod(p)

1-Loop Determinants :
We calculate the 1-Loop determinants from the quadratic fluctuations of the fields about their BPS configurations. Specifically, Plugging these values in Eqn.(3.7), we obtain the terms in the action proportional to t −1 as The integration over D can trivially be done and it just alters the overall normalization constant sitting in front. To deal with the Vector and Fermionic fields, we decompose the gauge field into a divergenceless part (X) and the rest as The integrals over φ and σ give determinants that cancel and we are left with a divergenceless Vector field and Fermionic fields. Next, we expand them in the Γ a of the Lie Algebra with the definition The action then becomes The 1-Loop Determinant is then, simply On Lens Spaces L(p, q), this result may be calculated as : where, α are the roots of G and q * is defined as q * q = 1 mod(p). For a detailed derivation of the result in Eqn.(3.24), we refer the reader to [22] 7 . We only draw our reader's attention to the fact that the above expression reduces to the 1-Loop determinant of the partition function evaluated on S 3 for the special case of p = 1 and q * = 0 7 Note, that supersymmetric CS theory admits Matter Multiplets (MM) too, in arbitrary representation Ri for the i-th multiplet, and indeed, in the literature, the full theory has been localized, but, for our purposes, we would not require any MM.
as it should as L(1, 0) = S 3 . Finally, we integrate over the BPS configurations, here, denoted by σ i 's and sum over the holonomies, identified by the components of the vector m. Using Weyl Integration formula, as always, the integral reduces from the vector space spanned by the entire Lie Algebra to a vector subspace spanned by just the Cartan Sub-Algebra (h). This, however, introduces a Vandermonde Determinant α>0 α(σ 0 ) 2 . This is exactly cancelled by the denominator in Eqn. (3.24). Also, to take into account the residual Weyl symmetry of the gauge group, we divide the final expression by the order of the Weyl group of the the gauge group. Explicitly, the expression for the partition function becomes We will evaluate the RHS of Eqn. (3.25) explicitly next.

Partition Function : Evaluation of the Matrix Model for spin-2 Gravity
As described in the section 2, the CS version of the spin-2 gravity we are interested in is based on the gauge group SU(2) × SU (2)  The partition function, for each saddle, identified by a value of p, receive contribution from two values of m ± . They are explicitly, For further details, we refer the reader to [9].
With the values of m ± 's in our hand, we can directly proceed to calculate the integral given in the RHS of Eqn. (3.25) explicitly As discussed, since the rank of SU(2) group is 1, the evaluation of the partition function reduces to the problem of solving a one dimensional integral, viz. : Fortunately, the integral given in Eqn. (3.27) is tractable. Since our chosen gauge group is a product group we have another flat connection, identified by m − . The corresponding CS level is denoted by k − and we obtain an equivalent expression for the second flat connection. Explicitly, As yet, the CS levels are arbitrary but we will choose a special parametrization, viz. , Here, γ is a tunable parameter, whose large limit, for e.g., reproduces k + + k − = 0 . However, we would focus on the small γ regime. The total contribution to the partition function is their product. Explicitly, Using the the values of m + and m − , the RHS of Eqn. (3.30) gives This is one of the most striking points in our analysis. Note that Eqn. (3.31) evaluates the contribution to the gravity partition function for a specific p and specific q. To calculate the total contribution of all the saddles, denoted, essentially by p, we have an overall "sum over geometries" . In short, the overall contribution to the gravity partition function Z gravity , we will have a sum over p and sum over q to accommodate the various contributions of all the saddles. In short, the gravity partition function will be obtained by : Z(σ 0 , m; p, q). (3.32) We observe an overall positive power of p multiplying the trigonometric terms. When summed over all topologies, ie. lens spaces, this p dependence might be a serious cause of divergence.
Interestingly, for the pure bosonic theory (for γ → ∞) [9] and (finite γ) [13] the overall p dependence was 1/p. Therefore the expectation that the supersymmetric theory should reproduce exactly the same result as the bosonic one (because fermions are non-dynamical), does not come out to be true. We will shortly come back to the detailed analytic structure of the sum and explore deeper in this aspect. We will express our result, after the sum over q's in terms of Kloosterman Sums S(x, y; p), which are tailor made for such sums. The Kloosterman sums are defined as In terms of these Kloosterman sums, the q sum in Eqn. (3.32) gives 8 : To carry out the summation over p, we expand the cosine and the exponential function in their respective Maclaurin series. We obtain an infinite series of Kloosterman Zeta Functions, defined as L(x, y; s) = ∞ p=1 p −2s S(x, y; p) (3.34) The Kloosterman Zeta functions are analytic for Re(s) > 1/2. Writing our result explicitly, in terms of these functions, will also help us isolate the divergent pieces in the gravity partition function, as explicitly those terms with Re(s) ≤ 1/2 . The final expression for Z gravity is then obtained as : Let us investigate the analytic structure of the partition function summed over all Lens spaces (3.35). From (3.34), ie the analyticity of the Kloosterman zeta function, it is easy to see a set of divergence is sourced from the terms for which m + 2n ≤ 2 in (3.35) and another set being originated from m ≤ 2 for n independent terms.

Higher Spin Case
Linearized higher spin fields can be coupled consistently to gravity in 3 dimensions with finite height of the higher spin tower, which is nicely captured by the Fronsdal action of symmetric traceless tensor fields. Interestingly, a first order version of the theory can also be formulated in terms of CS gauge fields. [23] gives an elaborate AdS counterpart of that exposition. At the level of corresponding Lie algebra for CS theory, going from AdS to dS background amounts to changing a sign in the structure constant. The CS theory that describes spin 3 fields, has a gauge group SU(3) × SU(3) [15]. We would like to calculate the exact partition function in this case so as to check whether supersymmetric version of the higher spins make the sum over topologies better in terms of convergence properties.
Let us now evaluate the partition function given by Eqn. ± , where, i running from 1 to 2, denotes the two components of m and ±, as before, denote the two gauge fields A ± corresponding to the two SU(3) groups of the gauge group G.
At this point, as in the case for spin-2 in Eq, (3.26), we will have to choose a pair of elements from the corresponding A 2 co-weight lattice. This choice is physically motivated by the fact that quantum fluctuations are considered over the background that describes dS geometry in terms of gravitons and zero excitations for higher spin degrees of freedom. The exact co-weight points are thus found by a principal embedding of su(2) in su (3). Thus the two components of m ± as m (i) With the values of m (i) ± 's in our hand, we can directly proceed to calculate the integral given in the RHS of Eqn. (3.25) explicitly.
The integral in Eqn. ( pγ (a(q+q * +2γ)−4(1+q)γ) × 824 terms involving exponentials which are expressed as Kloosterman Functions (3.39) Again, following similar arguments as before, the gravity partition function is given by a sum over the topologies, which classify the various saddles, and is obtained as Even without knowing the explicit structure of the terms in the right hand side of (3.39), just from the pre-factor p 2 we can conclude as in the spin-2 case that (3.40) will diverge because of terms appearing in the non-analytic domain of Kloosterman zeta function. We conclude by a comparative remark with the purely bosonic theory. In that case, the generic structure of the partition function for a tower of spin-N fields on L(p, q) is Z spin−N ∼ p −N +1 , which makes the sum over topologies more convergent for higher spins. In the supersymmetrized version however: Z spin−N ∼ p N −1 due to the presence of fermions we observe a completely reverse phenomenon as the divergence in the full partition function gets worse with higher spins.

Conclusions and future directions
To conclude, let us recall what we have achieved. We have calculated the exact quantum gravity partition function on the static patch of Euclidean de Sitter space-time. In trying to do so, we have argued that the quantum gravity path integral receives contributions from all the classical saddles, which we have obtained as the quotient spaces of S 3 by the abelian group, Z p . This, we have been identified with the Lens Space L(p, q) and we expect a formal sum over p and q, the parameters of the space to capture the contributions from the saddles.
To evaluate the quantum gravity partition function exactly, we have worked in the CS formulation of 3d Gravity. This has proved immediately helpful in calculating the exact partition function by supersymmetric localization technique. We have calculated the partition function for both spin-2 Gravity and higher spin cases. We observe that the Kloosterman Zeta Functions arise naturally in the result of the partition functions from where, we identify explicitly the divergent pieces. We also observe that our result, being exact, reproduces the known result in large k limit, apart from an overall factor. That contribution has been ascribed to the effect of introduction of non-dynamical fermionic degrees of freedom.
For the higher spin cases, we have propose a set of saddles which are points in the A 2 co-root lattice. With this prescription for m, we calculate the partition function and observe that the divergence is indeed worse. We have made some comments on the partition function.
As a future direction, we set aside the task of evaluating the quantum gravity partition function for the N = 2 supergravity theory, instead of the above purely Einstein gravity using the CS formulation. In that case, the fermions would be dynamical and we expect non-trivial contributions to the partition function, coming directly from the fermionic sector. It would also be interesting to see if the addition of dynamical fermions takes care of the divergences in the partition function, as one might expect from boson-fermion loop contribution cancellations.