{'fraction_non_alphanumeric': 0.05293078706258969, 'fraction_numerical': 0.0162821503477205, 'mean_word_length': 4.035853251806559, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the complete analytic structure of the massive gravitino propagator in four-dimensional de Sitter space

With the help of the general theory of the Heun equation, this paper completes previous work by the authors and other groups on the explicit representation of the massive gravitino propagator in four-dimensional de Sitter space. As a result of our original contribution, all weight functions which multiply the geometric invariants in the gravitino propagator are expressed through Heun functions, and the resulting plots are displayed and discussed after resorting to a suitable truncation in the series expansion of the Heun function. It turns out that there exist two ranges of values of the independent variable in which the weight functions can be divided into dominating and sub-dominating family.


I. INTRODUCTION
The investigation of Green functions has always been at the heart of important developments in quantum field theory and quantum gravity [1]. On the other hand, in recent years, developments in cosmology and string theory have led to renewed interest in supergravity theories in anti-de Sitter [2] and de Sitter space [3].
Thus, in our recent paper [4], we performed a two-component spinor analysis of geometric invariants leading to the gravitino propagator in four-dimensional de Sitter spacetime, following the two-spinor language [5] pioneered by Penrose. In that paper we also wrote down all the 10 different weight functions multiplying the invariants which occur in the massive gravitino propagator, relying upon the work by Anguelova and Langfelder [6]. It was also found there, that algebraically one can write down 8 weight functions denoted by α, β, γ, δ, ε, θ, τ, ω in terms of a pair denoted by (π, κ), in case of de Sitter space. Going one step further, we also expressed κ in terms of π and π ′ where π was defined in this fashion: π(z) = √ zπ(z) andπ(z) satisfies the Heun differential equation [7,8], whose solutions, denoted by Heun(a, q; b, c, d, e; z), with properly defined arguments, have in general four singular points, i.e. z 0 = 0, 1, a, ∞.
In this paper we have explicitly written down, first, the algebraic expression of all the 9 weight functions κ, α, β, γ, δ, ε, θ, τ, ω in terms of π(z) and π ′ (z), where z is defined as z = cos 2 µ 2R , with µ(x, x ′ ) being the geodesic distance between x and x ′ as defined in [4]. Finally, we will draw a few two-dimensional plots of these weight functions and classify their parameter space with respect to z in the region of our choice.
The plan of this paper is as follows. In Sec. II we set up all symbols, basically recalling all relevant definitions of use in this paper from our previous one [4]. Sec. III contains the explicit massive spin-3/2 propagator in four dimensions with all the ten invariant structures properly defined, along with the multiplicative weight functions written in terms of the (π, κ) pair. In Sec. IV we give a crash course on Heun's differential equations and write down several properties of the Heun function before showing thatπ(z) satisfies a Heun equation with properly defined arguments that we will list there. Sec. V and the appendix are devoted to build a dictionary of all the 9 weight functions κ, α, β, γ, δ, ε, θ, τ, ω written in terms of π(z) and π ′ (z) only, where prime denotes derivative with respect to z instead of being the derivative with respect to µ, the geodesic distance function. Then in the last Sec.
VI we display several two-dimensional plots showing the functional behavior of each of the 10 weight functions with respect to z. These show that there exist two ranges of values of z in which the weight functions can be divided into dominating and sub-dominating family.
Moreover, it appears helpful to have the result of a lengthy calculation completely worked out. Eventually, we give further details on the plots in the section devoted to concluding remarks.
In light of recent mathematical developments in [9], it might be possible to expand the Heun functions in the gravitino propagator as a combination of finitely or infinitely many hypergeometric functions, which in turn occur in the more familiar formulae for bosonic propagators in de Sitter space [10]. Thus, our work might help relating fermionic and bosonic propagators in four-dimensional de Sitter space through special-function techniques, double-checking the expectations from supersymmetry.

II. A REVIEW OF A FEW USEFUL DEFINITIONS
It has been more than two decades since Allen and co-authors used intrinsic geometric objects to calculate correlation functions in maximally symmetric spaces; their results, here exploited, were presented in two papers [11,12]. In this section we would like to review, first, the elementary maximally symmetric bi-tensors which have been discussed previously by Allen and Jacobson [11]. More recently, the calculation of the spinor parallel propagator has been carried out in arbitrary dimension [13].
A maximally symmetric space is a topological manifold of dimension n, with a metric which has the maximum number of global Killing vector fields. This type of space looks exactly the same in every direction and at every point. The simplest examples are flat space and sphere, each of which has 1 2 n(n + 1) independent Killing fields. For S n these generate all rotations, and for R n they include both rotations and translations.
We consider a maximally symmetric space of dimension n with constant scalar curvature n(n − 1)/R 2 . For the space S n , the radius R is real and positive, whereas for the hyperbolic space H n , R = il with l positive, and in the flat case, R n , R = ∞. If we further consider two points x and x ′ , which can be connected uniquely by a geodesic, with µ(x, x ′ ) being the geodesic distance between x and x ′ , then n a (x, x ′ ) and n a ′ (x, x ′ ) are the tangents to the geodesic at x and x ′ , and are given in terms of the geodesic distance as follows: (2.1) Furthermore, on denoting by g a b ′ (x, x ′ ) the vector parallel propagator along the geodesic, one can then write n b ′ = −g b ′ a n a . Tensors that depend on two points, x and x ′ , are bitensors [14]. They may carry unprimed or primed indices that live on the tangent space at x or x ′ .
These geometric objects n a , n a ′ and g a b ′ satisfy the following properties [11]: where A and C are functions of the geodesic distance µ and are given by [11] for de Sitter spacetime and thus they satisfy the relations Last, with our convention the covariant gamma matrices satisfy the property In our previous work [4], we followed the conventions for two-component spinors, as well as all signature and curvature conventions, of Allen and Lutken [12], and hence we used dotted and undotted spinors instead of the primed and unprimed ones of Penrose and Rindler [5].
In our work a primed index indicates instead that it lives in the tangent space at x ′ , while the unprimed ones live at x. The fundamental object to deal with is the bispinor D A ′ A (x, x ′ ) which parallel transports a two-component spinor φ A at the point x, along the geodesic to the point x ′ , yielding a new spinor χ A ′ at x ′ , i.e.
Complex conjugate spinors are similarly transported by the complex conjugate of A were listed in Sec. IV of [4]. It is worth mentioning that the covariant derivatives of the spinor parallel propagator were defined to be where A and C are defined in (2.3).
The two basic massive two-point functions for spin-1/2 particle, were defined by and in de Sitter space they turned out to be [4]: where (F ) stands for the Feynman Green functions with f DS and g DS defined in this fashion: (2.14) We also note that F (a, b; c; Z) and F (a, b; c + 1; Z) are two independent solutions of the Hypergeometric equation [15,16] :

III. MASSIVE SPIN-3/2 PROPAGATOR
In this section we consider the propagator of the massive spin-3/2 field. Let us denote the gravitino field by Ψ α λ (x). In a maximally symmetric state | s the propagator is The field equations imply that S satisfies

A. The ten gravitino invariants
It is very convenient to decompose the gravitino propagator in terms of independent structures constructed out of n µ , n ν ′ , g µν ′ and Λ α β ′ [4]. Thus, the propagator can be written in geometric way following Anguelova et al. [6] (see also [17]): The weight functions multiplying the invariants A rather tedious but straightforward calculation gives a system of 10 equations for the 10 coefficient functions α, ..., κ in (3.3) as found in (see equations (3.6)-(3.15) in [6]). It was also found there that one can easily express the algebraic solutions for α, β, γ, δ, ε, θ, τ, ω in terms of the (π, κ) pair in case of de Sitter space, i.e. (hereafter we set n = 4 in the general formulae of [6], since only in the four-dimensional case the two-component-spinor formalism can be applied) where we have used the relation Furthermore, from (3.4) we can immediately see that

IV. HEUN'S DIFFERENTIAL EQUATION: A PRIMER
The canonical form of the general Heun differential equation is given by ([8], [18]) The four regular singular points of the equation are located at z = 0, 1, a, ∞. Here a ∈ C, the location of the fourth singular point, is a parameter (a = 0, 1), and α, β, γ, δ, ǫ ∈ C are exponent-related parameters.
The solution space of the Heun differential equation is specified uniquely by the following Riemann P -symbol: This does not uniquely specify the equation and its solutions, since it omits the accessory parameter q ∈ C. The exponents are constrained by This is a special case of Fuchs's relation, according to which the sum of the 2n characteristic exponents of any second-order Fuchsian equation on CP 1 with n singular points must equal n − 2 [19].
There are 2 × 4 = 8 local solutions of (4.1) in all: two per singular point. If γ is not a nonpositive integer, the solution at z = 0 belonging to the exponent zero will be analytic.
When normalized to unity at z = 0, it is called the local Heun function, and is denoted Hl(a, q; α, β, γ, δ; z) [8]. It is the sum of a Heun series, which converges in a neighborhood of z = 0 [8,20]. In general, Hl(a, q; α, β, γ, δ; t) is not defined when γ is a nonpositive integer.
If ǫ = 0 and q = αβd, the Heun equation loses a singular point and becomes a hypergeometric equation. Similar losses occur if δ = 0, q = αβ, or γ = 0, q = 0. This paper will exclude the case when the Heun equation has fewer than four singular points. The case, in which the solution of (4.1) can be reduced to quadratures, will also be ruled out. If αβ = 0 and q = 0, the Heun equation (4.1) is said to be trivial. Triviality implies that one of the exponents at z = ∞ is zero (i.e., αβ = 0), and is implied by absence of the singular point

A. Reducing Heun to hypergeometric
The transformation to Heun (H) or hypergeometric (h) of a linear second-order Fuchsian differential equation with singular points at z = 0, 1, d, ∞ (resp. z = 0, 1, ∞), and with arbitrary exponents, is accomplished by certain linear changes of the dependent variable, called F-homotopies (see [16] and [8, § A2 and Addendum, § 1.8].) If an equation with singular points at z = 0, 1, a, ∞ has dependent variable u, carrying out the substitutioñ will convert the equation to a new one, with the exponents at z = 0, 1, d reduced by ρ, σ, τ respectively, and those at z = ∞ increased by ρ + σ + τ .
By this technique, one exponent at each finite singular point can be shifted to zero.
Similarly, the hypergeometric equation has a group of F-homotopic automorphisms isomorphic to (Z 2 ) 2 . These groups act on the 6 and 3-dimensional parameter spaces, respectively.

D. Application of Heun's equation to our problem
Finally we come to the punch line, why do we need these all and how does the Heun equation indeed find an application to our problem? The answer to this goes along the following line: On using (3.5) the differential equations for κ and π, the equations (3.14) and (3.15) of [6], acquire the form where θ and ω are given in (3.4). Clearly one can solve algebraically the second equation for κ. By differentiating the result one obtains also κ ′ in terms of π, π ′ and π ′′ , and substitution of these in the first equation yields a second order ODE for π(µ). Now let us look at the system (4.12) in case of de Sitter spacetime. On inserting A and C from (5.1) below and passing to the globally defined variable z = cos 2 µ 2R (see Sec. III), we obtain the following differential equation for π: where P 2 in (4.13) is a quartic polynomial in z, i.e. (4.14) Similarly, P 1 in (4.13) is a cubic polynomial in z, P 1 = 16 m 2 R 2 + 1 z 3 − 12 2m 2 R 2 + 5 z 2 + 8 m 2 R 2 + 2 z. Last, P 0 in (4.13) is a quadratic polynomial in z, i.e. P 0 = 4m 4 − 19m 2 + 32m 2 R 2 + 9 z 2 − 4m 4 − 14m 2 + 32m 2 R 2 + 21 z−3m 2 R 2 −6. (4.16) On making the substitution π(z) = √ zπ(z), (4.13) becomes an equation of the type Written in canonical form it reads as follows: where the parameters in (4.18) take the values The equation (4.17) is known as Heun's differential equation [7,8]. Its solutions, here denoted by Hl(a, q; b, c, d, e; z), have in general four singular points as we said before, i.e. z 0 = 0, 1, a, ∞. Near each singularity the function behaves as a combination of two terms that are powers of (z − z 0 ) with the following exponents: We now insert into the second of Eq. (4.12) the first of Eq. (3.4), finding eventually where f ≡ m(m 2 + R −2 + 2C(C − A)), (4.21) and π and π ′ are meant to be expressed through the Heun function Hl(a, q; b, c, d, e; z).
Eventually, we will show in the next section that all weight functions can be therefore expressed through such Heun function. The material covered in the present section and in the previous two is not new, and most of it is appropriate only for a physics-oriented choice of four-dimensional de Sitter space.

V. DICTIONARY OF WEIGHT FUNCTIONS FOR THE GRAVITINO PROPA-GATOR
Here we will explicitly list all the weight functions as functions of z = cos 2 µ 2R , in order to analyze their qualitative behavior as a function of z and de Sitter radius R in the next section. Let us recall a few definitions in de Sitter space, where A and C are functions of the geodesic distance µ and are given by [11] Since all other weight functions α, β, γ, δ, ε, θ, τ, ω can be written in terms of the (π, κ) pair, and in the last section we have seen κ can also be expressed in a form like (4.20), it is evident that all other 9 weight functions including κ, i.e. α, β, γ, δ, ε, θ, τ, ω, κ can be expressed in terms of π(µ) and π ′ (µ) only.
Similarly, by using a few of the familiar trigonometric identities, one can transform π ′ (µ) as One can also write down the expressions of (A + C) and (A − C) in terms of z and R only as follows: Another function appearing quite frequently in our evaluation of all the weight functions is f , which can be also expressed as a function of z and R only as follows: Now we start by listing all the weight functions in terms of π(z) and π ′ (z), bearing in mind thatπ (z) = Hl(a, q; b, c, d, e; z), (5.5) π(z) = √ z Hl(a, q; b, c, d, e; z), (5.6) where Hl(a, q; b, c, d, e; z) is the Heun function with arguments as defined before. One has therefore the lengthy formulae for all other weight functions written down in Eqs. (A1)-(A8) of the appendix.

VI. QUALITATIVE BEHAVIORS OF THE WEIGHT FUNCTIONS
Now using the series expansion (4.8) defined before one can numerically study the behavior of each weight function, by taking the first 10 terms of the infinite series (4.8). Indeed, dealing with an infinite number of terms is impossible, and one has therefore to resort to approximations, by truncating such a series. On taking less than 10 terms, we have found minor departures from the pattern outlined below in figures 1 to 9, whereas on taking 15 terms, the pattern in such figures is essentially confirmed.
We draw for example all these weight functions in a two-dimensional plot vs z, in the range (0,1). The plots, which also includeπ(z), are as follows. where it also intersects the π(z) curve. Beyond the point z = 0.8 the π andπ curves become virtually indistinguishable. At z = 1 they both have a vertical asymptote.

VII. CONCLUDING REMARKS
Our paper has obtained the complete analytic structure of massive gravitino propagators in de Sitter space. In Sec. VI we have plotted all weight functions α, β, γ, δ, ǫ, θ, τ = ω, π, κ occurring in the gravitino propagator (jointly withπ) as a function of z in a two-dimensional plot where z = cos 2 (µ/2R), µ being the geodesic distance between the points x and x ′ , and R is the de Sitter radius. Although the series (4.8) has been truncated, it remains true that Sec. VI is the first attempt to display a supersymmetric propagator in de Sitter via The plot range is between 0 and 1 for z, which is indeed the only admissible region, since the squared cos function lies always between 0 and 1. Note that the plot ofπ is basically nothing but the plot of the Heun function with properly defined coefficients, and the plot of π is √ z times the Heun function.
The numerical analysis of Sec. VI, as we already said therein, has been performed by taking only the first 10 terms of the infinite series representing the Heun function, by applying the Frobenius' method. If one goes on by taking more terms, one can get even more accurate results, but roughly the qualitative features remain the same. The task of plotting Heun functions is technical but not easy, since the modern computer packages still run into difficulties. Thus, our efforts can be viewed as preparing the ground for a more systematic use of Heun functions in fundamental theoretical physics. The flat-space limit is instead a considerable simplification, since the functions A and C in (5.1) are then found to reduce to A = 1 µ , C = − 1 µ , and the formulae in the appendix are therefore considerably simplified. It also remains to be seen whether the familiarity acquired with Heun functions will prove useful in studying gravitino propagators in other backgrounds relevant for modern high energy physics.