Standard Model Plethystics

We study the vacuum geometry prescribed by the gauge invariant operators of the MSSM via the Plethystic Programme. This is achieved by using several tricks to perform the highly computationally challenging Molien-Weyl integral, from which we extract the Hilbert series, encoding the invariants of the geometry at all degrees. The fully refined Hilbert series is presented as the explicit sum of 1422 rational functions. We found a good choice of weights to unrefine the Hilbert series into a rational function of a single variable, from which we can read off the dimension and the degree of the vacuum moduli space of the MSSM gauge invariants. All data in Mathematica format are also presented.


Introduction and Summary
The Standard Model of particle theory containing specific gauge interactions is expected to have more structures when extended to energies above 1-10 TeV, where supersymmetry might be incorporated. Indeed, to derive the Standard Model as an effective theory from a unified theory containing gravity is one of the chief prospects of theoretical particle physics. One of the most important aspects of a supersymmetric gauge theory is that its vacuum, due to the omnipresence of scalars in the theory, can be highly non-trivial, as parametrized by the vacuum expectation values (VEVs) of gauge invariant operators (GIOs) composed of these scalar fields [1][2][3]. This vacuum moduli space (VMS) can be explicitly obtained as solution of constraints coming from F-flatness and D-flatness and be realized, in the language of algebraic geometry, as an algebraic variety [4].
Supersymmetric extensions of the Standard Model clearly constitute one of the central subjects in particle phenomenology. In particular, the minimal extension, the MSSM, and its variants, have been subject to intense investigations. The flat directions of the MSSM have been identified in [5]. Combining these directions of thought, a long programme was launched to study the vacuum geometry of the MSSM and its relatives [6][7][8][9][10][11]: under the guiding principle that "interesting geometry is coextensive with interesting physics", the ultimate goal is to use geometric and topological properties of VMS as a selection rule for operators in the Standard Model Lagrangian. Specifically, if the VMS were to be found to have some special form in the mathematical sense, which (1) cannot be explained in terms of symmetries relating the relevant degree of freedom in the low energy effective field theory; and (2) is very unlikely to have occurred by chance, then this special form should be regarded as a consequence of some unknown physics. In this setting, we take special to mean non-trivial properties of algebraic geometry, such as exhibited by interesting topological invariants or emergence of special holonomy.
Under such a spirit, the presence of any special geometry would be a collective consequence of factors such as gauge group, particle spectrum and the interactions within the theory. Therefore, if a special geometry is found within the low energy sector of a theory and this geometry is very unlikely to have arisen by chance, then the existence of such geometry in the VMS should be a fundamental property across all energy scales. Hence, the addition of higher dimensional operators to our theory can only occur when they are compatible with this structure. In such sense, we are placing very restrictive constraints on allowed physical processes that are mediated by certain operators.
Already, many interesting features have been found, such as the VMS of the electro-weak sector being an affine cone over the classical Veronese surface, a structure ruined by addition of R-parity-violating operators, or the sensitive dependence of the geometry on the number of generations, or the appearance of Calabi-Yau varieties, etc [9][10][11]. Supersymmetry and the VMS thus provide us with a low energy window of how geometry can guide certain phenomenological questions. A good analogy would be the study of complex numbers: many unforeseen and standard procedure in algebraic geometry adapted to calculate the VSM [10], is beyond the computational power of even the most sophisticated computers by direct means. Therefore, investigations thus far have focused on the electro-weak sector wherein, as discussed, so much have already been uncovered. It is indeed expected that the geometry of the full MSSM would have far richer and salient features.
Rather fortuitously, there has been a parallel programme in studying supersymmetric gauge theories: this is the so-called Plethystic Programme [12,13]. It originated in the study of quiver gauge theories which arise from string theory, as world-volume theories on D-branes probing Calabi-Yau singularities [14], which have become the playground for the AdS/CFT correspondence [15]. Here, the VMS of the gauge theory is, by construction, the affine Calabi-Yau variety transverse to the D-branes, and was in part the initial motivation for the VMS/phenomenology programme. It allows one to build criteria to rule out certain top-down string model building to obtain desired low-energy outcomes: if the VMS is not Calabi-Yau, then one cannot use a direct top-down method.
The central object to the Plethystic Programme is the Hilbert series, well-known to algebraic geometry as the generating function for counting the dimension of graded pieces of the coordinate ring. Harnessing this analogy with the super-conformal index [16-21, 39, 40], the original motivation was to study the chiral ring of BPS operators in supersymmetric gauge theories: the Hilbert series the counts the single-trace operators, whilst its plethystic exponential counts the multi-trace, and its plethystic logarithm encodes the generators and relations of the variety.
A natural question therefore arises as to whether the two programmes can come to a useful syzygy. Specifically, can certain properties of VMS for the MSSM be obtained without recourse to the computationally expensive elimination algorithm, but be deduced from the Hilbert series, which may be calculated via other means? Luckily, this is indeed the case. When the algebraic variety has extra symmetries, such as precisely in our cases, when they come from certain symplectic quotients of Lie-group invariants, there is a classic method of Molien-Weyl integration [44] to obtain the Hilbert series. The purpose of this paper is to perform this, albeit difficult, integral and obtain the Hilbert series explicitly for the MSSM, whence one can further deduce relevant geometrical quantities.

79630506164699.
The organization of the paper is as follows. Section 2 reviews some elements of the Plethystic Programme and we can see therein how the it establishes the connection between Hilbert series and the geometry of VMS. Section 3 gives examples illustrating the programme both in sQCD and Abelian gauge theory. Section 4 establishes the scene for the plethystic integral for MSSM using the related characters of SU (3), SU (2) and U (1) as well as corresponding charges for the matter content thereof. Section 5 gives the description of obtaining the Hilbert series for MSSM with certain subtleties and the main obstacles within this procedure. The results are also presented with more details in this section as well.
Lastly, the VMS obtained here is not constrained by the superpotential W of MSSM, i.e. the relations from requiring ∂W/∂φ i = 0 with φ i being the scalar component of the chiral fields in MSSM, are not imposed on reaching the VMS. The case of non-trivial superpotential W = 0 is therefore left for future work.

The Plethystic Programme
In this section, we review some aspects of the Plethystic Programme. The reader is also referred to [41] for a rapid review of the programme and its context within quiver representations and gauge theory.

Elimination Algorithm for VMS
We first briefly recall the algorithm for computing the VMS of a generic N = 1 supersymmetric gauge theory with gauge group G, fields whose scalar components are φ i , and a polynomial superpotential W therein. The most efficient method to obtain the VMS is as follows.

Consider the ideal
3. Eliminate all variables φ i from I ⊂ R, giving the ideal M in terms of y j .
• OUTPUT: M corresponds to the vacuum moduli space as an affine variety in C[y 1 , . . . , y k ].
In the ensuing, we will address general varieties X though ultimately we will specialize to when X is obtained as the VMS M from the above.

The Hilbert Series
Now we define the protagonist of our investigations.
where X i the i-th graded piece of the coordinate ring for X and can be regarded as the number of independent degree i (Laurent) polynomials on X .
Note that the Hilbert series is not a topological invariant and it depends on the embedding of X .
Of course, H(t) can be generalized to be multi-variate H(t 1 , . . . , t n ) by considering the multigraded pieces X i 1 ,...,in . The dummy variables t i are called fugacities in the physics literature.
When there is more than one variable t i , the Hilbert series is called refined, otherwise it is often called unrefined.
There are two important forms of Hilbert series which will be used later in this paper for obtaining the degree and dimension of the underlying variety. We have that (cf. [42,43]) The Hilbert series H(t) is a rational function in t and can be written in two ways: Hilbert series of the first kind ; , Hilbert series of the second kind .

(2.2)
Here both P (t) and Q(t) are polynomials with integer coefficients and the dimension of the embedding space is given by the power of the denominators. Moreover, P (1) = degree(X ).
Thus we have a convenient way to obtain the degree of the variety ¶ .
Furthermore, since the Hilbert series is a rational function, ¶ Recall that when an ideal is a single polynomial, i.e., X is a hypersurface, the degree of the variety is simply the degree of the polynomial. For multiple polynomials, the degree is a generalisation of this notion. It then becomes the number of intersection points between a generic line and the variety. Theorem 2.2 H(t) affords a partial-fraction expansion around t = 1 [45,46] Thus, the coefficient of the leading pole gives the degree of the variety while the order of the pole is the dimension.
Indeed, for Calabi-Yau varieties, the coefficient of the leading pole can also be interpreted as the volume of base Sasaki-Einstein manifold, which in the AdS/CFT context is related to the central charges of the supersymmetric gauge theory [28,[46][47][48].

Remark:
We remark that when X is a quotient variety, i.e., X ≃ C n /Γ for some discrete finite group Γ acting on the n coordinates of C n , the problem of computing H(t; C n /Γ) reduces to counting the number of algebraically independent polynomials of each degree that are invariant under group action. This problem was solved by Molien [42,49] and the corresponding Hilbert series is the well-known Molien series, which can be computed by a sum over group elements:

Molien-Weyl Formula
The case of our principle interest is when X is not a finite quotient, but of the form of a symplectic quotient by a (continuous) Lie group coming from gauge symmetry. Luckily, there is a generalization of (2.4) into a so-called Molien-Weyl Integral [44] (cf. [8,22,23]). The problem of finding invariants under continuous gauge group is at the heart of invariant theory that can be traced back to 19-th century and we present a rapid review of the origin of Hilbert series and Gröbner basis in the context of commutative algebra in appendix C.
For our incarnation in physics, we wish to compute the Hilbert series for X = M coming from the algorithm in §2.1, whose coordinate ring R is the projection of the quotient ring R/I Since the gauge symmetry commutes with global symmetry, all elements of R with given charge t m should form a representation χ m of G c : In the last step, we have decomposed χ m , the representation on the elements of charge t m , into irreducible representations χ (i) . Therefore, the generating function for invariants is given by the projection onto the trivial representation with character χ (0) = 1, The projection is done by averaging H(t; z) on the gauge group with Haar measure dµ(z), Explicitly, a group G of rank r has its Haar measure in terms of contour integral where |W | is the order of the Weyl group and α is a root, or the weight of the adjoint representation such that α i is the i-th entry of the weight vector in the Dynkin basis.
Putting all the above together, the Molien-Weyl formula for the Hilbert series of the variety M whose coordinate ring is R reads Note here the integration requires knowledge about the Hilbert series of the coordinate ring R.

Plethystics and Syzygies
The next crucial concept needed is that of Plethystics. It is easy to show (q.v., [13]) that (being an exponential) the plethystic exponential is multiplicative in additive arguments, and furthermore The product form is particularly useful and it is usually called Euler form.
It is a non-trivial fact [13,44] that this has an analytic inverse function called the plethystic logarithm is the Möbius mu-function.
Remarkably, the plethystic logarithm can be used to find the defining relation (syzygies) of the generators of an algebraic variety [12,13].
where all b n ∈ Z and a positive b n corresponds to a generator in coordinate ring of X and a negative b n , a relation. In particular, if X is complete intersection, then P E −1 [H(t; M)] is a finite polynomial.
We illustrative this proposition in detail with concrete examples in Appendix ??.

Summary
To summarise, we recall that the plethystic exponential (PE) is defined to be where χ G R (t i , z a ) is the character for representation R of group G and it is expanded into monomials of complex variables z i . Note that the number of complex variables z i is equal to the rank of group G. The expansion of PE gives the complete set of combinations of "fugacities" t i . To find the generating function of gauge invariant opertors under group G, we need to project the representation generated by PE onto trivial subrepresentations of G. This is can be carried out by integrating over the whole group. This is precisely the Hilbert series H inv R in eq. (2.10) whose Molien-Weyl formula for Plethystic Integral is given by where dµ G is the Haar measure for group G. With these data at hand, we conveniently package them into the following theorem Haar measure G dµ G and corresponding plethystic exponential defined in eq. (2.15), the Hilbert series is computed by the following formula is the character for representation R of group G and it is expanded into monomials of complex variables z i and the Haar measure is given by

18)
where W is the Weyl group and α is a root, or the weight of the adjoint representation such that α i is the i-th entry of the weight vector in the Dynkin basis.
The remainder of this paper will be to evaluate this integral explicitly, first for some warm-up cases, and ultimately for the MSSM itself.

Warm-up Examples
Our goal is to apply the technology introduced in §2 to the MSSM, with gauge group G = the SQCD sector and (2) a single Abelian gauge theory. This will give us a more concrete understanding of all the previous definitions from physical side.

sQCD
Let us look at the example of SQCD with N c colours and N f flavours, but without superpotential [8]. (t i , z a ). We further introduce two more variables for counting number of quarks and anti-quarks t andt respectively. The plethystic exponential from (2.11) is precisely the object which constructs symmetric products of quarks and anti-quarks: Expanding the character more explicitly as gives us Here we associated dummy variables t andt to stand for quarks and anti-quarks counting the global U (1) charges in the maximal torus of the global symmetry. Therefore, we should restrict the values of t i to be |t i | < 1 for all i.
As described in §2.3, we want gauge invariant quantities, therefore, it is important that we project these representations onto trivial subrepresentations that are made up by quantities invariant under the action of gauge group. The Molien-Weyl integral from (2.10) thus gives the requisite Hilbert series (generating function) for (N f , N c ) as The Haar measure dµ SU (Nc) can be explicitly written using Weyl's integration formula as (see, Finally, let us construct the characters in the plethystic exponential. First we take the weights of the fundamental representation of SU (N c ) to be where all L's are (N c − 1)-tuples, and L k (2 ≤ k ≤ N c − 1) has −1 in the (k − 1)-th position and 1 in the k-th position. With this particular choice of weights, the coordinates on the maximal torus of SU (N c ) are given by Hence, the characters of the fundamental and anti-fundamental represen- Thus, we have that [8]: The final expression for the Hilbert series for SQCD is the ordinary integral Note that this a refined Hilbert series in the 2N f variables t i ant i .

An Abelian Gauge Theory
We have reviewed in the previous subsection, a rather formal and general example to elucidate the contents of Molien-Weyl formula for SQCD with SU (N ) gauge group. However, the spirit of the integral can be captured by a simple example using U (1) without loss of generality [36].
First, consider a single complex scalar field charged under a U (1) symmetry, i.e., φ → e iθ φ, Clearly, the gauge invariants are now (φφ * ) n and there is only one such operator for each n. We can then define a formal series as where c n = 1 counts the number of different invariants of a given n (since there is clearly only one per degree), so when expanded, it is If (φ, φ * ) are formally treated as numbers less than one, it is simply a geometric series in variables Here, we obtain a refined Hilbert series in two variables; because the field itself is a complex scalar, we can identify the field with its own corresponding fugacity.
Introducing another variable θ, the same variable that parametrises the U (1), (3.12) can be re-written as the integral .
This re-parametrisation can be seen by series expanding ( . By multiplying out and collect terms according to powers of e iθ , we see that the terms that are free of e iθ are exactly the formal series we started with, i.e., 1 + (φφ * ) + (φφ * ) 2 + (φφ * ) 3 + · · · . The terms with any number of factors of e iθ or e −iθ vanish upon the θ integration.
Making substitution z = e iθ , the dθ integral becomes a contour integral around |z| = 1. . (3.14) We can further reframe the second part of the integrand To understand the previous lines, let us expand the LHS of Eq.3.15, with φ and φ * being small complex numbers. To cubic order in both fields we have Indeed, we see the natural emergence of the plethystic exponential, as the generating function of all symmetric combination of its argument. Second, the integration over dθ = dz iz is the integration over the group manifold U (1). This makes sense as we want group invariant quantities, so we have to "average" over group elements. When integrated over dθ, any terms with non-trivial powers of z = e iθ become integrals dθe inθ for some integer n. This is identically 0 since integral is from 0 to 2π. Hence, terms with no powers of z remain and are U (1) invariant.

Molien-Weyl Integral for the MSSM
In this section, we set up the scene for performing the Molien-Weyl integral to obtain the Hilbert series of MSSM. Then, we will use the result to interpret geometrical properties for the VMS for the MSSM. We emphasize that the analysis will be done with the superpotential W = 0.
The group G under consideration for MSSM is, of course, the product gauge group SU (3) × The corresponding character for product group is then also a product for individual factor group, following from the very definition of a group character. Indeed, for a given group G, we can associate any representation R with a character χ R : G → C, where the map is defined to be the trace of any group element g in representation R. Under this definition, the character for direct sum and products for representation is given by Thus equipped, we simply need to input the particle contents with corresponding representation for the product gauge group of MSSM, along with appropriate Haar measure for each factor group, to construct the integral in Theorem 2.3.
The particle content for MSSM are well known and recapitulated in Table 1. The characters Right-handed up-type anti-quark Right-handed down-type anti-quark Left-handed lepton doublet Higgs Table 1: Minimal Supersymmetric Standard Model particle contents are given in the table, where the Representation column entries give information how each particle transform under the product gauge group. For example, the first row means quark Q transforms in fundamental representation (2) and has charge 1 6 under U (1). In addition,3 means anti-fundamental representation.
for each factor group are taken from [22] and the relevant ones are presented in Table 2. Finally, the Haar measures for each group [22] are presented in Table 3.
With the above data, we now proceed to explicitly construct the Molien-Weyl integral in its  Plethystic form, which we will call PI, with full MSSM contents. This simply involves putting each chiral field into its correct representation and input its quantum number for associated character. To do this, we first tabulate the characters of the particle content within MSSM in table 4. We can then use the formula of Plethystic exponential in eq. (2.11) to obtain the integrand. For example, the exponent of the Plethystic exponential for left-handed quarks is which upon taking exponential gives us the argument inside the logarithm. The Plethystic exponentials for the rest of the particle content can be obtained in the similar fashion. Thus we have the following proposition.
Right-handed up-type anti-quarks where the fugacities t i ∈ {Q i , L i , u i , d i , e i , H u , H d } are taken to be |t i | < 1 due to the fact that they count the U (1) charges inside the maximal torus of the global symmetries.
As can be seen, even though with the help of the Molien-Weyl formula, we have reduced the problem of computing the Hilbert series to an ordinary contour integral, the result is still a formidable integral, involving an integrand which is a rational function with 8 factors in the numerator and 49 factors in the denominator! The remainder of this paper is concerned with simplifying this integral explicitly and obtaining geometrical information therefrom. Moreover, we remark that there are fractional powers in the integration variables which might upset the reader: after all, the final answer is a Hilbert series, which must be a rational function. We will show in the ensuing section that all fractional powers actually cancel or disappear in the course of the integration, as is required.

Obtaining the MSSM Hilbert Series
In previous section, we constructed the contour integral to obtain Hilbert series of MSSM in

Finding Poles and Residues
The integration procedure can be carried out as follows. The intermediate results are too complicated to be presented in the text, or even in an appendix, but is available at following link for the repository.
For each of these 14 poles, the residue can be readily obtained. Normally, we would sum these 14 separate residues, put them under the same denominator and cancel any common factors between the inal denominator and numerator. However, this direct approach is already beyond computer package such as Mathematica. To get a taste of the complexities of the rational functions under discussion, let us present 2 of the 14 residues, all of which are complicated rational functions of similar complexity (again, the reader is referred to the above url for the full expressions as well as the Mathematica code). The residue for pole at and the residue for pole at y = L 1 / √ x is We can see from the above expressions that the denominators have over 40 terms and the current built-in functions from the likes Mathematica have difficulties in finding the common denominator and summing over the numerators even for these 2 terms, let along summing over all 14. The reason is that with 40 terms, when brought to the same denominator and expanded, we are confronted with 2 40 ∼ 10 12 monomial terms; factoring a polynomial with this many terms is clearly hopeless. It is remarkable that we could forge ahead and obtain a final answer, as we shall see. Therefore, we can simply make a choice for the fugacities. The ultimate answer cannot depend on this choice by construction. We take the following choice:  With this choice of variables at hand, we can fully determine whether a pole is inside or outside of the contour, thus we know whether the pole should be included when residues are collected. For example, we have a pole for the first of the 14 expression as z 1 = L 2 Q 1 x 2/3 . If we only use the condition from Eq. 4.3, we will not be able to decide this should be included in the residue or not. However, with the choice from 5.4, it is clear this should be discarded since it is outside the contour of |z 1 | = 1.
Using this choice of fugacities, we arrive at a total of 198 poles that are inside the contour for collecting residues. We obtain the integral for each individual rational function using Mathematica built-in function. After performing the 198 integrals, we clean up the results to reduce the amount of work for later integrals. This is done by collecting the terms sharing the same denominator and combining them into a single term. After these procedures, we arrive at a total of 114 terms (again, available at the aforementioned URL) that need to be integrated separately.

The z 2 Integral
Using the results from previous step, we proceed to perform these 114 integrals over variable z 2 . Using the choice of fugacities (5.4), the number of poles found the be located inside the contour |z 2 | = 1 is 1622. This amount of computation requires under 1 hour to complete on a laptop/PC with 4 cores using Mathematica built-in function. However, there are large amount of redundancies within these computation. This can be seen that there are terms that simply cancel when we sum all the terms and the number of terms is reduced to 838. In addition, we can use the same method in the previous paragraph to collect together terms that share the same denominator. The number of terms is now reduced to 574 to enter our final integration over x, which is quite reasonable.

The x Integral
First, the number of poles that are inside contour |x| = 1 with fugacity choice 5.4 is 3106.
We proceed normally with the integral as before. To clean up the redundancies within these results, we sum up all the terms so that some of the terms will just cancel as they are simply negative of each other. In addition, terms sharing common denominator are collected. Finally, we obtain a list of 1538 terms. One important aspect of these results comes from the fact that even we started with a plethystic integral with fractional power 1/6 in variable x, we still end up with all terms have integer exponents and coefficients. However, the raw results of 1538 terms contain terms with fractional exponents in some variables. Remarkably, these fractional exponents combine into integer ones when summed up. Of course, the final answer for the Hilbert series is a rational function and cannot contain any fractional powers. These extraordinary cancellations give us confidence that we are indeed doing the right thing.
To get a flavour of these terms, we present two of these terms which combine to give integer exponents, viz., As one can see, both expressions are sprinkled with troubling terms involving √ u i and u 3/2 i . Summing the expression (5.5) and (5.6) gives the common denominator to be and all fractional powers disappear! It is indeed reassuring that of the 3106 terms, any term with a fractional power therein has exactly 1 partner which cancels it upon summation. This is guaranteed by representation theory (characters) in the Molien-Weyl formula. The numerator is expanded to check for integer coefficients and exponents (note that this expansion gives us over 4 millions terms). The integer criterion indeed checks out for this example and the first few terms are

Unrefining the Hilbert Series
After the previous section, we now have a list of 1538 rational expressions that should be combined into a single rational function, which is the Hilbert Series for MSSM. However, due to the complexity of each rational expression, it is impossible to combine even two terms under a common denominator using common computer packages such as Mathematica or Mccaulay2.
To show the complexity of each term, we present an example below As we can see, a typical rational expression has roughly 33 factors in the denominator, thus combining them implies finding Lowest Common Multiple between each denominator with roughly 30 factors. That is, we need to compute resultants between all pairs from 1538 multivariate polynomials each with about 2 30 ≃ 10 9 monomial terms, rendering the process impractical on an average PC/Laptop. Nevertheless, each of the 1538 rational functions is, as seen from the above expression, is not too complicated. Therefore, we have It is difficult to extract geometrical information directly from this full Hilbert series. Happily, we can "unrefine", i.e., force the Hilbert series to be uni-variate by setting all 18 variables to a single one, say t, but to different powers. That is, the unrefinement is simply a substitution of variables by expression t α , where t is also a fugacity and α is the weight for particular variable that is being substituted. This weight normally corresponds to some particular U (1) charges and some particular choice should render the common denominator non-zero when unrefining (cf. [13]). This particular choice of weight we make is as follows: t 512 t 256 t 128 t 512 t 512 t 512 t 512 t 512 t 256 t 128 t 64 t 32 t 16 t 8 t 4 t 2 t

Simplifying the Unrefined Hilbert Series
After the unrefinement with weights in table 5, the Hilbert series is simplified into a rational function, with a polynomial of degree 816,890 as the numerator and a denominator with 994 factors of total degree 824,397. Note that the factors are already in Euler form and should correspond to GIOs which parametrize the VMS of the MSSM. Even this rational function looks unmanageable at first sight, we can still extract useful information out of it. Importantly, as a preliminary step, we need to perform a Taylor series in t for the Hilbert series in order to know how many independent generators there are in each degree. In terms of the supersymmetric gauge theory, this counts the number of independent 1/2-BPS single-trace operators at each U (1) charge [12,13]. Doing so we obtain: H(t) = 1 + 2t 2 + 4t 3 + 6t 4 + 10t 5 + 16t 6 + 20t 7 + 28t 8 + 38t 9 + 48t 10 + 64t 11 + 84t 12 + 104t 13 + 134t 14 + 168t 15 + 202t 16 + 250t 17 + 304t 18 + 360t 19 + 436t 20 + O t 21 . (5.10) It is very assuring that all coefficients are non-negative, as it is a requirement in the series development of the Hilbert series (since it counts the number of independent monomials in the polynomial ring corresponding to the variety). This requires highly non-trivial conspiracy between the numerator and denominator since each contains many terms with explicitly negative coefficients. Furthermore, the leading term is 1, as is also required. The coeffi- As a technical aside, we have to ensure that there are no terms like (1+t w ) in the product in the denominator since the Euler must have all terms strictly with the minus sign. We can guarantee this by multiplying, each time a term such as (1 + t w ) appear, numerator and denominator by (1 − t w ) so as to contribute a legitimate (1 − t 2w ) factor in the denominator. This actually happens only thrice: (1 + t 8 )(1 + t 16 )(1 + t 24 ) in our case. * * With this 991 we are in fact familiar. There are 991 generators of gauge invariants to the MSSM [6,7] However, by more recent re-calculations, this number is slightly higher than the correct value [10]. But we shall see that after more simplification, the number of factors in the denominator reduces to 445. Thus, at this stage, it seems to be a curious coincidence. We now need to extract as much of the list of factors in Q(t) from the numerator P (t). First, we know this is going to be possible because we can check that P (1) = 0 (even on Mathematica this is still doable as this is simply the sum over all coefficients) so that it must divide (1 − t) at least once (and as we will see, many times). To efficiently perform factorization, we will use a so-called extended synthetic division algorithm [51] for mono-variate polynomials. Luckily, there is an available Python/Sage implementation(c.f. Appendix B for a detailed discussion of this algorithm) of whose liberal use we will take advantage.
Our strategy is to first go over the 991 factors (with multiplicity this amounts to 1477 factors) of the form 1 − t a and try synthetic division into the numerator, this will cancel any such Euler factors therefrom. Doing so (and even with Python, it still takes on the order of 3 days on a regular laptop due to the large degree of the dividend), we find that the numerator now reduces to a polynomial P 1 (t), of degree 816, 890 − 259, 498 = 557, 392, (significantly reduced from the 816,890 of P (t)), likewise, the denominator reduces to Q 1 (t)m with only 445 unique factors (and with multiplicity, 684 factors), in the shorthand notation for the Euler form, Q 1 (t) is 3. To get the degree, we put the HS into the form where Q(t) is in Euler form and P 1 (1) = 0. Now the degree is P 1 (1) with factors of 2 being pulled out and ignored as they come from our choice of weights. This is a rather large number about −2.24 × 10 1633 and we present it in section 1.
Summarising what we have obtained so far, we start with the plethystic integral proposi- A Illustrative Examples for the Plethystic Programme The first part of this appendix reviews the application of plethystics in converting between single-and multi-trace partition functions that count BPS operators.
Single-Trace at N → ∞ To familiarise ourselves with the definitions in eq. (2.11), we take .
This result becomes exact when we take the limit N → ∞. In the next paragraph, we direct out attention to the relation between single-and multi-trace generating function via plethystic exponential.
Multi-Trace at N → ∞ For the case of a single D3-brane on C 3 , the adjoint fields x, y and z are simple complex numbers and thus any product of these fields are multi-trace operators.
Therefore, we only have 4 single trace operators: the identity, x, y and z. So the generating function for single-trace becomes Now we look at the single-trace generating function for N → ∞, which is eq. (A.13). Each of such operators is represented by a monomial t i 1 t j 2 t k 3 , which can be interpreted as a multi-trace operator for just N = 1 or one D3-brane. Therefore, this means g 1 , the generating function for multi-trace operators on a single D3-brane is the same as f ∞ , the generating function for single-trace operators for infinite D3 branes: g 1 = f ∞ . Now we find the relation between f 1 and g 1 : (A.14) We can see from the above that the function g 1 is the Plethystic Exponential of f 1 and this relation in fact generalise to any value of N .
After this short review on plethystic exponential, we see that it is a combinatoric tool for generating the Hilbert series or simply a generating function of all symmetric combination of its argument. It is interesting to see that the inverse of the exponential also contains certain geometric information as we shall shortly cover. The definition of plethystic logarithm is as follows: where µ(k) is the Möbius function The Molien series is given by M (t; ∆(27)) = −1 + t 3 − t 6 (−1 + t 3 ) 3 = 1 + 2t 3 + 4t 6 + 7t 9 + 11t 12 + 16t 15 + 22t 18 + · · · First we need to construct its invariant generators and syzygies using a technique from Reynolds and Gröbner basis. Then we can check these results against those from plethystic logarithm.
We also find a single relation in C[m, n, p, q]: So we find that C 3 /∆(3 · 3 2 ) is a complete intersection given by a single hypersurface in C 4 .
Let us turn to plethystic logarithm, we find it for ∆(3 · 3 2 ) to be We see that the RHS terminates and it can be interpreted as follows: there are 2 degree 3 invariants, 1 degree 6 and 1 degree 9 invariant, these 4 invariants obey a single relation of total degree 18. Comparing this with eq. (A.18), we indeed see that this is the defining relation for C 3 /∆(3 · 3 2 ). In fact, the finiteness of plethystic logarithm indicates that the underlying variety is a complete intersection, i.e. the number of defining equation is equal to the codimension of the variety in the embedding space. The story for non-complete intersection has more content to it. Now let us look at the abelian orbifold C 3 /Z 3 , which is toric and also dP 0 as a cone over P 2 . For the group action (x, y, z) → ω 3 (x, y, z), we can construct the Molien series to be 19) * Specifically, this is a theorem due to Nöther: The polynomial ring of invariants is finitely generated and the degree of the generators is bounded by the order of the group |G|.
where we can get the plethystic logarithm to be This agrees with known facts that the equation for this orbifold is 27 quadrics in C 10 , that is 10 degree three invariants satisfying 27 relations of degree 6. However, these information are only included in the first two terms in the series and the rest of the terms are a reflection of the fact that we no longer have a complete intersection. Therefore, the plethystic logarithm of the Hilbert series is no longer a polynomial and continues ad infinitum. In this final paragraph, let us explain why the plethystic logarithms for non-complete intersections are infinite. Firstly, the Poincaré series is always a rational function when simplified and collected. Particularly, the denominator of the series is of the form of products of (1 − t k ) with possible repeats of k while the numerator being some complicated polynomial. We call this the Euler form. When taking plethystic logarithm, we are essentially trying to solve the following problem: find integers b n such that where we used the identity Now we find the solution: the denominator contributes terms 2t 3 , t 6 and t 9 and the numerator contributes the terms −t 18 where we have 10t 3 from (1 − t 3 ) 10 and −27t 6 from (1 − t 6 ) 27 . However, for higher degree invariants, i.e., 28 degree 6 and 55 degree 9 invariants, etc, we need further expansion on both top and bottom for eq. (A.21). Using computer package such as Mccaulay2, we can find 595 relations for 10 degree 3 and 28 degree 6 invariants: 55 of degree 6, 225 of degree 9 and 315 of degree 12. This thus reads For higher degree invariants and relations, we can correct the coefficients for higher order terms such as t 9 and t 12 .

B Extended Synthetic Division
In this section, we review some basic materials of Extended Synthetic Division with the Python implementation codes presented. Synthetic division is a method of performing Euclidean division of polynomials with less calculation than regular polynomial long division. It is first developed for division by monomial of the form x − a, but later generalized to division by any monomials and polynomials. The advantage of this method is that it allows one to calculate division without writing out variables and it uses less calculations. Let us first look at a simple example: The steps are as follows 1. We negate them as before and write every coefficients but the first on to the left of the bar in an upward.
2. We copy the first coefficient and multiply the diagonal by the copied number and place them diagonally to the right from the copied entry.
3. We sum up the next column until we go past the entries at the top with the next diagonal multiplication.
4. We sum up the remaining column. Since there are two entries to the left of the bar, so the remainder is of degree 1. We then mark the separation with a vertical bar as so we have the final quotient and remainder as We present the division here for the conveinence of the reader More specifically, we present a Python implementation of the algorithm here In this appendix, we review some of the foundation of Hilbert series and see how it is constructed for specific counting purposes. Firstly, we are most interested in polynomial ring K[x 1 , . . . , x n ] consisting polynomials in variables x 1 , . . . , x n with coefficients in the ring K. We typically take K to be a field, such as real numbers R. We also have monomials in the form x α 1 1 · · · x αn n , whose linear combination gives a polynomials. So monomials serve as building blocks for polynomials via addition. Since we are ultimately interested in counting things in polynomial ring using Hilbert Series, we would like to simplify this counting to monomial level. Therefore, the notion of grading is introduced for this counting purpose. On the physical side story, the grading is usually from the charges of certain global symmetries. Let us look at some natural choice for grading, the degree of a polynomial, defined as deg(x α 1 1 · · · x αn n ) = α 1 + · · · + α n . Adding up monomials of the same degree gives us a homogeneous polynomial. Using this notation, we can decompose a set of all homogeneous, degree k polynomials R k into direct sum R = k∈N R k .
Mathematically, variables x 1 , . . . , x n are said to form a N graded algebra.
The dimension of R k is defined to be the number of independent degree k monomials. A Hilbert Function is defined as HF (R, k) = dim(R k ). The Hilbert series is then naturally Usually, we would like to construct the quotient ring M by finding a typical ideal, which is usually generated by a few polynomials f 1 , ..., f s . Then questions such as inclusion of a polynomial inside the ideal and non-trivial relations among generators, arise in this process.
The answers to these questions are computational and generally quite hard. However, a special set of basis of the ideal, called Gröbner basis can be constructed most efficiently to describe the polynomial sequence f 1 , ..., f s . Now we denote the set of polynomials in Gröbner basis by g 1 , ..., g r , where r = s in general. Since f i are taken as basis vectors for the ideal, we can change the basis to the new Gröbner basis, which simply generate the same ideal. A more thorough treatment for this topic can be found in [42]. To construct the Gröbner basis, we need to define an ordering of monomial first. † This monomial ordering ">" determines whether x α > x β , x α = x β or x α < x β for two monomials x α = x α 1 1 · · · x αn n and x β = x β 1 1 · · · x βn n . With this ordering, we can then find the † The common choices of monomial ordering are lexographic, graded lexographic and graded reverse lexographic ordering. Take two monomials x α 1 ...x αn n and x β 1 ...x βn n of total degree α = α1 + · · · + αn and β = β1 + · · · + βn. We take x α > x β if α > β; if α = β, then x α > x β if α1 > β1; if α = β and α1 = β1, then x α > x β if α2 > β2 and so on.
"largest" monomial inside a polynomial h ∈ K[x 1 , ..., x n ]. This is defined to be the initial monomial of h, denoted by in(h). ‡ For every polynomial in I = f 1 , ..., f s , we take their initial monomial and denote this set to be in(I). In general, in(I) is not equal to the set generated by initial monomials of the f i . But the defining property of Gröbner basis is that in(I) = in(g 1 ), ..., in(g r ) .
With the above abstract definition, we shall benefit from a few concrete examples. First let us take the polynomial ring of two variables with coefficients in the real numbers, R = R[x, y].
Here we take monomial ordering to be graded reverse lexographic ordering, which is the default setting for computer package Macaulay2. Example 2 Let R = R[x, y] and I =< x 2 , y 3 >. A monomial of the form x α y β is in the ideal for α ≥ 2 and β ≥ 3. Therefore, the monomials for the quotient ring are 1, x, y, xy, y 2 , xy 2 .
This finite polynomial hints us that it can be written as a rational function with both numerator and denominator being in Euler form. This is actually where the denominator is the Hilbert series of free ring R[x, y], while the numerator reflects the relation among generators of the ideal. ‡ This is also commonly defined as the leading term of h and denoted by LT(h).