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Advances in High Energy Physics
Volume 2011 (2011), Article ID 217035, 12 pages
http://dx.doi.org/10.1155/2011/217035
Review Article

A Simple Introduction to Gröbner Basis Methods in String Phenomenology

Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK

Received 3 February 2011; Accepted 13 February 2011

Academic Editor: Yang-Hui He

Copyright © 2011 James Gray. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

I give an elementary introduction to the key algorithm used in recent applications of computational algebraic geometry to the subject of string phenomenology. I begin with a simple description of the algorithm itself and then give 3 examples of its use in physics. I describe how it can be used to obtain constraints on flux parameters, how it can simplify the equations describing vacua in 4D string models, and lastly how it can be used to compute the vacuum space of the electroweak sector of the MSSM.

1. Introduction

There is currently a great deal of interest in applying the methods of computational algebraic geometry to string phenomenology and closely related subfields of theoretical physics. For some examples of recent work see [1, 2, 68, 1118, 21, 22] and references therein. These papers utilise advances in algorithmic techniques in commutative algebra to study a wide range of subjects including various aspects of globally supersymmetric gauge theory [1, 2, 68], finding flux vacua in string phenomenology [1016], studying heterotic model building on smooth Calabi-Yau in non-standard embeddings [17, 18], and more besides [1923].

Despite the wide range of physical problems which have been addressed within this context, the computational tools which are being used are all based, finally, on the same algorithm. The Buchberger algorithm [24, 25] is at once what lends these methods their power and also the rate limiting step-placing bounds on the size of problem that can be dealt with. The recent burst of activity in this field has been fueled, in part, by the advent of freely available, efficient implementations of this algorithm [26, 27]. There are also interfaces available between the commutative algebra program [27] and Mathematica [1114, 28], with [1114] being particularly geared towards physicist's needs. The aim of this paper is to give an elementary introduction to the Buchberger algorithm and some of its recent applications.

In order to give an idea of how one simple algorithm can make so much possible, I will present the Buchberger algorithm and then show how it may be applied to physics in 3 elementary examples. Firstly, I will describe how it can be used to obtain constraints on the flux parameters in four-dimensional descriptions of string phenomenological models which are necessary and sufficient for the existence of certain types of vacuum [1114]. Secondly, I will describe how the Buchberger algorithm can be used to simplify the equations describing the vacua of such systems making problems of finding minima much more tractable [1114]. Finally, I will describe how the same simple algorithm can be used to calculate the supersymmetric vacuum space geometry of the electroweak sector of the MSSM [1, 2].

The remainder of this paper is structured as follows. In Section 2, I take a few pages to explain the algorithm and the few mathematical concepts that we will require. In the three sections following that, I then describe the three examples mentioned above. I will conclude by making a few final comments about the versatility and scaling of the Buchberger algorithm.

2. A Tiny Bit of Commutative Algebra

Two pages of simple mathematics will suffice to achieve all of the physical goals mentioned in the introduction. First of all we define the notion of a polynomial ring. In this paper we will call the fields of the physical systems we study 𝜙𝑖 and any parameters present, such as flux parameters, 𝑎𝛼. The polynomial rings [𝜙𝑖,𝑎𝛼] and [𝑎𝛼] are then simply the infinite set of all polynomials in the fields and parameters and the infinite set of all polynomials in the parameters, respectively.

Another mathematical concept we will require is that of a monomial ordering. This is simply an unambiguous way of stating whether any given monomial is formally bigger than any other given monomial. We may denote this in a particular case by saying 𝑚1>𝑚2, where 𝑚1, 𝑚2[𝜙𝑖,𝑎𝛼] are monomials in the fields and parameters. It is important to say what is not meant by this. We are not saying that we are taking values of the variables such that the monomial 𝑚1 is numerically larger than the monomial 𝑚2. We are rather saying that, in our formal ordering, 𝑚1 is considered to come before 𝑚2.

For our purposes we will require a special type of monomial ordering called an elimination ordering. This means that our formal ordering of monomials has the following property: 𝜙𝑃𝑖,𝑎𝛼𝑎,LM(𝑃)𝛼𝑎𝑃𝛼.(2.1) In words this just says that if the largest monomial in 𝑃 according to our ordering, LM(𝑃), does not depend on 𝜙𝑖, then 𝑃 does not depend on the fields at all. The monomial ordering classes all monomials with fields in them as being bigger than all of those without such constituents.

Given this notion of monomial orderings, we can now present the one algorithm we will need to use—the Buchberger algorithm [24, 25]. The Buchberger algorithm takes as its input a set of polynomials. These may be thought of as a system of polynomial equations by the simple expedient of setting all of the polynomials to zero. The algorithm returns a new set of polynomials which, when thought of as a system of equations in the same way, has the same solution set as the input. The output system, however, has several additional useful properties as we will see.

The Buchberger Algorithm
(1)Start with a set of polynomials call this set 𝒢. (2)Choose a monomial ordering with the elimination property described above. (3)For any pair of polynomials 𝑃𝑖, 𝑃𝑗𝒢, multiply by monomials, and form a difference so as to cancel the leading monomials with respect to the monomial ordering: 𝑆=𝑝1𝑃𝐼𝑝2𝑃𝐽s.t.𝑝1𝑃LM𝐼,𝑝2𝑃LM𝐽cancel.(2.2)(4) Perform polynomial long division of 𝑆 with respect to 𝒢; that is, form =𝑆𝑚3𝑃𝑘, where 𝑚3 is a monomial and 𝑃𝑘𝒢 such that 𝑚3LM(𝑃𝑘) cancels a monomial in 𝑆. Repeat until no further reduction is possible. Call the result . (5) If =0, then consider the next pair. If 0, then add to 𝒢 and return to step (3).

The algorithm terminates when all S-polynomials which may be formed reduce to 0. The final set of polynomials is called a Gröbner basis.

As mentioned above, the resulting set of polynomials has several nice properties. The feature which is often taken as defining is that polynomial long division with respect to this new set of polynomials always gives the same answer—it does not matter in which order we divide the polynomials out by.

For us, however, the important point about our Gröbner basis 𝒢 is that it has what is called the elimination property. The set of all polynomials in 𝒢 which depend only upon the parameters, 𝒢[𝑎𝛼], gives a complete set of equations on the 𝑎𝛼 which are necessary and sufficient for the existence of a solution to the set of equations we started with. The reason why this is so is actually very straightforward. Our elimination ordering says that any monomial with a field in it is greater than any monomial only made up of parameters. Looking back at step (3) of the Buchberger algorithm we see that we are repeatedly canceling off the leading terms of our polynomials—those containing the fields—as much as we can. Thus, if it is possible to rearrange our initial equations to get expressions which do not depend upon the fields 𝜙𝑖, then the Buchberger algorithm will do this for us. Clearly, while we have interpreted the 𝑎𝛼 as parameters and the 𝜙𝑖 as fields in the above, as this is what we will require for Section 3, this was not necessary. The Buchberger algorithm can be used to eliminate any unwanted set of variables from a problem, in the manner we have described.

This completes all of the mathematics that we will need for our entire discussion, and we may now move on to apply what we have learnt.

3. Constraints

The first physical question we wish to answer is the following. Given a four-dimensional 𝒩=1 supergravity describing a flux compactification, what are the constraints on the flux parameters which are necessary and sufficient for the existence of a particular kind of vacuum? This question can be asked, and answered [1114], for any kind of vacuum, but in the interests of concreteness and brevity let us restrict ourselves to the simple case of supersymmetric Minkowski vacua.

Here is the superpotential of a typical system, taken from [29]. It describes a nongeometric compactification of type IIB string theory𝑊=𝑎03𝑎1𝜏+3𝑎2𝜏2𝑎3𝜏3+𝑆𝑏0+3𝑏1𝜏3𝑏2𝜏2+𝑏3𝜏3𝑐+3𝑈0+̂𝑐1+𝑐1+̃𝑐1𝜏̂𝑐2+𝑐2+̃𝑐2𝜏2𝑐3𝜏3.(3.1) This system has some known constraints on its parameters which are necessary for the existence of a permissible vacuum. These come from, for example, tadpole cancellation conditions: 𝑎0𝑏33𝑎1𝑏2+3𝑎2𝑏1𝑎3𝑏0𝑎=16,0𝑐3+𝑎1𝑐2+̂𝑐2̃𝑐2𝑎2𝑐1+̂𝑐1̃𝑐1𝑎3𝑐0𝑐=0,0𝑏2̃𝑐1𝑏1+̂𝑐1𝑏1𝑐2𝑏0=0,𝑐0̃𝑐2𝑐21+̃𝑐1̂𝑐1̂𝑐2𝑐0=0,𝑐1𝑏3̂𝑐2𝑏2+̃𝑐2𝑏2𝑐3𝑏1=0,𝑐3̃𝑐1𝑐22+̃𝑐2̂𝑐2̂𝑐1𝑐3𝑐=0,0𝑏3̃𝑐1𝑏2+̂𝑐1𝑏2𝑐2𝑏1=0,𝑐3𝑐0𝑐2̂𝑐1+̃𝑐2̆𝑐1̂𝑐1̃𝑐2=0,𝑐1𝑏2̂𝑐2𝑏1+̃𝑐2𝑏1𝑐3𝑏0=0,̂𝑐2̃𝑐1̃𝑐1𝑐2+𝑐1̂𝑐2𝑐0𝑐3=0.(3.2)

We also have the same constraints with the hats and checks switched around. In this example the fields, which we have been calling 𝜙𝑖, are 𝑆, 𝜏, and 𝑈, and everything else is a “flux” parameter, or an 𝑎𝛼 in our notation.

In total, the equations which must be satisfied if a supersymmetric Minkowski vacuum is to exist are 𝑊=0, 𝜕𝑆𝑊=0, 𝜕𝜏𝑊=0, 𝜕𝑈𝑊=0, and the constraints on the flux parameters given above. To extract a set of constraints solely involving the parameters which are necessary and sufficient for the existence of a solution to these equations, we simply follow the procedure outlined in the previous section.

We can carry out this calculation trivially in Stringvacua [1114] and, in fact, this example is provided for the user in the help system. The result is as follows:0=̃𝑐1=̃𝑐2=̂𝑐1=̂𝑐2=𝑐1=𝑐2=𝑐0=𝑐3,0=16+𝑎3𝑏03𝑎2𝑏1+3𝑎1𝑏2𝑎0𝑏3,0=16𝑎23𝑏2096𝑎2𝑎3𝑏0𝑏1288𝑎22𝑏21+432𝑎1𝑎3𝑏21+54𝑎32𝑏3181𝑎1𝑎2𝑎3𝑏31+27𝑎0𝑎23𝑏31+432𝑎1𝑎3𝑏0𝑏227𝑎22𝑎3𝑏20𝑏2+48𝑎1𝑎23𝑏20𝑏2288𝑎0𝑎3𝑏1𝑏218𝑎1𝑎2𝑎3𝑏0𝑏1𝑏245𝑎0𝑎23𝑏0𝑏1𝑏254𝑎1𝑎22𝑏21𝑏2+81𝑎21𝑎3𝑏21𝑏227𝑎0𝑎2𝑎3𝑏21𝑏2+54𝑎0𝑎2𝑎3𝑏0𝑏22+27𝑎0𝑎1𝑎3𝑏1𝑏2227𝑎20𝑎3𝑏32288𝑎1𝑎2𝑏0𝑏332𝑎0𝑎3𝑏0𝑏3+27𝑎32𝑏20𝑏345𝑎1𝑎2𝑎3𝑏20𝑏3+432𝑎0𝑎2𝑏1𝑏327𝑎1𝑎22𝑏0𝑏1𝑏3+54𝑎21𝑎3𝑏0𝑏1𝑏3+48𝑎0𝑎2𝑎3𝑏0𝑏1𝑏3+18𝑎0𝑎22𝑏21𝑏381𝑎0𝑎1𝑎3𝑏21𝑏3144𝑎0𝑎1𝑏2𝑏3+27𝑎21𝑎2𝑏0𝑏2𝑏354𝑎0𝑎22𝑏0𝑏2𝑏351𝑎0𝑎1𝑎3𝑏0𝑏2𝑏3+27𝑎0𝑎1𝑎2𝑏1𝑏2𝑏3+45𝑎20𝑎3𝑏1𝑏2𝑏327𝑎0𝑎21𝑏22𝑏3+27𝑎20𝑎2𝑏22𝑏3+16𝑎20𝑏2327𝑎31𝑏0𝑏23+45𝑎0𝑎1𝑎2𝑏0𝑏23+27𝑎0𝑎21𝑏1𝑏2348𝑎20𝑎2𝑏1𝑏23+3𝑎20𝑎1𝑏2𝑏23.(3.3)

The reader will note that the result is a somewhat lengthy set of equations. In principle one has to find quantized solutions to these expressions, an obviously intractable Diophantine problem, and therefore it might be asked why this result is of any use. In fact, knowledge of such constraints on the flux parameters is hugely useful for a number of reasons.(i)Firstly, we note that, while the full result of this process is often complex, some of the constraints can give us simple information about the system. In the current case, for example, it can be seen that ̃𝑐2=0 is required for the existence of a supersymmetric Minkowski vacuum. (ii)Secondly, if one is scanning over a range of flux parameters and trying to numerically solve the equations to find vacua, one can speed up one’s analysis by first substituting any given set of flux parameters into the constraints we have obtained. If the constraints are not satisfied, then vacua do not exist and there is no point in searching numerically for a solution. This turns what would be a time-consuming numerical process giving inconclusive results (no solution was found) into a quick analytic conclusion (no solution exists). (iii)Lastly, knowledge of such constraints can greatly speed up algebraic approaches to finding vacua such as those outlined in [1114].

4. Simplifying Equations for Vacua

Another use for the mathematics we learnt in Section 2 is the so-called “splitting tools” used in work such as [1114]. The physical idea here is simple. Consider trying to solve the equations 𝜕𝑉/𝜕𝜙𝑖=0 to find the vacua, including those which spontaneously break supersymmetry, of some supergravity theory. These equations are often extremely complicated. One way of viewing why this is so is that the equations for the turning points of the potential contain a lot of information. They describe not only the isolated minima of the potential which are of interest but also lines of maxima, saddle points of various sorts, and so forth. A useful tool to have, therefore, would be an algorithm that takes such a system as an input and returns a whole series of separate sets of equations, each individually describing fewer turning points. Since each separate equation system would then contain less information, one might expect them to be easier to solve. It would be beneficial to choose a division of these equations which has physical interest. The choice we will make here, and which programs like Stringvacua implement [1114], is to split up the equations for the turning points according to how they break supersymmetry—that is, according to which 𝐹-terms vanish when evaluated on those loci.

The ability that packages such as Stringvacua have to split up equations in this manner is based upon the following splitting tool (see [30] for a nice set of more detailed notes on this kind of mathematical technique). Say that one of the 𝐹-terms of our theory is called 𝐹. Then we can split the equations describing turning points of the potential into two pieces: 𝜕𝑉𝜕𝜙𝑖=0,𝐹=0,(4.1)𝜕𝑉𝜕𝜙𝑖=0,𝐹0.(4.2)

The first of these expressions is a set of equations which is easier to solve, in general, than 𝜕𝑉/𝜕𝜙𝑖=0 alone. We can use the 𝐹-term to simplify the equations for the turning points of the potential. On the other hand, expression (4.2) is not even a set of equations—it contains an inequality. We can convert (4.2) into a system purely involving equalities by making use of the mathematics we learned in Section 2.

Consider the following set of equations, including a dummy variable 𝑡: 𝜕𝑉𝜕𝜙𝑖=0,𝐹𝑡1=0.(4.3)

The second equation in (4.3) has a solution if and only if 𝐹0, simply 𝑡=1/𝐹. If 𝐹=0, then the equation reduces to 1=0 which clearly has no solutions. Equations (4.3), then, have a solution whenever the set of equalities and inequalities (4.2) do. Unfortunately they also depend upon one extra, and unwanted, variable—𝑡. This is not a problem as we already know how to remove unwanted variables from our equations. We can simply eliminate them, as we did the fields in Section 2. This will leave us with a necessary and sufficient set of equations in 𝜙𝑖 and 𝑎𝛼 for a solution to (4.3) and thus to (4.2).

So we can split the equations for the turning points of our potential into two simpler systems. One describes the turning points of 𝑉 for which 𝐹=0 and the other, those for which 𝐹0. We can of course perform such a splitting many times—once for each 𝐹-term! In fact, using additional techniques from algorithmic algebraic geometry [1114, 3133], which are essentially based upon the same trick, one can go much further. One can split the equations for the turning points up into component parts gaining one set of equations for every separate locus. Because we know which 𝐹-terms are nonzero on each of them, these are classified according to how they break supersymmetry. The researcher interested in a certain type of breaking can therefore select the equations describing the vacua of interest and throw everything else away.

The above process of splitting up the equations for the vacua of a system can be very simply carried out in Stringvacua. Numerous examples can be found in the Mathematica help files which come with the package [1114]. Here, let us consider the example of M-theory compactified on the coset (𝑆𝑈(3)×𝑈(1))/(𝑈(1)×𝑈(1)). The Kähler and superpotential for this coset, which has 𝑆𝑈(3) structure, has been presented in [34]𝐾=4log𝑖𝑈𝑈𝑇log𝑖1𝑇1𝑇2𝑇2𝑇3𝑇3,1𝑊=8𝑇4𝑈1+𝑇2+𝑇3+2𝑇2𝑇3𝑇1𝑇3𝑇1𝑇2.+200(4.4) Even this, relatively simple, model results in a potential of considerable size. Defining 𝑇𝑖=𝑖𝑡𝑖+𝜏𝑖 and 𝑈=𝑖𝑥+𝑦, we find 1𝑉=256𝑡1𝑡2𝑡3𝑥440000+𝑡23𝜏21400𝜏1𝜏24𝑡23𝜏1𝜏2+4𝑡23𝜏22+𝜏21𝜏22400𝜏1𝜏3+800𝜏2𝜏3+2𝜏21𝜏2𝜏34𝜏1𝜏22𝜏3+𝜏21𝜏234𝜏1𝜏2𝜏23+4𝜏22𝜏2324𝑡2𝑡3𝑥2+4𝑡23𝑥224𝑡1𝑡2+𝑡3𝑥2+4𝜏21𝑥2+8𝜏1𝜏2𝑥2+4𝜏22𝑥2+8𝜏1𝜏3𝑥2+8𝜏2𝜏3𝑥2+4𝜏23𝑥2+1600𝜏1𝑦8𝑡23𝜏1𝑦+1600𝜏2𝑦+16𝑡23𝜏2𝑦8𝜏21𝜏2𝑦8𝜏1𝜏22𝑦+1600𝜏3𝑦8𝜏21𝜏3𝑦+16𝜏22𝜏3𝑦8𝜏1𝜏23𝑦+16𝜏2𝜏23𝑦+16𝑡23𝑦2+16𝜏21𝑦2+32𝜏1𝜏2𝑦2+16𝜏22𝑦2+32𝜏1𝜏3𝑦2+32𝜏2𝜏3𝑦2+16𝜏23𝑦2+𝑡21𝑡22+𝑡23+𝜏22+2𝜏2𝜏3+𝜏23+4𝑥28𝜏2𝑦8𝜏3𝑦+16𝑦2+𝑡224𝑡23+𝜏214𝜏1𝜏3𝜏+2𝑦+423+𝑥2+4𝜏3𝑦+4𝑦2.(4.5)

To find the turning points of this potential we naively need to take eight different derivatives of (4.5) and solve the resulting set of intercoupled equations in eight variables. This is clearly prohibitively difficult. Using the techniques described in this section, however, Stringvacua can separate off parts of the vacuum space for us with ease. Consider, for example, the vacua which are isolated in field space and for which the real parts of all of the 𝐹-terms are nonzero, with the imaginary parts vanishing. To find these, the package tells us, we need only to solve the equations 9𝑥2500=0,5𝑡12𝑥=0,𝑡2𝑥=0,𝑡3𝑥=0,𝜏1=𝜏2=𝜏3=𝑦=0.(4.6)

Because they only describe a small subset of all of the turning points of the full potential, these equations are extremely simple in form and may be trivially solved. For this particular example the physically acceptable turning point that results is a saddle—something which can be readily ascertained once its location has been discovered.

5. Geometry of Vacuum Spaces

As a final example of what we can do with the simple techniques introduced in Section 2, we will show how to calculate the vacuum space of a globally supersymmetric gauge theory. It is a well-known result (see [35] and references therein) that the supersymmetric vacuum space of such a theory, with gauge group 𝐺, can be described as the space of holomorphic gauge invariant operators (GIOs) built out of 𝐹-flat field configurations. What does this space look like? Consider a space, the coordinates of which are identified with the GIOs of the theory. If there were no relations amongst the gauge invariant operators, then this space would be the vacuum space. However, there frequently are relations because of the way in which the GIOs are built out of the fields. For example, if we have three gauge invariant operators 𝑆1, 𝑆2, and 𝑆3 which are built out of the fields as 𝑆1=(𝜙1)2, 𝑆2=(𝜙2)2, 𝑆3=𝜙1𝜙2, then we have the relation 𝑆1𝑆2=(𝑆3)2. If we take these GIOs to be built out of the 𝐹-flat field configurations, then there will be still further relations among them. The vacuum space of the theory is the subspace defined by the solutions of these equations describing relations amongst the gauge invariant operators, once 𝐹-flatness has been taken into account.

How can we calculate such a thing? The holomorphic gauge invariant operators of a globally supersymmetric gauge theory are given in terms of the fields𝑆𝐼=𝑓𝐼𝜙𝑖.(5.1)

Here 𝑆𝐼 are our GIOs, and the 𝑓𝐼 are the functions of the fields that define them. Let us write the 𝐹-terms of the theory as 𝐹𝑖. Consider the following set of equations: 𝐹𝑖=0,𝑆𝐼𝑓𝐼𝜙𝑖=0.(5.2) These equations have solutions whenever the 𝑆𝐼 are given by functions of the fields in the correct way and when those field configurations which are being used are 𝐹-flat. However, according to the proceeding discussion, we wish to simply have equations in terms of the GIOs to describe our vacuum space. As in previous sections, we can eliminate the unwanted variables in our problem, in this case, the fields 𝜙𝑖, using the algorithm of Section 2 to obtain the equations describing the vacuum space.

As a simple example, let us take the electroweak sector of the MSSM [1, 2] (with right-handed neutrinos). Given the field content of the left-handed leptons, 𝐿𝑖𝛼, the right-handed leptons, 𝑒𝑖 and 𝜈𝑖, and the two Higgs, 𝐻 and 𝐻, one can build the elementary GIOs given in Table 1. The indices 𝑖, 𝑗 run over the 3 flavours, and the indices 𝛼,𝛽 label the fundamental of 𝑆𝑈(2).

tab1
Table 1: The set of elementary gauge invariant operators for the electroweak sector of the MSSM.

To compute the 𝐹-terms we require the superpotential. Let us take the most general renormalizable form which is compatible with the symmetries of the theory and R-parity𝑊minimal=𝐶0𝛼,𝛽𝐻𝛼𝐻𝛽𝜖𝛼𝛽+𝑖,𝑗𝐶3𝑖𝑗𝑒𝑖𝛼,𝛽𝐿𝑗𝛼𝐻𝛽𝜖𝛼𝛽+𝑖,𝑗𝐶4𝑖𝑗𝜈𝑖𝜈𝑗+𝑖𝐶5𝑖𝑗𝜈𝑖𝛼,𝛽𝐿𝑗𝛼𝐻𝛽𝜖𝛼𝛽.(5.3) Here 𝜖 is the invariant tensor of 𝑆𝑈(2) and 𝐶0, 𝐶3𝑖𝑗, 𝐶4𝑖𝑗, and 𝐶5𝑖𝑗 are constant coefficients.

We now just follow the procedure outlined at the begining of this section. We calculate the 𝐹-terms by taking derivatives of the superpotential, we label the gauge invariant operators 𝑆1 to 𝑆23, we form (5.2), and then we simply run the elimination algorithm given in Section 2.

The result is, upon simplification, given by six quadratic equations in 6 variables. It is a simple description of an affine version of a famous algebraic variety—the Veronese surface [1, 2]. What can be done with such a result? The first observation we can make is that this vacuum space is not a Calabi-Yau. This means, for example, that one can say definitively that it is not possible to engineer this theory by placing a single D3 brane on a singularity in a Calabi-Yau manifold, without having to get into any details of model building.

Secondly one can study such vacuum spaces in the hope of finding hints at the structure of the theory's higher energy origins. In the case we have studied in this section, for example, we can “projectivize” (pretend the GIOs are homogeneous coordinates on projective space rather than flat space coordinates) and study the Hodge diamond of the result. The structure of supersymmetric field theory tells us that this Hodge diamond should depend on 4 arbitrary integers, but there is nothing at low energies which prevents us from building theories with any such integers we like. Interestingly, in the case of electroweak theory, these integers are all zero or one:𝑝,𝑞=0,00,10,10,21,10,20,10,10,0=110001000.(5.4)

Whether this structure is indeed a hint of some high energy antecedent or just a reflection of the simplicity of the theory is debatable. This example does, however, demonstrate the idea of searching for such evidence of new physics in vacuum space structure. We should also add here that similar techniques can be used to show that the vacuum space of SQCD is a Calabi-Yau [68].

6. Final Comments

To conclude we will make several points—one of which is a note of caution, with the rest being more optimistic. The first point which we will make is that we should be careful lest the above discussion makes the algorithm we have been describing sound like an all-powerful tool. There is, as ever, a catch. In this case it is the way the algorithm scales with the complexity of the problem. A “worst case” upper bound for the degree of the polynomials in a reduced Gröbner basis can be found in [36]. If 𝑑 is the largest degree found in your original set of equations, then this bound is 2𝑑22+𝑑2𝑛1,(6.1)

where 𝑛 is the number of variables. This worst case bound is therefore scaling doubly exponentially in the number of degrees of freedom. These very high-degree polynomials are an indication that the problem is becoming very complex and thus computationally intensive. Despite this, physically useful cases can be analysed using this algorithm quickly, as demonstrated in this paper and in the references. This scaling does mean that one is not likely to gain much by putting one's problem on a much faster computer. One good point about (6.1) is that if you can find a way, using physical insight, to simplify the problem under study, then what you can achieve may improve doubly exponentially. Such a piece of physical insight was one of the keystones of the application of these methods to finding flux vacua [1114].

We finish by commenting that the methods of computational commutative algebra which we have discussed here are extremely versatile. We have been able to perform three very different tasks simply utilizing one algorithm in a very simple manner. These methods are of great utility in problems taken from the literature, and their implementation in a user friendly way in Stringvacua means that they may be tried out on any given problem with very little expenditure of time and effort by the researcher. Many more techniques from the field of algorithmic commutative algebra could be applied to physical systems than those described here or indeed in the physics literature. We can therefore expect that this subject will only increase in importance in the future.

Acknowledgments

The author is funded by STFC and would like to thank the University of Pennsylvania for generous hospitality while some of this document was being written. In addition he would like to thank the organisers of the 2008 Vienna ESI workshop “Mathematical Challenges in String Phenomenology,” where the talk upon which these notes are based was first given. The author would like to offer heartfelt thanks to his collaborators on the various projects upon which this paper is based. These include Lara Anderson, Daniel Grayson, Amihay Hanany, Yang-Hui He, Anton Ilderton, Vishnu Jejjala, André Lukas, Noppadol Mekareeya, and Brent Nelson.

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