String/M-theories About Our World Are Testable in the Traditional Physics Way∗ Gordon L. Kane † Michigan Center for Theoretical Physics (MCTP), Department of Physics, University of Michigan, 6 Ann Arbor, MI 48109 USA 1 0 2 January 28, 2016 n a J 7 2 Abstract ] h Some physicists hope to use string/M-theory to construct a comprehensive underlying the- t ory of our physical world – a “final theory”. Can such a theory be tested?A quantum theory - p of gravity must be formulated in 10 dimensions, so obviously testing it experimentally requires e projecting it onto our 4D world (called “compactification”). Most string theorists study the- h ories, including aspects such as AdS/CFT, not phenomena, and are not much interested in [ testing theories beyond the Standard Model about our world. Compactified theories generi- 1 callyhavemanyrealisticfeatureswhosenecessarypresenceprovidessometests,suchasgravity, v Yang-Mills forces like the Standard Model ones, chiral fermions that lead to parity violation, 1 1 softlybrokensupersymmetry,Higgsphysics,families,hierarchicalfermionmassesandmore. All 5 testsoftheoriesinphysicshavealwaysdependedonassumptionsandapproximatecalculations, 7 and tests of compactified string/M-theories do too. String phenomenologists have also formu- 0 lated some explicit tests for compactified theories. In particular, I give examples of tests from . 1 compactified M-theory (involving Higgs physics, predictions for superpartners at LHC, electric 0 dipolemoments, andmore). Itisclearthatcompactifiedtheoriesexistthatcandescribeworlds 6 like ours, and it is clear that even if a multiverse were real it does not prevent us from finding 1 comprehensivecompactifiedtheorieslikeonethatmightdescribeourworld. Ialsodiscusswhat : v we might mean by a final theory, what we might want it to explain, and comment briefly on i X multiverse issues from the point of view of finding a theory that describes our world. r a ∗Thiswriteupisbasedonaninvitedtalkatthemeeting“WhyTrustaTheory? ReconsideringScientificMethod- ology in Light of Modern Physics”, LMU, Munich, December 2015, and some related talks. †Email: [email protected] 1 1 Outline • Testing theories in physics – some generalities. • Testing 10-dimensional string/M-theories as underlying theories of our world obviously re- quires compactification to four space-time dimensions. • Testing of all physics theories requires assumptions and approximations, hopefully eventually removable ones. • Detailed example: existing and coming tests of compactifying M-theory on manifolds of G 2 holonomy, in the fluxless sector, in order to describe/explain our vacuum. • How would we recognize a string/M-theory that explains our world, a candidate “final the- ory”? What should it describe/explain? • Comments on multiverse issues from this point of view. Having a large landscape clearly does not make it difficult to find candidate string/M-theories to describe our world, contrary to what is often said. • Final remarks 2 Long Introduction The meeting ”Why Trust a Theory” provided an opportunity for me to present some comments and observations and results that have been accumulating. The meeting was organized by Richard Dawid, author of String Theory and the Scientific Method, a recent and significant book. Near the end I will comment on some of Dawid’s points. String/M-theory is a very promising framework for constructing an underlying theory that incorporates the Standard Models of particle physics and cosmology, and probably addresses all the questions we hope to understand about the physical universe. Some people hope for such a theory. A consistent quantum theory of gravity must be formulated in 10 or 11 dimensions. The differences between 10 and 11 D are technical and we can ignore them here. (Sometimes the theory can be reformulated in more dimensions but we will ignore that too.) Obviously the theory must be projected onto four space-time dimensions in order to test it experimentally. The jargon for such a projection is “compactification”. Remarkably, many of the compactified string/M-theories generically have the kinds of proper- ties that characterize the Standard Models of particle physics and cosmology and their expected extensions! These include gravity, Yang-Mills gauge theories (such as the color SU(3) and the elec- troweak SU(2)×U(1)), softly broken supersymmetry, moduli, chiral fermions (that imply parity violation), families, inflation and more. For some theorists the presence of such features is sufficient to make them confident that the string/M-theories will indeed describe our world, and they don’t feel the need to find the particular one that describes our vacuum. For others, such as myself, the presence of such features stimulates me to pursue a more complete description and explanation of our vacuum. Thecosmologicalconstantproblem(s)(CC)remainmajorissues,ofcourse. Wewillassumethat theCCissuesareeffectivelyorthogonaltotherestofthephysicsinfindingadescription/explanation 2 of our world. That is, solving the CC problems will not help us find our underlying theory, and not solving the CC problems will not make it harder (or impossible) to find our underlying theory. The evidence we have is consistent with such an assumption. Ultimately, of course, it will have to be checked. The value of the CC may be environmental, a point of view attractive to many. Some physicists have advocated that a number of the constants of the Standard Model and beyond are environ- mental. In particular, their values are then not expected to be calculable by the normal methods of particle theory such as a compactified string/M-theory. One problem with that attitude is that apparently not all of the constants are environmental. Perhaps the strong CP angle, or the proton lifetime, or the top quark mass are not, and perhaps more. In particular, note that some advocate that the Higgs boson mass is environmental, but (as described below) we argue that in the com- pactified M-theory the HIggs mass (at least the ratio of the Higgs mass to the Z mass and perhaps the Higgs vev) is calculable. It seems possible that the strong CP angle and the proton lifetime are also calculable. If so, it is still necessary to explain why some parameters seem to have to be in certain somewhat narrow ranges or the world would be very different. Note that the allowed ranges are quite a bit larger than the often stated ones - e.g. if the forces are unified one must change their strengths together, weakening typical arguments (ph/0408169). But the issue remains that some parameters need to lie in rather narrow ranged, and need to be understood. In a given vacuum, one can try to calculate the CC. Much has been written and said about whether string/M-theories are testable. Much of what has been said is obviously not serious or interesting. For example, obviously you do not need to be somewhere to test a theory there. No knowledgeable person doubts our universe had a big bang, although no one was there to observe it. There are several very compelling pieces of evidence or relics. One is the expansion and cooling of the universe, a second the properties of the cosmic microwave background radiation, and a third the nucleosynthesis and helium abundance results. Amazingly, scientists probably have been able to figure out why dinosaurs became extinct even though almost everyone agrees that no people were alive 65 million years ago to observe the extinction. You don’t need to travel at the speed of light to test that it is a limiting speed. Is the absence of superpartners at LHC a test of string/M-theory, as some people have claimed? What if superpartners are found in the 2016 LHC run – does that confirm string/M-theory? Before a few years ago there were no reliable calculations of superpartner masses in well-defined theories. An argument, called “naturalness”, that if supersymmetry indeed solves the problems it is said to solve then superpartners should not be much heavier than the Standard Model partners, implies that superpartners should have been found already at LEP or FNAL, and surely at Run I of LHC. All predictions up to a few years ago were based on naturalness, rather than on actual theories, and were wrong. It turns out that in some compactified string/M-theories one can do fairly good generic calculations of some superpartner masses, and most of them (but not all) turn out to be quite a bit heavier than the heaviest Standard Model particles such as the top quark, or W and Z bosons. Generically scalars (squarks, sleptons, Higgs sector masses except for the Higgs boson) turn out to be a few tens of TeV (25-50 TeV), while gauginos (partners of gauge bosons such as gluinos, photinos, winos, binos) tend to be of order one TeV. I will illustrate the mechanisms that produce these results later for a compactified M-theory. Thus actual compactified string/M-theories generically predict that superpartners should not have been found at LHC Run I. Typically some lighter gauginos do lie in the mass region accessible at Run II. It seems odd to call the results of good theories “unnatural”, but the historical (mis-)use 3 of natural has led to that situation. Interestingly, many string theorists who work on gravity, black holes, AdS/CFT, amplitudes, andsoondonotknowthetechniquestostudycompactifiedstring/M-theories, andtheircomments maynotbeuseful. Oldertheoristsmayremember. Muchofwhatiswrittenonthesubjectoftesting string theory does not take into account the need for compactification, particularly in blogs and somepopularbooks,whichisquitemisleading. That’softenalsothecaseforwhatsupposedexperts say – in 1999 a well known string theorist said at a conference “string theorists have temporarily given up trying to make contact with the real world”, and clearly that temporary period has not ended. Sadly, string theory conferences have few talks about compactified string/M-theories, and the few that might be there are mainly technical ones that do not make contact with experiments. Stringtheoristsseldomreadpapersabout,orhaveseminarsattheiruniversitiesabout,compactified string/M-theories that connect to physics beyond the Standard Model. But I want to argue that string/M-theory’s potential to provide a comprehensive underlying theory of our world is too great to ignore it. String/M-theory is too important to be left to string theorists. Before we turn to actual compactified theories, let’s look a little more at the meaning of testing theories. In what sense is “F=ma” testable? It’s a claim about the relation between forces and particlepropertiesandbehavior. Itmighthavenotbeencorrect. Itcanbetestedforanyparticular force, but not in general. The situation is similar for the Schr¨odinger equation. For a given Hamiltonianonecancalculatethegroundstateofasystemandenergylevels,andmakepredictions. WithoutaparticularHamiltonian,notests. Whatdowemeanwhenweaskto“teststringtheory”? The situation for string/M-theories is actually quite similar to F=ma. One can test predictions of particular compactified string theories, but the notion of “testing string theory” doesn’t seem to have any meaning. If you see something about “testing string theory”, beware. Quantum theory has some general properties, basically superposition, that don’t depend on the Schr¨odinger equation (or equivalent). Similarly, quantum field theory has a few general tests that don’tdependonchoosingaforce,suchasthatallelectronsareidentical(becausetheyareallquanta of the electron field), or the connection between spin and statistics. Might string/M-theory have some general tests? Possibly calculating black hole entropy, but I don’t want to discuss that here, because it is not a test connected to data. Otherwise, there don’t seem to be any general tests. Compactified string/M-theories generically give 4D quantum field theories, and they also imply particular Yang-Mills forces and a set of massless zero-modes or particles, so it seems unlikely they will have any general tests. In all areas of physics normally one specifies the “theory” by giving the Lagrangian. It’s importanttorecognizethat physical systems are described not by the Lagrangian but by the solutions to the resulting equations. Similarly, if string/M-theory is the right framework, our world will be described by a compactified theory, the projection onto 4D of the 10/11D theory, the metastable (or stable) ground state, called our “vacuum”. One also should recognize that studying the resulting predictions is how physics has always proceeded. All tests of theories have always depended on assumptions – from Galileo’s using inclined planes to slow falling balls so they could be timed, or assuming air resistance could be neglected in order to get a general theory of motion, to assuming what corner of string theory and compactification manifold should be tried. Someday there may be a way to derive what is the correct corner of string theory, or the compactification manifold, but most likely we will first find ones that work to describe/explain our world, and perhaps later find whether they are inevitable. 4 Similarly, in a given corner and choice of manifold we may first write a generic superpotential and K¨ahler potential and gauge kinetic function and use them to calculate testable predictions. Perhaps soon someone can calculate the K¨ahler potential or gauge kinetic function to a sufficient approximation that predictions are known to be insensitive to corrections. It’s very important to understand that the tests are tests of the compactified theory, but they do depend on and are affected by the full 10/11D theory in many ways. The curled-up dimensions contain information about our world – the forces, the particles and their masses, the symmetries, thedarkmatter,thesuperpartners,electricdipolemoments,andmore. Therearerelationsbetween observables. The way the small dimensions curl up tells us a great deal about the world. Just as a Lagrangian has many solutions (e.g. elliptical planetary orbits for a solar system), so the string/M-theory framework will have several or many viable compactifications, many solutions. This is usually what is meant by “landscape”. The important result to emphasize here is that the presence of large numbers of solutions is not necessarily an obstacle to finding the solution or set of solutions that describe our world, our vacuum. (See also the talk of Fernando Quevedo at this meeting.) That is clear because already people have found, using a few simple guiding ideas, a number of compactified theories that are like our world. Some generic features were described above, and the detailed compactified M-theory described below will illustrate this better. It is not premature to look for our vacuum. 3 What might we mean by “final theory”? How would we recog- nize it? In each vacuum perhaps all important observables would be calculable (with enough grad students and postdocs encouraged and supported to work in this area). What would we need to understand andcalculatetothinkwehadanunderlyingtheorythatwasastrongcandidatefora“finaltheory”? We don’t really want to calculate hundreds of QCD and electroweak predictions, or beyond the Standard Model ones. We probably don’t need an accurate calculation of the up quark mass, since at its MeV value there may be large corrections from gravitational and other corrections, but it is very important to derive m < m , and that m is not too large. It’s interesting and fun to up down up make such a list. Here is a good start. (cid:88) What is light? • What are we made of? Why quarks and leptons? • Why are there protons and nuclei and atoms? Why SU(3)×SU(2)×U(1)? • Are the forces unified in form and strength? • Why are quark and charged lepton masses hierarchical? Why is the down quark heavier than the up quark? • Why are neutrino masses small and probably not hierarchical? • Is nature supersymmetric near the electroweak scale? • How is supersymmetry broken? 5 • How is the hierarchy problem solved? Hierarchy stabilized? Size of hierarchy? • How is the µ hierarchy solved? What is the value of µ? • Why is there a matter asymmetry? • What is the dark matter? Ratio of baryons to dark matter? • Are protons stable on the scale of the lifetime of the universe? • Quantum theory of gravity? • What is an electron? (cid:66) Why families? Why 3? (cid:66) What is the inflaton? Why is the universe old and cold and dark? ♦ Which corner of string/M-theory? Are some equivalent? ♦ Why three large dimensions? ♦ Why is there a universe? Are there more populated universes? ♦ Are the rules of quantum theory inevitable? ♦ Are the underlying laws of nature (forces, particles, etc) inevitable? ♦ CC problems? The first question, what is light, is answered – if there is an electrically charged particle in a worlddescribedbyquantumtheory, thephaseinvariancerequiresafieldthatistheelectromagnetic field, so it has a check, (cid:88). The next set, with the bullet, are all addressed in the compactified M- theory; some are answered. They are all addressed simultaneously. The next two, with the (cid:66), are probably also addressed in compactified string/M-theory. The last six are still not addressed in any systematic way, though there is some work on them. Thelistispresentedsomewhattechnically,buttheideaisprobablycleartomostreaders. Other readers might have a somewhat different list. I’d be glad to have suggestions. The most important point is that compactified string/M-theories do address most of the questions already, and will do better as understanding improves. ThecompactifiedM-theorydescribedbelowassumedthatthecompactificationwastothegauge- matter content of the SU(5) MSSM, the minimal supersymmetric Standard Model. Other choices could have been made, such as SO(10),E ,E . So far there is no principle to fix that content. 6 8 There are probably only a small number of motivated choices. 4 Three new physics aspects: In compactified theories three things emerge that are quite important and may not be familiar. 6 • The “gravitino” is a spin 3/2 superpartner of the (spin 2) graviton. When supersymmetry is broken, the gravitino gets mass via spontaneous symmetry breaking. The resulting mass sets the scale for the superpartner masses and associated phenomena. For the compactified M-theory on a G manifold the gravitino mass is of order 50 TeV. That is the scale of the 2 soft-breaking Lagangian terms, and thus of the scalars and trilinear couplings, as well as M and M . It also can contribute to the dark matter mass. Two different mechanisms Hu Hd (described below) lead to suppressions of the gaugino masses from 50 TeV to ∼ 1 TeV, and the Higgs boson. • Thesecondis“moduli”,whichhavemanyphysicseffects,includingleadingtoa“non-thermal” cosmological history. The curled up dimensions of the small space are described by scalar fields that determine their sizes, shapes, metrics and orientations. The moduli get vacuum expectation values, like the Higgs field does. Their vacuum values determine the coupling strengths and masses of the particles, and they must be “stabilized” so the laws of nature will not vary in space and time. The number of moduli is calculable in string/M-theories (the thirdBettinumber), andistypicallyofordertenstoevenover200. IncompactifiedM-theory supersymmetrybreakinggeneratesapotentialforallmoduli,andstabilizesthem. Themoduli fields (like all fields) have quanta, unfortunately also called moduli, with calculable masses fixed by fluctuations around the minimum of the moduli potential. In general inflation ends when they are not at the minimum, so they will oscillate, and dominate the energy density of the universe soon after inflation ends. Themoduliquantacoupletoallparticlesviagravity, sotheyhaveadecaywidthproportional to the particle mass cubed divided by the Planck mass squared. Their lifetime is long, but one can show that generically the lightest eigenvalue of the moduli mass matrix is of order the gravitino mass, which guarantees they decay before nucleosynthesis and do not disrupt nucleosynthesis. Their decay introduces lots of entropy, and therefore washes out all earlier dark matter, matter asymmetry, etc. They then decay into dark matter and stabilize the matter asymmetry. That the dark matter and matter asymmetries both arise from the decay of the lightest modulus can provide an explanation of the ratio of matter to dark matter, though so far only crude calculations have been done along these lines. When moduli are ignoredtheresultinghistoryoftheuniverseafterthebigbangisasimplecoolingasitexpands, dominated by radiation from the end of inflation to nucleosynthesis, called a thermal history. Compactified string/M-theories predict instead a “non-thermal” history, with the universe matter dominated (the moduli are matter) from the end of inflation to somewhat before nucleosynthesis. All the above results were derived from the compactified M-theory before 2012. Compactified string theories give us quantum field theories in 4D, but they give us much more. They predict generically a set of forces, the particles the forces act on, softly broken supersymmetrictheories,andthemodulithatdominatecosmologicalhistoryandwhosedecay generates the dark matter and possibly the matter asylmmetry. • The third aspect is that because there are often many solutions we look for “generic” results. We have already used ”generic” several times above. Generic results are probably not a the- orem, or at least not yet proved. They might be avoided in special cases. One has to work at constructing non-generic examples. Importantly, predictions from generic analyses are gener- ically not subject to qualitative changes from small input changes. Most importantly, they 7 generically have no adjustable parameters, and no fine tuning of results. Many predictions of compactified string/M-theories are generic, and thus powerful tests. When non-generic K¨ahler potentials are used, the tests become model-dependent and much less powerful. 5 Compactified M-theory on a G Manifold (11-7=4) 2 Now we turn to looking at one compactified theory. Of course, I use the one I have worked on. The purpose here is pedagodical, not review or completeness, so references are not complete, and I apologize to many people who have done important work similar to what is mentioned. References are only given so those who want to can begin to trace the work. From 1995 to 2004 there was a set of results that led to establishing the basic framework. • In 1995 Witten discovered M-theory. • Soon after, Papadopoulos and Townsend (th/9506150) showed explicitly that compactifying 11D M-theory on a 7D manifold with G holonomy led to a 4D quantum field theory with 2 N=1 supersymmetry. Thus the resulting world is automatically supersymmetric – that is not an assumption! • Acharya (th/9812205) showed that non-Abelian gauge fields were localized on singular 3- cycles. The 3-cycles can be thought of as “smaller” manifolds within the 7D one. Thus the resulting theory automatically has gauge bosons, photons and Z’s and W’s, and their gaugino superpartners. • Atiyah and Witten (th/0107177) analyzed the dynamics of M-theory on G manifolds with 2 conical singularities and their relations to 4D gauge theories. • Acharya and Witten (th/0109152) showed that chiral fermions were supported at points with conical singularities. The quarks and leptons of the Standard Model are chiral fermions, with left-handed and right-handed ones having different SU(2) and U(1) assignments, and giving the parity violation of the Standard Model. Thus the compactified M-theory generically has the quarks and leptons and gauge bosons of the Standard Model, in a supersymmetric theory. • Witten (ph/0201018) showed that the M-theory compactification could be to an SU(5) MSSM, and solve the doublet-triplet splitting problem. He also argued that with a generic discrete symmetry the µ problem would have a solution, with µ = 0. • Beasley and Witten (th/0203061) derived the generic K¨ahler form. • Friedmann and Witten (th/0211269) worked out Newton’s constant, the unification scale, proton decay and other aspects of the compactified theory. However, in their work supersym- metry was still unbroken and moduli not stabilized. • Lucas and Morris (th/0305078) worked out the generic gauge kinetic function. With this and the generic K¨ahler form of Beasley and Witten one had two of the main ingredients needed to calculate predictions. • Acharya and Gukov brought together much of this work in a Physics Reports (th/0409191) 8 To extend previous work, we explicitly made five assumptions: (cid:73) Compactify M-theory on a manifold with G holonomy, in the fluxless sector. This is well 2 motivated. The qualitative motivation is that fluxes (the multidimensional analogues of electro- magnetic fields) have dimensions, so are naturally of string scale size. It is very hard to get TeV physics from such large scales. There are still few examples of generic TeV mass particles emerging from compactifications with fluxes. Using the M-theory fluxless sector is robust (see Acharya ref- erenced above, and recent papers by Halverson and Morrison, arxiv:1501.05965; 1412.4123), with no leakage issues. (cid:73)Compactify to gauge matter group SU(5)−MSSM. We followed Witten’s path here. One could try other groups, e.g. SU(3)×SU(2)×U(1),SO(10),E6,E8. There has been some recent work on the SO(10) case, and results do seem to be different (Acharya et al 1502.01727). Someday hopefully there will be a derivation of what manifold and what gauge-matter group to compactify to, or perhaps a demonstration that many results are common to all choices that have SU(3)× SU(2)×U(1)−MSSM. (cid:73) Use the generic K¨ahler potential and generic gauge kinetic function. (cid:73) Assume the needed singular mathematical manifolds exist. We have seen that many results do not depend on the details of the manifold (see below for a list). Others do. There has been considerable mathematical progress recently, such as a Simons Center semester workshop with a meeting, and proposals for G focused activities. There is no known reason to be concerned about 2 whether appropriate manifolds exist. (cid:73) We assume that cosmological constant issues are not relevant, in the sense stated earlier, that solving them does not help find the properties of our vacuum, and not solving them does not prevent finding our vacuum. Of course, we would like to actually calculate the CC in the candidate vacuum or understand that it is not calculable. We started in 2005 to try to construct a full compactification. Since the LHC was coming, we focused first on moduli stabilization, how supersymmetry breaking arises, calculating the gravitino mass and the soft-breaking Lagrangian for the 4D supergravity quantum field theory, which led to Higgs physics, LHC physics, dark matter, electric dipole moments, etc., leaving for later quark and lepton masses, inflation, etc. Altogether this work has led to about 20 papers with about 500 arXiv pages in a decade. Electric dipole moments are a nice example of how unanticipated results can emerge - when we examined the phases of the terms in the soft-breaking Lagrangian all had the same phase at tree level, so it could be rotated away (as could the phase of µ), so there were no EDMs at the compactification scale. Then the low scale phase is approximately calculable from known RGE running, and indeed explains why EDMs are much smaller than naively expected (0906.2986; 1405.7719), a significant success. One can write the moduli superpotential (see below). It is a sum of exponential terms, with exponents having beta functions and gauge kinetic functions. Because of the axion shift sym- metry only non-perturbative terms are allowed in the superpotential, no constant or polynomial terms. One can look at early references (Acharya et al, th/0606262, th/0701034, arxiv:0801.0478, arxiv:0810.3285) to see the resulting terms. We were able to show that in the M-theory compactification supersymmetry was spontaneously broken via gaugino and chiral fermion condensation, and simultaneously the moduli were indeed all stabilized, in a de Sitter vacuum, unique for a given manifold. We calculated the soft-breaking Lagrangian, and showed that many solutions had electroweak symmetry breaking via the Higgs 9 mechanism. So we have a 4D effective softly-broken supersymmetric quantum field theory. It’s important to emphasize that in the usual “effective field theory” the coefficients of all operators are independent, and not calculable. Here the coefficients are all related and are all calculable. This theory has no adjustable parameters. In practice, some quantities cannot be calculated very accurately, so they can be allowed to vary a little when comparing with data. 6 Some Technical Details Here for completeness and for workers in the field we list a few of the most important formulae, in particular the moduli superpotential, the K¨ahler potential, and the gauge kinetic function. Readers who are not working in these areas can of course skip the formulae, but might find the words somewhat interesting. The moduli superpotential is of the form W = A eib1f1 +A eib2f2. (1) 1 2 The b’s are basically beta functions. Precisely, b = 2π/c where the c are the dual coxeter k k k numbers of the hidden sector gauge groups. W is a sum of such terms. Each term will stabilize all the moduli – the gauge kinetic functions are sums of the moduli with integer coefficients (written below) so expanding the exponentials a potential is generated for all the moduli. With two (or more)termsonecanseeincalculationsthatthemoduliarestabilizedinaregionwheresupergravity approximations are good, while with one term that might not be so. With two we can also get some semi-analytic results that clarify and help understanding, so we mostly work with two terms, thoughsomefeaturesarecheckednumericallywithmoreterms. Thisisnota“racetrack”potential; the relative sign of the terms is fixed by axion stabilization. The generic K¨ahler potential is K = −3ln(4π1/3V ) (2) 7 where the 7D volume is N (cid:88) V = sai (3) 7 i i=1 with the condition N (cid:88) a = 7/3. (4) i i=1 The gauge kinetic function is N (cid:88) f = Nkz . (5) k i i i=1 with integer coefficients. The z = t +is are the moduli, with real parts being the axion fields i i i and imaginary parts the zero modes of the metric on the 7D manifold; they characterize the size and shape of the manifold. 10