DDAMTP-2016-65
Complex Geometry of Nuclei and Atoms
M. F. Atiyah School of Mathematics, University of Edinburgh,James Clerk Maxwell Building,Peter Guthrie Tait Road, Edinburgh EH9 3FD, U.K.
N. S. Manton Department of Applied Mathematics and Theoretical Physics,University of Cambridge,Wilberforce Road, Cambridge CB3 0WA, U.K.
Abstract
We propose a new geometrical model of matter, in which neutral atoms aremodelled by compact, complex algebraic surfaces. Proton and neutron numbers aredetermined by a surface’s Chern numbers. Equivalently, they are determined bycombinations of the Hodge numbers, or the Betti numbers. Geometrical constraintson algebraic surfaces allow just a finite range of neutron numbers for a given protonnumber. This range encompasses the known isotopes.
Keywords: Atoms, Nuclei, Algebraic Surfaces, 4-Manifolds email: [email protected] email: [email protected] a r X i v : . [ h e p - t h ] S e p Introduction
It is an attractive idea to interpret matter geometrically, and to identify conserved at-tributes of matter with topological properties of the geometry. Kelvin made the pioneeringsuggestion to model atoms as knotted vortices in an ideal fluid [1]. Each atom type wouldcorrespond to a distinct knot, and the conservation of atoms in physical and chemicalprocesses (as understood in the 19th century) would follow from the inability of knots tochange their topology. Kelvin’s model has not survived because atoms are now knownto be structured and divisible, with a nucleus formed of protons and neutrons boundtogether, surrounded by electrons. At high energies, these constituents can be separated.It requires of order 1 eV to remove an electron from an atom, but a few MeV to removea proton or neutron from a nucleus.Atomic and nuclear physics has progressed, mainly by treating protons, neutrons andelectrons as point particles, interacting through electromagnetic and strong nuclear forces[2]. Quantum mechanics is an essential ingredient, and leads to a discrete spectrum ofenergy levels, both for the electrons and nuclear particles. The nucleons (protons andneutrons) are themselves built from three pointlike quarks, but little understanding ofnuclear structure and binding has so far emerged from quantum chromodynamics (QCD),the theory of quarks. These point particle models are conceptually not very satisfactory,because a point is clearly an unphysical idealisation, a singularity of matter and chargedensity. An infinite charge density causes difficulties both in classical electrodynamics [3]and in quantum field theories of the electron. Smoother structures carrying the discreteinformation of proton, neutron and electron number would be preferable.In this paper, we propose a geometrical model of neutral atoms where both the protonnumber P and neutron number N are topological and none of the constituent particlesare pointlike. In a neutral atom the electron number is also P , because the electron’selectric charge is exactly the opposite of the proton’s charge. For given P , atoms (or theirnuclei) with different N are known as different isotopes.A more recent idea than Kelvin’s is that of Skyrme, who proposed a nonlinear fieldtheory of bosonic pion fields in 3+1 dimensions with a single topological invariant, whichSkyrme identified with baryon number [4]. Baryon number (also called atomic massnumber) is the sum of the proton and neutron numbers, B = P + N . Skyrme’s baryons aresolitons in the field theory, so they are smooth, topologically stable field configurations.Skyrme’s model was designed to model atomic nuclei, but electrons can be added toproduce a model of a complete atom. Protons and neutrons can be distinguished in theSkyrme model, but only after the internal rotational degrees of freedom are quantised [5].This leads to a quantised “isospin”, with the proton having isospin up ( I = ) and theneutron having isospin down ( I = − ), where I the third component of isospin. The1odel is consistent with the well-known Gell-Mann–Nishijima relation [6] Q = 12 B + I , (1.1)where Q is the electric charge of a nucleus (in units of the proton charge) and B is thebaryon number. Q is integral, because I is integer-valued (half-integer-valued) when B is even (odd). Q equals the proton number P of the nucleus and also the electron numberof a neutral atom. The neutron number is N = B − I . The Skyrme model has hadconsiderable success providing models for nuclei [7, 8, 9, 10]. Despite the pion fields beingbosonic, the quantised Skyrmions have half-integer spin if B is odd [11]. But a feature ofthe model is that proton number and neutron number are not separately topological, andelectrons have to be added on.The Skyrme model has a relation to 4-dimensional fields that provides some motivationfor the ideas discussed in this paper. A Skyrmion can be well approximated by a projectionof a 4-dimensional Yang–Mills field. More precisely, one can take an SU(2) Yang–Millsinstanton and calculate its holonomy along all lines in the (euclidean) time direction [12].The result is a Skyrme field in 3-dimensional space, whose baryon number B equals theinstanton number.So a quasi-geometrical structure in 4-dimensional space (a Yang–Mills instanton inflat R ) can be closely related to nuclear structure, but still there is just one topologicalcharge. A next step, first described in the paper [13], was to propose an identificationof smooth, curved 4-manifolds with the fundamental particles in atoms – the proton,neutron and electron. Suitable examples of manifolds were suggested. These manifoldswere not all compact, and the particles they modelled were not all electrically neutral.One of the more compelling examples was Taub-NUT space as a model for the electron.By studying the Dirac operator on the Taub-NUT background, it was shown how thespin of the electron can arise in this context [14]. There has also been an investigationof multi-electron systems modelled by multi-Taub-NUT space [15, 16]. However, thereare some technical difficulties with the models of the proton and neutron, and no wayhas yet been found to geometrically combine protons and neutrons into more complicatednuclei surrounded by electrons. Nor is it clear in this context what exactly should be thetopological invariants representing proton and neutron number.A variant of these ideas is a model for the simplest atom, the neutral hydrogen atom,with one proton and one electron. This appears to be well modelled by CP , the complexprojective plane . The fundamental topological property of CP is that it has a generating2-cycle with self-intersection 1. The second Betti number is b = 1, which splits into b +2 = 1 and b − = 0. A complex line in CP represents this cycle, and in the projective CP had a different interpretation in [13]. R . The 3-sphere is a twistedcircle bundle over a 2-sphere (the Hopf fibration) and this is sufficient to account for theelectron charge.In this paper, we have a novel proposal for the proton and neutron numbers. The 4-manifolds we consider are compact, to model neutral atoms. Our previous models alwaysrequired charged particles to be non-compact so that the electric flux could escape toinfinity, and this is an idea we will retain. We also restrict our manifolds to be complexalgebraic surfaces, and their Chern numbers will be related to the proton and neutronnumbers. There are more than enough examples to model all currently known isotopesof atoms. We will retain CP as the model for the hydrogen atom. Complex surfaces [17] provide a rich supply of compact 4-manifolds. They are principallyclassified by two integer topological invariants, denoted c and c . For a surface X , c and c are the Chern classes of the complex tangent bundle. c is an integer because X has real dimension 4, whereas the (dual of the) canonical class c is a particular 2-cyclein the second homology group, H ( X ). c is the intersection number of c with itself, andhence another integer.There are several other topological invariants of a surface X , but many are related to c and c . Among the most fundamental are the Hodge numbers. These are the dimensionsof the Dolbeault cohomology groups of holomorphic forms. In two complex dimensionsthe Hodge numbers are denoted h i,j with 0 ≤ i, j ≤
2. They are arranged in a Hodgediamond, as illustrated in figure 1. Serre duality, a generalisation of Poincar´e duality,requires this diamond to be unchanged under a 180 ◦ rotation. For a connected surface, h , = h , = 1.Complex algebraic surfaces are a fundamental subclass of complex surfaces [18, 19].A complex algebraic surface can always be embedded in a complex projective space CP n ,and thereby acquires a K¨ahler metric from the ambient Fubini–Study metric on CP n .For any K¨ahler manifold, the Hodge numbers have an additional symmetry, h i,j = h j,i .For a surface, this gives just one new relation, h , = h , . Not all complex surfaces arealgebraic: some are still K¨ahler and satisfy this additional relation, but some are not3igure 1: The Hodge diamond for a general complex surface (left) and its entries in termsof Betti numbers for an algebraic surface (right).K¨ahler and do not satisfy it.Particularly interesting for us are the holomorphic Euler number χ , which is an alter-nating sum of the entries on the top right (or equivalently, bottom left) diagonal of theHodge diamond, and the analogous quantity for the middle diagonal, which we denote θ .More precisely, χ = h , − h , + h , , (2.1) θ = − h , + h , − h , . (2.2)(Note the sign choice for θ .) The Euler number e and signature τ can be expressed interms of these as e = 2 χ + θ , (2.3) τ = 2 χ − θ . (2.4)The first of these formulae reduces to the more familiar alternating sum of Betti numbers e = b − b + b − b + b , because each Betti number is the sum of the entries in thecorresponding row of the Hodge diamond. The second formula is the less trivial Hodgeindex theorem. τ is more fundamentally defined by the splitting of the second Bettinumber into positive and negative parts, b = b +2 + b − . Over the reals the intersectionform on the second homology group H ( X ) is non-degenerate and can be diagonalised. b +2 is then the dimension of the positive subspace, and b − the dimension of the negativesubspace. The signature is τ = b +2 − b − .The Chern numbers are related to χ and θ through the formulae c = 2 e + 3 τ = 10 χ − θ , c = e = 2 χ + θ . (2.5)Their sum gives the Noether formula χ = ( c + c ), which is always integral.For an algebraic surface, there are just three independent Hodge numbers and theyare uniquely determined by the Betti numbers b , b +2 and b − . The Hodge diamond must4igure 2: Hodge diamonds for the projective plane CP (left) and for a K3 surface (right)take the form shown on the right in figure 1, which gives the correct values for b , e and τ . Note that b must be even and b +2 must be odd. χ and θ are now given by χ = 12 (1 − b + b +2 ) , (2.6) θ = 1 − b + b − . (2.7)If X is simply connected, which accounts for many examples, then b = 0. Hodge di-amonds for the projective plane CP and for a K3 surface, both of which are simplyconnected, are shown in figure 2. For the projective plane χ = 1 and θ = 1, so e = 3 and τ = 1, and for a K3 surface χ = 2 and θ = 20, so e = 24 and τ = − χ as proton number P , and θ as baryon number B . So neutron number is N = θ − χ .This proposal fits with CP having P = 1 and N = 0. We will see later that for eachpositive value of P there is an interesting, finite range of allowed N values.In terms of e and τ , P = 14 ( e + τ ) , B = 12 ( e − τ ) , N = 14 ( e − τ ) . (2.8)Note that for a general, real 4-manifold, these formulae for P and N might be fractional,and would need modification. It is also easy to verify that in terms of P and N , c = 9 P − N , (2.9) c = e = 3 P + N , (2.10) τ = P − N . (2.11)The simple relation of signature τ to the difference between proton and neutron numbersis striking. If we write N = P + N exc , where N exc denotes the excess of neutrons overprotons (which is usually zero or positive, but can be negative), then τ = − N exc .5f an algebraic surface X is simply connected then b = 0, and in terms of P and N , b +2 = 2 P − , b − = P + N − P − N exc . (2.12)These formulae will be helpful when we consider intersection forms in more detail.The class of surfaces that we will use, as models of atoms, are those with c and c non-negative. Many of these are minimal surfaces of general type. Perhaps the mostimportant results on the geometry of algebraic surfaces are certain inequalities that theChern numbers of minimal surfaces of general type have to satisfy. The basic inequalitiesare that c and c are positive. Also, there is the Bogomolov–Miyaoka–Yau (BMY)inequality which requires c ≤ c , and finally there is the Noether inequality 5 c − c +36 ≥
0. These inequalities can be converted into the following inequalities on P and N : P > , ≤ N < P , N ≤ P + 6 . (2.13)All integer values of P and N satisfying these are allowed. The allowed region is shownin figure 3, and corresponds to the allowed region shown on page 229 of [17], or in thearticle [20].There are also the elliptic surfaces (including the Enriques surface and K3 surface)where c = 0 and c is non-negative, and we shall include these among our models. Here, P ≥ N = 9 P , so c = 12 P and τ = − P . CP is also allowed, even though it isrational and not of general type, because c and c are positive. In addition to CP , thereare further surfaces on the BMY line c = 3 c [21], which have P > N = 0.Physicists usually denote an isotope by proton number and baryon number, whereproton number P is determined by the chemical name, and baryon number is P + N .For example, the notation Fe means the isotope of iron with P = 26 and N = 30. Thecurrently recognised isotopes are shown in figure 4.The shape of the allowed region of algebraic surfaces qualitatively matches the regionof recognised isotopes, and this is the main justification for our proposal. For example,for P = 1, the geometric inequalities allow N to take values from 0 up to 9. Thiscorresponds to a possible range of hydrogen isotopes from H to H. Physically, the well-known hydrogen isotopes are the proton, deuterium and tritium, that is, H, H and Hrespectively, but nuclear physicists recognise isotopes of a quasi-stable nature (resonances)up to H, with N = 6.The minimal models for the common isotopes, the proton alone, and deuterium, eachbound to one electron, are CP and the complex quadric surface Q. The quadric is theproduct Q = CP × CP , with e = 4 and τ = 0. We shall say more about its intersectionform below. 6 ●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● Figure 3: Proton numbers P , and neutron numbers N , for atoms modelled as algebraicsurfaces. The allowed region is limited by inequalities on the Chern numbers, asdiscussed in the text. Note the change of slope from 9 to 7 at the point P = 3, N = 27 on the boundary. The line N = P corresponds to surfaces with zerosignature, i.e. τ = 0. 7igure 4: Nuclear isotopes. The horizontal axis is proton number P ( Z in physics nota-tion) and the vertical axis is neutron number N . The shading (colouring online)indicates the lifetime of each isotope, with black denoting stability (infinite life-time). 8or P = 2, N is geometrically allowed in the range 0 to 18. The correspondingalgebraic surfaces should model helium isotopes from He to He. Isotopes from Heup to He are physically recognised. All of these potentially form neutral atoms withtwo electrons. The helium isotope He with no neutrons is not listed in some nucleartables, but there does exist an unbound diproton resonance, and diprotons are sometimesemitted when heavier nuclei decay. The most common, stable helium isotope is He, withtwo protons and two neutrons, but He is also stable. He nuclei are also called alpha-particles, and play a key role in nuclear processes and nuclear structure. It is importantto have a good geometrical model of an alpha-particle, which ideally should match thecubically symmetric B = 4 Skyrmion that is a building block for many larger Skyrmions[22, 23, 8, 10]. Running through the nuclear isotopes is the valley of stability [2]. In figure 4, this isthe irregular curved line of stable nuclei marked in black. On either side, the nucleiare unstable, with lifetimes of many years near the centre of the valley, reducing tomicroseconds further away. Sufficiently far from the centre are the nuclear drip lines,where a single additional proton or neutron has no binding at all, and falls off in a timeof order 10 − seconds.For small nuclei, for P up to about 20, the valley is centred on the line N = P . Inthe geometrical model, this line corresponds to surfaces with signature τ = 0. For larger P , nuclei in the valley have a neutron excess, N exc , which increases slowly from just afew when P is near 20 to over 50 for the quasi-stable uranium isotopes with P = 92, andslightly more for the heaviest artifically produced nuclei with P approaching 120.In standard nuclear models, the main effect explaining the valley is the Pauli principle.Protons and neutrons have a sequence of rather similar 1-particle states of increasingenergy, and just one particle can be in each state. For given baryon number, the lowest-energy state has equal proton and neutron numbers, filling the lowest available states. Ifone proton is replaced by one neutron, the proton state that is emptied has lower energythan the neutron state that is filled, so the total energy goes up. An important additionaleffect is a pairing energy that favours protons to pair up and neutrons to pair up. Mostnuclei with P and N both odd are unstable as a result.For larger values of P , the single-particle proton energies tend to be higher than thesingle-particle neutron energies, because in addition to the attractive, strong nuclear forceswhich are roughly the same for protons and neutrons, there is the electrostatic Coulomb9epulsion that acts between protons alone. This effect becomes important for nuclei withlarge P , and favours neutron-rich nuclei. It also explains the instability of all nuclei with P larger than 83. These nuclei simply split up into smaller nuclei, either by emittingan alpha-particle, or by fissioning into larger fragments. However, the lifetimes can bebillions of years in some cases, which is why uranium, with P = 92, is found in nature inrelatively large quantities.Note that if N = P , then the electric charge is half the baryon number, and accordingto formula (1.1), the third component of isospin is zero. By studying nuclear ground statesand excited states, one can determine the complete isospin, and it is found to be minimalfor stable nuclei. So nuclei with N = P have zero isospin. When the baryon number isodd, the most stable nuclei have N just one greater than P (if P is not too large), and theisospin is . Within the Skyrme model, isospin arises from the quantisation of internaldegrees of freedom, associated with an SO(3) symmetry acting on the pion fields. There isan energy contribution proportional to the squared isospin operator I , analogous to thespin energy proportional to J . In the absence of Coulomb effects, the energy is minimisedby fixing the isospin to be zero or . The Coulomb energy competes with isospin, andshifts the total energy minimum towards neutron-rich nuclei.These are the general trends of nuclear energies and lifetimes. However there is alot more in the detail. Each isotope has its own character, depending on its proton andneutron numbers. This is most clear in the energy spectra of excited states, and the spinsof the ground and excited states. Particularly interesting is the added stability of nucleiwhere either the proton or neutron number is magic. The smaller magic numbers are 2, 8,20, 28, 50. It is rather surprising that protons and neutrons can be treated independentlywith regard to the magic properties. This appears to contradict the importance of isospin,in which protons and neutrons are treated as strongly influencing each other.Particularly stable nuclei are those that are doubly magic, like He, O, Ca and Ca. Ca is the largest stable nucleus with N = P . Ca is also stable, and occurs insmall quantities in nature, but is exceptionally neutron-rich for a relatively small nucleus.The important issue for us here is to what extent our proposed geometrical modelbased on algebraic surfaces is compatible with these nuclear phenomena, not forgettingthe electron structure in a neutral atom. There are some broad similarities. First thereis the “geography” of surfaces we have discussed above, implying that the geometricalinequalities restrict the range of neutron numbers. Algebraic geometers also refer to“botany”, the careful construction and study of surfaces with particular topological in-variants. The patterns are very complicated. Some surfaces are simple to construct,others less so, and their internal structure is very variable. This is analogous to the com-plications of the nuclear landscape, and the similar complications (better understood) ofthe electron orbitals and atomic shell structure.10ather remarkable is that the line of nuclear stability where N = P corresponds tothe simple geometrical condition that the signature τ is zero. We have not yet triedto pinpoint an energy function on the space of surfaces, but clearly it would be easyto include a dominant contribution proportional to τ , whose minimum would be inthe desired place. Mathematicians have discovered that it is much easier to constructsurfaces on this line, and on the neutron-rich side of it, where τ is negative, than on theproton-rich side. There are always minimal surfaces on the neutron-rich side which aresimply connected, but not everywhere on the proton-rich side. The geometry of surfacestherefore distinguishes protons from neutrons rather clearly. This is attractive for thephysical interpretation, as it can be regarded as a prediction of an asymmetry betweenthe proton and neutron. In standard nuclear physics it is believed that in an ideal worldwith no electromagnetic effects, there would be an exact symmetry between the protonand neutron, but in reality they are not the same, partly because of Coulomb energy,but more fundamentally, because their constituent up (u) and down (d) quarks are notidentical in their masses, making the proton (uud) less massive than the neutron (udd),despite its electric charge.The geometrical model would need an energy contribution that favours neutrons overprotons for the larger nuclei and atoms. One possibility has been explored by LeBrun[24, 25]. This is the infimum, over complex surfaces with given topology, of the L normof the scalar curvature. For surfaces with b even, including all surfaces that are simplyconnected, this infimum is simply a constant multiple of c . The scalar curvature can bezero for surfaces on the line c = 0, for example the K3 surface, which is the extremeof neutron-richness, with P = 2 and N = 18. It would be interesting to consider morecarefully the energy landscape for an energy that combines τ and a positive multiple ofthe L norm of scalar curvature. A complex surface X is automatically oriented, so any pair of 2-cycles has an unambiguousintersection number [26]. Given a basis α i of 2-cycles for the second homology group H ( X ), the matrix Ω ij of intersection numbers is called the intersection form of X . Ω ij ≡ Ω( α i , α j ) is the intersection number of basis cycles α i and α j , and the self-intersectionnumber Ω ii is the intersection number of α i with a generic smooth deformation of itself. Ωis a symmetric matrix of integers, and by Poincar´e duality it is unimodular (of determinant ± −
1. The numbers of each of these are b +2 and b − , respectively, and wehave already given an interpretation of them for simply-connected algebraic surfaces X in terms of P and N in equation (2.12) above.11owever, diagonalisation over the reals does not make sense for cycles, because onecan end up with fractional cycles in the new basis. One may only change the basis ofcycles using an invertible matrix of integers, whose effect is to conjugate Ω by such amatrix. The classification of intersection forms is finer over the integers than the reals.For almost all algebraic surfaces, Ω is indefinite. b +2 is always positive, and b − ispositive too, except for surfaces with b = 0 and B = θ = 1. So the only surfacesfor which the intersection form Ω is definite are CP , and perhaps additionally the fakeprojective planes, for which we have not found a physical interpretation. For CP , with P = 1 and N = 0, the intersection form is the 1 × ii is odd, or more invariantly, Ω( α, α ) is odd for some 2-cycle α . An oddform can always be diagonalised, with entries +1 and − α, α ) is even for any cycle α . The simplestexample is Ω = (cid:18) (cid:19) . (4.1)This is the intersection form of the quadric Q, with the two CP factors as basis cycles, α and α . If α = xα + yα then Ω( α, α ) = 2 xy , so is always even. Over the reals this formcan be diagonalised and has entries +1 and − − E . This is the negative of the Cartan matrix of theLie algebra E (with diagonal entries − l (cid:18) (cid:19) ⊕ m ( − E ) , (4.2)with l > m ≥ l must be odd, and the Betti numbers are b +2 = l and b − = l + 8 m .The signature is τ = − m .For most surfaces, the signature is not a multiple of 8, so the intersection form is odd.If the signature is a multiple of 8, it may be even. For given Betti numbers, there couldbe two distinct minimal surfaces (or families of these), one with an odd intersection form,and the other with an even intersection form. We do not know if surfaces with both typesof intersection form always occur. 12e can reexpress these conditions in terms of the physical numbers P and N . If N exc = N − P is neither zero nor a positive multiple of 8, then the intersection formmust be odd. If N = P , then the intersection form can be of the hyperbolic plane type l (cid:18) (cid:19) , with l = 2 P −
1, or it might still be odd. Notice that l is odd, as it must be.The isotopes for which even intersection forms are possible therefore include all those with N = P . These are numerous. In addition to the stable isotopes with N = P that occur upto Ca, with P = 20, there are many that are quasi-stable, like Fe, with P = 26. Theheaviest recognised isotopes with N = P are Sn and perhaps
Xe, with P = 50 and P = 54. Our geometrical model suggests that the additional stability of these isotopes isthe result of the nontrivial structure of an even intersection form.If N exc = 8 m then the intersection form can be of type (4.2), again with l = 2 P − l = 1 and m = 1,and the K3 surface, for which l = 3 and m = 2. The potential isotopes correspondingto these surfaces are H and He. These are both so neutron-rich that they have notbeen observed, but there are many heavier nuclei (and corresponding atoms) for whichthe neutron excess N exc is a multiple of 8.There is some evidence that nuclei whose neutron excess is a multiple of 8 have addi-tional stability. The most obvious example is Ca, but this is conventionally attributedto the shell model, as P = 20 and N = 28, both magic numbers. A more interesting andless understood example is the heaviest known isotope of oxygen, O, with 8 protons and16 neutrons. This example and others do not obviously fit with the shell model. Themost stable isotope of iron is Fe, whose neutron excess is 4, but it is striking that Fe,whose neutron excess is 8, has a lifetime of over a million years. Here P = 26 and N = 34. Ni, also with a neutron excess of 8, is one of the stable isotopes of nickel. There are alsostriking examples of stable or relatively stable isotopes with neutron excesses of 16 or 24.Some of these are outliers compared to the general trends in the valley of stability. Anexample is
Sn, the heaviest stable isotope of tin, with N exc = 24. A more careful studywould be needed to confirm if the additional stability of isotopes whose neutron excess isa multiple of 8 is statistically significant.There is no evidence that a neutron deficit of 8 has a stabilising effect. In fact, almostno nuclei with such a large neutron deficit are recognised. The only candidate is Ni,with the magic numbers P = 28 and N = 20.13 Other Surfaces
In addition to the minimal surfaces of general type there are various other classes ofalgebraic surface. Do these have a physical interpretation?On a surface X it is usually possible to “blow up” one or more points. The result isnot minimal, because a minimal surface, by definition, is one that cannot be constructedby blowing up points on another surface. Blowing up one point increases c by 1 anddecreases c by 1. This is equivalent, in our model, to increasing N by 1, leaving P unchanged. In other words, one neutron has been added. Topologically, blowing up isa local process, equivalent to attaching (by connected sum) a copy of CP . This adds a2-cycle that has self-intersection −
1, but no intersection with any other 2-cycle. The rank(size) of the intersection form Ω increases by 1, with an extra − . The result is the Hirzebruchsurface H , which is a non-trivial CP bundle over CP . Its intersection form is (cid:18) − (cid:19) .The Hirzebruch surface and quadric are both simply connected and have the same Bettinumbers, b +2 = b − = 1, corresponding to P = 1 and N = 1, but the intersection form isodd for the Hirzebruch surface and even for the quadric. The proposed interpretation isthat the Hirzebruch surface represents a separated proton, neutron and electron, whereasthe quadric represents the deuterium atom, with a bound proton and neutron as itsnucleus, orbited by the electron.There is an inequality of LeBrun for the L norm of the Ricci curvature supporting thisinterpretation [24, 25]. The norm increases if points on a minimal surface are blown up,the increase being a constant multiple of the number of blown-up points. This indicatesthat both the norm of the Ricci curvature and the norm of the scalar curvature, possiblywith different coefficients, should be ingredients in the physical energy.So far, we have not considered any surfaces X that could represent a single neutron, ora cluster of neutrons. Candidates are the surfaces of Type VII. These have c = − c , with c positive, equivalent to P = 0 and arbitrary positive N . These surfaces are complex, but14re not algebraic and do not admit a K¨ahler metric. They are also not simply connected.It is important to have a model of a single neutron. The discussion of blow-ups suggeststhat CP is another possible model. In this case a single neutron would be associated witha 2-cycle with self-intersection −
1, mirroring the proton inside CP being represented bya 2-cycle with self-intersection +1.A free neutron is almost stable, having a lifetime of approximately 10 minutes. Thereis considerable physical interest in clusters of neutrons. There is a dineutron resonancesimilar to the diproton resonance. Recently there has been some experimental evidencefor a tetraneutron resonance, indicating some tendency for four neutrons to bind [27].Octaneutron resonances have also been discussed, but no conclusive evidence for theirexistence has yet emerged. Neutron stars consist of multitudes of neutrons, accompaniedperhaps by a small number of other particles (protons and electrons), but their stabilityis only possible because of the gravitational attraction supplementing the nuclear forces.Standard Newtonian gravity is of course negligible for atomic nuclei.Products of two Riemann surfaces (algebraic curves) of genus 2 or more are examplesof minimal surfaces of general type, but they are certainly not simply connected. Theirinterpretation as atoms should be investigated. Other surfaces, for example ruled sur-faces, may have some physical interpretation, but our formulae would give them negativeproton and neutron numbers. They do not model antimatter, that is, combinations ofantiprotons, antineutrons and positrons, because antimatter is probably best modelledusing the complex conjugates of surfaces modelling matter. Also bound states of protonsand antineutrons, with positive P and negative N , do not seem to exist. We have proposed a new geometrical model of matter. It goes beyond our earlier proposal[13] in that it can accommodate far more than just a limited set of basic particles. Inprinciple, the model can account for all types of neutral atom.Each atom is modelled by a compact, complex algebraic surface, which as a realmanifold is four-dimensional. The physical quantum numbers of proton number P (equalto electron number for a neutral atom) and neutron number N are expressed in terms ofthe Chern numbers c and c of the surface, but they can also be expressed in terms ofcombinations of the Hodge numbers, or of the Betti numbers b , b +2 and b − .Our formulae for P and N were arrived at by considering the interpretation of justa few examples of algebraic surfaces – the complex projective plane CP , the quadricsurface Q, and the Hirzebruch surface H . Some consequences, which follow from the15nown constraints on algebraic surfaces, can therefore be regarded as predictions of themodel. Among these are that P is any positive integer, and that N is bounded belowby 0 and bounded above by the lesser of 9 P and 7 P + 6. This encompasses all knownisotopes. A most interesting prediction is that the line N = P , which is the centre of thevalley of nuclear stability for small and medium-sized nuclei, corresponds to the the line τ = 0, where τ = b +2 − b − is the signature. Surfaces with τ positive and τ negative areknown to be qualitatively different, which implies that in our model there is a qualitativedifference between proton-rich and neutron-rich nuclei.For simply connected surfaces with b = 0 (or more generally, if b is held fixed)then an increase of P by 1 corresponds to an increase of b +2 by 2. The interpretationis that there are two extra 2-cycles with positive self-intersection, corresponding to theextra proton and the extra electron. This matches our earlier models, where a protonwas associated with such a 2-cycle [13], and where multi-Taub-NUT space with n NUTsmodelled n electrons [15, 16]. On the other hand, an increase of N by 1 corresponds toan increase of b − by 1. This means that a neutron is associated with a 2-cycle of negativeself-intersection, which differs from our earlier ideas, where a neutron was modelled bya 2-cycle with zero self-intersection. It appears now that the intersection numbers arerelated to isospin (whose third component is for a proton and − for a neutron) ratherthan to electric charge (1 for a proton and 0 for a neutron).Clearly, much further work is needed to develop these ideas into a physical model ofnuclei and atoms. We have earlier made a few remarks about possible energy functionsfor algebraic surfaces. Combinations of the topological invariants and non-topologicalcurvature integrals should be explored, and compared with the detailed information onthe energies of nuclei and atoms in their ground states. It will be important to accountfor the quantum mechanical nature of the ground and excited states, their energies andspins. Discrete energy gaps could arise from discrete changes in geometry, for example,by replacing a blown-up surface with a minimal surface, or by considering the effect ofchanging b while keeping P and N fixed, or by comparing different embeddings of analgebraic surface in (higher-dimensional) projective space. In some cases there shouldbe a discrete choice for the intersection form. There are also possibilities for finding ananalogue of a Schr¨odinger equation using linear operators, like the Laplacian or Diracoperator, acting on forms or spinors on a surface. Alternatively, the right approach maybe to consider the continuous moduli of surfaces as dynamical variables, and then quantisethese. The moduli should somehow correspond to the relative positions of the protons,neutrons and electrons. Some of the ideas just mentioned have already been investigated inthe context of single particles, modelled by the Taub-NUT space or another non-compact4-manifold [14, 28]. Further physical processes, for example, the fission of larger nuclei,and the binding of atoms into molecules, also need to be addressed.16efore these investigations can proceed, it will be necessary to decide what metricstructure the surfaces need. Previously, we generally required manifolds to have a self-dual metric, i.e. to be gravitational instantons, but this now seems too rigid, as there arevery few compact examples. Requiring a K¨ahler–Einstein metric may be more reasonable,although these do not exist for all algebraic surfaces [29, 30]. We plan to pursue the manyissues raised here in a subsequent paper. Acknowledgements
We are grateful to Chris Halcrow for producing figures 1-3, and Nick Mee for supplyingfigure 4.
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