QMW-PH/00-01 SWAT-TH/00-257 ITP-UU-00/05 hep-th/0003155 Spin and abelian electromagnetic duality on four-manifolds David I. Olive* Department of Physics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK e-mail: [email protected] 0 and 0 0 Marcos Alvarez 2 Department of Physics, Queen Mary and Westfield College r a Mile End Road, London E1 4NS, UK M e-mail: [email protected] 7 1 Abstract 1 We investigate the electromagnetic duality properties of an abelian gauge theory on v 5 a compact oriented four-manifold by analysing the behaviour of a generalised partition 5 function under modular transformations of the dimensionless coupling constants. The true 1 partition function is invariant under the full modular group but the generalised partition 3 0 function exhibits more complicated behaviour depending on topological properties of the 0 four-manifold concerned. It is already known that there may be “modular weights” which 0 / arelinearcombinations oftheEulernumber and Hirzebruch signatureofthefour-manifold. h t But sometimes the partition function transforms only under a subgroup of the modular - p group (the Hecke subgroup). In this case it is impossible to define real spinor wave- e functions on the four-manifold. But complex spinors are possible provided the background h : magneticfluxesareappropriatelyfractionalratherthanintegral. Thisgivesrisetoasecond v i partition function which enables the full modular group to be realised by permuting the X two partition functions, together with a third. Thus the full modular group is realised r a in all cases. The demonstration makes use of various constructions concerning integral lattices and theta functions that seem to be of intrinsic interest. * On leave at Instituut voor Theoretische Fysica, University of Utrecht, Postbus 80.006, 3508 TA Utrecht 1 1. Introduction It is now widely accepted that electromagnetic duality provides a powerful and useful new principle which is valid in a large class of physically interesting quantum field theories in a Minkowski space-time of four dimensions. The prototype for the quantum version of this idea was furnished by a proposal of Montonen and one of the present authors, [Montonen and Olive 1977], who considered the context of a special sort of spontaneously broken SU(2) gauge theory, later realised to be naturally supersymmetric [D’Adda, Di Vecchia and Horsley 1978, Witten and Olive 1978, Osborn 1979]. This can be regarded as a semi-realistic theory of unified particle interactions as it is the same sort of theory as the standard model even though it differs in some crucial respects. The manner in which duality is realised on particle states requires magnetically charged states arising as solitons solutions and, in addition, quantum bound state of these [Sen 1994]. This acceptance, which now extends to superstring theories, where solitons occur as higher branes, has been achieved despite the fact that no sort of proof has been found, even in the case of the SU(2) gauge theory with the highest allowable degree of supersymmetry, N = 4. It is this situation for which most evidence has steadily accumulated that the idea is exactly true. This supporting evidence has been facilitated bythe acquisition of new mathematical techniques that have enhanced our understanding of quantum field theory. For example, the Atiyah-Singer index theorem, which is related to the theory of axial anomalies, plays a ubiquitous role. A modifed version of this theorem is crucial in determining both classical and quantum properties of self-dual monopole solutions. Since the quantum electromagnetic duality transformations combine to form a group related to the modular group (or its generalisations) it is reasonable to expect some version of the theory of modular forms to become increasingly important. Conversely the idea of electromagnetic duality has led to breakthroughs in the classification theory of four- manifolds (playing the role of Euclidean space-times for twisted supersymmetric gauge theories). It would be nice to have a simpler, toy model which, although less realistic physically, could, in compensation, be more tractable mathematically. Indeed, such a theory exists. It is simply free Maxwell theory with only putative couplings, not realised in practice. Thus particle states possess what could be called a “Cheshire cat” existence, that will be become clearer later. Such a theory would be too trivial on flat space time and, in order to obtain some worthwhile structure, the ambient space-time manifold has to be allowed to be fairly general. It is taken to be smooth, compact and oriented. Hence it is a four- manifold which obeys the topological symmetry known as Poincar´e duality, and this will be a crucial ingredient forestablishing electromagnetic duality here. This toy model follows the original proposals by E Verlinde and E Witten in 1995. What is interesting about this model is that much of the same sort of mathematical structure as mentioned above again comes into play. In particular there enter modular invariantthetafunctionsofarathergeneralnature. TheAtiyah-Singerindextheoremplays a mysterious role interlocked with the modular group and its subgroups of index three and the nature of space-time and its possible spin structures. This is despite the significant differences between the two situations (no hint of supersymmetry, curved Riemannian space-time in one case, N = 4 supersymmetry, flat, non-compact Minkowski space-time in 2 the other). This points to the conclusion that electromagnetic duality is indeed a rather general phenomenon. Moreoveritislikelythatthispictureextends toanyspace-time withMinkowskimetric and dimension which isa multiple offour. If thisdimension isdenoted 4k then theputative coupling of Maxwell field strengths with charged particles is replaced by a coupling of a 2k-form field strength to (2k 2)-branes. − In section 2 we start with the naive idea of electromagnetic duality as a classical symmetry of the energy-momentum tensor of Maxwell theory in Minkowski space with respect to rotations between the electric and magnetic fields. It is shown how this idea can be extended to a largergroupof SL(2,R)transformations acting onthe energy momentum tensor. Itisexplained how thisleadsinturn toconsideration oftheFeynmanpath integral, of, rather surprisingly, the exponential of the Euclidean action. A special case of this Euclidean path integral is the Minkowski space partition function. The effect of the Dirac quantisation condition for magnetic fluxes exhibits some extra subtleties in space-times of four dimensions, particularly when complex spinor wave func- tions are considered. This is reviewed in section 3. The effect is to break the continuous SL(2,R) group to a discrete subgroup related to the modular group in a way that is ex- plained in sections 4 and 5. It is noteworthy that nothing like the Zwanziger-Schwinger quantisation condition for dyonic charges plays any role. Owing to the complicated topology of the four-manifold of space-time, the possible magnetic fluxes are related to a lattice formed by the free part of the second homology group. This lattice is unimodular by virtue of Poincar´e duality and its even or oddness properties are related to the presence or absence of spin structures on the space-time four manifold. Infactfour-manifoldscanbeseparatedintothreedistincttypeswhoseproperties are described in section 3. In section 4 properties of integral lattices are reviewed. Starting from an odd unimod- ular lattice a general construction is given of an even integral lattice, leading sometimes to new unimodular lattices which can be both odd or even. Furthermore, also in section 4 the relevance to the Dirac quantisation condition and the question of spin structures is explained. Insection 5, followingthearguments ofEVerlindeandE Witten[1995], the“extended partition functions” are evaluated explicitly using the Dirac quantisation conditions and the semiclassical method. The results are proportional to generalised theta functions asso- ciated with the unimodular lattice formed by the free part of the second homology group. Particular attention is paid to the different possibilities afforded by the compatibility of either scalar or spinor complex wave functions on four-manifolds of type II, i.e. when the relevant lattice is odd. In section 6 a more general construction is presented that associates theta functions with any integral lattice, not necessarily unimodular, whether or not the scalar product is positive definite. Such integral lattices are contained as subgroups of the lattices reciprocal to them and so define a finite number of cosets, to each of which corresponds a theta function. An action of the modular group is defined on these theta functions when the lattice is even and of the Hecke subgroup when the lattice is odd. Careful analysis of the self-consistency of this action furnishes a proof of “Milgram’s formula”, valid for any even 3 integral lattice. This expresses the signature of the lattice, mod eight, in terms of coset properties. In section 7, the theta function construction of section 6 is applied to the even integral lattice associated with an odd unimodular lattice by the construction of section 4. The result is toassociateup to four theta functions witha givenodd unimodularlattice, though there are usually linear dependences. When the odd unimodular lattice corresponds to that associated with any type II four-manifold, two of these theta functions enter the distinct Maxwell partition functions for fluxes supporting either scalar or spinor complex wave functions, provided the electric charges they carry coincide. Although each partition function is individually covariant with respect to only a subgroup of SL(2,Z) of index three, they are related to each other by the missing transformations. In this way the full SL(2,Z) group of electromagnetic duality transformations is restored for space-time four-manifolds of type II. 2. Abelian gauge fields and electromagnetic duality WeshallusuallybeconsideringasingleabelianMaxwellfieldstrengthdescribed, inexterior calculus notation, by a closed two form F on a space-time manifold which is closed, 4 M compact, connected, smooth and oriented. Nevertheless much of the argument extends to higher dimensional space-times of the same type, (which will be denoted ), as long as 4k M their dimension is a multiple of four and the field-strength F is a closed 2k-form, that is a mid-form. It would then be a generalised Kalb-Ramond field, [Kalb and Ramond 1974], that could couple to the world-volume of a 2k 2-brane. − For reasons that will become clear it is important to allow the space-time manifold to be topologically complicated. Sometimes we shall suppose that be such that it can 4k M be endowed with a Minkowski metric (that is with one time component) and this would require that its Euler number χ( ) vanish. We may further require that this metric 4k M can always be “Wick rotated” to a Euclidean metric (with no time components) by an analytic continuation. This may impose further constraints on the topological properties of . 4k M With either sort of metric the Hodge star operation, , can be defined, converting ∗ p-forms on to 4k p-forms. Acting on 2k-forms, that is, mid-forms, the repeated 4k M − action of the Hodge star operation yields = ( 1)t, (2.1) ∗∗ − where t equals the number of time components, that is, one or zero. So, in the Minkowski case, has eigenvalues i and hence no real eigenfunctions. It is this Minkowski situation, ∗ ± rather than the Euclidean one, in which electromagnetic duality can apply. The basic idea of a resemblance between the parts of F thought of as the electric and magnetic fields, E and B, is very old and was reinforced by Maxwell’s discovery of his equations governing their behaviour in vacuo. “Duality rotations” between E and B provide a symmetry of the Minkowski energy density (E2 + B2)/2 (this particular expression applies when the space-time is flat), and more generally the complete energy momentum tensor. Moreover the rotations map between solutions of the equations even though the action does change. 4 This idea wasextended in thecontext ofsupergravity theories by Gaillardand Zumino [1981], building on ideas of Cremmer and Julia [1979]. The following action can be defined on any of the space-times mentioned: 1 W = F τˆF, (2.2) 2τ ∧ 2 ZM4 where, in Minkowski space, θ 2π¯h τˆ = τ + τ = + , (2.3) 1 ∗ 2 2π ∗ q2 so that this action, (2.2), is indeed real. In suitably extended supergravity theories τ will depend upon scalar fields related to the metric tensor by supersymmetry transformations. As far as this paper is concerned, there are no scalar fields and no supersymmetry. τ and 1 τ are simply dimensionless quantities parametrising the theory. It will be convenient to 2 combine them as real and imaginary parts of a single complex variable θ 2π¯h τ = τ +iτ = +i (2.4) 1 2 2π q2 The conventional Maxwell term is 1 F F. The other term, called the theta term, has 2 ∧∗ no apparent effect classically as it affects neither the Euler-Lagrange equations, nor the R value of the energy-momentum tensor. However it can affect the quantum phase when the topology of the background space-time is sufficiently non-trivial. In circumstances to be explained, (τ )−1 F F is quantised so that the exponentiated quantum action, 2 M4 ∧ exp(iW), depends upon the parameter θ in a periodic manner. But the dependence upon h¯ R θ disappears altogether when the topology is too trivial, that is, when the second Betti number of vanishes. 4 M Following Gaillard and Zumino we consider the effect of the linear transformations: τˆF τˆ′F′ = AτˆF +BF (2.5a) → F F′ = CτˆF +DF, (2.5b) → where A,B,C and D are real constants. Then the dimensionless complex coupling constant variable, τ, (2.4) undergoes the fractional linear transformation Aτ +B τ τ′ = . (2.6a) → Cτ +D while its imaginary part τ undergoes 2 τ (AD BC) τ τ′ = 2 − . (2.6b) 2 → 2 Cτ +D 2 | | 5 The transformations (2.5) evidently provide symmetries of the two equations of motion dF = 0 and dτˆF = 0. Now we consider the effect on the symmetric energy momentum tensor T , obtained from (2.2) by variation of the metric and written in the Sugawara µν form, 1 T = (F gλσF +∗ F gλσ∗F ), (2.7) µν µλ σν µλ σν 2 where F = F dxµ dxν/2 and ∗F =∗ F dxµ dxν/2. The result is µν µν ∧ ∧ T T′ = Cτ +D 2T . µν → µν | | µν If we restrict the transformations (2.5) to the subgroup leaving τ unchanged in (2.6), evidently τ is also unchanged and so by (2.6b), the result is 2 T T′ = (AD BC)T . µν → µν − µν Hence the energy momentum tensor is invariant under the subgroup if AD BC = 1 (2.8) − and this yields a U(1) subgroup comprising the duality rotations previously mentioned. The symmetry of the energy momentum tensor (2.7) can be enlarged from this U(1) to the full SL(2,R) if the transformations (2.5) are modified in the following natural way. Let us substitute for the physical field strengths F more geometrical quantities G, by F = h¯G/q. G is more geometrical in the sense that its fluxes, unlike those of F are dimensionless. In terms of G the dimensionless action is given by W 1 = G τˆG ¯h 4π ∧ ZM4 and so is explicitly a function of τ. Now the energy momentum tensor (2.7) reads ¯hτ T = 2(G gλσG +∗ G gλσ∗G ). (2.7′) µν µλ σν µλ σν 4π so that a dependence on τ is made explicit. Now substitute G for F in (2.5), thereby 2 defining new duality transformations, still leading to (2.6). Under these transformations acting on both G and τ in the energy momentum tensor T in the form (2.7’), the µν preceding calculations show that T , is invariant providing only that (2.8) holds. µν Thus now the dimensionless complex coupling τ changes whilst preserving the positive nature of τ . This is appropriate as τ is the inverse of the fine structure constant and 2 2 hence intrinsically positive. These transformations form the three dimensional non compact group SL(2,R), or what is the same by a group theory isomorphism, the symplectic group Sp(2,R). The transformations also mapbetween solutionsof thefree Maxwell equationsand the key question will be to what extent these transformations continue to provide symmetries whenthereisapossibilityofelectricallychargedparticles(orbranes)beingpresent, subject 6 to the rules of quantum theory. Because it is the Minkowski energy, rather than the action, which is invariant classically, the natural quantity to consider in the quantum theory is the partition function constructed with this energy: Z(τ) = Tr e−E(τ) . (2.9) (cid:16) (cid:17) It is this partition function that will be the candidate for quantum electromagnetic duality, just as it is the partition function that displays the Kramers-Wannier duality of the Ising model [Kramers and Wannier 1941]. Indeed it will be found that (2.9) is invariant under thetransformations (2.6)provided they arerestricted to adiscrete subgroup, isomorphicto the modular group. The discreteness is a consequence of the Dirac quantisation condition that the magnetic fluxes have to satisfy in order to permit complex wave functions. For rather general, nonlinear dynamical systems with a finite number of degrees of freedom and the property that the action includes only terms quadratic, linear and inde- pendent of velocities there is a Feynman path integral expression for the partition function (2.9). Z(τ) = ... δAeh¯iWEUCLIDEAN. (2.10) Z Z The Euclidean action W is obtained from the original action by what can be EUCLIDEAN thought of as a “Wick rotation” whereby velocities are multiplied by i and time by i. As − a result, iW has an imaginary part linear in velocities, and a real part that EUCLIDEAN is negative definite if the original energy is positive. Consequently the path integral is highly convergent. Because of the trace in (2.9), the paths integrated over are closed paths traversed in configuration space in unit time with distinguished end points. This result is known as the Feynman-Kac formula [Feynman and Hibbs, Feynman]. Thepresence ofacomplexphasefactorin(2.10)duetotermslinearinvelocityappears to contradict the manifest reality of the partition function as defined in (2.9). But this is illusory because the space of closed paths in configuration space that are integrated over possess a Z symmetry with respect to the interchange of pairs of identical paths 2 differing only in the sense of time evolution along the path. Under this interchange the two contributions to exp iW are related by complex conjugation. As a result h¯ EUCLIDEAN the sum of these two complex contributions is indeed real. Because it is quadratic in field strengths, something similar happens with the more complicated action (2.2) under consideration here. As a result of a similar argument, the Maxwell partition function can be expressed in the form (2.10) where now W EUCLIDEAN is obtained from W, (2.2), by a “Wick rotation” of the metric tensor. This tensor only enters the Maxwell term as the theta term is “topological” and hence independent of the metric, and so unaffected by the Wick rotation. The result is that W is given EUCLIDEAN by the same expression as before, (2.2), when it is understood that τ is replaced by θ 2π¯h τ = τ +i τ = +i . (2.11) EUCLIDEAN ∗ 2 2π ∗ q2 The metric dependence is encoded in the Hodge operator which, by (2.1), now has unit ∗ square and hence eigenvalues 1. This has two consequences. One is that iW EUCLIDEAN ± 7 is complex when the field strengths are real and that its real part is negative definite, thereby ensuring convergence of the integral over gauge potentials A in (2.9). The other consequence is that if is regarded as imaginary in Minkowski space-time and real in ∗ Euclidean space, in view of its eigenvalues, then τ has the same complex structure in either case. In this sense it is unaffected by the Wick rotation. Accordingly the complex variable τ given by the expression (2.4) not involving the Hodge is equally relevant with ∗ either metric. In evaluating this partitionfunction the space-time four manifold has to be considered as = S , where “time” is the coordinate around the circle, (periodic because 4 1 3 M × M of the trace), and an appropriate section of . If the metric on S factorises 3 4 1 3 M M ×M correspondingly it is easy to see that the partition function will be again real because reversing the sense of time around this circle will effect a complex conjugation of the two quantum amplitude contributions. It will turn out to be highly instructive to consider what, by abuse of terminology, is often also called a “partition function”. This expression is given by the path integral (2.10) but with the integral in the action being over the four manifold given by the full space- time , instead of S . This path integral can be defined even for four manifolds 4 1 3 M ×M with non-vanishing Euler number, that is ones for which a Minkowski metric is impossible. There isno reason forthisnew quantityto bereal but it willturnout to havean interesting response to the electromagnetic duality transformations (2.6) (as pointed out by E Witten and E Verlinde). So these extended partition functions do have interesting mathematical properties as we shall see in more detail and it would be interesting to understand what, if any, physical significance they have. We shall henceforth refer to the real partition functions associated with S as “strict partition functions”. 1 3 ×M We shall see that both the Euler number, χ( ), and the Hirzebruch signature, 4 M η( ), vanish for manifolds S . These are the two topological invariants of a 4 1 3 4k M ×M M manifoldthatare“local”inthesense thattheycanbeexpressedasintegralsofclosedforms over the manifold. Linear combinations of these topological invariants, namely (χ η)/2, ± will specify in a precise way how the extended partition functions deviate from satisfying exact electromagnetic duality. These conclusions will depend on the explicit evaluation of the functional integral expression (2.10) for the partition function on any , and this is facilitated by taking 4 M account of another aspect of the quantum theory. If electrically charged particles are to be treated quantum mechanically, the background field strengths must satisfy certain Dirac flux quantisation conditions in order to allow the possible presence of complex wave functions for them. It is this that imposes a discrete structure that converts (2.10) to a sum rather than an integral, at least in the semiclassical approximation, which is very likely exact. As we shall see, the resultant expression for the partitionfunction (2.9) is proportional to an infinite sum forming a generalised sort of theta function associated with the lattice of homology classes of two-cycles in the space-time four-manifold . As explained below, 4 M this lattice is what is known as the free part of H ( ,Z) and is intimately connected to 2 4 M the Dirac quantised fluxes. If is orientable, smooth, closed and compact, its topological structure satisfies the 4 M 8 symmetry known as Poincar´e duality. This implies that the following relation between the five Betti numbers of : 4 M b = b , b = b . (2.12) 0 4 1 3 Hence the Euler number is given by χ( ) = 2(b b )+b . (2.13) 4 0 1 2 M − Furthermore, the aforementioned lattice, which has dimension b , is unimodular with re- 2 spect to the scalar product furnished by the intersection number. It is this which is the origin of the covariance of (2.9) with respect to the S-transformation of electromagnetic duality: 0 1 S = − (2.14) 1 0 (cid:18) (cid:19) sending τ to 1/τ, a especially interesting example of (2.6). Before explaining this we − must review the Dirac quantisation condition for fluxes in more detail, paying particular attention to the extra subtleties associated with spinning particles carrying electric charge. 3. The quantisation condition on four-manifolds As already explained, we consider Maxwell theory in space-times that are compact, con- nected and oriented four-dimensional manifolds , whether or not their Euler number 4 M vanishes. These spaces are particularly convenient because they satisfy Poincar´e duality, which is a topological property closely related to electromagnetic duality. The main math- ematical toolneeded inunderstanding theimplicationsof theglobal topologyof on the 4 M duality properties of Maxwell theory is homology and cohomology theory. This discipline is described in many textbooks (see for example [Schwarz 1994]), and a short introduction to the relevant ideas was given in our earlier paper [M Alvarez and Olive 1999]. We shall follow the notations of the latter without full explanation. The physical relevance of homology and cohomology theory is that it provides the natural mathematical language for the ideas of Faraday, Maxwell and, later, Dirac con- cerning electromagnetic theory. Important physical quantities are the (magnetic) fluxes of the field strength F through a complete set of two-cycles Σ ,...Σ within the space-time 1 b2 four manifold . 4 M According to Poincar´e’s lemma, the gauge potential A, satisfying F = dA, can be constructed by integration locally, in topologically trivial neighbourhoods of , up to a 4 M gaugetransformation. A isneeded to define theelectromagnetic coupling to complex wave- functions of electrically charged particles. If the wave function is scalar, corresponding to a boson with charge q , quantum mechanical consistency of the patching procedure for it B requires the fluxes to be quantised [Dirac 1931, Wu and Yang 1975, O Alvarez 1985]: q B F = m(Σ) Z (3.1) 2π¯h ∈ ZΣ By virtue of ordinary Stokes’ theorem and that the facts that Σ and F are both closed, the value of the flux is unchanged either if Σ is replaced by Σ+∂Π, with Π a three-dimensional 9 chain, or if F is replaced by F +dB. Also, integer linear combination of cycles that satisfy the quantisation condition Σ also satisfy the same condition. These statements can be summarised by saying that the cycles Σ are free elements of the integer homology class H ( ,Z), and F is in a cohomology class H2( ,Z). 2 4 4 M M Now H ( ,Z) is an abelian group (with respect to the natural addition operation) and 2 4 M it possesses a unique subgroup built of elements of finite order, called the torsion group T ( ,Z). The quotient group 2 4 M F ( ,Z) H ( ,Z)/T ( ,Z) (3.2) 2 4 2 4 2 4 M ≡ M M is “free” and consists of b copies of the integers, where b is the second Betti number. 2 2 This is the same as saying that F ( ,Z) is a lattice of dimension b . 2 4 2 M A slightly more general version of (3.1) is the “quantum Stokes’ relation”: iqB F iqB A e h¯ Σ = e h¯ ∂Σ . (3.3) R R Now Σ is allowed to have a non-vanishing boundary ∂Σ and hence be a two-chain rather than a two-cycle. In the limiting case when the boundary vanishes, (3.1) is recovered (and so is necessary for the validity of (3.3)). The quantity on the right hand side of (3.3) is Dirac’s path dependent phase factor [Dirac 1955]. It is well defined in the situation described even though the exponent is not. Such phase factors are relevant in several different contexts [Bohm-Aharonov, Wilson] and go by several other names (Wilson loop, U(1) holonomy etc). However, many electrically charged particles, such as the electron, also carry spin and, as a result (3.1) and (3.3) may have to be modified if the topology of space-time is sufficiently complicated. If the complex wave function to which A couples is spinor rather than scalar, and the associated fermionic particle carries charge q , (3.3) is modified by F the presence of a possible minus sign [M Alvarez and Olive 1999]. eiqh¯F ΣF = ( 1)w(Σ) eiqh¯F ∂ΣA, (3.4) − R R at least for two-chains Σ whose boundary is an even cycle: ∂Σ = 2α. (3.5) There are essentially two possibilities for this when Σ is odd. Either α vanishes and Σ is a closed surface, or not. If the latter, Σ could be the real projective plane in two dimensions, andso notorientable. Noticethatalthough2α isclosed, α itselfisnot. Hencetheone-cycle α is what is known as a torsion cycle. The sign factor ( 1)w(Σ) in (3.4) arises unambiguously in the procedure of patching − together the neighbourhoods that make up Σ, precisely when Σ satisfies (3.5), [M Alvarez and Olive 1999]. However, when Σ is closed, w(Σ) can be constructed independently as the integer specifying the self-intersection number of Σ with itself. This is possible because Σ is a two-cycle in a closed oriented four-manifold. The equivalence of the two notions (mod 2) can be deduced from the Atiyah-Singer index theorem on . When Σ is neither closed 4 M 10