aa r X i v : . [ m a t h - ph ] J a n ON S-DUALITY AND GAUSS RECIPROCITY LAW
AN HUANG
Abstract.
We review the interpretation of Tate’s thesis by a sort of conformalfield theory on a number field in [1]. Based on this and the existence of ahypothetical 3-dimensional gauge theory, we give a physical interpretation ofthe Gauss quadratic reciprocity law by a sort of S-duality.
Introduction
The Gauss quadratic reciprocity law is arguably one of the most famous theoremsin mathematics. Trying to generalize it from different directions has been a centraltopic in number theory for centuries. In the form of Artin reciprocity law, or classfield theory, abelian reciprocity laws are well-understood and has been a beautifulpart of algebraic number theory for a long time. However, the sought for generalreciprocity laws is still in a very unclear stage, for example the Langlands program,regarded as a nonabelian generalization of class field theory, is far from settled.On the other hand, S-duality (or strong-weak duality) is a common name formany amazing stories in physics, which can be traced back at least to the theo-retical work on electric-magnetic duality and magnetic monopoles. S-duality forclassical or quantum U (1) gauge theory is more or less well-understood theoreti-cally. However, S-duality for nonabelian gauge theories remains mysterious, or atleast largely conjectural. In recent years, people are making huge progress in in-terpreting the geometric Langlands program by nonabelian S-dualities. One mayconsult [9] for such an example.However, as far as we know, there is little work on trying to interpret numbertheory reciprocity laws by S-duality of some physical theory. One of the reasonsis, of course, gauge theories are geometrical in nature, so it seems not natural totry to directly relate S-dualities with reciprocity laws. However, as we mentionedabove, the geometric Langlands program, which is the geometric counterpart of thenumber theory Langlands program, now has good physical interpretations. So wehope, at least, that one can find some sort of physical interpretations of abelianreciprocity laws in number theory. What we will try to do in this paper, is to showthat a sort of hypothetical abelian S-duality gives rise to the quadratic reciprocitylaw with possible generalizations to some higher power reciprocity laws. Of course,because number theory has a different nature, in order to do this in a somewhatdirect way, probably one has to leave aside most existing geometric stories, but onlyto keep in mind basic principles of quantum field theory, and to make use of thegeometric picture of number theory to take analogues. Then, one can try to getsome sort of physics models to describe some number theory. This is the philosophyof discussion in this paper. Date : October 08, 2009.
In section 1, we review section 5 of [1], which is an interpretation of Tate’s thesisby a sort of conformal field theory on a number field. One may consult [1] for it,but we will include the discussion here for convenience. Moreover, we will get abetter understanding of this reinterpretation in the present paper by considerationscoming from the 3-dimensional theory. This 2-dimensional conformal field theory ona number field, will be used as the central tool for us to get a physical interpretationof the quadratic reciprocity law.In [3], Witten proposed to study some reciprocity laws for function fields bystudying quantum field theories on algebraic curves. (One can see remarks at theend of section IV in [3]) This is one of the sources of our ideas. Moreover, we will useideas from [6], and assume that our 2-dimensional theory is originated from a GL(1)’gauge theory’ living on
Spec O K , or the number field K , regarded as 3-dimensionalin the point of view of the etale cohomology, which is expected to be the arithmeticcounterpart of 3-dimensional Chern-Simons theory with G = U (1) with some sortof S-duality. Then this S-duality should reflect itself in some natural way in thepath integral of our 2-dimensional theory living on the same number field regardedas 2-dimensional, as we will see. (In physics terminology, probably we should namethe above description as a form of AdS/CFT correspondence. But we will not tryto discuss the exact nature of this 3-dimensional theory in this paper, especially wewon’t discuss anything about gravity, so we avoid such words to prevent possiblemisunderstandings.) Discussions of such kind of dualities of 3-dimensional Chern-Simons theory already exist in physics literature. Although we don’t know how todefine such a 3-dimensional arithmetic gauge theory, we will provide several piecesof evidence to show that this is something plausible: we will see that its relationwith our ’current group’ on number fields, closely mimics the relation between 3-dimensional Chern-Simons theory and 2-dimensional current algebra, as discussedin [6]. Also, we will see that one can get natural physical interpretations of severalimportant but somewhat mysterious ingredients of the 2-dimensional theory fromthe point of view of this 3-dimensional theory, which is hard to see from the 2-dimensional theory itself.In section 2, we start from describing in certain detail some background andreferences of our ideas. We will take analogues of things we already know in or-der to get hints, and we make things precise whenever we touches number theory.Then we show that the S-duality of this 3-dimensional theory reflected in the pathintegral of our two dimensional theory, gives us the Gauss quadratic reciprocitylaw. Things will become rather concrete when we actually get to the quadraticreciprocity law, despite that the physics picture is quite conjectural. We will indi-cate possible generalizations of this idea aiming to give the same sort of physicalinterpretations of some higher power reciprocity laws, but we also point out sometechnical difficulties. Our discussion on this topic is hypothetical, and of course pre-liminary at best. However, along the way, we will see how some basic but intricatealgebraic number theory come up from physical considerations, and furthermore alot subtle ingredients of quantum field theory and number theory mingle together.The best hope is that our attempt may lead to a framework of providing physicalinterpretations of number theory reciprocity laws from which physicists will find iteasier to understand algebraic number theory, and we may be able to make newconjectures in number theory from the framework. N S-DUALITY AND GAUSS RECIPROCITY LAW 3 A physical Interpretation of Tate’s Thesis
For introductory material on Tate’s thesis, one can see for example [2].In [3], Witten formulated several quantum field theories on an (smooth, com-plete) algebraic curve over an algebraically closed field. Here we will try to formu-late a simplest possible conformal field theory on an algebraic number field from asomewhat different point of view. We will use some ideas of [3], of course, especiallywe will take some analogues of these ideas to apply to the case of number fields forguidance. We have no intention to make our discussion here rigorous or complete,however. Our goal here is to tentatively explore this possible connection betweennumber theory and physics. We will see that much of Tate’s thesis come out fromphysical considerations.Let K be a number field, and O K its ring of integers. Let A K be the ring ofadeles, and I K the idele group, C K the idele class group, and I K the ideal classgroup. We denote by τ the diagonal embedding of K × into I K . We fix a globaladditive character ψ of A K , trivial on K . For any local embedding F of K , let d × x denote the multiplicatively invariant Haar measure on F normalized so that the(local) units have volume 1. Also we denote by dx the self-dual additively invariantHaar measure with respect to the local component of ψ . By abuse of notation, wealso denote by d × x (and dx ) the multiplicatively (and additively) invariant Haarmeasure on I K given by multiplying the local Haar measures. Now we will attemptto describe what one may call the GL(1) ’current group’ on a number field.First of all, for a commutative ring, we have at our hand the geometric objectgiven by the prime spectrum of the ring, to be used to take analogue with thegeometric case. For any place v of K , local operators are in Hom(Spec K v , GL(1)) =GL(1 , K v ). By taking analogue of the discussion on multiplicative Ward identities in[3], if the local operator f v has negative valuation, then physically it corresponds toa positive energy excitation at v . So globally, quantum fields live in Q v GL(1 , K v ),with the restriction that just like ordinary conformal field theory, for all but finitelymany places v , f v lives in GL(1 , O v ). So, in other words, quantum fields are elementsof the idele group I K .Next, any two quantum fields differing by an element of τ ( K × ) should be re-garded as the same. We have reasons for imposing this requirement: one mayconsult section V of [3]. Multiplying by elements of τ ( K × ) is the analogue ofconformal symmetry transformation.So the path integral should be on the idele class group C K . To integrate, weneed a measure which should be an analogue of what physicists call the Feynmanmeasure on the space of fields. In ordinary quantum field theory on flat spacetime,this (undefined) concept of Feynman measure should be translational invariant,which can be regarded as a consequence of the symmetries of flat spacetime. Inour multiplicative case, the analogue of this is the requirement that the measureshould be multiplicatively translational invariant. So this measure has to be theHaar measure on C K with an ambiguity of a scalar, which makes perfect sense.Next, in path integral formulation of ordinary quantum mechanics and quantumfield theory, expressions like e iHt , e i R Ldt , or e R Ldx show up essentially because ofthe Schrodinger equation, which itself can be regarded more or less as a consequenceof the basic principles of quantum mechanics and the flat spacetime Lorentz symme-try. (There are many discussions on this issue, and we won’t discuss it here. Notethat the Schrodinger equation itself is not Lorentz invariant.) Here on the ideles,
AN HUANG we have the multiplicative translational symmetry for the Haar measure, so whatsubstitutes e R Ldx in the path integral should be a multiplicative function on C K (Note that formally, e iHt is a quasicharacter on the additive group of t , which is aconsequence of the Schrodinger equation.), which is nothing but ωω s in general (onphysics grounds, we assume that this function should be continuous. Furthermorewe will provide physical interpretations of ω and ω s in the next section), where ω is a Hecke character on I K , and ω s is the quasicharacter given by(1.1) ω s ( x ) = | x | s for any x ∈ I K . Where s is a complex number.Note that ω and ω s can be factorized as products of local characters, and thisis consistent with integrating the Lagrangian density over spacetime in ordinaryquantum field theory (or over the worldsheet in two dimensional conformal fieldtheory).Before we go any further, let us stop and make an observation which gives usa hint of why our construction possibly can come from a gauge theory. For anordinary U (1) gauge theory, the path integral should sum over all possible U (1)principal bundles over the base manifold. Here, we have the canonical isomorphism(1.2) Pic( Spec O K ) ∼ = I K Where the Picard group Pic(
Spec O K ) classifies the isomorphism classes of invertiblesheaves on Spec O K . In fact, our path integral on C K somehow sums over I K :In number theory, there is a canonically defined surjective group homomorphismfrom the idele class group to the ideal class group: π : C K → I K with Kerπ = I ( S ∞ ) /τ ( R × )where I ( S ∞ ) = Y archimedean places K × v × a nonarchimedean places R × v where R × v is the group of (local) units in O v , and R × is the group of global units of O K . So integration over C K already includes a summation over I K . π refines theinformation in the ideal class group, whose usefulness is illustrated by global classfield theory. For us, it’s usefulness is revealed by the path integral.Before we can write down the path integral, we still have to consider the insertionof local operators. In ordinary quantum field theory, we have expressions like(1.3) Z φ ( x ) e R L ( φ ( x )) d D x Dφ However, it is hard to make sense of it unless one makes the inserted operatorshave good decaying properties, and thinks of the measure as a linear map fromsome space of functions to R . See [4] for discussions on this issue.To integrate over C K , the integrand should be functions on C K . Of course, theinsertion of local operators should carry appropriate physical meaning. To thinkabout what is the form our insertion should look like, here we consult the formof Polyakov path integral. See for example, [5], equation (3.5.5): for the inclusionof a particle, one inserts in the path integral a local vertex operator given by the N S-DUALITY AND GAUSS RECIPROCITY LAW 5 state-operator correspondence. Furthermore, to make the vertex operator insertionsdiff-invariant, one integrates them over the worldsheet.To mimic this process of insertion of vertex operators, we start from an unkownfunction f ( x ) on I K which is a product of local functions with suitable decayingproperties, and carrying appropriate physical meaning. Then we sum over K × to make it K × invariant (so we insist that f ( x ) should make the following sumconvergent): X α ∈ K × f ( αx ) Remark . From the above, it is not correct to say that K × should be the analogueof the string worldsheet, since the Polyakov path integral is intended to calculatestring S-matrices, whereas our path integral is to be regarded as a path integralin conformal field theory. Rather, it makes some sense to regard Spec O K as theanalogue of the worldsheet. But as we will see, archimedean places will also matter,so one should really say that the theory is to live on number fields.Finally we can write down our path integral:(1.4) Z I K /τ ( K × ) X α ∈ K × f ( αx )( ωω s )( x ) d × x Furthermore, note that ( ωω s )( αx ) = ( ωω s )( x ), for any α ∈ K × . So the aboveequals(1.5) Z I K f ( x )( ωω s )( x ) d × x which is exactly the global zeta integral z ( s, ω ; f ) for the test function f . So wepropose that the allowed functions should be in S ( A K ), the space of Schwartz-Bruhat functions on A K .Note that in the above integral, we have only one parameter s which can becontinuously varied. So it’s tempting to regard s as coming from the ’couplingconstant’. We will see what this means as a coupling constant in the next section.If we allow f to vary, the global zeta integral z ( s, ω ) becomes a distribution,which is well known to be convergent for ℜ ( s ) >
1, and has a meromorphic analyticcontinuation to the whole s plane and satisfies the functional equation(1.6) \ z (1 − s, ω − ) = z ( s, ω )Where the global Fourier transform b is defined after we fix the additive character ψ of A K . We also have the functional equation for the complete global L function(1.7) Λ( s, ω ) = ǫ ( s, ω )Λ(1 − s, ω − )which is independent of the choice of ψ .(1.6) tells us that we can use analytic continuation to define our quantum theoryfor any value of the coupling constant s . We will make use of this fact in the nextsection discussing reciprocity laws. Remark . If we switch from number fields to global function fields (namely, func-tion fields over a finite field), since Tate’s thesis works for both cases, all the abovediscussion is essentially valid, except that we don’t need to worry any more aboutarchimedean places, and also we don’t need to take analogues between number
AN HUANG fields and function fields. (For the global function field case, we also have a canon-ical group homomorphism from the idele class group to the divisor class group,which should replace our discussion above around (1.2). ) It is interesting to notethat [3] discusses quantum field theories on curves over an algebraically closed field,where Witten uses algebraic constructions relying on the algebraically closednessof the ground field, and he also remarks: ” While one would wish to have an ana-logue of Lagrangians and quantization of Lagrangians in this more general setting,such notions appear rather distant at present.” On the other hand, if we applyour discussion to global function field case, we are actually discussing conformalfield theory on curves over a finite field. What we were trying to do, was just towrite down a path integral which mimics a path integral in ordinary quantum fieldtheory. But our discussion is not valid for curves over algebraically closed fields.2.
Toward Gauss reciprocity law and beyond
First of all, let us briefly recall the relation between 3 dimensional Chern-Simonstheory and 2 dimensional current algebra as discussed in [6]:For the simplest case without Wilson loops, in order to solve the quantum Yang-Mills theory with Chern-Simons action on an arbitrary three manifold M , we firstchop M into pieces, then solve the problem on the pieces, and then glue things backtogether. On a piece Σ × R , where Σ is a closed surface, quantization of the theoryis tractable by canonical quantization. With a certain gauge choice, solutions ofthe classical equation of motion gives us a finite dimensional phase space M , themoduli space of flat connections on Σ modulo gauge transformations. One knowsthat M has finite volume with respect to its natural symplectic structure, and inparticular this implies that the quantum Hilbert space is finite dimensional. Onthe other hand, to actually get the quantum Hilbert space, one may first pick acomplex structure J on Σ, together with a linear representation of the gauge group G . For G = SU ( N ) together with its fundamental representation, the moduli space M , written as M J , can be reinterpreted as the moduli space of all stable rank N holomorphic vector bundles of vanishing first chern class. M J is a complex Kahlerprojective variety, and the quantum Hilbert space is the space of global holomorphicsections of certain line bundle on M J . Furthermore, one has a prescription to getrid of the choice of J by a canonical flat connection on certain vector bundles onmoduli space of J .On the other hand, if one considers current algebra on a Riemann surface, witha symmetry group G at some level, then the Ward identities uniquely determinethe correlation functions for descendants of the identity operator in genus zero case.However, if the genus of the Riemann surface is greater than zero, then in generalthe space of solutions of the Ward identities for descendants of the identity is a finitedimensional vector space called the ’space of conformal blocks’, which is the sameas the quantum Hilbert space obtained by quantizing the 3 dimensional theory asrecalled above, as is shown by works of Segal and Witten. Witten remarked thatthis is the secret of the relation between these two theories!To make analogies of these for number fields, we first recall some work on formalanalogies between number fields and three manifolds started with the work of B.Mazur and others. Spec O K should be regarded as at least 3-dimensional from thepoint of view of etale cohomology. In [7], it is shown that the etale cohomologygroups H net ( Spec O K , G m ) vanish for n >
3, and they are equal to Q / Z for n = N S-DUALITY AND GAUSS RECIPROCITY LAW 7
3. Furthermore these cohomology groups satisfy Artin-Verdier duality which isreminiscent of 3-dimensional Poincare duality. From these and other evidencespeople are suggesting analogies between
Spec O K and three manifolds. Points in Spec O K (prime ideals) can be viewed as 1-dimensional objects and are comparedto knots in a 3-manifold. In particular the absolute Galois group of a finite field isisomorphic to the profinite completion of Z , the Fundamental group of a circle.Having said all these, our postulate is that our 2-dimensional theory originatesfrom a 3-dimensional gauge theory on Spec O K from the point of view of etaletopology, which is the arithmetic counterpart of the 3-dimensional Chern-Simonstheory with G = U (1) with some sort of S-duality, as we have said. In the fol-lowing, we will provide evidences for our postulation, and provide explanations ofsome mysteriously looking ingredients (for example the unexplained origin of thequasicharacter in section 1) in the 2-dimensional path integral from considerationsof the 3-dimensional theory.Let’s first look at what the ’classical phase space’ should be of such a 3-dimensionalgauge theory. The moduli space of gauge equivalence classes of flat connections onΣ corresponds to equivalence classes of homomorphisms(2.1) φ : π (Σ) → G , up to conjugation. Obviously, the arithmetic counterpart of π (Σ) is the Galoisgroup of the maximal unramified extension of K . With G = U (1), abelian char-acters of this Galois group factors through the quotient by its commutator, whichis the Galois group of the maximal abelian unramified extension of K , and is iso-morphic to the ideal class group of K by Hilbert class field theory. So, the classicalphase space of our 3-dimensional gauge theory as a group should be isomorphic tothe dual group of the ideal class group I K of K . So heuristically, the finiteness ofthe class number of an algebraic number field, seems to be the arithmetic coun-terpart of the finite volume property of M , or the finite dimensional property ofthe quantum Hilbert space of 3d Chern-Simons on Σ × R . The finiteness of thedimension of the latter space in presence of certain Wilson operator insertions isclosely related with the skein relations in knot theory as revealed by section 4 of[6].Let’s look at the same phase space from another point of view. As we recalledabove, upon picking a complex structure on Σ, for G = U (1), the phase space hasanother interpretation as the moduli space of stable holomorphic line bundles onΣ with vanishing first chern class. The analogue for this in our setting is the finitePicard group Pic( Spec O K ), which is isomorphic to the ideal class group I K for K .So either way, by taking formal analogues, we see that the classical phase space canbe identified as a group with the ideal class group.Quantization of 3-dimensional Chern-Simons theory on Σ × R , as we recalled, isby taking the space of global holomorphic sections of some line bundle on M J . Here M J is substituted by the finite group I K , so the quantum Hilbert space is just afinite dimensional complex vector space with dimension equal to the class numberof K .Next, let us consider what should be the ’space of conformal blocks’ for ourtheory of current group on K . Since we have a path integral, we can easily get thequantum equation of motion, and Ward identities for symmetries by certain changeof variables in the path integral (for introductory material on this topic, the readercan refer to chapter 9 of [8]): For any idele α ∈ I K , we make a change of variables AN HUANG x → αx in the path integral, the translational invariance of the Haar measure tellsus that(2.2) Z I K f ( x )( ωω s )( x ) d × x = Z I K f ( αx )( ωω s )( αx ) d × αx = ( ωω s )( α ) Z I K f ( αx )( ωω s )( x ) d × x This should be regarded as the quantum equation of motion, and the (multiplica-tive) Ward identity in our setting is the same equation with the restriction that α ∈ τ ( K × ).To get the description of the space of conformal blocks, we need to look at theclass number from the point of view of the canonical homomorphism φ from theidele group to the ideal group as follows: We consider an equivalence relation onthe ideles I K defined as: two elements v and v are said to be equivalent if andonly if valuations of them at every nonarchimedean place are equal. In other words,if they are mapped to the same element in the ideal group by φ . The group τ ( K × )acts on the quotient I K /Ker ( φ ) by ’pointwise’ multiplications. The number oforbits in I K /Ker ( φ ) with respect to this action is equal to the class number of K . In ordinary conformal field theory, we need to study to what extent the Wardidentities determine correlation functions for all descendants of the unit, insertedat any allowed combinations of points on the Riemann surface. Now if our Heckecharacter ω is trivial, then the counterpart of all descendants of the unit inserted atany allowed combinations of points should be the set of all possible f ( x ), which are(restricted) products of local characteristic functions for some π sp O v , where s is theprescribed valuation at p . (Since it’s reasonable to say that local excitations of ourquantum field are measured by local evaluations, as we have said in the previoussection. Moreover this point will become clearer when we discuss insertions ofWilson loop operators.) By examining the Ward identity in our setting, it is clearthat the dimension of the complex vector space of conformal blocks should be thenumber of orbits for the action of τ ( K × ) on this set, where the action is given by(2.3) α ∈ τ ( K ) × → ( f ( x ) → f ( αx ))This action is the same as the action of τ ( K × ) on the quotient I K /Ker ( φ ) as we justdiscussed, so the number of orbits equals the class number of K . In other words, thecomplex vector space of conformal blocks for our theory of current group on K fora trivial Hecke character, has dimension equal to the class number of K , which, aswe have seen above, is also equal to the dimension of the quantum Hilbert space ofour hypothetical 3-dimensional gauge theory. This mimics nicely the ’secret’ of therelation between 3-dimensional Chern-Simons theory and 2-dimensional currentalgebra as we recalled! Furthermore, the condition that ω being trivial has thecounterpart that there being no insertion of Wilson loops in 3d Chern-Simons theory(recall that all the above discussion of formal similarities is in the absence of Wilsonloops). Being succeeded at this stage, let us next consider possible insertions ofWilson or t’Hooft operators in our theory to get something more interesting.In [9], Kapustin and Witten considered certain topological Wilson-t’Hooft opera-tors in some 4 dimensional supersymmetric topological Yang-Mills theories reducedto two dimensions, and interpreted geometric Langlands as coming from the S-duality of the underlying 4 dimensional gauge theory switching Wilson and t’Hooftoperators. Note that although we described our current group on a number field bythe global zeta integral in section 1, we know very little about how to choose the N S-DUALITY AND GAUSS RECIPROCITY LAW 9 test function f ( x ), and know nothing about how to choose the quasicharacter. Inthe following, we will argue that to talk about charges, we need to make a choiceof s , and insertion of certain t’Hooft operators in the hypothetical 3-dimensionaltheory should reflect itself in the 2-dimensional theory as insertion of certain Heckecharacters in the path integral (to be more precise, our following arguments on thispoint for number fields other than Q are incomplete for some reasons, as we willsee. But we expect some refinements will work.), and insertion of certain Wilsonoperators in the hypothetical 3-dimensional theory should reflect itself in the 2-dimensional theory as certain changes of f ( x ). Then, the quadratic reciprocity lawcomes to surface if we switch the Wilson and t’Hooft operators by the hypotheticalS-duality of the 3-dimensional theory!In the following we first restrict ourselves to quadratic reciprocity, and the dis-cussions will be carried out on K = Q . We will first see how the simplest case ofquadratic reciprocity comes out as quickly as we can, then we refine our discussionto get the full quadratic reciprocity law. After that, we will say a bit about ourideas for algebraic number fields other than Q .In ordinary quantum field theory, the effect of including a Wilson loop operatorin the path integral is to add an external charge in a certain representation of thegauge group, whose trajectory in spacetime is the loop; and the effect of including at’Hooft operator is to instruct a certain singularity of the fields localized along thesupport of the t’Hooft operator. From the point of view of etale topology, we areinterested in Wilson and t’Hooft operators supported along ’circles’ Spec F q , where q is a prime. So in the 2-dimensional theory, the effect of these operators are justinsertions of some local operators.We first consider about t’Hooft operators supported on Spec F q . To detect thenature of a singularity of a connection field like the Dirac magnetic monopole, onecan trace the field along loops in space where there is no singularity, and look at theresulting holonomy. From this point of view, we know how to describe the effect of at’Hooft operator supported at Spec F q in our settings: we need to take a covering of Spec Z , or more precisely, a field extension of Q , which is only ramified at q , and lookat the ’monodromy’ of other circles Spec F p for primes p = q . Since here we are onlyinterested in the quadratic reciprocity, we restrict ourselves to the case of a doublecovering, thus a quadratic extension. Note that when q ≡ mod Q → Q ( √ q ). We notice that very similar mathematical questionhas been considered in the very interesting article [10], where it is shown that themonodromy should correspond to the Frobenius element of the Galois group at p .(Also the authors interpret the Legendre symbols as linking numbers, and the Gaussreciprocity law as the interchanging symmetry of linking numbers.) For the caseof a quadratic extension, this monodromy only depend on whether p splits or not.Elementary number theory then tells us that the monodromy of an odd prime p isgiven by ( qp ), Where ( ) denotes the Legendre symbol. While for p = 2 (so q = 2),it’s given by q − , which equals the Kronecker symbol ( D ). (Kronecker symbol atthe place 2 is not multiplicative, however it is multiplicative on 1 + 4 Z . Moreoverit’s the same as the Legendre symbol at any other places. Roughly speaking, thespecialness of 2 comes from the fact that we are considering quadratic symbols,so 2 is the only special prime where the effect of Hensel’s lemma for quadraticpolynomials is different from the case of other primes. But the physics picture ofthe monodromy is the same anyway.) Then, how do we incorporate this into our ω q defined on ideles with component at q equaling to1 as(2.4) ω q ( x ) = Y nonarchimedean places p = q ( qp ) − V p ( x ) Where we need the power − V p ( x ) because we need ω q to be multiplicative in ourmultiplicative theory. Moreover, because of conformal symmetry, we need to requirethe function ω q to come from a function on the idele class group. In other words,we require the τ ( K × ) invariance of ω q . Once this restriction is put, ω q is now amultiplicative function defined on all the ideles, trivial on τ ( K × ), and thus is aHecke character for which the conductor we do not know a priori without assumingquadratic reciprocity law. (There is a slight cheating to directly say that ω q is aHecke character, since we haven’t shown that ω q is continuous with respect to thetopology of ideles. Of course if we are allowed to use the quadratic reciprocity law,there is no problem of showing this. But the point here is that we want to interpretquadratic reciprocity law by physics without first assuming it. On physics grounds,we say that ω q is a Hecke character by pretending that it is continuous.)Next let’s decide how to include a Wilson loop operator at p . First of all, to talkabout charges, we need to choose a (one dimensional) continuous representation ofGL(1 , A K ) which is a product of local representations, and restricts to the trivialrepresentation on τ ( K × )(because of conformal symmetry of the 2-dimensional the-ory). In other words, we need to choose a quasicharacter of the idele group whoserestriction to τ ( K × ) is 1. Furthermore, it’s unnatural for this quasicharacter tohave any nontrivial conductor a priori (and in fact, as we have seen in the above,the insertion of t’Hooft operators secretly takes care of such a choice). So, whatwe need to choose is exactly an ω s , which is uniquely determined by the complexnumber s . It is also for this reason, that we may regard s as the GL(1) couplingconstant. (In fact, we will see that the choice of s has zero total effect for the inser-tion of Wilson loop operators, once they are correctly done. But it does have crucialrole in the realization of S-duality in our 2-dimensional path integral, because of(1.6).) Once the representation is chosen, we can decide how to insert a Wilsonloop operator at p : it instructs us to evaluate the monodromy at the ’loop’ p inthe 2-dimensional path integral, in our chosen representation. In the 2-dimensionalpath integral, to include such an effect of a Wilson loop operator at p , the procedurethen is to change f ( x ) to f ( α p x ), where α p = ( p, , , ..., p, , , ... ) is an idele withnorm one such that the nonarchimedean valuation of α p is equal to 1 at p , and 0at any other primes. Also, we should require positive valuation at the real place,since − α p to be 1 at any other nonarchimedean place, and tobe just p at p , to avoid complications coming from unknown Hecke characters, ort’Hooft operator insertions. Then, it is clear that α p is the unique idele satisfyingall these restrictions. (Let’s explain the norm one requirement: the reason of thisis the same as it is explained in [6]: the total charge of a closed universe should bezero, since the electric flux has nowhere to go. So we need to require α p to havenorm 1 in order that the representation ω s takes it to 1. Later we will discuss in-sertions of multiple Wilson loop operators. For that case, it suffices that the norm N S-DUALITY AND GAUSS RECIPROCITY LAW 11 of the product of corresponding idele equals one, since only the total charge shouldbe zero.)Let’s denote the Wilson operator at p as w p , and use a subscript q to indicatethe inclusion of a t’Hooft operator at q , or in other words, we insert a ω q in thepath integral. Let’s evaluate the ’amplitude’ < w p > q for the Wilson loop at p inthe presence of a t’Hooft operator at q . According to the usual formula of quantumfield theory, we have(2.5) < w p > q = R I K f ( α p x )( ω q ω s )( x ) d × x R I K f ( x )( ω q ω s )( x ) d × x Where f ( x ) is a function to be inserted for the vacuum in the presence of t’Hooftoperator insertions given by ω q . But we will see that we can get quadratic reci-procity without any knowledge of f ( x ) other than that the denominator of the righthand side of the above equality is not zero.By using the quantum equation of motion (2.2), (2.5) gives us(2.6) < w p > q = ω q ( p ) − = ( qp )We see that this expression is independent of s . From this one sees an effect of thenorm one requirement.To get to special cases of quadratic reciprocity as quickly as possible, we firstrequire that p ≡ mod q ≡ mod ω q and α p are transformed into ω p and α q .After S-duality, we are able to describe the 2-dimensional physics in the same way,as we have seen in (1.6), but we need to replace ω p by ω − p . Furthermore, theFourier transform has the effect of transforming the idele α q into its inverse: to bemore explicit, let f ( x ) = f ( α q x ), we have [ f ( x ) = Z A K f ( y ) ψ ( xy ) dy (*) = Z A K f ( α q y ) ψ ( xy ) dy = Z A K f ( y ) ψ ( xα − q y ) dy = b f ( α − q x )One notices that the fact that α q has norm 1 is crucial in the derivation of theabove. So from the 2-dimensional point of view, S-duality tells us that(2.7) < w q − > p − = < w p > q By (2.6), the above equality is(2.8) ( pq ) = ( qp )which is nothing but the Gauss quadratic reciprocity law for p and q for the specialcase when both p and q are congruent to 1 mod 4 !Next, we refine the above discussion, and consider the insertion of t’Hooft andWilson operators in more detail in order to get the full quadratic reciprocity law. Ifwe intend to insert a t’Hooft operator at q for an odd prime q ≡ mod are in trouble since there is no quadratic extension of Q that only ramifies at q . (Weneed also to take into account the ramifications of the field extension at archimedeanplaces.) So in fact, if q is congruent to 3 mod 4, to insert a t’Hooft operator at q , wehave to secretly include some other insertions of the t’Hooft operator at some otherplaces. As an example let’s consider the field extension Q → Q ( q ( − q − q ) = L ,for the case when q is an odd prime. When q ≡ mod q . But when q ≡ mod q and infinity, respectively. As another examplewe consider the field extension Q → Q ( √ q ) = L for the case when q ≡ mod q and 2, with ramification indexes both equal to 2.However, there is something different at places q and 2: the discriminant D = 4 q ,and so the exponents of q and 2 in the discriminant are 1 and 2, respectively. (Forlatter purpose, this can be equivalently stated as: considering the different of thefield extension L / Q , the differential exponent of the prime above q is 1, and ofthe prime above 2 is 2.) We need to include the information of these exponentsas the multiplicities of the local insertion of t’Hooft operators. (This should beclear when we think of the algebraic-geometric picture of the effective differentdivisor, where differential exponent is interpreted as multiplicity in a definite way.From a differential-geometric point of view, this multiplicity corresponds to thefirst Chern class of a U (1) bundle representing the field singularity.) So for the fieldextension L / Q , we have inserted a t’Hooft operator at q with multiplicity 1, anda t’Hooft operator at 2 with multiplicity 2. If q = 2, we can take the extension Q → Q ( √
2) = L . Then we get a single t’Hooft operator insertion at 2 withmultiplicity 3. In any of these cases, elementary number theory again tells us thatthe combined monodromy of a prime p which is unramified for a quadratic fieldextension is given by ( Dp ), Where ( ) denotes the Kronecker symbol, and D is thediscriminant of the field extension. Again, to simply get the correct monodromy,the effect of the inclusion of a set of such t’Hooft operators (We don’t need tospecify the order of the insertions of t’Hooft operators, since we are consideringan abelian theory.) given by a quadratic field extension M/ Q in the 2-dimensionalpath integral, is to add to the path integral a multiplicative function ω M/ Q definedon ideles as: For ideles whose components equal 1 at all ramified places(2.9) ω M/ Q = Y nonarchimedean places p prime to D ( Dp ) − V p ( x ) One may wonder why we don’t include the possible monodromy coming from thereal place, as for the place p = ∞ , the monodromy is − ∞ , and otherwise it is equal to 1. The reason we can ignore thisissue is that we forbid the coexistence of Wilson and t’Hooft operators at the sameplace, which excludes the possibility for a nontrivial monodromy coming from thereal place. Next we should require that ω M/ Q becomes trivial when restricted to τ ( K × ), as before. Again, on physics grounds, we say that ω q is a Hecke characterby pretending that it is continuous. (Actually one can show that it is continuous ifone is allowed to use the quadratic reciprocity law.)Next we consider the insertion of Wilson operators in more details. First notethat since we have to include in the theory insertion of t’Hooft operators at infiniteprimes, by S-duality we also have to think of the problem of insertion of Wilson N S-DUALITY AND GAUSS RECIPROCITY LAW 13 loop operators at these primes (even though they won’t have observable effectsin the 2-dimensional path integral for the reason stated above, we still have toinclude them as they are required by S-duality). We can straightforwardly extendour discussion for the nonarchimedean cases, and see that Wilson loop operatorsinserted at the real place has the effect of changing f ( x ) to f ( α ∞ x ), where theidele α ∞ = ( − , , , , ... ), with the − p , againas required by S-duality. This is easy: since we are working with a multiplicativetheory, we just need to raise the idele α p to the power given by the multiplicity m , and make change to f ( x ) using this new idele: f ( x ) → f ( α mp x ). This has theeffect of raising the monodromy to the power given by the multiplicity. Also, aswe mentioned before, to satisfy the total charge zero requirement, one has to makesure that the norm of the products of all ideles α mp is equal to 1. Then, one hasobvious generalization of equation (*), which tells us how to operate with S-dualitytransformation. In practice, it makes no difference if we further require that thenorm of each α p is equal to 1 (Sometimes it makes things easier to state). Lastly,the insertion of Wilson operators should always come in a set with appropriatemultiplicities in order that after S-duality transformation, they give a legal set ofinsertions of t’Hooft operators.Now we apply S-duality to several different cases to get the quadratic reciprocitylaw for primes p and q that are different from each other:First of all, it’s obvious that ( − q ) = − q ≡ mod − p ) for p ≡ mod Q → Q ( √− p with multiplicity 1. Then as before, after applying S-duality, we getWilson operators at ∞ and 2 with multiplicities 1 and 2, respectively. Also we havea t’Hooft operator at p with multiplicity 1. So the path integral gives(2.10) ( − p ) = ( p This equality tells us that ( − p ) = 1.For the case when both p and q are odd, and at least one of them is congruentto 1 mod 4, say p ≡ mod p with multiplicity 1, and t’Hooft operator insertions given by L/ Q . Then afterapplying S-duality, we get Wilson operators at q and ∞ both with multiplicity 1,and a t’Hooft operator at p with multiplicity 1. S-duality transformation of thepath integral gives(2.11) ( Dp ) = ( pq )This is exactly the quadratic reciprocity law for p and q in this case.Next, consider the case when q is an odd prime, and p = 2. We consider t’Hooftoperator insertions given by L/ Q , and a single Wilson operator insertion at 2 with multiplicity 3. Then as before, S-duality gives(2.12) ( D = ( 8 q )which obviously reduces to the quadratic reciprocity law for 2 and q .Finally, we consider the case when both p and q are odd primes congruent to3 mod 4. This time, t’Hooft operator insertions are given by L/ Q , and there areWilson operators at p and 2 with multiplicities 1 and 2, respectively. So S-dualitygives(2.13) ( Dp )( D = ( 4 pq )again this equality reduces to the quadratic reciprocity law for p and q in this case. Remark . Note that the specific choice of f ( x ) doesn’t matter in this story.Next, we want to try to generalize our discussion to an arbitrary algebraic numberfield K , with the hope of getting more reciprocity laws. However, we will seethat although Wilson and t’Hooft operator insertions seem to generalize withoutdifficulty, there are other difficulties one should overcome before one can get anyhigher power reciprocity laws.For a number field K , we fix a uniformizer π p at each place p . An insertion of aset of t’Hooft operators is given by a finite abelian extension L/K of a prime degree(for simplicity). As usual, we denote by δ L/K the different of the field extension,and by D L/K the relative discriminant. Let R L/K denote the finite set of places of K that are ramified in this extension. Then we have t’Hooft operator insertions atplaces q in R L/K with multiplicities given by the differential exponent of any primeabove q (this is equal to the exponent of q in D L/K ). Furthermore, the monodromyof an unramified loop (place) p should be given by the local Frobenious element,or the local norm residue symbol ( π p ,L/Kp ). (This is a well defined element in theGalois group since L/K is abelian.) A Wilson operator with multiplicity m ata place p is to change f ( x ) to f ( α mp x ), where α p is any idele given by: if p isnonarchimedean, then the component at any nonarchimedean place other than p is1, and the component at p is π p , the component at any real place is positive, and thenorm equals 1; if p is a real place, then the component at p equals −
1, and at anynonarchimedean place equals 1, and has positive valuation at any other archimedeanplace, and the norm equals 1. If p is a complex place, α p has component 1 at anynonarchimedean place, and has positive valuation at any archimedean place, andhas norm 1. Note that α p is not uniquely determined at archimedean places, butit’s obvious that any two choices of α p will have exactly the same effect in the pathintegral. Furthermore, from the above description it is easy to see that Wilsonoperators at complex places have no effect at all in the path integral, so we canjust instead put the restriction that there are no insertion of Wilson operators atcomplex places. In fact this is required by S-duality: since no complex place canpossibly be ramified, no t’Hooft operators can be inserted at complex places, soS-duality tells us no Wilson operators can be inserted at complex places. Also, werequire that no Wilson and t’Hooft operators can be inserted simultaneously at asingle place.Note that the monodromy lives in the Galois group. In order to discuss higherpower reciprocity laws, we need to first decide how to take the values of mon-odromies living in different Galois groups in the 2-dimensional path integral. For N S-DUALITY AND GAUSS RECIPROCITY LAW 15 quadratic extensions, there is only one way to identify the Galois group with thegroup Z / Z , so we don’t need to worry about different identifications. This is oneof the reasons why quadratic reciprocity law is much easier to get. However, forhigher order extensions, such naive unique identifications no longer exist. In thefollowing, we will explain that an alternative idea also fails:Kummer theory is one central ingredient for the definition of Hilbert symbols andhigher reciprocity laws. One may try to use Hilbert symbols to define the values ofmonodromies, and hope to get some higher reciprocity laws. So next, we let K to bethe cyclotomic field K = Q ( ζ p ), where p is any fixed odd prime. For any Kummerextension L/K , Kummer theory and local class field theory combined gives us thelocal Hilbert symbol ( π p , L/K ) p of order p , taking values in the group of p throots of unity. Let’s try to take this as the value of the monodromy detected by p in the presence of the set of t’Hooft operator insertions given by L/K , and see if itworks. Then as before, the t’Hooft operator insertions given by
L/K , should havethe effect in the 2-dimensional path integral of inserting a multiplicative function ω L/K on the idele group given by(2.14) ω L/K ( α ) = Y p/ ∈ R L/K ( π p , L/K ) − v p ( α ) p For ideles α whose component at any place in R L/K is equal to 1. The product inthe above equation is well defined since for all but finitely many places, the localHilbert symbol equals 1. Again, we require that the restriction of ω L/K to τ ( K × ) istrivial, and it is straightforward to check that ω L/K is uniquely defined. On physicsgrounds we pretend it is continuous, and so ω L/K defines a Hecke character.Now we pick two different prime elements p and q in the ring of integers Z [ ζ p ] of K , both being coprime to p . We consider two Kummer extensions L = K ( p √ p ) /K ,and M = K ( p √ q ) /K . For the cyclic extension K/ Q , only p ramifies, and we have( p ) = (1 − ζ P ) p − as ideals in K . Let’s denote the ideal (1 − ζ p ) by v . Kummertheory tells us that the only possible ramifications of L/K are at p and v , andthe only possible ramifications of M/K are at q and v . We need to make therequirement that v is unramified for at least one of L/K or M/K , and under thisrequirement, we hope to see that the p th power reciprocity law for p and q followsfrom S-duality.Note that p is tamely ramified for L/K with exponent p − q is tamely ramified for M/K with exponent q − v is unramified for L/K .Then we consider t’Hooft operator insertions given by
M/K , and an insertion ofWilson operator at p with multiplicity p − p . So there is a t’Hooft operatorat q with multiplicity q −
1, and a possible t’Hooft operator at v with multiplicitygiven by the exponent exp D M/K v of v in the relative discriminant D M/K .Therefore, the S-duality transformation of the 2-dimensional path integral shouldgive us(2.15) ( π p , M/K ) p − p = ( π q , L/K ) p − q (1 − ζ p , L/K ) exp D M/K v By the skew symmetry of the local Hilbert symbol, we know that the above isequivalent to(2.16) ( p, q ) p ( p, q ) q = (1 − ζ p , L/K ) − exp D M/K v Since v is unramified for L/K , and q is coprime to p , we have ( p, q ) v = ( q, p ) − v = 1,Hilbert reciprocity law then tells us that if (2.16) is true, we should have(2.17) (1 − ζ p , L/K ) exp D M/K v = 1However, obviously it’s possible choose certain p such that (1 − ζ p , L/K ) is nontriv-ial. Furthermore we may choose q such that exp D M/K v is not divisible by p . (Anexplicit example is: q = 3, q = 5, then exp D M/K v = 4, not divisible by 3. One candemonstrate this by calculating the relative discriminant explicitly by using localand global conductors, and solving some explicit congruence equations. But wewill skip the details since it seems irrelevant with our main discussion.) This meansthat our naive attempt to get the value of monodromy by using Hilbert symbols isfailed. Remark . Despite the difficulties, it seems reasonable to expect that, possiblywith some more effort, one may get more reciprocity laws from this S-duality.Moreover exploration in this direction may also help clarify the unknown physicspicture of the hypothetical 3-dimensional gauge theory if it does exist as expected.Although our discussion is hypothetical, one can see that a lot interesting and subtleingredients in both quantum field theory and number theory show up and mingletogether. At least, as we have seen, one does get the full quadratic reciprocity lawfrom S-duality. So we hope that at least some of the ideas presented here will beof interest for further explorations.
Acknowledgments
I would like to thank Richard Borcherds, Shenghao Sun, Chul-hee Lee, andKenneth Ribet for useful discussions.
References
1. A. Huang,
On Twisted Virasoro operators and number theory , arxiv.org, math-ph/0909.07952. S.S. Kudla et al.,
An Introduction to the Langlands Program , Birkhauser Boston (2003)3. E. Witten,
Quantum Field Theory, Grassmannians, and Algebraic Curves , Commun. Math.Phys. 113, 529-600 (1988)4. R. Borcherds,
R. Borcherds QFT lecture notes compiled by A. Geraschenko et. ,http://math.berkeley.edu/ ∼ anton/index.php?m1=writings5. J. Polchinski, String Theory , Vol. 1, Cambridge Univ. Press (2001)6. E. Witten,
Quantum Field Theory and the Jones Polynomial , Commun. Math. Phys. 121,351-399 (1989)7. B. Mazur,
Notes on etale cohomology of number fields , Annales scientifiques de L’E.N.S., 4 e serie, tome 6, n An Introduction to Quantum Field Theory , Westview press, 19959. A. Kapustin, E. Witten,
Electric-Magnetic Duality And The Geometric Langlands Program ,arxiv.org, hep-th/060415110. M. Kapranov, A.Smirnov,
Cohomology Determinants And Reciprocity Laws: Number FieldCase , unpublished notes
Department of Mathematics, UC Berkeley
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