GAUGE THEORY ANDGEOMETRIC LANGLANDS

Edward WittenStrings 2006, Beijing

The Langlands program is an attemptto unify many old and new results innumber theory

– ranging from quadratic reciprocity,proved around 1800, to the modernproof of Fermat’s Last Theorem.

Number theory is difficult becausecalculus is powerful ….

And so mathematicians have sought to findgeometric analogs of problems in numbertheory.

Number field --- Riemann surface C

Prime number --- point on C

The Langlands program, too, has ageometric analog, which has beenintensively developed.

Even in its geometric form, the Langlandsprogram involves statements that at firstsight are likely to look unrecognizable tophysicists.

But, if one probes a little more deeply, theLanglands program has many obvious andless obvious analogies with quantum fieldtheory.

For one thing, Langlands introduced (ca.1970) a “duality” between a simple Lie

group G and a “dual” group often called LG. However, this relationship, which

pairs SU(N) with SU(N)/ZN, E8 with itself,SO(2n+1) with Sp(n), etc., also plays avery distinguished role in four-dimensional

gauge theory…

In fact, the Langlands dual group LG isprecisely the magnetic group introduced in1976 by Goddard, Nuyts, and Olive toclassify magnetic monopoles:

Their idea was that if electric charges areclassified by representations of a group G,then the corresponding magnetic chargesare classified by a representation of thedual group, which in fact coincides with thedual LG of Langlands.

The GNO work was of course the jumpingoff point for Montonen-Olive duality, anextremely fruitful idea in which the twogroups G and LG enter in a completelysymmetric fashion.

By contrast, in the Langlands program theroles of G and LG are at first sightbafflingly unlike. An object of one typeassociated with LG is classified by acompletely different type of objectassociated with G.

That is a bit disconcerting at first sight,but on the positive side, the objects ofstudy on the two sides are bothobjects that are familiar in QFT – or atleast they are once we make thetranslation from number theory togeometry.

On the left hand side of the Langlandscorrespondence, we have a flatconnection, on a Riemann surface C,with gauge group LG. In gaugetheory, flat connections are those withleast energy, most supersymmetry,etc.

The right hand side of the Langlandscorrespondence involves an“automorphic representation” of G, anotion which when suitably translatedto geometry is very closely related tothe “conformal blocks” of currentalgebra, with symmetry group G, on aRiemann surface C.

Moreover, in recent yearsmathematicians (Beilinson, Drinfeld,Frenkel, Gaitsgory…) working ongeometric Langlands have reliedheavily on two-dimensional currentalgebra – albeit with a focus onaspects (such as the exceptionalbehavior at level k=-h) that appearextremely exotic from a physical pointof view.

I will be reporting on a recent paper

Electric-Magnetic Duality And The GeometricLanglands Program.Anton Kapustin (Caltech) , Edward Witten (Princeton,Inst. Advanced Study) . Apr 2006. 225pp.e-Print Archive: hep-th/0604151

in which we tried to understand exactly whatthe geometric version of the Langlandsprogram means from the point of view ofquantum field theory.

To simplify a bit, this paper is based on six ideas:

(1) A certain twisting of N=4 super Yang-Millstheory gives a family of TQFT’s parameterizedby CP1. S-duality acts on this CP1

(2) If this theory is compactified to two dimensionson a Riemann surface C, one gets a sigmamodel whose target is “Hitchin’s moduli space”;S-duality turns into, roughly speaking, mirrorsymmetry of the sigma model.

(3) Wilson and ‘t Hooft operators of the four-dimensional gauge theory act on branes ofthe sigma model. A brane that is mappedto a multiple of itself by the Wilson or ‘tHooft operators is called an electric ormagnetic eigenbrane, so S-dualityautomatically maps electric eigenbranes tomagnetic eigenbranes.

(4) Electric eigenbranes correspond torepresentations of the fundamental groupin LG. This is the left hand side of thegeometric Langlands correspondence.

(5) ‘t Hooft operators of gauge theorycorrespond to the Hecke operators of thegeometric Langlands program.

(6) Magnetic eigenbranes can be givenanother interpretation because of theexistence of a certain “coisotropic A-brane” on Hitchin’s moduli space. Theyare associated with the “D-modules” thatappear on the right hand side of thegeometric Langlands program.

(1) A certain twisting of N=4 super Yang-Mills theory gives a family of TQFT’sparameterized by CP1. S-duality acts onthis CP1

In general, twisting is achieved, starting withthe theory on flat R4, by defining a newLorentz group – a combination of thestandard Lorentz group and a subgroupof the group of R-symmetries, which forN=4 SYM is SU(4), such that one of thesupercharges, which I will call Q,becomes Lorentz-invariant

Lorentz-invariance typically implies thatQ2=0, and if one works relative to thecohomology of Q, one typically gets atopological quantum field theory.

The key step is to show that – by virtueof the underlying supersymmetryalgebra – the stress tensor T is trivialas an element of the cohomology ofQ.

There are three ways to pick a suitablesubgroup of SU(4)R and thereby “twist”N=4 super-Yang-Mills. Two of the twistsgive theories qualitatively similar toDonaldson theory of four-manifolds (andone was studied by Vafa and EW in 1994).

The third twist, though studied by e.g.Marcus 1995, has had no knownapplications until now. It is the onerelevant to the geometric Langlandsprogram.

When one makes this third twist, one findsthat there is not one Lorentz-invariantsupercharge Q, but two of them, say Q1and Q2 . So we can define Q to be anarbitrary complex linear combination ofthese two

Q = u Q1 + v Q2

with coefficients u,v that we require to benot both zero.

So working relative to the cohomology of

Q = u Q1 + v Q2

we get a family of topological field theories, parameterized by a copy of CP1 with

homogeneous coordinates u,v.

There are essentially no trivial equivalencesamong these theories, but there are non-trivial Montonen-Olive dualities that lead tothe geometric Langlands program.

There actually is one further trick: theparameter t = v/u combines with the gaugecoupling parameter τ = θ/2π +4π i/e2 tomake a parameter ψ which is the reallyimportant one.

Montonen-Olive duality acts on ψ byΨ → (a Ψ+b)/(c Ψ+d)and the basic statement of geometric

Langlands relates Ψ=0 to Ψ= 1

(2) If this theory is compactified to twodimensions on a Riemann surface C, onegets a sigma model whose target is“Hitchin’s moduli space”; S-duality turnsinto, roughly speaking, mirror symmetry ofthe sigma model.

This step largely follows BJSV and HMS(1995). We take the four-manifold

M4 to be a product of Riemann surfaces Σand C. In the limit that Σ is very largecompared to C, we get an effective twodimensional σ-model on Σ.

The target space of the σ-model isHitchin’s moduli space of stable Higgsbundles, which I will call MH.

MH is, roughly speaking, the cotangentbundle of the moduli space of flatbundles on C. It is a hyper-Kahlermanifold, and the sigma model withtarget MH has (4,4) supersymmetry.

Our four-dimensional topological field theoryreduces in two dimensions essentially to afamiliar type of A- or B-model of the sigmamodel (or a hybrid made possible by thefact that in this case the target is hyper-Kahler).

In one of the complex structures, the S-duality reduces essentially to a mirrorsymmetry.

(3) Wilson and ‘t Hooft operators of the four-dimensional gauge theory act on branes ofthe sigma model. A brane that is mappedto a multiple of itself by the Wilson or ‘tHooft operators is called an electric ormagnetic eigenbrane, so S-dualityautomatically maps electric eigenbranes tomagnetic eigenbranes.

In the twisted topological field theory relatedto Donaldson theory, the importantoperators are local operators and their

“descendants,” but in the present problem…

The important operators are not localoperators, but rather Wilson and ‘t Hooft“line operators”

S

associated with an open or closed loop S inspacetime.

To perform a gauge theory calculation with aWilson loop, one simply performs the pathintegral with an insertion of an extra factor,which is essentially the trace of theholonomy around the loop S, taken in arepresentation R of the gauge group G.

In particular, a Wilson loop is labeled by arepresentation R of G.

By contrast, the ‘t Hooft operator is a“disorder operator.” It is defined not bygiving a classical expression (such as theholonomy) that should be quantized, butby giving a recipe, by describing asingularity that the fields should havealong (in this case) a curve in spacetime

We require that along the specified curve, S,the fields should have a singularity which is

most easily described by saying that in thedirections transverse to S, one has a Diracmonopole singularity:

To be more exact, what I’ve written is thebasic Dirac monopole for U(1)…

We need to supersymmetrize and toembed it from U(1) into the gaugegroup G. The basic observation ofGoddard,Nuyts, and Olive (orLanglands) is in effect that theembeddings of U(1) in G, up toconjugacy, are classified by a choiceof representation LR of the dual groupLG. So ‘t Hooft operators areclassified by a choice of such arepresentation LR.

Electric-magnetic duality maps LG to Gwhile also mapping Wilson operatorsto ‘t Hooft operators.

So a Wilson operator of LG, classifiedby a representation LR, is mapped toan ‘t Hooft operator of G, alsoclassified by the representation LR.

(4) Electric eigenbranes correspond torepresentations of the fundamental groupin LG. This is the left hand side of thegeometric Langlands correspondence.

Now to get to the Langlands program, weneed to consider branes in the gaugetheory – or in the effective two-dimensional sigma model.

The branes are defined in the four-dimensional gauge theory by localboundary conditions at the boundary ofthe four-manifold M.

In two dimensions, they reduce to theordinary branes of the sigma model,including the usual A-branes and B-branesof the appropriate topological fieldtheories.

Now the key fact is that line operatorsbehave as operators acting on

branes… A brane (the solid line) with a line operator (the dashed line) gives us a new “composite” brane.

This gives an action of line operators on “theories,” more abstract than the usual action of operators on states.

If L is a line operator and B is a brane, wesay that B is an “eigenbrane” of L if Lacting on B gives us back several copiesof B.

In formulas

where V is a vector space.

An eigenbrane of the Wilson operators iscalled an electric eigenbrane, and aneigenbrane of the ‘t Hooft operators iscalled a magnetic eigenbrane.

Clearly, S-duality will map electriceigenbranes to magnetic eigenbranes.

The electric eigenbranes turn out to besimply zerobranes … (which are B-branesin each complex structure) … Since the

target space of the sigma model is theHitchin moduli space MH, a zerobrane,being a point in MH , corresponds to a flatbundle on our Riemann surface C withgauge group , the

complexification of the gauge group

The complexification comes in because ofthe way MH parameterizes pairs

Such a homomorphism of the fundamentalgroup of C into appears on the lefthand side of the usual geometricLanglands correspondence.

The right hand side involves whatever weare going to get by applying electric-magnetic duality to the zerobranes

Here, some more ideas are involved…

Once we know the electric eigenbranes, wecan find what the magnetic eigenbranesare by applying duality (which acts via anSYZ torus fibration)…

They are A-branes of the usual sort(supported on a Lagrangian submanifoldthat is endowed with a flat connection)

But to understand what it means concretelyfor a brane to be a magnetic eigenbranetakes more work ….

This is a time zero slice that looks like I times C where I is aninterval and C is the Riemann surface on which we compactified …P is the point at which a time-independent ‘t Hooft operator isinserted….

Reading the picture from right to left, westart with a “brane” in which the gaugebundle on C is specified – this determinesa particular holomorphic G-bundle E overC.

Proceeding to the left, nothing happens (tothe holomorphic structure) until it “jumps”when one crosses the position of the ‘tHooft operator.

Mathematically, this jump in the bundle iscalled a “Hecke transformation,” so wearrive at our fifth main point:

(5) ‘t Hooft operators of gauge theorycorrespond to the Hecke operators of thegeometric Langlands program.

Accordingly, the magnetic eigenbranes arecalled “Hecke eigensheaves” in themathematical literature.

There is one more key step to arrive at theusual statement of the geometricLanglands duality:

(6) Magnetic eigenbranes can be givenanother interpretation because of theexistence of a certain “coisotropic A-brane” on Hitchin’s moduli space. Theyare associated with the “D-modules” thatappear on the right hand side of thegeometric Langlands program.

The key point here is that in general, in two-dimensional A-models, in addition to theusual A-branes (supported on aLagrangian submanifold) there can beadditional “coisotropic” A-branes, firstdescribed by Kapustin and Orlov, whosesupport is above the middle dimension.

They don’t exist on Calabi-Yau threefolds,but they do exist on hyper-Kahlermanifolds such as MH

On MH, there is a “canonical coisotropicbrane,” which I will call β, with the property

that the β – β strings are described by a (sheaf of) noncommutative algebras – the algebra of differential operators on MH

The mechanism by whichnoncommutativity comes in is similarto the usual mechanism with the B-field

Now something funny happens: Justbecause the brane β exists, if we aregiven any other A-brane α, we canconsider the

“sheaf” of β-α strings, and this gives us a“module” for the β-β strings, i.e. a “D-

module”:

α β β

So every representation of thefundamental group, i.e. electriceigenbrane, gives us a D-module,as claimed in the usual statementof geometric Langlands duality.

GAUGE THEORY ANDGEOMETRIC LANGLANDS

Edward WittenStrings 2006, Beijing