Moduli spaces of Higgs bundles in mathematics and physicsIn physics A corresponds to a gauge eld on...

Moduli spaces of Higgs bundles in mathematics andphysics

Yan Soibelman

KANSAS STATE UNIVERSITY

November 19, 2013

Yan Soibelman (KANSAS STATE UNIVERSITY)Moduli spaces of Higgs bundles in mathematics and physicsNovember 19, 2013 1 / 33

Higgs fields and Hitchin equations

In 1987 British mathematician Nigel Hitchin published “The self-dualityequation on a Riemann surface” (Proc. London Math. Soc. (3), 55,(1987), 59-126)


In that paper he introduced equations which are now called Hitchinequations:

∂AΦ = 0,

F (A) + [Φ,Φ∗] = 0.

Here A is a connection on a principal G -bundle P over the given Riemannsurface C , G is a given compact, say, simple Lie group, (e.g. G = SU(2)),and we use in the first equation the anti-holomorphic part of the covariantderivative: dA = ∂A + ∂A.The second ingredient Φ is a (1, 0)-form on C with values in the(complexified) Lie-algebra bundle of P (i.e. in ad(P)⊗ C, wheread(P) = P ×G Lie(G )). In physics A corresponds to a gauge field on P,while Φ corresponds to a scalar field (the space-time is 2-dimensional).Hitchin called Φ the Higgs field.


The person

Peter Ware Higgs, CH, FRS, FRSE (born 29 May 1929) is a Britishtheoretical physicist, Nobel Prize laureate and emeritus professor at theUniversity of Edinburgh.


And the particle

One possible signature of a Higgs boson from a simulated collisionbetween two protons. It decays almost immediately into two jets ofhadrons and two electrons, visible as lines


Why mathematicians should bother?

From Wikipedia:The Standard Model of particle physics is a theory concerning theelectromagnetic, weak, and strong nuclear interactions, which mediate thedynamics of the known subatomic particles.The detection and verification of the Higgs boson in 2013 completed theset of predicted particles. Because of its success in explaining a widevariety of experimental results, the Standard Model is sometimes regardedas a ”theory of almost everything”.Mathematically, the standard model is a quantized Yang-Mills theory.But: mathematically rigorous construction of the quantum Yang-Millstheory is one of the (open) Millennium Prize Problems of the ClayMathematics Institute.


Yang-Mills equations

Recall that classical Yang-Mills equations is a system of PDE obtained byextremizing the Yang-Mills functional

S(A) =1

4g2

∫Tr(F ∧ ∗F ),

where F := F (A) = dA + A ∧ A is the curvature of the G -valuedconnection A on the Euclidean 4-dimensional space-time, Tr means thetrace associated with the Killing form, and∗ : Ω2(R4, ad(P))→ Ω2(R4, ad(P)) is the Hodge operator.Classical YM equations have the form

dA(F ) = dA(∗F ) = 0.

They are non-abelian analogs of the Maxwell’s equations.


Self-dual Yang-Mills equations

An important class of solutions of the classical YM equations consists ofself-dual ones: ∗F = F .Physically relevant solutions (those which have finite action and henceextends to the sphere S4) are called instantons. Moduli spaces ofinstantons were introduced to mathematics by Simon Donaldson in hiswork on differential structures on R4

Imposing invariance with respect to one variable we arrive to self-dual YMequations in 3d . Solutions with finite action are “magnetic monopoles”.They were studied e.g. by Atiyah and Hitchin in their book “The Geometryand Dynamics of Magnetic Monopoles” Princeton University Press, 1988).


2-dimensional self-dual Yang-Mills

Hitchin equations describe solutions of the self-dual 4d YM in R4 whichare invariant with respect to the translation group R2.Using the translation invariance Hitchin converts coefficients A3,A4 of theconnection A = A1dx1 + A2dx2 + A3dx3 + A4dx4 into auxiliary fieldsϕ1, ϕ2. Introducing complex coordinate z = x1 + ix2 (and hence theoperator ∂) he obtains his equations.Although 2d self-dual YM does not have an obvious physical meaning (itis conformally invariant and does not have a finite action solution), it isrelated to many deep mathematical structures, e.g. to the moduli spacesof stable holomorphic vector bundles on curves. The (non-self-dual) 2dYM was studied before Hitchin by M. Atiyah and R. Bott in their seminalpaper “The Yang-Mills Equations over Riemann Surfaces”, PhilosophicalTransactions of the Royal Society of London. Series A, Math. and Phys.Sci., v. 308, no. 1505 (1983).


Atiyah, Bott and Donaldson


From PDE to algebraic geometry

Recall Hitchin equations:∂AΦ = 0,

F (A) + [Φ,Φ∗] = 0.

Let us assume for simplicity that G = U(n), so we can speak about vectorbundles instead of principal G -bundles.One can show that Hitchin equations mean that the Higgs field Φ is aholomorphic map Φ : E → E ⊗ Ω1

C , where E is a holomorphic vectorbundle (connection defines the holomorphic structure) and Ω1

C is a sheaf ofholomorphic differential 1-forms on C . Therefore we can move from PDEand differential geometry to complex analysis. Conformal invariance meansthat we may consider Hitchin equations for the compact complex curve C .In that case every holomorphic object is algebraic, and hence we can applypowerful methods of algebraic geometry. E.g. solutions with Φ = 0correspond to flat unitary connections, which are (by Narasimhan-Seshadritheorem) the same as stable holomorphic vector bundles on C .


Higgs bundles

Definition

A Higgs bundle on an algebraic curve C is given by a pair (E ,Φ), whereE → C is an algebraic vector bundle and Φ : E → E ⊗ Ω1

C is a morphismof vector bundles.

Although the name “Higgs field” was proposed by Hitchin himself, theterm “Higgs bundle” was suggested later by Carlos Simpson who developedthe theory of Higgs bundles as a non-abelian analog of Hodge theory.


Spectral curves

We can treat the Higgs field Φ as a family over C of holomorphic matriceswith entries which are holomorphic 1-forms on C . Eigenvalues of Φ areparametrized by the spectral curve S := det(y · id − Φ(x)) ⊂ T ∗C . Thenatural projection T ∗C → C when restricted to S gives rise to the n : 1ramified covering S → C , where n = rk(E ). Different branches of thecovering can be identified with eigenvalues of the variable matrixΦx , x ∈ C .


Moduli of Higgs bundles and Hitchin map

Consider the problem of deforming a given vector bundle E → C (meaningthat we deform its complex structure, keeping the complex structure on Cfixed). The tangent space at E to the space of such deformations can beidentified with H1(C ,End(E )). Assuming C is compact, by Serre dualitythe latter space is dual to H0(C ,End(E )⊗ Ω1

C ). Hence we can identifythe cotangent space at E with the space of all Higgs fields on E . LetBunGL(n) be the moduli space (in fact stack) of vector bundles over C .Varying E in the above consideration we can identify T ∗BunGL(n) with themoduli space (stack) MHiggs of Higgs bundles consisting of pairs (E ,Φ).The map h which assigns to a pair (E ,Φ) the spectral curve of Φ is calledHitchin map. It can be thought of as assignment to (E ,Φ) coefficientsTr(Φk), 1 ≤ k ≤ n of the characteristic polynomial. Hence the Hitchinmap h :MHiggs → B is in fact a holomorphic map to a vector space B(although the vector structure is not canonical).


Hitchin integrable system

I will assume for simplicity that MHiggs is a manifold.Since MHiggs is a cotangent bundle, it carries a holomorphic symplecticform. It turns out that the generic fiber of the Hitchin map h is a complexLagrangian submanifold (i.e. maximal isotropic complex submanifold withrespect to the symplectic form). Moreover, the fiber h−1(b) over genericspectral curve b ∈ B can be naturally identified with the Jacobian Jac(Sb)of the corresponding spectral curve Sb ⊂ T ∗C . Therefore we have a(complex) symplectic manifold MHiggs fibered by complex Lagrangian tori.By analogy with classical Liouville integrability we can say theh :MHiggs → B is a complex integrable system. The base B consists ofspectral curves. Since B '

∏1≤k≤n H

0(C , (Ω1C )⊗k), we can choose a basis

in this vector space and identify h with a collections (H1, ...,Hn) ofPoisson-commuting holomorphic functions. They are called Hitchinhamiltonians. Many classical integrable systems are either special cases ofthe Hitchin systems or degenerations of them (especially if the Higgs fieldis allowed to have poles).


Geometry of Hitchin integrable system for mathematicians

Generic Higgs bundle (E , φ)⇔ (S ,L) spectral curve + line bundle


Key figures in Geometric Langlands: mathematics

Geometric Langlands Correspondence of A. Beilinson and V. Drinfeldamong other things gives rise to a quantization of Hitchin hamiltonians(i.e. replacing them by commuting differential operators).

(see their paper at http://www.math.uchicago.edu/ mitya/langlands.html)Yan Soibelman (KANSAS STATE UNIVERSITY)Moduli spaces of Higgs bundles in mathematics and physicsNovember 19, 2013 17 / 33

Key figures in Geometric Langlands: physics

The role of GLC in N = 4, d = 4 SYM was later explained by A. Kapustinand E. Witten (see ArXiv:hep-th/0604151).

Here N = 4 refers to the number of supersymmetries (the latter meansmathematically that Grassmann i.e. anticommuting variables are used inthe extension of the conventional YM Lagrangian). Many supersymmetricQFTs admit “topological twists” giving a new way to interpret geometricinvariants of the underlying space-time as correlators of the correspondingTFTs (e.g. Witten’s approach to Donaldson invariants of 4d manifolds).


Hitchin integrable systems and Donaldson-Thomas theory

I start with mathematics and later discuss the related physics.Recall the spectral curve S ⊂ T ∗C . Then we can “upgrade” S to acomplex 3-dimensional manifold XS . Roughly, if S is defined by theequation f (x , y) = 0, then we add two new variables u, v and defineX := XS by the equation uv = f (x , y) (coordinate-free description is givenin terms of conic bundles). The non-compact manifold X belongs to aremarkable class of 3-folds: it Calabi-Yau manifold. In other words, it is acomplex Kahler manifold, which carries a nowhere vanishing holomorphic3-form Ω3,0. A simpler example is given by C3 with the (3, 0)-formdz1 ∧ dz2 ∧ dz3. Being Kahler, X is also real symplectic, so we can speakabout compact Lagrangian submanifolds L ⊂ X . Special Lagrangiansubmanifolds are such L for which Arg(Ω3,0)|L = const. SLAGs, in asense, are “harmonic objects”: they minimize the volume.Donaldson-Thomas-type problem: for fixed γ ∈ H3(X ,Z) computethe “number” Ω(γ) of SLAGs L with [L] = γ.


General Donaldson-Thomas problem

Given: a triangulated category C, a Bridgeland stability condition on C,and a class γ ∈ K0(C).Problem: compute the “number” Ω(γ) of semistable objects E such that[E ] = γ.Example: counting semistable vector bundles with fixed Chern classes. Or:counting semistable representations of a quiver which have fixeddimension. In all cases the notion of stability condition (Tom Bridgeland,2000) is crucial.


General DT-problem was formulated in 2008 by M. Kontsevich and myselfin “Stability structures, motivic Donaldson-Thomas invariants and clustertransformations”, ArXiv:0811.2435. We also suggested a generalwall-crossing formula which shows how the collection Ω(γ) depends on thestability condition.


Quadratic differentials and WKB curves

I am going to illustrate all that in the case of SL(2) Hitchin system.I will assume that the Higgs field Φ has finitely many first order poles withsemisimple residues. Then Φ defines a quadratic differential φ on C withfinitely many second order poles: φ = ci

(z−zi )2 dz2 + .... Thus we have a

singular metric on an open subset of C (we remove poles and zeros of φ).There are “basic” SLAGs L ⊂ X = XS which correspond to geodesicsbetween two zeros of φ. More general SLAGs correspond to piecewisegeodesic lines. Homology classes γ of SLAGs are the same as the relativehomology classes of the piecewise geodesics. Therefore theDonaldson-Thomas invariant Ω(γ) counts piecewise geodesics with fixedclass γ. Alternatively, we can lift everything to the spectral curve S andcount closed “WKB curves” which are mapped to piecewise geodesics onC . The argument Arg(Ω3,0)|L is equal to the argument of the “centralcharge” Z (γ) :=

∫γ ydx where γ is the class of the closed WKB curve on

S corresponding to L, and ydx is the Liouville form on T ∗C .


SLAG spheres and geodesics


Triangulation of spectral curve


Wall-crossing formulas

Integers Ω(γ) counting geodesics depend on the “central charge”Z : γ 7→ Z (γ). Indeed, the holomorphic volume form Ω3,0 on X is uniqueup to a non-zero scalar only. This corresponds to rotation Z 7→ e iθZ . Ifwe change the angle θ a bit, the numbers Ω(γ) := Ωe iθZ (γ) generically donot change . But for some “critical values” of θ they can jump (and thetriangulation changes). These critical central charges form “walls” of realcodimension one. This change of Ω(γ)’s is described by the wall-crossingformulas: ∏

γ

TΩbefore(γ)γ =

∏γ

TΩafter (γ)γ ,

where Ωbefore and Ωafter corresponds to θ < θcrit and θ > θcrit . Theproduct is taken in the clockwise order, and Tγ is a birationaltransformation of the torus (C∗)n, n = rk H3(X ,Z) := rk Γ (also Γ is thefirst relative homology of C ).More precisely, in the monomial basis eγ , γ ∈ Γ we haveTγ(eµ) = (1− eγ)〈γ,µ〉, and 〈•, •〉 is the pairing on homology.


Physical interpretation

It is related to N = 2, d = 4 supersymmetric gauge theories. It wasworked out in a series of papers by D. Gaiotto, G. Moore, A. Neitzke.

Mathematically most understandable is “Wall-crossing, Hitchin Systems,and the WKB Approximation”, ArXiv:0907.3987.


DT invariants as BPS invariants

In few phrases, there is a remarkable class of observables in generalN = 2, d = 4, called BPS states. Their mass is equal to |Z |, where Z isthe central charge of the supersymmetry algebra acting on the space ofstates. BPS states are stable in the sense of physics (e.g. they surviveunder Mirror Symmetry). Mathematically they should correspond tosemistable objects in the appropriate categories. The total space ofHitchin system is interpreted as the space of expectation values ofso-called vector multiplets. By general physics arguments it must behyperkahler. Furthermore, since it is an integrable system, then genericallyit should carry a “semiflat” Kahler metric (the one which is flat alongtorus fibers). In order to get the “right” metric, one should correct thesemiflat one by “instanton corrections” coming from BPS states. Thelatter are interpreted geometrically as Lagrangian discs or cylinders in T ∗Cwith WKB-boundary on spectral curves S ⊂ T ∗C . In this way theDT-invariants Ω(γ) are interpreted as BPS invariants (known in physicsfor years), and WCFs imply that the corrected metric is smooth.


How physicists see the geometry of DT-invariants

(Figure and explanation from GMN paper):

An M2-brane wrapped on a cylinder, stretched between two sheets of thespectral curve (in physics the latter supports the M5-brane). Under theprojection T ∗C → C , the disc projects to a closed string in the (2, 0)theory.


Illustration of wall-crossing phenomenon


Conclusion: mathematical advances

Mathematics:1) GLC is still a very active area with many talented mathematiciansinvolved (e.g. D. Arinkin, R. Bezrukavnikov, A. Braverman, D. Gaitsgory.Drinfeld himself still publishes papers on the subject).2) Ngo used Hitchin systems in his proof of Fundamental Lemma (FieldsMedal, 2010).3) Kac conjecture (counting of absolutely indecomposable representationsof a quiver) was recently proved by T. Hausel with collaborators using hisresults about Hodge polynomials of of Hitchin system.4) Wall-crossing formulas have been connected with Mirror Symmetry forHitchin systems as well as with the tropical geometry of the base ofHitchin system and WKB-approximations of differential equations(M.K+Y.S, ArXiv:1303.3253).


Conclusion: physics advances

Physics1) Generalization by Gaiotto-Moore-Neitzke of WKB curves to higherranks (new objects, called spectral networks, which are also related togauge theories with defects).2) Application of wall-crossing formulas to construction of the modulispace of hypermultiplets in supergavity (Pioline and others). Here twistorspace for Hitchin system is upgraded to the twistor space of aquaternion-Kahler manifold.3) Relation between 2-dimensional integrable systems and N = 2, d = 4gauge theories (N.Nekrasov, S.Shatashvili). In this story quantumcohomology of fibers of Hitchin system play an important role.4) Work of GMN, S. Cecotti, C. Vafa with collaborators on the combined2d/4d wall-crossing formulas. Refined (=motivic) version is the subject ofwork in progress by Gaiotto-Moore-Witten.


And much, much more...

The above two slides of recent advances are tiny pieces of the researchactivity around the moduli of Higgs bundles and Hitchin integrablesystems. Here are few more puzzles.Hitchin systems are related via the recent work of Witten to knot theory(Khovanov homology), and via the work of A. Oblomkov, V. Shende andZ. Yun to representation theory of Cherednik algebras. There should be arelation between two approaches. Hitchin systems are related via the workof S. Mulase with collaborators to WKB asymptotics of Shrodingerequations, matrix integrals and Eynard-Orantin topological recursion. Thisshould be related to a quantization of certain Lagrangian submanifold inMHiggs × C∗ × T ∗C depending on triples (b ∈ B, λ ∈ C∗, z ∈ Sb).


We have seen that some of the best physicists and mathematicians workedon the subject. Some of the best young mathematicians and physicists stilldo.The notion of the moduli space of Higgs bundles is due to Nigel Hitchin(mathematician). It was motivated by the work of Peter Higgs (physicist).And the long history of the subject demonstrates the usefulness ofexchange of ideas between the two communities.


Moduli spaces of Higgs bundles in mathematics and physicsIn physics A corresponds to a gauge eld on...

Documents

Transcript of Moduli spaces of Higgs bundles in mathematics and physicsIn physics A corresponds to a gauge eld on...