Bernard A. Mares [email protected] Research Statement

e primary focus of my current research program is fourdimensional dierential topology and nonabelian gauge theory. For my thesis project, I studied the VafaWitten equations (..) over a general oriented Riemannian fourmanifold. My eventual goal is to use these equations to establish new invariants of smooth fourmanifolds, analogous to the tremendously successful invariants of Donaldson and SeibergWitten. I explain some background material in Section , and describe some connections with various elds in math and physics including algebraic geometry, number theory, quantum eld theory, and Sduality. Although these invariants currently lack a rigorous mathematical basis, physicists have conjectured that the generating functions for these invariants should be modular forms. Numerous independent computations from algebraic geometry support this conjecture, providing compelling evidence that these invariants are welldened. As described in ., the VafaWitten equations were recently extended by Haydys and Witten to a ve dimensional gauge theory which is being used to study Khovanov homology and FukayaSeidel categories [Hay, Wit]. In Section I state my specic research goals. I describe how the VafaWitten equations present new challenges which are not present in other similar equations. Consequently, several analytical obstacles must be overcome before the VafaWitten invariants can be rigorously dened, namely compactness and transversality. As explained in ., the partial compactness result of my thesis is one step towards this aim. Continuing this work, my immediate goal is to establish a complete compactication of the moduli space, and to understand under which circumstances the VafaWitten equations lead to welldened fourmanifold invariants. Another topic I plan to investigate is moduli spaces of singular solutions to the VafaWitten equations, which I discuss in .. e theory of singular connections (which I review in .) was developed by Kronheimer and Mrowka [KM, KMb] to allow singularities of prescribed holonomy along an embedded surface. Singular connections were rst used to understand constraints on the genus of embedded surfaces, and later used to determine the structure of Donaldson invariants over manifolds of “simple type.” e singular VafaWitten moduli spaces are widely believed to be related to Jacobi forms, but surprisingly little work has been done from the analytical perspective.

Background

. e Donaldson invariants

During the s and the early s, the moduli space of antiselfdual (ASD) instantons was the primary

tool for studying smooth fourmanifolds. Let M, g be an oriented Riemannian fourmanifold, and let G

> A > be a compact Lie group. Given a principal Gbundle P M, a connection A P, has curvature FA

2 2, 



M; adP . e Hodge star operator splits twoforms into metricdependent 1 eigenspaces M; adP .

 

 Correspondingly, curvature decomposes as FA FA FA. For a specic choice of M, g and P, we dene ASD moduli space as

P,g 

¢ > A G ÅASD A P FA 0 P ,

P,g Å

where G P is the group of automorphisms of P. When nonempty, ASD typically is a nitedimensional

G submanifold of A P P. Roughly speaking, Donaldson showed how for a principal bundle P with G SU 2 ,

P,g

` A G

the moduli space ÅASD P P determines a homology class. Numerical invariants of manifolds then

G arise by pairing this homology class with the cohomology classes of A P P [Don, DK]. ese invariants

Bernard A. Mares [email protected] have many applications including distinguishing smooth structures on fourmanifolds, and understanding connectedsum decompositions. e proof that Donaldson’s invariants are welldened is notorious for the analytic challenges involved. For P,g example, sometimes ÅASD has singularities and is noncompact. Usually the singularities can be eliminated P,g via metric perturbations, and there is a natural Uhlenbeck compactication ÅASD. To dene invariants of the underlying smooth structure, one must ensure that the construction is independent of the choice of metric P,g g. Roughly, this amounts to showing that dierent choices of metric lead to homologous ÅASD.

. Supersymmetric YangMills theory, Sduality, and the VafaWitten equations

Sduality, also known as MontonenOlive duality, electricmagnetic duality, or strongweak duality, is a conjectural duality between quantum eld theories (or string theories), which relates respective behaviors at strong and weak coupling. Since the mathematical foundations of quantum eld theory have yet to be established, for this subsection we adopt the physicist’s level of rigor without further qualication.

4

Æ

e Æ 2 and 4 supersymmetric YangMills theories over R can be “twisted” so that when generalized to a Riemannian fourmanifold X, g , the observables become independent of the choice of metric g. Assuming that such observables are welldened, they should be dierentialtopological invariants for the

smooth structure of X.

In the Æ 2 case, there is a unique such twist called “DonaldsonWitten theory” [Wit], and the observables are the Donaldson invariants. e supersymmetric Lagrangian corresponds to a MathaiQuillen

path integral, which computes the characteristic numbers for the Euler class of the gaugeequivariant map 

A ´ FA [AJ]. Electricmagnetic duality leads to a formula expressing the Donaldson invariants in terms

of the SeibergWitten invariants, known as Witten’s conjecture [Wit]. anks to the SO3 monopole program of Feehan and Leness (a.k.a. the PU2 monopole program), many cases of Witten’s conjecture now

have a rigorous proof [FL].

With Æ 4 supersymmetry, there are three inequivalent topological twists [Yam, Mar]. Once again, each twist has a MathaiQuillen expression [LL], so we obtain equations for each twist. e “VafaWitten

twist” analyzed in [VW] corresponds to the equations A A > P,

 1

  F  B B C, B 0,

A 2 2, 

>

where B M; ad , (..)

‡ P

 0,

dAB dAC 0

C > M; adP .

2,  3

> 

(e product denoted by β1  β2 for βi M; R is pointwiseequivalent to the cross product on R .)

e virtual dimension of the corresponding moduli space is zero, so for each topological type P of principal

Gbundle P X, one expects a single numerical invariant VW X, P dened as a regularized Euler number. Under certain restrictions on the Riemannian curvature of X, [VW] explained how the resulting moduli

P,g P,g Å space ÅVW of solutions to the VafaWitten equations coincides with the ASD moduli space ASD, and how the VafaWitten invariant should be

P,g

Å Ǒ VW X, P χ ASD , (..)

P,g where Å ASD denotes some compactication, and χ is the Euler characteristic. Note that while the moduli P,g space Å ASD depends on the metric up to cobordism, the Euler characteristic is not cobordisminvariant.

Bernard A. Mares [email protected]

P,g

However, the aforementioned restrictions on Riemannian curvature ensure that the topology of ÅASD cannot change. For any simple, simplylaced Lie group G, one can assemble the VafaWitten invariants for principal G bundles into a generating function

k

É VW X, k q (..) k according to the instanton number. Vafa and Witten observed that Sduality predicts a modular transforma tion law, which relates the generating functions corresponding to G and its Langlands dual group G ¹ . e full details can be found in [VW, Wu].

. Singular connections

Kronheimer originally proposed the use of singular connections as a tool for understanding the genus of

embedded surfaces in fourmanifolds [Kro]. One chooses an embedded surface Σ ` X and a parameter

1

α > 0, 2 . A singular SU 2 connection is then allowed to have holonomy which is conjugate to

e2πiα 0

0 e 2πiα

along any innitesimal loop around Σ. When applied to the ASD equations, singular connections are a

powerful tool because of how they interpolate between an ordinary SU 2 moduli space as α 0, and an

1

SO3 moduli space as α 2 . Moreover, this interpolation incorporates the topology of Σ. Kronheimer’s proposed program was successfully carried out by Kronheimer and Mrowka [KM, KMb]. eir results also managed to prove an extremely powerful structure theorem for the Donaldson invariants [KM, KMa]. Roughly, their structure theorem implies (under certain restrictions) that the innitely many Donaldson invariants are determined by just a nite collection of data. According to the mostlyproven Witten conjecture, this data turns out to be the SeibergWitten invariants.

. Recent developments

Since their discovery in , the SeibergWitten invariants [Wit] essentially replaced the role of the far more dicult Donaldson invariants. However the past few years have seen a resurgence of activity involving the ASD equations and other nonabelian gauge theories. In particular, there have been several recent applications of instanton Floer homology and/or singular connections to knot theory, e.g. [KM, HK]. e VafaWitten equations also recently appeared as a fourdimensional reduction of a particular ve dimensional gauge theory, independently discovered by Haydys [Hay, ()] and Witten [Wit, (.)]. According to Haydys, these equations are expected to produce numerical invariants of vemanifolds, Floer homology groups in four dimensions which “categorify” the VafaWitten invariants, and a FukayaSeidel category in three dimensions [Hay]. According to Witten, these equations are expected to compute Khovanov homology under appropriate boundary conditions [Wit]. One of these boundary conditions involves elds which are asymptotic to solutions of the VafaWitten equations. Currently the aforementioned results from this new HaydysWitten gauge theory are speculative, due in large part to the same analytic diculties plaguing the VafaWitten invariants. A better understanding of the Vafa Witten equations would certainly help towards understanding the properties of this vedimensional theory.

Bernard A. Mares [email protected]

Figure – An idealized view of ÅVW: while deforming the metric gt on X, the Euler

Å

characteristic of ÅASD can change. Any change to χ ASD should balance with the creation

Å or destruction of points in Å VW ASD.

Research objective: a mathematical denition of the VafaWitten invariants

. e problem

While the conjectural VafaWitten invariant VW X, P has a geometric interpretation under particular P,g circumstances as the Euler characteristic of ÅASD, as in (..), this denition is not viable in general. Dierent choices of the metric g lead to cobordant moduli spaces, and Euler characteristic is not invariant under cobordism! Instead we must look to the full VafaWitten equations to formulate a proper denition.

When ÅVW is a union of points cut out transversely by the equations (..), the VafaWitten invariant should

be a signed count 

є ˆ x

 VW X, P É 1 .

P,g Å

x > VW



Å However, ÅVW contains a copy of ASD because if FA 0, then A, B, C A,0,0 is a solution. When

ÅASD has positive dimension, the above summation makes no sense. Instead, consider the complement

Å Å Å

VW ¢ VW ASD.

Å

If VW P, g is cut out transversely by nitely many points, then we could try to dene 

є ˆ x

Å  

VW X, P χ ASD P, g É 1 .

È

Å ˆ 

x > VW P, g

is would require proving compactness and transversality for ÅVW P, g . en one must show that the resulting answer is independent of g, as illustrated in Figure .

Unfortunately, these assumptions about ÅVW are overly optimistic. For a concrete example, consider the

1  threedimensional reduction X S Y to some oriented Riemannian threemanifold Y. If the principal bundle P pulls back from Y, then the VafaWitten equations reduce to Corlette’s equations [Cor]. When C C G is semisimple with complexication G , Corlette proves that the moduli space is the G character variety, C and B corresponds to the imaginary part of a at G connection.

Bernard A. Mares [email protected]

Since character varieties are ane varieties, oen with positive dimension, they can be noncompact. While noncompactness due to bubbling of ASD connections is easily handled via the Uhlenbeck compactication [DK], the surprising new feature is noncompactness arising from the extra eld B. Indeed, compactness

properties due to spinor elds are precisely what make the analysis of the SeibergWitten equations [KM] and PU 2 monopole equations [Tel, FL] so tractable. However for the VafaWitten equations, the Uhlenbeck compactication is no longer sucient. ere are two obvious methods of attack on the compactication problem. One possibility would be to nd a perturbation from which both compactness and transversality follow. Since the moduli space has virtual dimension zero, it’s conceivable that some perturbation can always produce a transverse and nite moduli space. However, perturbations seem to hurt a priori estimates, making compactness more dicult. A more promising approach is to examine the MorganShalen compactication of the character variety [MS, MS]. If I can nd the appropriate fourdimensional generalization, the same mechanism may lead to a useful compactication of the VafaWitten moduli space.

. Analytic progress e rst step toward proving a compactness theorem for the ASD, SeibergWitten, or PU2 monopole 2 equations is to obtain an a priori L1 bound for all solutions. is results from expanding and rearranging the L2 norm of the equations. Proceeding in the same way for the VafaWitten equations, one computes

2 ‡ 

 1 2 1

       FA B B C, B dAB dAC B s 6W B

2 12 ∫X

non-negative, and zero for solutions Riemannian curvature term

1 2 1 2 1 ‡ 2 2 2 1 2 1

  ©       ¸ FA AB dAB dAC C, B B B FA FA . (..)

2 4 2 4 2 ∫X

2  essentially equivalent to the L1 norm of ˆ A,B,C topological

Here s is scalar curvature, and W  is the selfdual Weyl tensor, which acts on selfdual twoformsasatraceless 2 symmetric endomorphism [AHS]. We wish to bound the “L1” terms for solutions. e le hand side vanishes for solutions. Since the “topological” term is constant for any xed principal bundle, the only obstacle

1 2

 is the “Riemannian curvature term,” which is quadratic in B. If the quartic term 4 B B were positive

denite, it would dominate the Riemann term, allowing us to control the Riemannian term by completing the square. Positivedeniteness of the quartic term occurs for the SeibergWitten equations and the PU 2 monopole equations, but not for the VafaWitten equations. Analytically, this the reason the VafaWitten moduli space fails to be Uhlenbeckcompact. In my thesis, I showed that compactness fails only in a rather tame way. In particular, I use a meanvalue

inequality to show that a sequence can fail to be compact only when the L2 norm of B approaches innity:

eorem ([Mar, eorem ..]). Let P X be a principal SU 2 bundle over an oriented Riemannian

P,g

> Å B fourmanifold X. For any constant b, the Uhlenbeck closure of A, B,0 VW s.t. B L2 b is compact.

. Singular solutions to the VafaWitten equations

Recall from (..) that when we assemble the VafaWitten invariants for each instanton number into a generating function, Sduality predicts that this generating function transforms as a modular form. For singular instantons, there is an additional topological number: the monopole number. e generating

Bernard A. Mares [email protected] function for singular instantons is therefore a function of two formal parameters. According to the work of [MNVW, Yos, Kap], Sduality predicts that this generating function transforms as a Jacobi form. When X is Kähler, there is a KobayashiHitchin correspondence between parabolic bundles and singular ASD connections. is led to some curious algebrogeometric computations of Jacobi forms in the aforementioned references. ese computations are carried out as in (..), under the assumption that the VafaWitten invariants are calculable from the moduli space of parabolic bundles. By keeping track of boundary terms in (..), it should be straightforward to determine under which conditions this assumption is justiable. Finally, on a more speculative note, it would be interesting to investigate the singular VafaWitten moduli spaces and understand the implications. In analogy with the program of Kronheimer and Mrowka, one could try to understand the interplay between embedded surfaces and the structure of VafaWitten invariants.

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