Searcharxiv⌕ Search

arXiv subjects

Nan-Kuo Ho

Publications and source records attributed to Nan-Kuo Ho.

18 recordsLinked to original sources
arXiv2026

Geometry of the tt*-Toda equations I: universal centralizer and symplectic groupoids

We investigate the geometry of a certain space of meromorphic connections with irregular singularities, and prove in particular that it is a (real) symplectic Lie groupoid. The connections have a physical meaning: they correspond to certain solutions of the topological-antitopological fusion (tt*) equations of Cecotti and Vafa, and hence to deformations of supersymmetric quantum field theories. The groupoid structure arises because we restrict ourselves to the tt* equations of Toda type, whose monodromy data has a Lie theoretic description. To obtain these results, we show first that the universal centralizer of a Lie group is a holomorphic symplectic groupoid over the Steinberg cross section.

math.SG↗
arXiv2021

Polytopes, supersymmetry, and integrable systems

We review some links between Lie-theoretic polytopes and field theories in physics, which were proposed in the 1990's. A basic ingredient is the Coxeter Plane, whose relation to integrable systems and the Stokes Phenomenon has only recently come to light. We use this to give a systematic mathematical treatment, which gives further support to the physical proposals. This article is based on a talk which was scheduled to be given at the workshop "Representations of Discrete Groups and Geometric Topology on Manifolds", Josai University, 12-13 March 2020.

math.DG↗
arXiv2021

Flat connections and the commutator map for SU(2)

We study the topology of the SU(2)-representation variety of the compact oriented surface of genus 2 with one boundary component about which the holonomy is a generator of the center of SU(2).

math.SG↗
arXiv2018

Hitchin's equations on a nonorientable manifold

We define Hitchin's moduli space for a principal bundle $P$, whose structure group is a compact semisimple Lie group $K$, over a compact non-orientable Riemannian manifold $M$. We use the Donaldson-Corlette correspondence, which identifies Hitchin's moduli space with the moduli space of flat $K^\mathbb{C}$-connections, which remains valid when M is non-orientable. This enables us to study Hitchin's moduli space both by gauge theoretical methods and algebraically by using representation varieties. If the orientable double cover $\tilde{M}$ of $M$ is a Kähler manifold with odd complex dimension and if the Kähler form is odd under the non-trivial deck transformation on $\tilde{M}$, Hitchin's moduli space of the pull-back bundle $\tilde{P}$ over $\tilde{M}$ has a hyper-Kähler structure and admits an involution induced by the deck transformation. The fixed-point set is symplectic or Lagrangian with respect to various symplectic structures on Hitchin's moduli space over $\tilde{M}$. We show that there is a local diffeomorphism from Hitchin's moduli space over (the nonorientable manifold) $M$ to the fixed point set of the Hitchin's moduli space over (its orientable double cover) $\tilde{M}$. We compare the gauge theoretical constructions with the algebraic approach using representation varieties.

math.DG↗
arXiv2018

Conditions of smoothness of moduli spaces of flat connections and of character varieties

We use gauge theoretic and algebraic methods to examine sufficient conditions for smooth points on the moduli space of flat connections on a compact manifold and on the character variety of a finitely generated and presented group. We give a complete proof of the slice theorem for the action of the group of gauge transformations on the space of flat connections. Consequently, the slice is smooth if the second cohomology of the manifold with coefficients in the semisimple part of the adjoint bundle vanishes. On the other hand, we find that the smoothness of the slice for the character variety of a finitely generated and presented group depends not only on the second group cohomology but also on the relation module of the presentation. However, when there is a single relator or if there is no relation among the relators in the presentation, our condition reduces to the minimality of the second group cohomology. This is also verified using Fox calculus. Finally, we compare the conditions of smoothness in the two approaches.

math.DG↗
arXiv2018

Kostant, Steinberg, and the Stokes matrices of the tt*-Toda equations

We propose a Lie-theoretic definition of the tt*-Toda equations for any complex simple Lie algebra $\mathfrak{g}$, based on the concept of topological-antitopological fusion which was introduced by Cecotti and Vafa. Our main result concerns the Stokes data of a certain meromorphic connection, whose isomonodromic deformations are controlled by these equations. Exploiting a framework introduced by Boalch, we show that this data has a remarkable structure, which can be described using Kostant's theory of Cartan subalgebras in apposition and Steinberg's theory of conjugacy classes of regular elements. A by-product of this is a convenient visualization of the orbit structure of the roots under the action of a Coxeter element. As an application, we compute canonical Stokes data of certain solutions of the tt*-Toda equations in terms of their asymptotics.

math.DG↗
arXiv2018

A Lie-theoretic description of the solution space of the tt*-Toda equations

We give a Lie-theoretic explanation for the convex polytope which parametrizes the globally smooth solutions of the topological-antitopological fusion equations of Toda type (tt$^*$-Toda equations) which were introduced by Cecotti and Vafa. It is known from [GL] [GIL1] [M1] [M2] that these solutions can be parametrized by monodromy data of a certain flat $SL_{n+1}\mathbb{R}$-connection. Using Boalch's Lie-theoretic description of Stokes data, and Steinberg's description of regular conjugacy classes of a linear algebraic group, we express this monodromy data as a convex subset of a Weyl alcove of $SU_{n+1}$.

math.DG↗
arXiv2017

The SU(2)-character variety of the closed surface of genus 2

We study the symplectic geometry of the SU(2)-representation variety of the compact oriented surface of genus 2. We use the Goldman flows to identify subsets of the moduli space with corresponding subsets of $\mathbb P^3(\mathbb C)$. We also define and study two antisymplectic involutions on the moduli space and their fixed point sets.

math.SG↗
arXiv2010

Orientability in Yang-Mills Theory over Nonorientable Surfaces

In arXiv:math/0605587, the first two authors have constructed a gauge-equivariant Morse stratification on the space of connections on a principal U(n)-bundle over a connected, closed, nonorientable surface. This space can be identified with the real locus of the space of connections on the pullback of this bundle over the orientable double cover of this nonorientable surface. In this context, the normal bundles to the Morse strata are real vector bundles. We show that these bundles, and their associated homotopy orbit bundles, are orientable for any n when the nonorientable surface is not homeomorphic to the Klein bottle, and for n<4 when the nonorientable surface is the Klein bottle. We also derive similar orientability results when the structure group is SU(n).

math.SG↗
arXiv2009

Anti-Perfect Morse Stratification

For an equivariant Morse stratification which contains a unique open stratum, we introduce the notion of equivariant antiperfection, which means the difference of the equivariant Morse series and the equivariant Poincare series achieves the maximal possible value (instead of the minimal possible value 0 in the equivariantly perfect case). We also introduce a weaker condition of local equivariant antiperfection. We prove that the Morse stratification of the Yang-Mills functional on the space of connections on a principal U(n)-bundle over a connected, closed, nonorientable surface is locally equivariantly Q-antiperfect when the rank n=2,3; we propose that it is actually equivariantly Q-antiperfect when n=2,3. Our proposal yields formulas of G-equivariant Poincare series of the representation variety of flat G-connections for the nonorientable surface where G=U(2), SU(2), U(3), SU(3). Our rank 2 formulas agree with formulas proved by T. Baird in arXiv:0806.1975. Baird verified our conjectural rank 3 formulas when the nonorientable surface is the real projective plane or the Klein bottle (arXiv:0901.1604); he proved our conjectural U(3) formula for any closed nonorientable surfaces by establishing equivariant Q-antiperfection in this case (arXiv:0902.4581).

math.SG↗
arXiv2008

Yang-Mills Connections On Orientable and Nonorientable Surfaces

In math.SG/0605587, we studied Yang-Mills functional on the space of connections on a principal G_R-bundle over a closed, connected, nonorientable surface, where G_R is any compact connected Lie group. In this sequel, we generalize the discussion in "The Yang-Mills equations over Riemann surfaces" by Atiyah and Bott, and math.SG/0605587. We obtain explicit descriptions (as representation varieties) of Morse strata of Yang-Mills functional on orientable and nonorientable surfaces for non-unitary classical groups SO(n) and Sp(n). It turns out to be quite different from the unitary case. we use Laumon and Rapoport's method in "The Langlands lemma and the Betti numbers of stacks of G-bundles on a curve" to invert the Atiyah-Bott recursion relation, and write down explicit formulas of rational equivariant Poincaré series of the semistable stratum of the space of holomorphic structures on a principal $SO(n,\bC)$-bundle or a principal $Sp(n,\bC)$-bundle.

math.SG↗
arXiv2008

Yang-Mills Connections on Nonorientable Surfaces

In "The Yang-Mills equations over Riemann surfaces", Atiyah and Bott studied Yang-Mills functional over a Riemann surface from the point of view of Morse theory. We generalize their study to all closed, compact, connected, possibly nonorientable surfaces. We introduce the notion of "super central extension" of the fundamental group of a surface. It is the central extension when the surface is orientable. We establish a precise correspondence between Yang-Mills connections and representations of super central extension. Knowing this exact correspondence, we work mainly at the level of representation varieties which are finite dimensional instead of the level of strata which are infinite dimensional.

math.SG↗
arXiv2006

The real locus of an involution map on the moduli space of flat connections on a Riemann surface

It is known that every nonorientable surface $Σ$ has an orientable double cover $\tildeΣ$. The covering map induces an involution on the moduli space $\tilde{\M}$ of gauge equivalence classes of flat $G$-connections on $\tildeΣ$. We identify the relation between the moduli space $\M$ and the fixed point set of the moduli space $\tilde{\M}$. In particular, $\M$ is isomorphic to the fixed point set of $\tilde{\M}$ if and only if the order of the center of $G$ is odd. One important application is that we give a way to construct a minimal Lagrangian submanifold of the moduli space $\tilde{\M}$.

math.SG↗
arXiv2005

The volume of the moduli space of flat connections on a nonorientable 2-manifold

We compute the Riemannian volume on the moduli space of flat connections on a nonorientable 2-manifold, for a natural class of metrics. We also show that Witten's volume formula for these moduli spaces may be derived using Haar measure, and we give a new proof of Witten's volume formula for the moduli space of flat connections on an orientable surface using Haar measure.

math.SG↗
arXiv2004

Connected Components of the Space of Surface Group Representations II

In math.SG/0303255, we discussed the connected components of the space of surface group representations for any compact connected semisimple Lie group and any closed compact (orientable or nonorientable) surface. In this sequel, we generalize the results in math.SG/0303255 in two directions: we consider general compact connected Lie groups, and we consider all compact surfaces, including the ones with boundaries. We also interpret our results in terms of moduli spaces of flat connections over compact surfaces.

math.SG↗
arXiv2004

On the Connectedness of Moduli Spaces of Flat Connections over Compact Surfaces

We study the connectedness of the moduli space of gauge equivalence classes of flat G-connections on a compact orientable surface or a compact nonorientable surface for a class of compact connected Lie groups. This class includes all the compact, connected, simply connected Lie groups, and some non-semisimple classical groups including U(n) and Spin^C(n).

math.SG↗
arXiv2003

Connected Components of The Space of Surface Group Representations

Let G be a connected, compact, semisimple Lie group. It is known that for a compact closed orientable surface $Σ$ of genus $l >1$, the order of the group $H^2(Σ,π_1(G))$ is equal to the number of connected components of the space $Hom(π_1(Σ),G)/G$ which can also be identified with the moduli space of gauge equivalence classes of flat G-bundles over $Σ$. We show that the same statement for a closed compact nonorientable surface which is homeomorphic to the connected sum of k copies of the real projective plane, where $k\neq 1,2,4$, can be easily derived from a result in A. Alekseev, A.Malkin and E. Meinrenken's recent work on Lie group valued moment maps.

math.SG↗