Publication list - detailed (short)
Papers listed in MathSciNet. Papers available at the Mathematics ArXiv and at the High Energy Physics ArXiv .29. Hausel, T.: S-duality in hyperkähler Hodge theory, arXiv:0709.0504
Abstract
Here we survey questions and results on the Hodge theory of hyperkaehler quotients, motivated by certain S-duality considerations in string theory. The problems include L^2 harmonic forms, Betti numbers and mixed Hodge structures on the moduli spaces of Yang-Mills instantons on ALE gravitational instantons, magnetic monopoles on R^3 and Higgs bundles on a Riemann surface. Several of these spaces and their hyperkaehler metrics were constructed by Nigel Hitchin and his collaborators.28. Hausel, T., Rodriguez-Villegas, F.: Mixed Hodge polynomials of character varieties, arXiv:math.AG/0612668
Abstract
We calculate the E-polynomials of certain twisted GL(n,C)-character varieties M_n of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n,F_q) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,C)-character variety. The calculation also leads to several conjectures about the cohomology of M_n: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious Hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n = 2.Citations
27. Hausel, T.: Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform, Proceedings of the National Academy of Sciences of the United States of America 103, no. 16, 6120--6124, arxiv:math.AG/0511163
Abstract
A Fourier transform technique is introduced for counting the number of solutions of holomorphic moment map equations over a finite field. This in turn gives information on Betti numbers of holomorphic symplectic quotients. As a consequence simple unified proofs are obtained for formulas of Poincaré polynomials of toric hyperkähler varieties (recovering results of Bielawski-Dancer and Hausel-Sturmfels), Poincaré polynomials of Hilbert schemes of points and twisted ADHM spaces of instantons on C^2 (recovering results of Nakajima-Yoshioka) and Poincaré polynomials of all Nakajima quiver varieties. As an application, a proof of a conjecture of Kac on the number of absolutely indecomposable representations of a quiver is announced.Citations
26. Hausel, T.: Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve, in Geometric Methods in Algebra and Number Theory Series: Progress in Mathematics, Vol. 235 Bogomolov, Fedor; Tschinkel, Yuri (Eds.) 2005, arXiv: math.AG/0406380
Abstract
This is a survey of results and conjectures on mirror symmetry phenomena in the non-Abelian Hodge theory of a curve. We start with the conjecture of Hausel-Thaddeus which claims that certain Hodge numbers of moduli spaces of flat SL(n,C) and PGL(n,C)-connections on a smooth projective algebraic curve agree. We then change our point of view in the non-Abelian Hodge theory of the curve, and concentrate on the SL(n,C) and PGL(n,C) character varieties of the curve. Here we discuss a recent conjecture of Hausel-Rodriguez-Villegas which claims, analogously to the above conjecture, that certain Hodge numbers of these character varieties also agree. We explain that for Hodge numbers of character varieties one can use arithmetic methods, and thus we end up explicitly calculating, in terms of Verlinde-type formulas, the number of representations of the fundamental group into the finite groups SL(n,F_q) and PGL(n,F_q), by using the character tables of these finite groups of Lie type. Finally we explain a conjecture which enhances the previous result, and gives a simple formula for the mixed Hodge polynomials, and in particular for the Poincare polynomials of these character varieties, and detail the relationship to results of Hitchin, Gothen, Garsia-Haiman and Earl-Kirwan. One consequence of this conjecture is a curious Poincare duality type of symmetry, which leads to a conjecture, similar to Faber's conjecture on the moduli space of curves, about a strong Hard Lefschetz theorem for the character variety, which can be considered as a generalization of both the Alvis-Curtis duality in the representation theory of finite groups of Lie type and a recent result of the author on the quaternionic geometry of matroids.Citations
25. Hausel, T., Proudfoot, N. : Abelianization for hyperkähler quotients, Topology, 44 (2005) 231-248, arXiv: math.SG/0310141
Abstract
We study an integration theory in circle equivariant cohomology in order to prove a theorem relating the cohomology ring of a hyperkähler quotient to the cohomology ring of the quotient by a maximal abelian subgroup, analogous to a theorem of Martin for symplectic quotients. We discuss applications of this theorem to quiver varieties, and compute as an example the ordinary and equivariant cohomology rings of a hyperpolygon space.Citations
24. Hausel, T.: Quaternionic Geometry of Matroids, Central European Journal of Mathematics, 3 (1), (2005), 26--38 arXiv: math.AG/0308146
Abstract
Building on a recent joint paper with Sturmfels, here we argue that the combinatorics of matroids is intimately related to the geometry and topology of toric hyperkaehler varieties. We show that just like toric varieties occupy a central role in Stanley's proof for the necessity of McMullen's conjecture (or g-inequalities) about the classification of face vectors of simplicial polytopes, the topology of toric hyperkaehler varieties leads to new restrictions on face vectors of matroid complexes. Namely in this paper we give two proofs that the injectivity part of the Hard Lefschetz theorem survives for toric hyperkaehler varieties. We explain how this implies the g-inequalities for rationally representable matroids. We show how the geometrical intuition in the first proof, coupled with results of Chari, leads to a proof of the g-inequalities for general matroid complexes, which is a recent result of Swartz. The geometrical idea in the second proof will show that a pure O-sequence should satisfy the g-inequalities, thus showing that our result is in fact a consequence of a long-standing conjecture of Stanley.
Citations
23. Hausel, T.,Swartz, E.: Intersection forms of toric hyperkähler varieties, Proceedings of the American Mathematical Society, 134, (2006), 2403-2409, arXiv: math.AG/0306369
Abstract
This note proves combinatorially that the intersection pairing on the middle dimensional compactly supported cohomology of a smooth toric hyperkaehler variety is always definite, providing a large number of non-trivial L^2 harmonic forms for toric hyperkaehler metrics on these varieties. This is motivated by a result of Hitchin about the definiteness of the pairing of L^2 harmonic forms on complete hyperkaehler manifolds of linear growth.
Citations
22. Etesi, G., Hausel, T.: New Yang-Mills-instantons on multi-centered metrics, arXiv: hep-th/0207196 , Communications in Mathematical Physics, 235 No. 2 , (2003) 275-288
Abstract
In this paper we explicitly calculate the analogue of the 't Hooft SU(2) Yang--Mills instantons on Gibbons--Hawking multi-centered gravitational instantons which come in two parallel families: the multi-Eguchi--Hanson, or A_k ALE gravitational instantons and the multi-Taub--NUT, or A_k ALF gravitational instantons. We calculate their action and find the reducible ones. Following Kronheimer we also exploit the U(1) invariance of our solutions and study the corresponding explicit singular SU(2) magnetic monopole solutions on Euclidean three-space.Citations
21. Hausel, T., Hunsicker, E., Mazzeo, R.: Hodge cohomology of gravitational instantons, Duke Mathematical Journal, 122 Issue 3, (2004) 485-548, arXiv: math.DG/0207169
Abstract
We study the space of L^2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type. These metrics generalize the geometric structures at infinity of several different well-known classes of metrics, including asymptotically locally Euclidean manifolds, the (known types of) gravitational instantons, and also Poincaré metrics on Q-rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in Kähler manifolds. The answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold. The L^2 signature formula implied by our result is closely related to the one proved by Dai [dai] and more generally by Vaillant [Va], and identifies Dai's tau invariant directly in terms of intersection cohomology of differing perversities. This work is also closely related to a recent paper of Carron [Car] and the forthcoming paper of Cheeger and Dai [CD]. We apply our results to a number of examples, gravitational instantons among them, arising in predictions about L^2 harmonic forms in duality theories in string theory.Citations
20. Etesi, G., Hausel, T.: Geometric construction of new Yang-Mills instantons over Taub-NUT space, Physics Letters B , 514 (1-2) (2001), 189-199 arXiv: hep-th/0105118
Abstract
In this paper we exhibit a one-parameter family of new Taub-NUT instantons parameterized by a half-line. The endpoint of the half-line will be the reducible Yang-Mills instanton corresponding to the Eguchi-Hanson-Gibbons L^2 harmonic 2-form, while at an inner point we recover the Pope-Yuille instanton constructed as a projection of the Levi-Civita connection onto the positive su(2) subalgebra of the Lie algebra so(4). Our method imitates the Jackiw-Nohl-Rebbi construction originally designed for flat R^4. That is we find a one-parameter family of harmonic functions on the Taub-NUT space with a point singularity, rescale the metric and project the obtained Levi-Civita connection onto the other negative su(2) subalgebra of so(4). Our solutions will possess the full U(2) symmetry, and thus provide more solutions to the recently proposed U(2) symmetric ansatz of Kim and Yoon.19. Hausel, T., Sturmfels, B.: Toric hyperkaehler varieties, Documenta Mathematica, 7 (2002), 495-534, arXiv: math.AG/0203096
Abstract
Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces by symplectic torus actions. A toric hyperkahler variety is a complete intersection in a Lawrence toric variety. Both varieties are non-compact, and they share the same cohomology ring, namely, the Stanley-Reisner ring of a matroid modulo a linear system of parameters. Familiar applications of toric geometry to combinatorics, including the Hard Lefschetz Theorem and the volume polynomials of Khovanskii-Pukhlikov, are extended to the hyperkahler setting. When the matroid is graphic, our construction gives the toric quiver varieties, in the sense of Nakajima.Citations
18. Hausel, T., Thaddeus, M.: Examples of mirror partners arising from integrable systems, Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique, 333 (4) (2001) 313-318, arXiv: math.AG/0106140
Abstract
In this note we present pairs of hyperkaehler orbifolds which satisfy two different versions of mirror symmetry. On the one hand, we show that their Hodge numbers (or more precisely, stringy E-polynomials) are equal. On the other hand, we show that they satisfy the prescription of Strominger, Yau, and Zaslow (which in the present case goes back to Bershadsky, Johansen, Sadov and Vafa): that a Calabi-Yau and its mirror should fiber over the same real manifold, with special Lagrangian fibers which are tori dual to each other. Our examples arise as moduli spaces of local systems on a curve with structure group SL(n); the mirror is the corresponding space with structure group PGL(n). The special Lagrangian tori come from an algebraically completely integrable Hamiltonian system: the Hitchin system.Citations
17. Hausel, T., Thaddeus, M.: Mirror symmetry, Langlands duality and Hitchin systems, Inventiones Mathematicae, 153, No. 1, 2003, 197-229 arXiv: math.AG/0205236
Abstract
We study the moduli spaces of flat SL(r)- and PGL(r)-connections, or equivalently, Higgs bundles, on an algebraic curve. These spaces are noncompact Calabi-Yau orbifolds; we show that they can be regarded as mirror partners in two different senses. First, they satisfy the requirements laid down by Strominger-Yau-Zaslow (SYZ), in a suitably general sense involving a B-field or flat unitary gerbe. To show this, we use their hyperkahler structures and Hitchin's integrable systems. Second, their Hodge numbers, again in a suitably general sense, are equal. These spaces provide significant evidence in support of SYZ. Moreover, they throw a bridge from mirror symmetry to the duality theory of Lie groups and, more broadly, to the geometric Langlands program.Citations
16. Etesi, G., Hausel, T.: Geometric interpretation of Schwarzschild Instantons, Journal of Geometry and Physics , 37 (2001) 126-136 arXiv: hep-th/0003239
Abstract
In this note we address the problem of finding Abelian instantons of finite energy on the Euclidean Schwarzschild manifold. This amounts to construct self-dual L^2 harmonic 2-forms on the space. Gibbons found a non-topological L^2 haRmonic form in the Taub-NUT metric, leading to Abelian instantons with continuous energy. We imitate his construction In the case of the Euclidean Schwarzschild manifold and find a non-topological self-dual L^2 harmonic 2-form on it. We show how this gives rise to Abelian instantons and identify them with SU(2)-instantons of Pontryagin number 2n^2 found by Charap and Duff in 1977. Using results of Dodziuk and Hitchin we also calculate the full L^2 harmonic space for the Euclidean Schwarzschild manifold.Citations
15. Hausel, T., Thaddeus, M.: Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles , Proceedings of the London Mathematical Society 88 (2004) 632-658 ,arXiv: math.AG/0003093
Abstract
The moduli space of stable vector bundles on a Riemann surface is smooth when the rank and degree are coprime, and is diffeomorphic to the space of unitary connections of central constant curvature. A classic result of Newstead and Atiyah-Bott asserts that its rational cohomology ring is generated by the universal classes, that is, by the Kunneth components of the Chern classes of the universal bundle. This paper studies the larger, non-compact moduli space of Higgs bundles, as introduced by Hitchin and Simpson, with values in the canonical bundle K. This is diffeomorphic to the space of all connections of central constant curvature, whether unitary or not. The main result of the paper is that, in the rank 2 case, the rational cohomology ring of this space is again generated by universal classes. The spaces of Higgs bundles with values in K(n) for n > 0 turn out to be essential to the story. Indeed, we show that their direct limit has the homotopy type of the classifying space of the gauge group, and hence has cohomology generated by universal classes. A companion paper treats the problem of finding relations between these generators in the rank 2 case.Citations
14. Hausel, T., Thaddeus, M. : Relations in the cohomology ring of the moduli space of rank 2 Higgs bundles, Journal of the American Mathematical Society, 16 (2003), 303-329, arXiv: math.AG/0003094.
Abstract
The moduli space of stable bundles of rank 2 and degree 1 on a Riemann surface has rational cohomology generated by the so-called universal classes. The work of Baranovsky, King-Newstead, Siebert-Tian and Zagier provided a complete set of relations between these classes, expressed in terms of a recursion in the genus. This paper accomplishes the same thing for the non-compact moduli spaces of Higgs bundles, in the sense of Hitchin and Simpson. There are many more independent relations than for stable bundles, but in a sense the answer is simpler, since the formulas are completely explicit, not recursive. The results of Kirwan on equivariant cohomology for holomorphic circle actions are of key importance. Together, Parts I and II describe the cohomology rings of spaces of rank 2 Higgs bundles at essentially the same level of detail as is known for stable bundles.Citations
13. Hausel, T.: Geometric quantization and Jones-Witten theory (in Hungarian) in Algebraic topology and geometry in Physics, (lecture notes of Summer school for Hungarian Physics students, Óbánya, 1997), MAFIHE, Budapest, 1999
Abstract
These lecture notes give, for an audience of partly physics and partly mathematics students, a solid introduction to geometric quantization and its relation to Jones-Witten theory.12. Hausel, T.: Geometry of the moduli space of Higgs bundles, thesis for Ph.D. in Pure Mathematics, DPMMS, Cambridge University, August 1998, arXiv:math.AG/0107040
Abstract
This thesis contains work which appeared in several papers. Additionally to the results in the papers it contains a detailed introduction and some further proofs and remarks. The dissertation gives a description of the topology and symplectic and algebraic geometry of Hitchin's hyperkaehler moduli space M of rank 2 Higgs bundles with fixed determinant of odd degree over a fixed Riemann surface. After the long introduction it describes a compactification of M in great detail, using symplectic cutting (math.AG/9804083). Examining the downward Morse flow of a natural circle action on M it shows the vanishing of intersection numbers (math.AG/9805071). Examining the upward Morse flow it explains a set of generators of the cohomology ring and a conjectured explicit description of the cohomology ring (which was proven in math.AG/0003094). Then finally it introduces the resolution tower for M, and shows that its direct limit is homotopically equivalent with the classifying space of the gauge group. In turn it yields another proof of the generation theorem (as in math.AG/0003093) and also yields a purely algebraic geometric proof of the Mumford conjecture about the cohomology ring of the moduli space of rank 2 stable bundles on curves. It finishes by proving homotopy stabilizations in the resolution tower analogously to the Atiyah-Jones conjecture.Citations
11. Hausel, T.: Vanishing of intersection numbers on the moduli space of Higgs bundles , Adv. Theor. and Math. Phys. , 2 (1998) 1011-1040, arXiv:math.AG/9805071
Abstract
In this paper we consider the topological side of a problem which is the analogue of Sen's S-duality testing conjecture for Hitchin's moduli space of rank 2 stable Higgs bundles of fixed determinant of odd degree over a Riemann surface. We prove that all intersection numbers in the compactly supported cohomology vanish, i.e. "there are no topological L^2 harmonic forms on Hitchin's space". This result generalizes the well known vanishing of the Euler characteristic of the moduli space of rank 2 stable bundles of fixed determinant of odd degree over the given Riemann surface. Our proof shows that the vanishing of all intersection numbers in the compactly supported cohomology of Hitchin's space is given by relations analogous to Mumford's relations in the cohomology ring of the moduli space of stable bundles.Citations
10. Hausel, T.: Compactification of moduli of Higgs bundles , Journal für die reine und angewandte Mathematik , Volume 503 (1998) 169-192, arXiv:math.AG/9804083,
Abstract
In this paper we consider a canonical compactification of Hitchin's moduli space of stable Higgs bundles with fixed determinant of odd degree over a Riemann surface, producing a projective variety by gluing in a divisor at infinity. We give a detailed study of the compactified space, the divisor at infinity and the moduli space itself. In doing so we reprove some assertions of Laumon and Thaddeus on the nilpotent cone.Citations
9. Hausel, T., Makai, E. jr.,Szûcs, A.: Inscribing cubes and covering by rhombic dodecahedrons via equivariant topology, Mathematika 47 (2000), 371-397 , arXiv: math.MG/9906066
Abstract
First, we prove a special case of Knaster's problem, implying that each symmetric convex body in R^3 admits an inscribed cube. We deduce it from a theorem in equivariant topology, which says that there is no S_4-equivariant map from SO(3) to S^2, where S_4 acts on SO(3) as the rotation group of the cube and on S^2 as the symmetry group of the regular tetrahedron. We also give some generalizations. Second, we show how the above non-existence theorem yields Makeev's conjecture in R^3 that each set in R^3 of diameter 1 can be cover ed by a rhombic dodecahedron, which has distance 1 between its opposite faces. This reveals an unexpected connection between inscribing cubes into symmetric bodies and covering sets by rhombic dodecahedra. Finally, we point out a possible application of our second theorem to the Borsuk problem in R^3. (Similar results were obtained recently by V.V. Makeev and independently by G. Kuperberg (cf. math.MG/9809165).)Citations
8. Hausel, T., Makai, E. jr.,Szûcs A.: Polyhedra inscribed and circumscribed to convex bodies, General Mathematics , 1997, Proc. of 3rd Internat. Workshop on Diff. Geom. and its Appls. and the 1st German-Romanian Seminar on Geometry, 1997, Sibiu, Romania
Abstract
In this talk we give a survey of older results and some new results about the following question: what type of polyhedra can be inscribed or circumscribed to convex bodies in R^n?Citations
7. Hausel, T., Moment map, toric varieties and mixed volumes, dissertation for diploma in Department of Mathematics, Eötvös Loránd University , December 1995
Abstract
We survey two proofs of Kushnirenko's theorem about the number of solutions of certain polynomial equations in this dissertation. One proof approaches from symplectic geometry. The second proof uses ideas from toric geometry. We give a detailed introduction to both subjects. Our aim in this dissertation is to show the power of new ideas in modern geometry by applying them to prove theorems which can be stated in elementary terms.6. Hausel, T.: On a Gallai-type problem for lattices. Acta Mathematica Hungarica (66) (1995), no.1-2, 127-145
Abstract
In this paper we prove that if any three elements of a finite collection of convex discs in the plane intersect in a lattice point, then there are two lattice points so that each convex set contains (at least) one of the two points.Citations
5. Bezdek K.,Hausel, T.: On the number of lattice hyperplanes which are needed to cover the lattice points of a convex body. Intuitive Geometry (Szeged,1991), 27-31, Colloq. Math. Soc. János Bolyai, 63, North-Holland, Amsterdam, 1994
Abstract
Let K be a convex body in E^d and L be a d-dimensional lattice of E^d. Assume that the union of n lattice hyperplanes of L covers the lattice points in the non-empty intersection of K and L. In this note we prove that there is a certain lower bound for n in terms of the lattice width of K with respect to L.Citations
4. Bezdek K.,Hausel, T.: Coating by cubes. Beiträge zur Algebra und Geometrie 35 (1994), no.1, 119-123
Abstract
In this paper we prove two results corresponding to the question: how many cubes are needed in R^n to "coat" a given cube.3. Hausel, T.: Transillumination of lattice packing of balls. Studia Sci. Math. Hungar 27 (1992), no.1-2, 241-242
Abstract
Here we prove the existence of a lattice packing of balls in R^n intersecting every affine subspace of R^n of codimension constant times square root of n.Citations
2. Hausel, T.: On a two dimensional problem in lattice geometry, (in Hungarian) KÖMAL (Journal of Mathematics and Physics for Secondary Schools) (1989), no. 3, 103-107
Abstract
This paper proves in a geometrical way that if a convex disc in the plane has n lattice points, then there are at least n-4 triples of colinear such lattice points.1. Hausel, T.: Pedal triangle and convergent sequences , (in Hungarian) KÖMAL (Journal of Mathematics and Physics for Secondary Schools) (1988) no. 10, 433-437
Abstract
Starting from a problem in the geometry of Euclidean triangles we solve a problem on certain convergent sequences.