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CTQM Problem List

The CTQM problem list is concerned with the focal point of the research interest of the center:

The Topology and Quantization of Moduli Spaces.


The creation of this problem list was suggested to us by Dennis Sullivan at the CTQM opening symposium.

This problem list is webbased and will evolve dynamically over time. All researchers in the field are invited to suggest both new problems to list and comments to existing problems on the list. This is done by sending a mail to the director of CTQM, Jørgen Ellegaard Andersen at .

  1. Find a cell decomposition of moduli space of surfaces of genus g and no marked points.
    (Dennis Sullivan, CUNY)
    Comments:
    1. The cell decompositions known for other moduli spaces (see Penner) use distinguished points. (DS)
    2. People have tried to use shortest closed geodesics in place of distinguished points.This leads to new interesting questions about the possible combinatorial configurations of collections of shortest closed geodesics. (DS)
    3. Perhaps one might use weierstrass points instead of distinguished points to solve this problem. (DS)

  2. Extend cell decompositions of the noncompact moduli spaces to their natural nodal compactifications.
    (Dennis Sullivan, CUNY)

  3. There are several genus zero subfamilies of moduli spaces whose cell complexes together with the induced maps between them on the level of chains or at the level of homology corresponding to glueing along boundary components describe universal algebraic structures. Examples include A-infinity algebras, BV algebras, formal Frobenius Manifolds, and infinity Lie bialgebras. Describe the algebraic structure tacitly defined by all moduli spaces and all possible glueings either at the level of chains or at the level of homology.
    (Dennis Sullivan, CUNY)

  4. Find geometric structures on surfaces which are parametrized by the Teichmüller component -- the distingushed contractible component of the moduli space of representations of a surface group into a split real form of a simple Lie group.
    (Nigel Hitchin, Oxford University)
    Comments:
    1. For PSL(3,R) Goldman and Choi showed that this space parametrizes convex RP^2 structures. (NH)
    2. Labourie has shown that the mapping class group acts properly on these spaces. (NH)
    3. Labourie has also shown that for PSL(n,R) there is a (in general non-smooth) curve in RP^(n-1) which describes the situation, but one would like a smooth structure. For example, in the case of PSL(3,R), Labourie's curve is the non-smooth boundary of a convex set, but a flat projective structure is a smooth concept. (NH)
    4. Fock and Goncharov have results about these spaces using tropical geometry. (NH)

  5. Geometrically quantize Teichmüller space using the complex structure from its embedding in the PSL(2,C) Higgs bundle moduli space, which is just C^(3g-3). The symplectic structure then gives a complete S^1-invariant Kähler metric.
    (Nigel Hitchin, Oxford University)

  6. The Fourier-Mukai transform takes a solution to the Higgs bundle equations on a Riemann surface to a bundle with a unitary connection compatible with the flat hyperkähler structure on the cotangent bundle of the Jacobian. Characterize this connection.
    (Nigel Hitchin, Oxford University)
    Comments:
    1. Bonsdorff's thesis work showed that one holomorphic structure extends to the projective bundle compactification of the cotangent bundle. (NH)
    2. Doing this for fixed points of the circle action on the moduli space would give another approach to determining the hyperbolic metric on a Riemann surface. (NH)

  7. It is beleived (although Dennis Sullivan is sceptical) that all mapping class groups are linear. For quite a long time I have a specific candidate for a faithful finite-dimensional representation. Precise problem: Try to make the plan stated below this work.
    (Maxim Kontsevich, IHES)
    Comments:
    1. Main idea: According to Thurston, a "generic" conjugacy class R in T_g is pseudo-Anosov and have a canonical representative which preserves two tranversal laminations and multiplies the transversal measures by lambda, 1/lambda, where lambda (exponent of entropy) is strictly >1 and an algebraic integer (appears as the largest eignevalue on 1st cohomology of the ramified double covering on which the qudartic differential becomes an abelian differential). Moreover, one may assume that the laminations have 4g-4 triple singular points. I would like to construct a representation in which R acts non-trivially, because there will be some eigenvalue of the corresponding matrix equal to a positive power of lambda.

      A representation of T_g is the same as a local system on M_g. Here is the description of it. The fiber of it at a complex curve C is a kind of middle (or total?) cohomology of the configuration space Conf_{4g-4}(C)=C^{4g-4}-Diagonals (the configuration space of 4g-4 points (x_1,...,x_{4g-4}) on C), with coefficients in the flat bundle, the 1st cohomology group of the double cover of C, doubly ramified at (x_1,...,x_{4g-4}).

      Why I beleive that this should work: Thurston's representative gives a map from Conf_{4g-4}(C) (together with the local system) to itself, with a prefered marked point such that the action on the fiber has the largest eignevalue equal to lambda.

      Can one use Lefschez fixed point formula, or maybe some dynamical reasoning, to see that this eigenvalue appear in the total cohomology of Conf_{4g-4}(C) with local coefficients as well? (MK)


  8. Quantize surfaces with hyperbolic metric.
    (Maxim Kontsevich, IHES)
    Comments:
    1. Deformation quantization gives in general only an algebar over formal power series in Planck constant. Nevertheless, for a flat symplectic torus T the Moyal product in the integral form gives an actual family of associative products on smooth functions on T, depending on a real parameter hbar. On the other side, it is well-known that it is not possible to quantize the round 2-dimensional sphere, if one asks for an SO(3)-invariant product.

      The question: can one actually quantize surface of negative curvature?

      A possible candidate: F.Berezin quantized equivariantly the hyperbolic plane, using Toeplitz operators. Can one make sense for the star-product of Gamma-invariant functions where Gamma is a cocompact subgroup of PSL(2,R)?

      It seems that some old calculaton of F.Radulescu confirms the convergence. If the answerČis yes, two more questions:

      1) Will PSL(2,Z) transformations on hbar induce Morita equivalences?

      2) for a given hbar, does there exists a flat connection on the corresponding bundle of algebras over the moduli space of complex sructures (=cocompact subgroups in PSL(2,R)? I expect a positive answer, it will give a new family of infinite-dimensional projective representations of the mapping class group T_g. (MK)


  9. Sums of products of q-factorials over rational polyhedra as fake quantum invariants of 3-manifolds.
    (Maxim Kontsevich, IHES)
    Comments:
    1. Let P be a rational polyhedron in R^n, contained in the standard unit cube. Fix also some integers (a_1,...,a_n) and an integral quadratic form Q in n variables. Such a gadget produces an sequence of algebraic numbers Z(k), in fact elements of cyclotomic fields.

      Definition: for a positive integer k let q be a primitive k-th root of 1 (e.g; exp(2 Pi i/k)), for m in {0,...,k-1} denote by (m]! the q-factorial of m, =(1-q)(1-q^2)...(1-q^m).

      Z(k):= sum over integer points m=(m_1,...,m_n) in kP of the product Prod_{i=1}^n ([m_i]!)^(a_i) times q^Q(m,m)

      (Better notation would be Z(q), not Z(k)).

      Let us consider a finite linear combinatioin with integer coefficients of such things. Obviusly, any quantum invariant of a 3-manifold is (as function of level) of such type.

      I claim that ANY function as above has an asymptotic behavior as k--> infinity of the same nature as quantulm invariant. I mean that it will have an asymptotic expansion with the exponent equal to the Beilinson-Borel regulator of an element of K_3 (number field), times a series in 1/k with algebraic coefficients.

      It leads to a bit non-precise problem: find linear relations between functions Z(q) arising from different polytopes P, vectors (a_1,..;,a_n) and quadratic forms Q.

      Is there some universal group U, like scissor group such that an invariant of 3-manifold and a compact Lie group G is an element of U, and actual numerical valuesn of quantum invariants at different levels appear as its images of certain homomorphisms from U to cyclotomic fields? (MK)


  10. Give a mathematical definition of the Gopakumar-Vafa invariants of a Calabi-Yau manifold.
    (Robbert Dijkgraaf, University of Amsterdam)
    Comments:
    1. There have been several attempts to define these invariants in terms of the Lefshetz action on the L^2-cohomology of an appropriate moduli space of sheaves (hep-th/9910181). (RD)
    2. The GV invariants can also be defined as certain combinations of Gromov-Witten invariants. Then the nontrivial conjecture is that they are integer, in contrats with the GW invariants, which are in general only rational. (RD)

  11. Define and calculate Donaldson-Thomas invariants for rank > 1.
    (Robbert Dijkgraaf, University of Amsterdam)
    Comments:
    1. In particular use localization techniques as in math.AG/0312059 to compute DT invariants in the case of toric manifolds. (RD)
    2. In the case of a bundle of surfaces S over a curve, relate these invariants to the quantum cohomology of the instanton moduli space on S. (RD)
    3. This could give a mathematical interpretation of the OSV conjecture hep-th/0405146, that relates DT invariants for large rank to GW invariants. (RD)

  12. Quantize the moduli space of complex structures on a CY manifold X, as regarded as a Lagrangian inside H^3(X,C).
    (Robbert Dijkgraaf, University of Amsterdam)
    Comments:
    1. In physics this procedure is known as quantum Kodaira-Spencer theory hep-th/9309140. (RD)
    2. The resulting wave function should equal the generation function of GW invariants on the moduli space of Kaehler metrics on the mirror manifold. (RD)

  13. Does there exist a non-trivial knot with trivial Jones polynomial?
    (Vaughan Jones, UC Berkeley)

  14. Decide whether or not the higher genus mapping class groups has Kazhdan's property T.
    (Vaughan Jones, UC Berkeley)
    Comments:
    1. One obtains Hilbert space representations of the higher genus mapping class groups by summing over all levels of the quantum SU(2) representations comming from TQFT. It would be natural to investigate these representations in this context. (VJ)

  15. Give a Toeplitz operator construction of the unitary structure which Hitchin's connection preserves.
    (Jørgen Ellegaard Andersen, CTQM)
    Comments:
    1. It is known abstractly by the work of Laszlo combined with the recent work of Ueno and Andersen, that Hitchin's connection is unitary. This however does not allow one to study any kind of asymptotic properties for this unitary structure. (JEA)

  16. Prove Witten's conjecture concerning the asymptotic expansion of the quantum SU(n) invariants of closed oriented 3-manifolds.
    (Jørgen Ellegaard Andersen, CTQM)
    Comments:
    1. The geometric quantizations of Lagrangian subvarieties on moduli spaces, the theory of Toeplitz operators and their asymptotic well behavedness with respect to Hitchin's connection should be usefull tools. (JEA)

  17. The moduli space of SU(2)-bundles on a Riemann surface has several desingularizations (due to Kirwan, Narasimhan-Ramanan, and Seshadri). Use Floer homology on one of these desingularizations to construct invariants of three-manifolds.
    (Ciprian Manolescu, Columbia)
    Comments:
    1. The Atiyah-Floer conjecture states that instanton homology should be equal to a version of Lagrangian Floer homology defined on the moduli space of SU(2)-bundles. The problem is that this space is singular, so it is natural to look at a desingularization. (CM)
    2. Wehrheim and Salamon have a project for proving the Atiyah-Floer conjecture, where on the unknown side they use a homology defined in terms of the symplectic vortex equations. (CM)
    3. Is any of the three desingularizations Fano? Probably not, but if the answer turns out to be yes, then Floer homology would have a nice behavior there (no bubbling). (CM)

  18. Khovanov (math.QA/0302060) defined a categorification of the colored Jones polynomial. Use Floer homology to construct a symplectic analogue of this.
    (Ciprian Manolescu, Columbia)
    Comments:
    1. Seidel and Smith defined an invariant of knots using Floer homology on a quiver variety. They conjectured it to be equal to Khovanov's categorification of the usual Jones polynomial. The problem asks to do a similar construction for the colored Jones. (CM)

  19. Define a bigrading on Seidel-Smith homology.
    (Ciprian Manolescu, Columbia)
    Comments:
    1. Khovanov homology is a bigraded theory such that taking the Euler characteristic in one direction gives the Jones polynomial. Seidel-Smith homology is a version of Floer homology conjectured to be equivalent to Khovanov's theory, but it only has one grading. (CM)
    2. The second grading was constructed on the chain complex in math.SG/0411015. The hope is to show that the differentials preserve this grading. If this is too hard, it would be good to at least prove that the second grading defines a filtration on the chain complex. (CM)

  20. Does the moduli space of curves admit a complete Riemannian metric of nonpositive curvature?
    (Peter Storm, Stanford)
    Comments:
    1. More precisely, one can ask does the Teichmueller space of a surface S admit a complete Riemannian metric of nonpositive sectional curvature such that the usual Mod(S) action is isometric? Can such a metric have finite covolume? Can such a metric be quasi-isometric to the Teichmueller metric? I learned this question from Benson Farb, who thinks the last two answers should be no. (PS)

  21. For a free group F and a connected non-solvable Lie group G, faithful representations form a dense subset of Hom(F,G). Let H be the fundamental group of a closed hyperbolic surface. For which Lie groups will faithful representations form a dense subset of Hom(H,G)?
    (Peter Storm, Stanford)
    Comments:
    1. This was recently answered in the affirmative for PSl(2,C) and PSl(2,R) by R. Kent and J. DeBlois (Duke 131, no. 2 (2006), 351 - 362). (See also recent work of Nikolay Gusevskii.) This is a question due to Goldman, originating in his thesis. (PS)

  22. Coisotropic A-Branes Of Higher Rank
    (Edward Witten, IAS)
    Comments:
    1. In the A-model of a symplectic manifold X (or alternatively in the Floer-Fukaya category of X) the usual branes are supported on a Lagrangian manifold L that is endowed with a flat vector bundle V that may have any rank.

      However, Kapustin and Orlov showed that under certain conditions the A-model also admits ``coisotropic branes'' whose support is on coisotropic submanifolds of greater than one-half the dimension of X. The differential-geometric conditions for a coisotropic brane of rank 1 were determined by Kapustin and Orlov, and have been elucidated in terms of generalized complex geometry by Gualtieri.

      Coisotropic branes of rank greater than 1 should also exist. The problem is to characterize them in terms of differential geometry. If M is a coisotropic submanifold of X, endowed with a complex vector bundle V of rank greater than 1 and a connection A on V, what are the conditions on M and A so that the triple (M,V,A) determine an A-brane or in other words an object in the Floer-Fukaya category?

      Among other possible applications, the question is significant for the A-model of hyper-Kahler manifolds and in particular for the geometric Langlands program. (EW)

  23. There are a number of categorifications of knot polynomials: Heegard-Floer homology for the Alexander polynomial, Khovanov homology for the Jones polynomial, Khovanov-Rozansky for HOMFLYPT at a particular value of n, and Khovanov-Rozansky for HOMFLYPT at a general value of n. Dunfield-Gukov-Rasmussen conjecture [math.GT/0505662] that there is a single theory relating all of these, relying on a triply-graded homology theory with a family of related differentials. Construct such a theory.
    (Dylan Thurston, Columbia)
    Comments:
    1. The triply-graded homology is presumably Khovanov-Rozansky at general n [math.QA/0505056]. The challenge is to construct the differentials. (DT)
    2. The construction could either be combinatorial or geometric (presumably counting pseudo-homolorphic curves). (DT)

  24. Give a combinatorial formula for the Seiberg-Witten invariants.
    (Dylan Thurston, Columbia)

  25. Using the Manolescu-Ozsvath-Sarkar combinatorial definition of Heegaard-Floer homology for knots, give a combinatorial proof that the span of HF homology equals the genus, that a knot is fibered if and only if its HF homology is monic, or some of the other results proved using this homology.
    (Jørgen Ellegaard Andersen, CTQM and Dylan Thurston, Columbia)

Revised 19.12.2006

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