Geometry, Quantum Topology and Asymptotics

Rinat Kashaev

The June 30–July 5, 2014 conference, during the period of one week, brought together mathematicians and theoretical physicists working at the border between mathematics and quantum physics. That means that the main topics of the conference were about interactions between these two.

To give an example of such an interaction to a non-specialist, one can recall the discovery of V. Jones of his famous polynomial invariant of knots approximately 30 years ago. A knot here means a mathematical object represented by a smooth embedding of the circle in space, and the main goal in knot theory is the classification of different types of knots, where two knots are considered equivalent if the corresponding embeddings are isotopic, i.e. if they can be transformed into each other in continuous manner.

The basic idea behind the classification of knot types is to construct invariants, that is to associate with each knot an algebraic quantity, for example a number, in such a way that two quantities associated with two knots are same whenever they are equivalent. The ultimate goal would be to find a perfect invariant in the sense that two quantities are the same if and only if both knots are of the same type.

Such an invariant exists from the time of H. Poincaré (about 100 years ago), and it is called “the fundamental group of the complement of the knot together with the peripheral system”, but unfortunately this invariant is too complicated to be useful in practice, because comparing two such quantities is a problem as difficult as the comparison of two knot types.

On the other hand, the Jones polynomial, being very useful in practical terms (it is easy to compare two polynomials), is not perfect in the sense that there are different knot types with identical Jones polynomials. Nonetheless, the Jones polynomial is a knot invariant which attracts a lot of interest due to its mysterious nature from the mathematical viewpoint: we know well how to calculate it but we do not know well what it measures and what is its mathematical origin.

In the construction of the Jones polynomial, profound ideas of quantum physics are used that are not very well studied by mathematicians, mainly because theoretical physicists often develop their ideas without complete mathematical rigor. Here we observe an important difference between mathematicians and theoretical physicists: the former appreciate proofs of theorems, while the latter try to arrive as quickly as possible to new results mostly in the form of new formulae. With such an attitude of a theoretical physicist it is difficult to always maintain the highest level of mathematical rigor.

The discovery of the Jones polynomial resulted in new research directions. In particular, researchers from both sides, mathematicians and theoretical physicists, have developed the theory of quantum groups that helped to construct generalizations of the Jones polynomial which are now called “quantum invariants”. Moreover, a new mathematical discipline called “quantum topology” has emerged. Despite all these developments, the problem of mathematical understanding of quantum invariants still persists.

One of the prominent contemporary mathematicians M. Atiyah, in his lecture given at Clay Research Conference on the Poincar conjecture, Paris 2010, pointed out that after the solution of the 3-dimensional Poincaré conjecture the most important problems which remain in low dimensional topology are the 4-dimensional smooth Poincaré conjecture and the problem of geometrical understanding of the quantum invariants. In this light, the subjects discussed at the Confucius Institute conference are very important.

One of the central topics at the conference was the mirror symmetry in the theory of quantum strings. It is a relationship between specific mathematical objects called Calabi–Yau manifolds which are used for handling the extra dimensions of string theory. The mirror symmetry establishes a very non-trivial equivalence between two Calabi–Yau manifolds which geometrically look very different. It started attracting the attention of mathematicians approximately 20 years ago when it was used for solving problems in enumerative geometry, e.g. for counting rational curves on Calabi–Yau manifolds. Similarly to the Jones polynomial, the mirror symmetry originally was based on physical ideas without precise mathematically formulations but some of its mathematical implications have since found mathematically rigorous understanding. One of the most important research subjects in this direction is called SYZ-mirror symmetry, a conjecture formulated by A .Strominger, S.-T. Yau, and E. Zaslow. A number of speakers discussed Calabi–Yau manifolds in the context of SYZ-mirror symmetry.

With 47 registered participants, there were given 24 talks. The conference provided an opportunity for fruitful interaction between specialists working in different subjects, mathematicians and theoretical physicists. One specific goal was to promote scientific collaboration and exchange between mathematicians from China and the West. Many participants appreciated the nice environment and the work done by the wonderful team of organizers from the Confucius Institute, namely the director Basile Zimmermann and his colleagues Alain-Vincent Hurlimann, Xavier Magnenat, Catherine Sattiva, Fabien Scotti, Bastien Stucky.

On the conference web-site, http://geneva2014.gatech.edu, video recordings of the most of the talks are available. Below is a short review for specialists of the contents of the talks.

Jørgen Ellegaard Andersen (Aarhus University) explained how the asymptotic behavior of the Hitchin connection near the Thurston boundary of Teichmüller space leads to the specification of a basis for the geometric quantization of the moduli spaces of semi-stable holomorphic bundles on a Riemann surface and discussed possible relations of this basis to SYZ-mirror symmetry.

Murad Alim (Harvard University) talked about his ongoing work in collaboration with H. Movasati, E. Scheidegger and S. T. Yau on construction of a Lie algebra by using a finitely generated differential ring of functions on the moduli space of Calabi–Yau threefolds and certain vector fields with the Gauss–Manin connection.

Vladimir Bazhanov (Australian National University) talked about Fateev’s correspondence between classical and quantum integrable systems which allows to derive functional and integral equations determining the full spectrum of finite volume massive QFT and which is of potential applications to the quantization problem of integrable non-linear sigma models.

Gaëtan Borot (Max Planck Institute) talked about calculation of SU (N ) Chern–Simons theory partition functions in the case of Seifert 3-manifolds with the so called basic knots by using the methods of matrix models.

Kwok Wai Chan (The Chinese University of Hong Kong) explained his joint work in progress with Daniel Pomerleano and Kazushi Ueda on the study of homological mirror symmetry for toric Calabi–Yau manifolds from the viewpoint of the SYZ conjecture.

Tudor Dimofte (Institute for Advanced Study) announced about solution of a long standing problem of quantizing Chern–Simons theory with complex gauge group SL(2, C).

Bertrand Eynard (Institut de Physique Théorique) discussed recent progress in applying the topological recursion to computation of the asymptotic expansion of the wave functions and their annihilators of “quantum curves”.

Hiroyuki Fuji (Tsinghua University) talked about his work with Sergei Gukov, Piotr Sulkowski, and Marko Stosic on the q-difference structure of the colored superpolynomial for knots which is the Poincare polynomial of the colored HOMFLY homology. He also discussed an analogue of the volume conjecture in this context.

Stavros Garoufalidis (Georgia Institute for Technology) explained some recent results on the calculation of asymptotics of q-series related to new quantum knot invariants, in particular those arising from the Teichmüller TQFT.

An Huang (Harvard University) talked about his work done in collaboration with Spencer Bloch, Bong Lian, Vasudevan Srinivas, Shing-Tung Yau, and Xinwen Zhu about a generalization of the GKZ system, called the tautological system that governs period integrals of a complete intersection family in certain ambient variety with a large group action.

Louis Hirsch Kauffman (University of Illinois at Chicago) reviewed the algebraic structures underlying Khovanov homology that arise from the geometry and topology of the category of surface cobordisms. He also discussed relationships of Khovanov homology with quantum statistical mechanics and quantum information theory.

Siu Cheong Lau (Harvard University) talked about the SYZ construction of mirrors of conifold transitions of toric Calabi–Yau manifolds based on the wallcrossing and computation of open Gromov–Witten invariants. This gives a geometric viewpoint to the classical correspondence between Minkowski decomposition of polytopes and factorization of polynomials.

Thang Le (Georgia Institute of Technology) explained the role of the Habiro ring in the unification of quantum invariants of 3-manifolds.

Wenxuan Lu (Tsinghua University) talked about the stability conditions on K3 surfaces from the perspective of mirror symmetry in the so called attractor backgrounds for certain black holes.

Feng Luo (Rutgers University) reported about an identity for closed hyperbolic surfaces whose terms depend on the dilogarithms of the lengths of simple closed geodesics in all 3-holed spheres and 1-holed tori in the surface.

Vladimir Mangazeev (Australian National University) discussed geometric consistency relations between angles of 3D circular quadrilateral lattices, their quantization, and new solutions of the tetrahedron equation (the 3D analog of the Yang–Baxter equation).

Rahul Pandharipande (ETH Zürich) explained his recent proof (with R. Thomas) of the Katz–Klemm–Vafa formula governing higher genus curve counting in arbitrary classes on K3 surfaces. The subject intertwines Gromov–Witten, Noether–Lefschetz, and Donaldson–Thomas theories.

Emanuel Scheidegger (University of Freiburg) discussed the relation between topological strings on Calabi–Yau threefolds and modular forms. A special differential ring of functions (also discussed in the talk of M. Alim) plays the analogous role of the ring of quasimodular forms in the case of elliptic curves, and the topological string amplitudes are polynomials in this ring.

Bernd Siebert (Universität Hamburg) talked about geometric quantization of semi-positive varieties inspired by mirror symmetry. The construction of maximal degenerations of Calabi–Yau pairs via wall structures comes with a distinguished basis of sections of the polarizing line bundle. In the case of abelian varieties these are classical theta functions.

Roland van der Veen (University of Amsterdam) dicussed relations between the classical topology of normal surfaces and quantum knot invariants. An important instance of such relation is the slope conjecture which states that the degree of the colored Jones polynomial detects boundary slopes of essential surfaces in the knot complement.

Shing-Tung Yau (Harvard University, Chinese University of Hong Kong, Tsinghua University) described the extension of mirror symmetry to non-Kaehler Calabi–Yau manifolds and some recent advances in understanding the structures and cohomologies of differential forms on the symplectic side.

Don Zagier (College de France, France and Max Planck Institute for Mathematics) talked about Kashaev invariants, Nahm sums, and modularity.

Yuguang Zhang (Tsinghua University) explained his result with Valentino Tosatti where they show that a fibered Calabi–Yau manifold is essentially a product, if there is no singular fiber.


KASHAEV, Rinat. « Geometry, Quantum Topology and Asymptotics ». In Blog Scientifique de l’Institut Confucius, Université de Genève. Permanent link: https://ic.unige.ch/?p=548, accessed 06/23/2024.