This conference is devoted to relations between quantum field theory and string theory one hand, and mathematical knot theory and random matrix models on the other hand. Surprising connections between these areas of research have been found in last years. In the conference we will summarize important recent developments in this context and try to set the goals for the future research. Topics considered in the conference include: supersymmetric gauge theories, BPS states, topological string theory, integrability, homological knot invariants, matrix models, topological recursion.

Speakers:

Murad Alim, Maciej Borodzik, Andrea Brini, Miranda Cheng, Tudor Dimofte, Tobias Ekholm, Sergei Gukov, Lotte Hollands, Rinat Kashaev, Albrecht Klemm, Piotr Kucharski, Andrei Mironov, Motohico Mulase, Satoshi Nawata, Boris Pioline, Elli Pomoni, Markus Reineke, Marko Stošić, Cumrun Vafa, Don Zagier

Registration:

To register for the conference please write an e-mail to Petr Vasko (petr.vasko@fuw.edu.pl) before August 20, 2018, stating your name and affiliation, and arrival and departure dates. Some support for travel or accommodation for participants may be available – to apply for it please send a request before August 1, 2018, to the same e-mail address (petr.vasko@fuw.edu.pl), explaining why and what kind of support you might need. Please also let us know if you would be interested in presenting a poster during the conference, or whether you wish to stay at the Hera Guest House (see below).

Organizers:

Jakub Jankowski, Ehsan Hatefi, Miłosz Panfil, Piotr Sułkowski (chair), Petr Vasko
Faculty of Physics, University of Warsaw

Special events

Public lecture
Prof. Cumrun Vafa (Harvard University) – "Physics, Math and Puzzles" [lecture website]
September 24 (Monday), 18:00

Conference dinner
September 26 (Wednesday), 19:00

Program

MONDAY, September 24
11:00-11:45 – Registration
11:45-12:45 – Cumrun Vafa, "String Theory and Homological Invariants for 3-Manifolds"
12:45-14:30 – Lunch break
14:30-15:30 – Rinat Kashaev , "Title"
15:30-16:00 – Coffee break
16:00-17:00 – Murad Alim , "From tt* flat connections to parabolic Higgs bundles and opers"
18:00 – Public lecture: Prof. Cumrun Vafa , "Physics, Math and Puzzles" [lecture website]

TUESDAY, September 25

8:30-9:00 – Wake-up coffee
9:00-10:00 – Maciej Borodzik, "Title"
10:00-10:30 – Coffee break
10:30-11:30 – Boris Pioline, "Attractor flow trees, BPS indices and quivers"
11:45-12:45 – Markus Reineke, "BPS state algebras of quivers"
12:45-14:30 – Lunch break
14:30-15:30 – Elli Pomoni , "Title"
15:30-16:00 – Coffee break
16:00-17:00 – Andrea Brini , "Chern-Simons theory and the higher genus B-model"

WEDNESDAY, September 26

8:30-9:00 – Wake-up coffee
9:00-10:00 – Tudor Dimofte, "Title"
10:00-10:30 – Coffee break
10:30-11:30 – Don Zagier, "Title"
11:45-12:45 – Tobias Ekholm, "Open Gromov-Witten skein module invariants and large N duality"
12:45-14:30 – Lunch break
14:30-15:30 – Satoshi Nawata , "Geometry and physics of DAHA"
15:30-16:00 – Coffee break
19:00 – Conference dinner

THURSDAY, September 27

8:30-9:00 – Wake-up coffee
9:00-10:00 – Motohico Mulase, "Ribbon graphs, Frobenius-Hopf duality, and CohFT"
10:00-10:30 – Coffee break
10:30-11:30 – Sergei Gukov, "Title"
11:45-12:45 – Andrei Mironov, "Title"
12:45-14:30 – Lunch break
14:30-15:30 – Lotte Hollands , "Title"
15:30-16:00 – Coffee break
16:00-17:00 – Piotr Kucharski , "Physics and geometry of knots-quivers correspondence"

FRIDAY, September 28

8:30-9:00 – Wake-up coffee
9:00-10:00 – Marko Stošić, "Knots-quivers correspondence, lattice paths, and rational knots"
10:00-10:30 – Coffee break
10:30-11:30 – Albrecht Klemm, "Title"
11:45-12:45 – Miranda Cheng, "Title"
12:45-14:30 – Lunch + farewell

Abstracts

Murad Alim , "From tt* flat connections to parabolic Higgs bundles and opers"
Abstract: I will discuss the tt* flat connection of the middle dimensional cohomology of mirror Calabi-Yau manifolds with one-dimensional moduli spaces. This data can be interpreted as parabolic Higgs bundles on the sphere, where the parabolic weights are determined from the associated variation of Hodge structure. The tt* flat connection becomes the non-abelian Hodge flat connection and can be shown to be gauge equivalent to a meromorphic oper in these cases. This is based on work in progress with Florian Beck and Laura Fredrickson.

Andrea Brini , "Chern-Simons theory and the higher genus B-model"
Abstract: I'll give an update on the relation of quantum invariants of knots and 3-manifolds with higher genus mirror symmetry and the topological recursion. For 3-manifolds, I'll give a general realisation of quantum invariants of spherical space forms via the Eynard-Orantin recursion on spectral curves of the relativistic Toda system on ADE root latices. For knots, I'll outline a conjecturally general correspondence between the coloured HOMFLY polynomial and the topological recursion, which rests on the identification of a natural group of piecewise linear transformations on both sides of the correspondence.

Tobias Ekholm, "Open Gromov-Witten skein module invariants and large N duality"
Abstract: We define open Gromov-Witten invariants with values in the skein module of a 3-dimensional Lagrangian, which in essence is an interpretation of the boundaries of holomorphic curves as defects in Chern-Simons theory on the brane. Together with a deformation of complex structures known as Symplectic Field Theory stretching this gives an explanation of large N duality. More generally, this leads to a definition of a certain ‘universal’ open Gromov-Witten of the cotangent bundle of any 3-dimensional manifold. The talk reports on joint work in progress with V. Shende.

Piotr Kucharski , "Physics and geometry of knots-quivers correspondence"
Abstract: Recently proposed knots-quivers correspondence seems to be a very interesting phenomenon, however very little is known about its meaning. In this talk I would like to change it by presenting a physical interpretation as a duality between 3d N=2 theories and adding a geometric perspective of holomorphic disks.

Motohico Mulase, "Ribbon graphs, Frobenius-Hopf duality, and CohFT"
Abstract: In this talk, I will outline a new formulation of CohFT utilizing categories of ribbon graphs and their duals. First I will show that the category of dual ribbon graphs is a universal Frobenius structure that contains information of all Frobenius algebras. At the same time, this category knows degeneration history of pointed algebraic curves through ribbon graphs. The duality of surface graphs then produces a Frobenius-Hopf duality, which is the key to the Givental-Teleman reconstruction/classification theorem of semi-simple CohFTs. CohFT is a theory of virtual fundamental classes of moduli spaces of stable maps from curves into a target space X. On this fundamental class, interactions between cohomology on the moduli of stable curves and cohomology of X take place. Gromov-Witten invariants are correlation functions of these interactions. Here, cohomology of the target space forms a Frobenius algebra, and CohFT is a collection of n-point functions on this Frobenius algebra with values in the cohomolgy of moduli of stable curves. The consistency of the theory indicates that there is a strong similarity between Frobenius algebra operations and degeneration of curves. Our new formalism brings to us a clear picture of this similarity through graph duality, and illustrates how this duality implies the reconstruction theorem. The talk is based on my joint work with Olivia Dumitrescu.

Markus Reineke, "BPS state algebras of quivers"
Abstract: We review the definition of BPS state algebras, aka Cohomological Hall algebras, of quivers (with stability, but without superpotential) following M. Kontsevich and Y. Soibelman. After stating basic properties and examples, we relate them to the geometry of moduli spaces of stable quiver representations following joint work with H. Franzen and S. Meinhardt, and speculate on quantum group realizations using "exotic" R-matrices.

Marko Stošić, "Knots-quivers correspondence, lattice paths, and rational knots"
Abstract: Knots-quivers correspondence relates the colored HOMFLY-PT invariants of a knot with the motivic Donaldson-Thomas invariants of a corresponding quiver. This correspondence is made completely explicit at the level of generating series. The motivation for this relationship comes from topological string theory, BPS (LMOV) invariants, as well as categorification of colored HOMFLY-PT polynomials and A-polynomials. In this talk I shall present the examples of knots-quivers correspondence in the case of the classes of torus knots and rational (2-bridge) links. In the particular case of torus knots this comes with surprising applications in combinatorics involving counting of lattice paths and number theory. One of the outcomes of this correspondence is that from the information of the colored HOMFLY-PT polynomials of torus knots we get novel explicit expressions for the classical combinatorial problem of counting lattice paths under the lines with rational slopes, as well as new integrality/divisibility properties. In another direction, we show that all rational links satisfy knots-quivers corresponds.
(based on joint works with P. Sulkowski, M. Reineke, P. Kucharski, M. Panfil and P. Wedrich).

Cumrun Vafa, "String Theory and Homological Invariants for 3-Manifolds"
Abstract: I discuss how 6d superconformal theories discovered in string theory can lead to new homological invariants for 3-manifolds.


Spacetime details

Dates & Time:

September 24-28, 2018


Venue:
Faculty of Physics, University of Warsaw
ul. Pasteura 5, 02-093 Warsaw, Poland
Lecture hall 0.06 (ground floor)

                 

Accommodation – travel – practical information:
We have secured a limited number of rooms at the Hera Guest House (affordable, student standard) owned by the University of Warsaw (to go from Hera to the campus, you can take the bus line 167 at the stop “Spacerowa” and get off at “Och-Teatr,” and walk from there; directions on jakdojade.pl and Google Maps). Another affordable hotel, a walking distance to the campus, is Ibis Warszawa Reduta. For information about many other hotels, how to travel to Warsaw, public transport in Warsaw, etc., see the webpage with practical information.

Poster


Participants

  1. Nezhla Aghaee (University of Bern)
  2. Murad Alim (University of Hamburg)
  3. Alexandra Anokhina (ITEP, Moscow)
  4. Florian Beck (University of Hamburg)
  5. Hector Benitez (University of Sao Paolo)
  6. Anzor Beridze (Shota Rustaveli State University, Batumi)
  7. Michael Bleher (Heidelberg University)
  8. Maciej Borodzik (University of Warsaw)
  9. Andrea Brini (Imperial College)
  10. Daniel Bryan (University of Hamburg)
  11. Miranda Cheng (University of Amsterdam)
  12. Sheng-Fu Chiu (Academia Sinica, Taiwan)
  13. Sungbong Chun (California Institute of Technology)
  14. Paweł Ciosmak (University of Warsaw)
  15. Tudor Dimofte (University of California Davis)
  16. Tobias Ekholm (Uppsala University)
  17. Mirte van der Eyden (University of Amsterdam)
  18. Angelo Raffaele Fazio (Universidad Nacional de Colombia, Bogota)
  19. Anthonny F. Canazas Garay (Pontificia Universidad Católica de Chile)
  20. Sergei Gukov (California Institute of Technology)
  21. Leszek Hadasz (Jagellonian University, Krakow)
  22. Lukas Hahn (Heidelberg University)
  23. Ehsan Hatefi (University of Warsaw)
  24. Lotte Hollands (Heriot-Watt University, Edinburgh)
  25. Jakub Jankowski (University of Warsaw)
  26. Tony Kakona (ICTP)
  27. Marek Wojciech Kalinowski (Polish Academy of Sciences, Warsaw)
  28. Rinat Kashaev (University of Geneva)
  29. Albrech Klemm (University of Bonn)
  30. Paweł Klimasara (University of Silesia, Katowice)
  31. Eren Kovanlikaya (University of Hamburg)
  32. Jerzy Król (University of Silesia, Katowice)
  33. Vadym Kurylenko (University of Hamburg)
  34. Piotr Kucharski (Uppsala University)
  35. Helder Larraguivel (University of Warsaw)
  36. Pietro Longhi (Uppsala University)
  37. Toru Masuda (Academy of Sciences, Prague)
  38. Andrei Mironov (ITEP, Moscow)
  39. Motohico Mulase (University of California Davis)
  40. Satoshi Nawata (Fudan University, Shanghai)
  41. Sebastian Nill (Heidelberg University)
  42. Nils Albin Nilsson (National Centre for Nuclear Research, Warsaw)
  43. Dmitry Noshchenko (University of Warsaw)
  44. Miłosz Panfil (University of Warsaw)
  45. Jacek Pawełczyk (University of Warsaw)
  46. Michal Pawelkiewicz (Institut de Physique Théorique, Saclay)
  47. Marcin Piątek (University of Szczecin)
  48. Boris Pioline (Université Pierre et Marie Curie - Paris 6 and CNRS)
  49. Grzegorz Plewa (National Centre for Nuclear Research, Warsaw)
  50. Elli Pomoni (DESY)
  51. Tomasz Radożycki (Cardinal Stefan Wyszynski University, Warsaw)
  52. Himal Rathnakumara (University of Hamburg)
  53. Markus Reineke (Bochum University)
  54. Błażej Ruba (Jagellonian University, Krakow)
  55. Fabio Schlindwein (Heidelberg University)
  56. Marko Stošić (IST, Lisbon)
  57. Artur Strąg (University of Warsaw)
  58. Piotr Sułkowski (University of Warsaw)
  59. Yaser Tavakoli (University of Warsaw)
  60. Paweł Traczyk (University of Warsaw)
  61. Cumrun Vafa (Harvard University)
  62. Petr Vasko (University of Warsaw)
  63. Martin Vogrin (University of Hamburg)
  64. Don Zagier (Max Planck Institute, Bonn)

Acknowledgments