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, Stavros Garoufalidis, Sergei Gukov, 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



Full size photo

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" [slides]
12:45-14:30 – Lunch break
14:30-15:30 – Rinat Kashaev, "The spectral problem of the modular oscillator in the strongly coupled regime"  [slides]
15:30-16:00 – Coffee break
16:00-17:00 – Stavros Garoufalidis, "Counting incompressible surfaces and the 3D-index"
18:00-19:30 – 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, "Khovanov homotopy type and periodic links" [slides]
10:00-10:30 – Coffee break
10:30-11:30 – Boris Pioline, "Attractor flow trees, BPS indices and quivers" [slides]
11:45-12:45 – Markus Reineke, "BPS state algebras of quivers"
12:45-14:30 – Lunch break
14:30-15:30 – Elli Pomoni, "Counting instantons in N=1 theories of class Sk"  [slides]
15:30-16:00 – Coffee break
16:00-17:00 – Andrea Brini, "Chern-Simons theory and the higher genus B-model"  [slides]

WEDNESDAY, September 26
8:30-9:00 – Wake-up coffee
9:00-10:00 – Tudor Dimofte, "Categories of line operators in 3d N=4 gauge theory"
10:00-10:30 – Coffee break
10:30-11:30 – Don Zagier, "Knots, state integrals, and the modular group"
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, "Logarithmic CFTs from three dimensions"
11:45-12:45 – Andrei Mironov, "Tangles, link invariants and topological vertex"
12:45-14:30 – Lunch break
14:30-15:30 – Murad Alim, "From tt* flat connections to parabolic Higgs bundles and opers"
15:30-16:00 – Coffee break
16:00-17:00 – Piotr Kucharski, "Physics and geometry of knots-quivers correspondence"  [slides]

FRIDAY, September 28
8:30-9:00 – Wake-up coffee
9:00-10:00 – Marko Stošić, "Knots-quivers correspondence, lattice paths, and rational knots" [slides]
10:00-10:30 – Coffee break
10:30-11:30 – Albrecht Klemm, "Periods and quasiperiods of modular forms and D-brane masses of the quintic"
11:45-12:45 – Miranda Cheng, "3d Modularity" [slides]
12:45-14:30 – Lunch and 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.

Maciej Borodzik, "Khovanov homotopy type and periodic links"
Abstract: Khovanov homology is an important link invariant. It assigns to a link bigraded homology groups. Recently, Lipshitz and Sarkar constructed a finite CW-complex, whose (co)homology is the Khovanov homology of a link. This complex is well-defined up to stable homotopy equivalence. It is called the Khovanov homotopy type. We show that if the underlying link is periodic, one can define a group action on Khovanov homotopy type. It is well defined up to stable equivariant homotopy equivalence. Borel homology of this space is related to the equivariant Khovanov homology defined by Politarczyk. This is a joint project with W. Politarczyk and M. Silvero.

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.

Miranda Cheng, "3d Modularity"
Abstract: I this talk I will discuss the unexpected appearance of various modular-like symmetry structures in problems of 3-manifold topology and physical 3d N=2 theories. In particular we will discuss the role of false theta functions, mock modular forms, SL(2,Z) Weil representations and quantum modular forms in the computation of half-indices of the 3d theories. This talk is based on joint work with Sungbong Chun, Francesca Ferrari, Sergei Gukov, and Sarah Harrison.

Tudor Dimofte, "Categories of line operators in 3d N=4 gauge theory"
Abstract: A 4d N=4 gauge theory (defined by a group G and a hyperkahler G-space X) has two basic types of BPS line operators, characterized in physical terms as Wilson lines and vortex lines. Each type acquires the structure of a (braided tensor) category, with morphisms in the category corresponding to spaces of local operators at a junction of lines. I will propose a definition of these two categories, generalizing previous work of Rozansky-Witten and Kapustin-Rozansky-Saulina; and I will discuss the conjectured equivalences of such categories induced by three-dimensional mirror symmetry. This is joint work with Niklas Garner, Mike Geracie, Justin Hilburn, and Philsang Yoo. Given time, I will mention a potential application to HOMFLY homology.

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.

Stavros Garoufalidis, "Counting incompressible surfaces and the 3D-index"
Abstract: I will explain some connections between the counting of incompressible surfaces in hyperbolic 3-manifolds with boundary and the 3D index of Dimofte-Gaiotto-Gukov. Joint work with N. Dunfield, C. Hodgson and H. Rubinstein, and, as usual, with lots of examples and patterns.

Sergei Gukov, "Logarithmic CFTs from three dimensions"
Abstract: In this talk I will describe a surprising structure of the new q-series invariants of 3-manifolds – called "Zed-hat" – which play the same role for 3-manifolds as the (colored) Jones polynomials do for knots. In particular, I will focus on representation theory behind this surprising structure, which will then be fully revealed in a companion talk by Miranda Cheng. This talk is based on a joint work with Miranda Cheng, Sungbong Chun, Francesca Ferrari, Sarah Harrison, and will use some background material from the talk of Cumrun Vafa as a prerequisite.

Rinat Kashaev, "The spectral problem of the modular oscillator in the strongly coupled regime"
Abstract: Motivated by applications for non-perturbative topological strings in toric Calabi-Yau manifolds through Grassi-Hatsuda-Mariño conjecture, I will talk about the spectral problem for a pair of commuting modular conjugate (in the sense of Faddeev) Harper type operators with complex values of Planck's constant. The eigenvectors are expressed in terms of a special entire function on the complex plane with the Taylor expansion coefficients given in terms of specific q-orthogonal polynomials, while the eigenvalues are solutions of transcendental Bethe type equations. This is a joint work with Sergey Sergeev.

Albrecht Klemm, "Periods and quasiperiods of modular forms and D-brane masses of the quintic"
Abstract: We consider one complex structure parameter mirror families $W$ of Calabi-Yau 3-folds with Picard-Fuchs equations of hypergeometric type. By mirror symmetry the even D-brane masses of the orginal Calabi-Yau $M$ can be identified with four periods w.r.t. to an integral symplectic basis of H_3(W,Z) at the point of maximal unipotent monodromy. It was discovered by Chad Schoen in 1986 that the singular fibre of the quintic at the conifold point gives rise to a Hecke eigen form of weight four $f_4$ on $\Gamma_0(25)$ whose Fourier coefficients $a_p$ are determined by counting solutions in that fibre over the finite field $\mathbb{F}_{p^k}$. The D-brane masses at the conifold are given by the transition matrix $T_{mc}$ between the integral symplectic basis and a Frobenius basis at the conifold. We predict and verify to very high precision that the entries of $T_{mc}$ relevant for the D2 and D4 brane masses are given by the two periods (or L-values) of $f_4$. These values also determine the behaviour of the Weil-Petersson metric and its curvature at the conifold. Moreover we describe a notion of quasiperiods and find that the two quasi period of $f_4$ appear in $T_{mc}$. We extend the analysis to the other hypergeometric one parameter 3-folds and comment on simpler applications to local Calabi-Yau 3-folds and polarized K3 surfaces.

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.

Andrei Mironov, "Tangles, link invariants and topological vertex"
Abstract: The recently suggested tangle calculus for knot and link polynomials is intimately related to topological string considerations and can help to build their invariants from the topological vertices. We will discuss this interplay in the simplest example of the Hopf link and its 2-cable.  It turns out that the resolved conifold with four different representations on the four external legs, on the topological string side, is described by a special projection of the four-component link L8n8, which reduces to the Hopf link colored with two composite representations.

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 cohomology 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.

Boris Pioline, "Attractor flow trees, BPS indices and quivers"
Abstract: Moduli spaces of quiver representations naturally appear in physics in the context of  BPS dyons  in D=4, N=2 field theories or string vacua. Based on physics intuition about BPS black holes, I will present two formulae for computing the Poincaré polynomial of quiver moduli spaces in terms of new invariants which do not depend on the stability parameters. The first formula, developed in joint work with J. Manschot and Ashoke Sen some years ago [arXiv: 1404.7154] expresses the Poincaré polynomial in terms of  recursively defined "quiver invariants", which are supposed to count single centered black holes but whose mathematical significance is still obscure. The second formula, inspired by the "split attractor conjecture" in supergravity and obtained in joint work with S. Alexandrov [arXiv: 1804.06928], expresses the same Poincaré polynomial in terms of "attractor invariants", which are Poincaré polynomial for subquivers for a stability condition specified by the dimension vector.

Elli Pomoni, "Counting instantons in N=1 theories of class Sk"
Abstract: In this talk we will explain how to obtain the instanton partition functions of a large class of N=1 superconformal theories (SCFTs), called Sk.
          We will begin by introducing this class of N=1 SCFTs, which is obtained from Gaiotto’s class S of N=2 SCFTs via orbifolding. We can study the Coulomb branch of these theories by constructing and analyzing their spectral curves. Employing our experience with the AGT correspondence we will search for a 2D/4D relation for the N=1 SCFTs in class Sk. From the curves we can identify the 2D CFT symmetry algebra and its representations, namely the conformal blocks of the Virasoro/W-algebra, that underlie the 2D theory and reproduce the spectral curves of the N = 1 SCFTs. These conformal blocks give a prediction for the instanton partition functions of the 4D N = 1 SCFTs of class Sk. Finally, we will present a completely independent, elliptic genus calculation, counting open string states on Dp/D(p-4) brane systems in type IIB string theory, which exactly reproduces our previous result for the instanton partition functions.

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 correspondence. Based on joint works with P. Sułkowski, 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.

Don Zagier, "Knots, state integrals, and the modular group"
Abstract: I will report on recent and ongoing work with Stavros Garoufalidis concerning the modularity properties of the Kashaev invariant of knots and its relationship to the asymptotic properties of various q-series arising from state integrals.

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

Acknowledgments