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Homological Algebra of the Infrared

红外同调代数研讨会

会议编号:  M230102

时间:  2023-01-08 ~ 2023-01-14

浏览次数:  3208

组织者:   Shing-Tung Yau (Yau MSC)、Will Donovan (Yau MSC)、 Mauricio Romo (Yau MSC)、 Luyu Zheng (Yau MSC)

会议介绍

    Speaker List

    Sibasish Banerjee 

    Alexey Bondal

    Yu-Wei Fan 

    Bohan Fang

    Jie Gu 

    Mikhail Kapranov

    Ludmil Katzarkov

    Ahsan Khan 

    Maxim Kontsevich

    Si Li

    Timothy Logvinenko 

    Pietro Longhi 

    Greg Moore

    David Nadler 

    Vadim Schechtman

    Yan Soibelman 

    Johannes Walcher


    Scientific Aims

    In 2015, D. Gaiotto, G. Moore and E. Witten [8] gave a ground-breaking analysis of certain two-dimensional supersymmetric quantum field theories, providing a new set of tools to study them via their low-energy data. These tools, and developments inspired by them, are known as the “algebra of the infrared”. The mathematical interpretation of these ideas, in particular as initiated in M. Kapranov, M. Kontsevich and Y. Soibelman [11], has already stimulated a new frontier in homological algebra, where objects in classical homological constructions are replaced with higher categorical analogs. Physically, the work of Gaiotto–Moore–Witten can be considered as part of the larger study of BPS states of certain supersymmetric gauge theories.

    On the mathematics side, a key piece of this picture is the introduction of “schobers” by M. Kapranov and V. Schechtman [12], giving a notion of perverse sheaves of categories. These objects have already found striking applications, in particular in mirror symmetry. They give new insights into how to define and study Fukaya categories following M. Kapranov, Y. Soibelman and L. Soukhanov [13]. Some recent developments of interest include the following.

     • Given a Landau–Ginzburg model, M. Kapranov, Y. Soibelman and L. Soukhanov [13] give an interpretation of certain infrared data in terms of a schober on C, building on Kapranov–Kontsevich–Soibelman [11]. This should lead to a construction of a Fukaya category, when the definition of the schober is augmented by certain higher data. 

    • S. Spenko and M. Van den Bergh [15] construct schobers in a mirror symmetry context for a large class of examples. These schobers are supported on stringy Kahler moduli spaces for certain quotient spaces associated to quasi-symmetric representations of reductive groups (in physics language, these are gauged linear σ-models). In a further paper [16] they develop the mirror symmetry interpretation by relating their schober to a perverse sheaf for a GKZ system.

    On the physics side, current developments in the study of BPS states which are very similar in spirit to Gaiotto–Moore–Witten include, but are not limited to, the following.

    • In recent work G. Moore and A. Khan [14] studied how the A∞ category of  1/2 BPS A-type branes (roughly characterized by Lefschetz thimbles) behaves under deformation of the complex structure parameters on massive LG models.

    • In [9] D. Galakhov studied the phenomena of B-brane transport using domain wall defects. Mathematically this corresponds to studying Fourier–Mukai kernels for different functors between triangulated categories corresponding to B-branes on 2d QFTs. This can be viewed as a B-model version of Gaiotto–Moore–Witten, in some cases. 

    • The development of exponential networks started in [6] and corresponds to an M-theory lift of the more established spectral networks. S. Banerjee, P. Longhi and M. Romo started in [1] the systematic study of the relation between exponential networks and BPS counting of 5d QFTs. Mathematically these BPS degeneracies have proven to be related to (generalized) DT invariants.

    The following related topics are also of current interest.

    • Recently several papers have appeared studying the relation between wall crossing phenomena and WKB analysis/resurgence. For instance, the work of A. Grassi, J. Gu and M. Marino [10]. 

    • A. Bondal, M. Kapranov and V. Schechtman [2] explain a program to associate schobers to “webs of flops” in birational geometry, and W. Donovan and T. Kuwagaki [3] have provided mirror constructions to this. 

    • T. Dyckerhoff [4] proves a categorification of the Dold–Kan correspondence relating chain complexes and simplicial abelian groups. Dyckerhoff–Jasso–Walde [5] use this result to give a new viewpoint on the “higher Auslander–Reiten” theory of Iyama. 

    • H. Fan, W. Jiang and D. Yang [7] have recently given a definition for the Fukaya category of Landau-Ginzburg model, satisfying expectations from Kapranov–Soibelman–Soukhanov, with implication for the development of the algebra of the infrared.

    This meeting will bring together physicists and mathematicians working in these and related areas, to share the state of the art, and address new questions.


    References

    [1] S. Banerjee, P. Longhi and M. Romo, “Exploring 5d BPS Spectra with Exponential Networks,” Annales Henri Poincare 20, no.12, 4055-4162 (2019) doi:10.1007/s00023-019-00851-x [arXiv:1811.02875 [hep-th]]. 

    [2] A. Bondal, M. Kapranov and V. Schechtman, “Perverse schobers and birational geometry,” [arXiv:1801.08286 [math]]. 

    [3] W. Donovan and T. Kuwagaki, “Mirror symmetry for perverse schobers from birational geometry,” [arXiv:arXiv:1903.11226 [math]]. 

    [4] T. Dyckerhoff, “A categorified Dold–Kan correspondence,” [arXiv:arXiv:1710.08356 [math]]. 

    [5] T. Dyckerhoff, G. Jasso and T. Walde, “Simplicial structures in higher Auslander–Reiten theory,” [arXiv:1811.02461 [math]]. 

    [6] R. Eager, S. A. Selmani and J. Walcher, “Exponential Networks and Representations of Quivers,” JHEP 08, 063 (2017) doi:10.1007/JHEP08(2017)063 [arXiv:1611.06177 [hep-th]]. 

    [7] H. Fan, W. Jiang and D. Yang, “Fukaya category of Landau–Ginzburg model,” [arXiv:1812.11748 [math]]. 

    [8] D. Gaiotto, G. Moore and E. Witten, “Algebra of the Infrared: String Field Theoretic Structures in Massive N = (2, 2) Field Theory In Two Dimensions,” [arXiv:1506.04087 [hep-th]]. 

    [9] D. Galakhov, “On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry,” [arXiv:2105.07602 [hep-th]]. 

    [10] A. Grassi, J. Gu and M. Marino, “Non-perturbative approaches to the quantum Seiberg-Witten curve,” JHEP 07, 106 (2020) doi:10.1007/JHEP07(2020)106 [arXiv:1908.07065 [hep-th]]. 

    [11] M. Kapranov, M. Kontsevich and Y. Soibelman, “Algebra of the infrared and secondary polytopes,” [arXiv:1408.2673 [math]]. 

    [12] M. Kapranov and V. Schechtman, “Perverse Schobers,” [arXiv:1411.2772 [math]]. 

    [13] M. Kapranov, Y. Soibelman and L. Soukhanov, “Perverse schobers and the Algebra of the Infrared,” [arXiv:2011.00845 [math]]. 

    [14] A. Z. Khan and G. W. Moore, “Categorical Wall-Crossing in Landau-Ginzburg Models,” [arXiv:2010.11837 [hep-th]]. 

    [15] S. Spenko and M. Van den Bergh, “A class of perverse schobers in Geometric Invariant Theory,” [arXiv:1908.04213 [math]]. 

    [16] S. Spenko and M. Van den Bergh, “Perverse schobers and GKZ systems,” [arXiv:2007.04924 [math]].




组织者

Shing-Tung Yau,Yau MSC / Harvard U
Will Donovan,Yau MSC
Mauricio Romo,Yau MSC
Luyu Zheng,Yau MSC

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