Modern Interactions between Algebra, Geometry andPhysics
Workshop
DateApril 18, 2016 - April 22, 2016Venue
TOKYO ELECTRON House of Creativity 3F, Lecture Theater, KatahiraCampus, Tohoku University

Invited Speakers
Sergei Gukov (Caltech and MPI, Bonn)
Kazuki Hiroe (Josai University)
Mikhail Kapranov (IPMU, Tokyo)
Maxim Kontsevich (IHES)
Takuro Mochizuki (RIMS, Kyoto)
Ryszard Nest (Copenhagen)Kyoji Saito (IPMU, Tokyo)
Pierre Schapira (Paris)
Yan Soibelman (Kansas State)
Yoshitsugu Takei (RIMS, Kyoto)
Boris Tsygan (Northwestern)
Daisuke Yamakawa (Tokyo Institute of Technology)1Time schedule
Date
18(Mon)19(Tue)20(Wed)21(Thu)22(Fri)Time
13:30 - 15:00
15:00 - 15:30
15:30 - 17:00
10:00 - 11:30
11:30 - 13:30
13:30 - 15:00
15:00 - 15:30
15:30 - 17:00
10:00 - 11:30
11:30 - 13:30
13:30 - 15:00
15:00 - 15:30
15:30 - 17:00
10:00 - 11:30
11:30 - 13:30
13:30 - 15:00
15:00 - 15:30
15:30 - 17:00
10:00 - 11:30

Speaker
Pierre Schapira (Paris)
Coﬀee break
Kyoji Saito (IPMU, Tokyo)
Maxim Kontsevich (IHES)
Lunch
Sergei Gukov (Caltech and MPI, Bonn)Coﬀee break
Yoshitsugu Takei (RIMS, Kyoto)
Daisuke Yamakawa (Tokyo Institute of Technology)
Lunch
Takuro Mochizuki (RIMS, Kyoto)
Coﬀee break
Yan Soibelman (Kansas State)
Boris Tsygan (Northwestern)
Lunch
Kazuki Hiroe (Josai University)
Coﬀee break
Ryszard Nest (Copenhagen)Mikhail Kapranov (IPMU, Tokyo)

Title and Abstract• Pierre Schapira (Paris)
Title: Subanalytic topologies and filtrations on the sheaf of holomorphic
functions
Abstract: In [KS01], we have constructed the subanalytic topology Msa ona real analytic manifold M and applied it to construct the sheafof holomorphic functions with temperate growth on a complex
manifold.
In [GS16], we introduce the linear subanalytic topology Msal andthe morphism of sites ρsal : Msa → Msal . The derived direct image functor Rρsal ∗ admits a right adjoint ρ!sal which allows us toassociate functorially a sheaf (in the derived sense) on Msa to a2

presheaf on Msa satisfying suitable properties, this sheaf having
the same sections as the presheaf on any open set with Lipschitz
boundary.
This construction applies to various presheaves on real manifolds,
such as the presheaves of functions with temperate growth of agiven order at the boundary or the Sobolev presheaves (a workof G. Lebeau). On a complex manifold, we get various sheavesof holomorphic functions with growth. As an application, we canendow the sheaf of holomorphic functions with various filtrations,in the derived sense, and, using the Riemann-Hilbert correspondence, we can endow functorially regular holonomic D-modules
with filtrations. (Joint work with Stéphane Guillermou.)
References[GS16] Stéphane Guillermou and Pierre Schapira, Construction ofsheaves on the subanalytic site, Astérisque 234 (2016), available atarXiv:1212.4326.
[Kas03] Masaki Kashiwara, D-modules and Microlocal Calculus, Translations of Mathematical Monographs, vol. 217, American Math. Soc.,
2003.
[KS01] Masaki Kashiwara and Pierre Schapira, Ind-sheaves, Astérisque,vol. 271, Soc. Math. France, 2001.

• Kyoji Saito (IPMU, Tokyo)
Title: Dual Artin monoids and zero loci of their skew-growth functions
Abstract: Associated with a finite root system of type P, we consider threecombinatorial structures: a dual Artin monoid, a lattice of noncrossing partitions and a cluster complex of finite type. For eachstructure, we associate certain generating functions: the skew
growth function for the dual Artin monoid, the generating function of Moebius invariants of the non-crossing partition lattice
and the generating function of dimensions of faces of the clustercomplex. Owing to several authors, it is known that all of them
give the same polynomial. Our interest and goal is the study ofthe zero loci of the polynomial. Namely, we show that it has exactly rank of P simple real roots on the interval (0,1]. The proof3

is based on a strange fact that the polynomial behaves similarly
to orthogonal polynomials, and sometimes may be expressed byusing Jacobi polynomials. But we don’t have conceptual understanding of this phenomenon (it resembles mirror symmetry). A
similar result on the zeros of polynomials for Artin monoids is also
observed and conjectured, but it is still open.
• Maxim Kontsevich (IHES)
Title: Resurgence and wall-crossing via complexified path integral
Abstract: I will explain how various instances of resurgence can be explained
through analysis of semi-infinite Lefschetz thimbles in complexified path integral in quantum mechanics. The case of zero action
give rise to Gaiotto-Moore-Neitzke calculus of spectral networks,
while for the case of action quadratic in momenta one obtains fineresurgence properties of heat kernels.
• Sergei Gukov (Caltech and MPI, Bonn)
Title: Mock modularity and categorification of 3-manifold quantum groupinvariants
Abstract: A long-standing problem of categorifying RTW invariants of 3manifolds faces an immediate challenge that, unlike invariants ofknots, these invariants exhibit no obvious integrality and are noteven valued in C[[q]]. Using insights from physics, we first turn
RTW invariants of 3-manifolds into q-series with integer coeﬃcients, providing a natural home to previous mysterious role ofEichler integral in Chern-Simons theory. This builds on work ofLawrence-Zagier, Hikami, Ohtsuki, and others. Then, the same
physical setup that in the past led to a realization of knot Floer
homology and Khovanov-Rozansky homology gives us the desired
categorification of Z[[q]]-valued transforms of RTW invariants.
Performing explicit computations is not a problem and will bedemonstrated in many concrete examples.
• Yoshitsugu Takei (RIMS, Kyoto)
Title: Exact WKB analysis for continuous and discrete Painlevé equations — Stokes geometry, connection formula and wall-crossing
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Abstract: Generalizing the exact WKB analysis for one-dimensional Schrödinger
equations established by Voros, Pham, Delabere and others, Aoki,
Kawai and I developed the exact WKB analysis for continuousPainlevé equations and clarified, in particular, their Stokes geometry and connection formula. Later Iwaki discussed the wallcrossing formula for Painlevé equations as well. In this talk, after
reviewing these previous works, I would like to talk about my recent research jointly done in part with N. Joshi (Sydney) on theexact WKB analysis for discrete Painlevé equations that are obtained from continuous Painlevé equations through the Bäcklundtransformation. In the analysis of such discrete Painlevé equations
both connection formula and wall-crossing formula for continuousPainlevé equations play the same role and appear as diﬀerent kindsof connection formula.
• Daisuke Yamakawa (Tokyo Institute of Technology)
Title: Twisted wild character varieties
Abstract: This is joint work with Philip Boalch. The Riemann-HilbertBirkhoﬀ correspondence gives a category equivalence between meromorphic connections on a compact Riemann surface with prescribed irregular classes and the so-called Stokes/monodromy data.
The moduli spaces of Stokes/monodromy data are called the wild
character varieties, and known to have canonical Poisson structures when the irregular classes are untwisted (shown by Boalch).
In this talk we will extend the construction to the twisted caseusing a slight extension of quasi-Hamiltonian geometry.
• Takuro Mochizuki (RIMS, Kyoto)
Title: Asymptotic behaviour of certain families of harmonic bundles on
Riemann surfaces
Abstract: Let (E, ∂ E , θ) be a stable Higgs bundle of degree 0 on a compact
connected Riemann surface. Once we fix a flat metric hdet(E) on
the determinant of E, we have the harmonic metrics ht (t > 0) for
the stable Higgs bundles (E, ∂ E , tθ) such that det(ht ) = hdet(E) .
In this talk, we will discuss two results on the behaviour of ht whent goes to ∞. First, we show that the Hitchin equation is asymptotically decoupled under some asumption for the Higgs field. We5

apply it to the study of the so called Hitchin WKB-problem. Second, we discuss the convergence of the sequence (E, ∂ E , θ, ht ) in
the case where the rank of E is 2. We explain a rule to determine
the parabolic weights of a “limiting configuration”, and we show
the convergence of the sequence to the limiting configuration in
an appropriate sense.
• Yan Soibelman (Kansas State)
Title: Riemann-Hilbert correspondence for diﬀerence equations in higherdimensions.
Abstract: I am going to discuss a conjectural Riemann-Hilbert correspondence for holonomic modules over quantum tori. The approach isbased on the ideas of Floer theory and tropical geometry. It is apart of the project ”Holomorphic Floer theory”, joint with Maxim
Kontsevich.
• Boris Tsygan (Northwestern)
Title: A microlocal category associated to a symplectic manifold
Abstract: For a symplectic manifold with some topological conditions, wedefine an infinity category enriched in infinity local systems ofmodules over the Novikov ring. The construction is based on thecategory of modules over an extension of the Fedosov deformation
quantization, and should be viewed as the De Rham side of themicrolocal category defined recently by Tamarkin.
• Kazuki Hiroe (Josai University)
Title: On additive Deligne-Simpson problem
Abstract: The ”multiplicative” Deligne-Simpson problem was proposed byP. Deligne and C. Simpson, which asks the existence of irreducible
linear monodromy representations of the Riemann sphere withholes after fixing isomorphic classes of local monodromies. V.
Kostov considered the additive analogue of this problem, so-called”additive” Deligne-Simpson problem which asks the existence ofirreducible Fuchsian systems on the Riemann sphere with fixed
local isomorphic classes. This additive analogue can be rephrased
as the nonemptiness of the moduli space of stable meromorphic
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connections with logarithmic singularities on the trivial bundle.
For the geometry of this moduli space, W. Crawley-Boevey’s work
giving a realization of the moduli space as a quiver variety wasremarkable and gave a complete answer to the additive DeligneSimpson problem. In this talk, we will discuss a generalization of
the additive DS problem for non-Fuchsian systems with unramified irregular singularities. After the works of Crawley-Boevey,
P. Boalch, D. Yamakawa, and so on, an embedding of the moduli space of these non-Fuchsian systems into a quiver variety will
be explained. Thanks to this embedding, the connectedness ofthe moduli space can be shown and also the nonemptiness of themoduli space, which is equivalent to our generalized additive DS
problem, is determined.
• Ryszard Nest (Copenhagen)Title: On analytic construction of the group three-cocycles
Abstract: One of the most important group cocycles is the two-cocycle on
the restricted general linear group of a polarised Hilbert space(H, H+ ). It has a wide range of applications, like the central
extensions of the loop group, theory of Toeplitz operators, gauge
theory or invariants of the K2alg . This two-cocycle can be seen asa two-cocycle associated to the action of the group GLres (H, H+ )on the category of subspaces K ⊂ H such that the product oforthogonal projections
PH+ PK : K → H+
is in L2 (H). Morphisms in this category are given by lines Det(PK1 PK2 ).
Similarly, given an action of a group G on an n-category satisfying
certain conditions, one can construct a (n + 1)-cocycle on G. A
well known example is the n-Tate space, essentially an algebraof the form K = k((s1))((s2)) . . . ((sn)), where the group is thegroup of invertibles in K and the n-category structure comes fromthe natural filtration of K.
alg
The corresponding cocycles, when evaluated on Kn+1(K), reproduce the Tate tame symbol. However, the constructions are purely
algebraic and do not seem to extend to the analytic context, as in
the case of n = 1.
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In this talk we will sketch a construction of a (family of) twocategory associated to a pair of commuting idempotents P and
Q on a Hilbert space and construct the associated three cocycleon the associated groups. For example, in the case of a twoTate space, this produces an extension of the Tate symbol fromC((z1))((z2)) to, say, C ∞ (T2 ), but also a corresponding invariantof K3alg of the non-commutative torus C ∞ (T2θ ).
The construction is based on the properties of the determinant ofFredholm operators, in particular on the existence of the canonicalperturbation isomorphism Det(T ) ≃ Det(S) whenever T and S
are two Fredholm operators satisfying T − S ∈ L1 (H).
This is a joint work with Jens Kaad and Jesse Wolfson.
• Mikhail Kapranov (IPMU, Tokyo)
Title: Some remarks on D-modules with a large parameter and their
Stokes geometry.
Abstract: The classical WKB analysis of linear PDE with a large parameter t amounts, algebraically, to studying modules over the ringDX [t] and equipping this ring with the non-standard filtration inwhich deg(t) = 1. This filtration re-defines principal symbols of
operators and characteristic varieties of modules which become(after setting t = 1), non-homogeneous coisotropic subvarieties
L ⊂ T ∗ X (“spectral curves”, if dim(X) = 1).
For holonomic DX [t]-modules these characteristic varieties are nonhomogeneous Lagrangian and dictate the shape of WKB solutions.
Higher-dimensional examples include Frobenius manifolds (pencilsof flat connections), A-hypergeometric systems and systems forcharacters and spherical functions in representation theory. (Thelatter two types of systems involve free parameters which can bemade to depend on t.)
I will discuss the possible geometry of Stokes “walls” for holonomic
DX [t]-modules, in particular, the higher-dimensional analogs of“new turning points” and their relations to syzygies among elementary matrices generating the group SLn (Z).

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