# D-modules, quantum geometry, and related topics

3-7 December 2018
RIMS Kyoto
Asia/Tokyo timezone

## Poster session will take place on Tuesday, December 4

### Maxim Kontsevich

Title: Inverse Mellin of ratios of Gamma's (joint with K. Penson)

File of the poster

### Davide Barco

Title: Topological computations of enhanced Fourier-Sato transform of an elementary D-module

Abstract: Let M be a holonomic D-module in the affine line V. The Fourier transform M^ of M is a holonomic D-module on the dual affine line V^∗. It is natural to study the Stokes structure underlying such module at infinity in V^∗ and several works and authors have dealt with this task (Malgrange, Hien-Sabbah, D’Agnolo-Hien-Morando- Sabbah).
In recent years, Mochizuki has given a recipe for a complete description of the Fourier transform of a general holonomic D_V-module using the rapid decay homology theory introduced by Bloch and Esnault.
A different point of view to the study of the Stokes phenomena is given by the Riemann-Hilbert correspondence as stated by D’Agnolo and Kashiwara.
This associates to M the enhanced ind-sheaf F of its enhanced solutions, moreover, by functoriality, the correspondence interchanges Fourier-Laplace transform for holonomic D-module with Fourier-Sato transform for enhanced ind- sheaves.
Our aim is to get a description of the Fourier transform for a particular kind of D_V-module, called elementary.
While a complete understanding of this case has already been achieved by Hien and Sabbah, our approach here is different since it makes use of the theory of enhanced sheaves and the explicit description given by Mochizuki.

### Huachen Chen

Title: Toward a F-structure on a space of Bridgeland stability conditions

File of the poster

Abstract: Recently, Sheridan and Smith proved that a particular Picard-rank-20 K3 surface is the mirror to some K3 categories associated to cubic fourfolds. We would like to present that on the space of stability conditions on this K3 surface, there exists a F-structure, away from a codimension one subset. Locally, this F-structure comes from (restriction of) the semiuniversal unfoldings of defining equations of smooth cubic fourfolds.

### Veronica Fantini

Title: The work of Chan, Leung and Ma on Maurer-Cartan equations and scattering diagrams, and possible extensions

File of the poster

Abstract: Recent work of Chan, Leung and Ma analyses the asymptotic behaviour of certain special solutions to the Maurer-Cartan equation which governs first order deformations of a semi-flat Calabi-Yau manifold, through a Fourier transform. Their main result is that the leading order asymptotics defines naturally a consistent scattering diagram in the sense of Kontsevich-Soibelman and Gross-Siebert. The poster attempts to explain this interesting construction and to point out some possible directions for development.

### Jacob Gross

Title: Homology of moduli spaces of complexes

File of the poster

Abstract: Joyce recently proved that the homologies of moduli spaces of certain dg-categories are graded vertex algebras–this poster is dedicated to some computations in examples. We prove that, for a smooth complex projective variety X, the rational homology of the derived stack Perf(X) of perfect complexes on X is the tensor product of the group algebra of the 0th semi-topological K-group of X with the free commutative-graded algebra on infinitely many (shifted) copies of the rational morphic cohomology L*H*(X,Q) of X. In certain cases, such as when X is a complex projective K3 surface, we identify the homology H_*(Perf(X),Q) with a lattice vertex algebra on H*(X). We prove a "Kirwan surjectivity" theorem: for a Riemann surface C, the inclusion Coh(C) –> Perf(C) induces an injection in rational homology. Pulling back the bicharacter on the homology of Perf(X) determines a unique graded vertex algebra structure on the homology of Coh(C). This is part of a larger programme which hopes to use the vertex-algebraic structures on the homologies of 2n-shifted symplectic derived stacks to prove wall-crossing formulae for Mochizuki invariants of algebraic surfaces, Donaldson–Thomas invariants on Fano 3-folds, and Donaldson–Thomas invariants of Calabi–Yau 4-folds.

### Masashi Hamanaka

Title: Noncommutative instantons and reciprocity

File of the poster

Abstract:  Noncommutative space (NC) is a space which coordinate ring is noncommutative. Let x,y be the spacial coordinates. The noncommutativity is reprsented by the following commutation relations: [x,y]=\sqrt{-1} \theta, where \theta is a real constant and called the noncommutative parameter. When \theta vanishes identically, the coordinate ring is commutative and the underlying space reduces to a commutative one. The commutation relations, like the canonical commutation relations in quantum mechanics, lead to “space-space uncertainty relation.” Singularities in commutative space could resolve in noncommutative space thereby. This is one of the prominent features of field theories on noncommutative space and yields various new physical objects such as U(1) instantons.

Instantons are finite-action solutions of the ASD Yang-Mills equation and have been studied from the several viewpoints of mathematical physics, particularly, integrable systems, geometry and field theories. become They can reveal non-perturbative aspects of the quantum field theories. Actually, the path-integrations, formulating the quantum theories, could reduce to finite-dimensional integrations over the instanton moduli spaces. The Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction is a powerful method to obtain the instantons. Furthermore, via the construction, the instanton moduli space is mapped to the set of quadruple matrices which are solutions of the ADHM equation and called the ADHM data. The aforementioned integration, being thereby an integration over the matrices, becomes tractable.

To evaluate the integration the use of noncommutative instantons is relevant so that a localization formula can be applied to the integration. In the procedure, various formulas and relations of the ADHM construction are required. Hence it is worthwhile to elucidate the one-to-one correspondence (reciprocity) between moduli spaces of the noncommutative instantons and the ADHM data and to present all the ingredients in the construction explicitly. There are two formalism to describe noncommutative gauge theories: the star-product formalism and the operator formalism.

In this poster, we would like to discuss the ADHM construction of noncommutative instantons together with a brief review of commutative ADHM construction from the viewpoint of the Fourier-Mukai-Nahm transformation [2]. We prove the reciprocity in both the star-product formalism [1] and the operator formalism. We reconsider origin of the instanton number by applying an idea of Atiyah and Hori to the noncommutative situation even for the U(1) case. This is based on collaboration with Toshio Nakatsu (Setsunan Univ.).
[1] Masashi Hamanaka and Toshio Nakatsu, Noncommutative Instantons and Reciprocity (to appear)
[2] Masashi Hamanaka and Toshio Nakatsu, ADHM Construction of Noncommutative Instantons arXiv:1311.5227

### Brian Hepler

Title: Hypersurface normalizations and numerical invariants

File of the poster

Abstract: We define a "new" perverse sheaf, the comparison complex, naturally associated to any locally reduced complex analytic space $X$ on which the (shifted) constant sheaf $\mathbb{Q}_X[dim X]$ is perverse. In the hypersurface case, this complex is isomorphic to the perverse eigenspace of the eigenvalue one for the Milnor monodromy action on the vanishing cycles; we also examine how the characteristic polar multiplicities of this complex behave in certain one-parameter families of deformations of hypersurfaces with codimension-one singularities and recover a classical formula for the Milnor number of a plane curve singularities in terms of double-points. In general, the vanishing of the cohomology sheaves of the comparison complex provide a criterion for determining if the normalization of the space X is a rational homology manifold. When the normalization is a rational homology manifold, we can also compute several terms in the weight filtration of the constant sheaf $\mathbb{Q}_X^\bullet[n]$ in those cases for which this perverse sheaf underlies a mixed Hodge module.

References:
• Hepler, B. Deformation Formulas for Parameterizable Hypersurfaces. ArXiv e-prints, 2017.  arXiv:1711.11134
• Hepler, B. Rational Homology Manifolds and Hypersurface Normalizations. ArXiv e-prints, 2018 Accepted for publication at Proceedings of the American Mathematical Society. arXiv:1804.09799
• Hepler, B. and Massey, D. Perverse Results on Milnor Fibers inside Parameterized Hypersurfaces. Publ. RIMS Kyoto Univ., 52:413–433, 2016

### Andreas Hohl

Title: D-modules of pure Gaussian type and topological Laplace transform

Abstract: We consider differential systems of pure Gaussian type, which are D-modules on the complex projective line with an irregular singularity at infinity, and as such are subject to the Stokes phenomenon. The aim is to describe the Stokes multipliers of the Laplace transform of such a system in terms of the Stokes multipliers of the original system. We use the theory of enhanced (ind-)sheaves and the Riemann-Hilbert correspondence of D'Agnolo-Kashiwara in order to formulate the problem in a purely topological way. Under an assumption on the parameters, we prove an explicit result about the topological Laplace transform of enhanced sheaves of pure Gaussian type. This also gives an alternative proof of a result of Sabbah, which has originally been proven in the context of Stokes-filtered local systems.

### Pengfei Huang

Title: Moduli spaces of parabolic Higgs bundles

File of the poster

Abstract: Parabolic structures for vector bundles were first introduced by Metha and Seshadri in 1980's when they studied the representation of the fundamental group of a punctured Riemann surface. They were later generalized by Maruyama and Yokogawa to the higher dimensional case, and they constructed a moduli space for these objects.
The Higgs field for a parabolic vector bundle was introduced by Carlos Simpson and he built the parabolic version of the non-abelian Hodge theory. Later this was studied and generalized by Takuro Mochizuki for the parabolic version of the Kobayashi-Hitchin correspondence.
During the development of this theory, there has been a very important observation by Biswas, Borne, Iyer and Simpson concerning the equivalence between parabolic vector bundles and vector bundles on the associated root stack.
We obtain the parabolic Higgs version of the Biswas-Borne-Iyer-Simpson correspondence.

### Yohei Ito

Title: Fourier transforms of regular holonomic D-modules in higher dimensions

File of the poster

Abstract: We study Fourier transforms of regular holonomic D-modules in  higher dimensions (this is a joint work with Kiyoshi Takeuchi). It is  well-known that Fourier transforms of D-modules preserve the  holonomicity. However they do not preserve the regularity in general. In 1986  Brylinski proved that if a regular holonomic D-module M is monodromic  then its Fourier transform is again regular. Here we consider the more  general case where the regular holonomic D-module M is not necessarily monodromic.

### Tomohiro Iwami

Title: An analogue of Miyaoka-Yau type inequality for certain threefold with regards to the associated third Chen classes

File of the poster

Abstract: Among types of extremal curve neighborhoods appearing in the classification of 3-dimesional flips (J. Kollar-S. Mori,1992), type (IIA) is an important class which gives a rationality criterion of Q-conic bundles (Y. Prokhorov-S. Mori, 2008, 2009) and some related del Pezzo fibrations (Y. Prokhorov-S. Mori, 2016, 2017). Configuration of singularities of Du Val elements in |-K_X| of type (IIA) is determined by a kind of small deformation (LG-deformation (S. Mori, 1988)) of filtrations as the form of gr^{n,i}(O,J) (ibid.) associated to such elements as a reflexive sheaves.In this process, codimension 3 supports on such elements as in QL(C) (ibid.) corresponding to c_3(-K_X) have important role in the above birational contractions except for flips, and also these filtrations as gr^{n,i}(O,J) associated to this type satisfy the abundance property for the related Euler characteristic as \chi(F^1/F^4) ≥ 0 (ibid.). Based on these situations, in this poster, we briefly report about our works on which we induce an analogue of Miyaoka-Yau type inequality for extremal contractions of type (IIA) with regarding to the associated third Chern classes as above.

### Arata Komyo

Title: A family of flat connections on the projective space having dihedral monodromy and algebraic Garnier solutions

File of the poster

Abstract: Algebraic solutions of the Painlev\'e VI equation and the Garnier systems have been studied. For example, Girand constructed an explicit two-parameter family of flat connections over the projective plane. These connections have dihedral monodromy and their polar locus is a prescribed quintic composed of a conic and three tangent lines. By restricting them to generic lines, we get an algebraic family of isomonodromic deformations of the five-punctured sphere. This yields algebraic solutions of a Garnier system. In this poster, we give a generalization of this construction. That is, we construct an explicit $n$-parameter family of flat connections over the projective space of dimension n.
Reference: arXiv:1806.00970

### Tatsuki Kuwagaki

Title: Riemann-Hilbert correspondence and Fukaya category

Abstract: Riemann-Hilbert correspondence translates differential equations into some topological data. For irregular singularities, the topological data is called Stokes structures. Some years ago, D'Agnolo-Kashiwara proposed a formalism treating all the Stokes structures simultaneously and proved Riemann-Hilbert correspondence for holonomic D-modules. In this poster, I will present a modified version of this formalism and also discuss a relationship with Fukaya category.
Reference: arXiv:1808.02760

### Yuki Matsubara

Title: On the cohomology of the moduli space of parabolic connections

File of the poster

Abstract: In this poster, we study the moduli space of logarithmic connections of rank 2 on the projective line P1 - {t_1, ..., t_5} with fixed spectral data. We compute the cohomology of such moduli space, and this computation will be used to extend the results of geometric Langlands corespondence due to D. Arinkin to the case where the parabolic connections have five simple poles on P1.

### Tao Su

Title: A Hodge-theoretic study of augmentation varieties associated to Legendrian knots/tangles

Abstract: In this poster, we give a tangle approach in the study of Legendrian knots (in the contact three-space). Generalizing those of Legendrian knots, one can define combinatorially the Legendrian Contact Homology (LCH) DGAs for Legendrian tangles. They are Legendrian isotopy invariants and satisfy a van-Kanpem property, reducing the study of which to a local problem. It's then interesting to study the geometry of "(rank 1) representation varieties" of the LCH DGAs, called augmentation varieties. Such an variety, hence its mixed Hodge structure on the compactly supported cohomology, is a Legendrian isotopy invariant up to a normalization. Moreover, its point-counting/E-polynomial is given by ruling polynomial, the Legendrian analogue of Jones polynomial. The tangle approach also leads naturally to a ruling decomposition of this variety into simple pieces, inducing a spectral sequence converging to the MHS. As some applications, we show that the variety is of Hodge-Tate type, show a vanishing result on its cohomology, and provide an example-computation of the MHSs.

Reference: arXiv:1707.04948
Ph.D thesis, which can be found here: https://math.berkeley.edu/~taosu/

### Szilard Szabo

Title: Perversity equals weight for Painlevé systems

File of the poster

Abstract: An important conjecture in non-Abelian Hodge theory by de Cataldo, Hausel and Migliorini asserts that the weight filtration on the cohomology spaces of a character variety agrees with the perverse Leray filtration on the cohomology spaces of the corresponding Dolbeault moduli space. We prove an analogous result for wild character varieties and the corresponding irregular Hitchin systems associated to the Painlevé cases. The proof is based on an earlier description of the wild character varieties arising in these cases by Marius van der Put and Masa-Hiko Saito on one hand, and on our study of the geometry of irregular Hitchin systems on the other hand.
Corresponding manuscripts on arXiv: 1802.03798, 1808.10125
Published/accepted papers: 1710.09922, 1604.08503

### Kouichi Takemura

Title: Middle convolution and the index of rigidity for linear differential equations with irregular singularities

File of the poster

Abstract: We investigate a linear algebraic formulation of the index of rigidity and the middle convolution for the linear system of differential  equations which may include irregular singularities, which was  introduced in arXiv:1002.2535.  In particular, we show that the index of rigidity is preserved by the  middle convolution in our formulation.