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We construct global Kuranishi charts for the moduli spaces of stable pseudoholomorphic maps to a closed symplectic manifold in all genera. This is used to prove a product formula for symplectic Gromov–Witten invariants. As a consequence we obtain a Künneth formula for quantum cohomology. [ABSTRACT FROM AUTHOR]

We present a method for constructing covariantly constant endomorphisms for the mod p equivariant quantum connection, using the quantum Steenrod power operations of Fukaya and Wilkins. The example of the cotangent bundle of the projective line is fully computed, and we discuss the relationship with the mod p solutions of trigonometric KZ equation recently constructed by Varchenko. As a byproduct, we compute the first examples of quantum Steenrod operations that are not a priori determined by ordinary Gromov–Witten theory and classical Steenrod operations, which may be of independent interest. [ABSTRACT FROM AUTHOR]

According to Zuo and an unpublished work of Bertola, there is a two-index series of Dubrovin–Frobenius manifold structures associated to a B-type Coxeter group. We study the relations between these structures for the different values of these indices. We show that part of the data of such Dubrovin–Frobenius manifold indexed by (k, l) can be recovered by the (k + r , l + r) Dubrovin–Frobenius manifold.Continuing the program of Basalaev et al. (J Phys A: Math Theor 54:115201, 2021) we associate an infinite system of commuting PDEs to these Dubrovin–Frobenius manifolds and show that these PDEs extend the dispersionless BKP hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we study the genus two G-function which was introduced by Dubrovin, Liu and Zhang for the cubic elliptic orbifold. As results, we first prove the quasi-modularity for the descendant correlation functions of certain type in all genus. Then we prove any derivatives of the genus two G-function of certain type are quasi-modular forms after a mirror transformation. In particular, we compute the explicit closed formula for its certain first derivative. Our proof mainly relies on two techniques: Givental quantization formalism for semisimple Frobenius manifold and the tautological relations on the moduli space of stable curves. [ABSTRACT FROM AUTHOR]

This paper considers the (negative) cyclic open–closed map O C - , which maps the cyclic homology of the Fukaya category of a symplectic manifold to its S 1 -equivariant quantum cohomology. We prove (under simplifying technical hypotheses) that this map respects the respective natural connections in the direction of the equivariant parameter. In the monotone setting this allows us to conclude that O C - intertwines the decomposition of the Fukaya category by eigenvalues of quantum cup product with the first Chern class, with the Hukuhara–Levelt–Turrittin decomposition of the quantum cohomology. We also explain how our results relate to the Givental–Teleman classification of semisimple cohomological field theories: in particular, how the R-matrix is related to O C - in the semisimple case; we also consider the non-semisimple case. [ABSTRACT FROM AUTHOR]

We describe the monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian in terms of the equivariant K -theory algebra of the cotangent bundle. This description is based on the hypergeometric integral representations for solutions of the equivariant quantum differential equation. We identify the space of solutions with the space of the equivariant K -theory algebra of the cotangent bundle. In particular, we show that for any element of the monodromy group, all entries of its matrix in the standard basis of the equivariant K -theory algebra of the cotangent bundle are Laurent polynomials with integer coefficients in the exponentiated equivariant parameters. [ABSTRACT FROM AUTHOR]

Dubrovin establishes a certain relationship between the GUE partition function and the partition function of Gromov–Witten invariants of the complex projective line. In this paper, we give a direct proof of Dubrovin's result. We also present in a diagram the recent progress on topological gravity and matrix gravity. [ABSTRACT FROM AUTHOR]

Our previous papers introduce topological notions of normal crossings symplectic divisor and variety, show that they are equivalent, in a suitable sense, to the corresponding geometric notions, and establish a topological smoothability criterion for normal crossings symplectic varieties. The present paper constructs a blowup, a complex line bundle, and a logarithmic tangent bundle naturally associated with a normal crossings symplectic divisor and determines the Chern class of the last bundle. These structures have applications in constructions and analysis of various moduli spaces. As a corollary of the Chern class formula for the logarithmic tangent bundle, we refine Aluffi's formula for the Chern class of the tangent bundle of the blowup at a complete intersection to account for the torsion and extend it to the blowup at the deepest stratum of an arbitrary normal crossings divisor. [ABSTRACT FROM AUTHOR]

We construct the quantum curve for the Baker–Akhiezer function of the orbifold Gromov–Witten theory of the weighted projective line ℙ[r]. Furthermore, we deduce the explicit bilinear Fermionic formula for the (stationary) Gromov–Witten potential via the lifting operator contructed from the Baker–Akhiezer function. [ABSTRACT FROM AUTHOR]

We explain how to construct a quantum deformation of a spectral curve associated to a tau-function of the KP hierarchy. This construction is applied to Witten–Kontsevich tau-function to give a natural explanation of some earlier work. We also apply it to higher Weil–Petersson volumes and Witten's r-spin intersection numbers. [ABSTRACT FROM AUTHOR]

In this paper, we extend the matrix-resolvent method to the study of the Dubrovin–Zhang type tau-functions for the constrained KP hierarchy and the bigraded Toda hierarchy of (M, 1)-type. We show that the Dubrovin–Zhang type tau-function of an arbitrary solution to the bigraded Toda hierarchy of (M, 1)-type is a Dubrovin–Zhang type tau-function for the constrained KP hierarchy, which generalizes the result in Carlet et al. (Mosc Math J 4:313–332, 2004) and Fu and Yang (J Geom Phys 179:104592, 2022) for the Toda lattice hierarchy and the NLS hierarchy corresponding to the M = 1 case. [ABSTRACT FROM AUTHOR]

The Gamma conjecture II for the quantum cohomology of a Fano manifold F, proposed by Galkin, Golyshev and Iritani, describes the asymptotic behavior of the flat sections of the Dubrovin connection near the irregular singularities, in terms of a full exceptional collection, if there exists, of D b (F) and the Γ ^ -integral structure. In this paper, for the smooth quadric hypersurfaces we prove the convergence of the full quantum cohomology and Gamma II. For the proof, we first give a criterion on Gamma II for Fano manifolds with semisimple quantum cohomology, by Dubrovin's theorem of analytic continuations of semisimple Frobenius manifolds. Then we work out a closed formula of the Chern characters of spinor bundles on quadrics. By the deformation-invariance of Gromov–Witten invariants we show that the full quantum cohomology can be reconstructed by its ambient part, and use this to obtain estimations. Finally we complete the proof of Gamma II for quadrics by explicit asymptotic expansions of flat sections corresponding to Kapranov's exceptional collections and an application of our criterion. [ABSTRACT FROM AUTHOR]

We continue our quest for real enumerative invariants not sensitive to changing the real structure and extend the construction we uncovered previously for counting curves of anti-canonical degree ⩽ 2 on del Pezzo surfaces with K 2 = 1 to curves of any anti-canonical degree and on any del Pezzo surfaces of degree K 2 ⩽ 3 . [ABSTRACT FROM AUTHOR]

Let X be a smooth projective variety with a simple normal crossing divisor D : = D 1 + D 2 + ⋯ + D n , where D i ⊂ X are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks X D , r → by constructing an I-function lying in a slice of Givental's Lagrangian cone for Gromov–Witten theory of multi-root stacks. We provide three applications: (1) We show that some genus zero invariants of X D , r → stabilize for sufficiently large r → . (2) We state a generalized local-log-orbifold principle conjecture and prove a version of it. (3) We show that regularized quantum periods of Fano varieties coincide with classical periods of the mirror Landau–Ginzburg potentials using orbifold invariants of X D , r → . [ABSTRACT FROM AUTHOR]

Generalizing a construction presented in Arsie and Lorenzoni (Lett Math Phys 107:1919–1961, 2017), we show that the orbit space of B 2 less the image of the coordinate lines under the quotient map is equipped with two Dubrovin-Frobenius manifold structures which are related respectively to the defocusing and the focusing nonlinear Schrödinger (NLS) equations. Motivated by this example, we study the case of B n and we show that the defocusing case can be generalized to arbitrary n leading to a Dubrovin-Frobenius manifold structure on the orbit space of the group. The construction is based on the existence of a non-degenerate and non-constant invariant bilinear form that plays the role of the Euclidean metric in the Dubrovin–Saito standard setting. Up to n = 4 the prepotentials we get coincide with those associated with the constrained KP equations discussed in Liu et al. (J Geom Phys 97:177–189, 2015). [ABSTRACT FROM AUTHOR]

In this paper we survey some results on the symmetric semi-perfect obstruction theory on a Deligne—Mumford stack X constructed by Chang—Li, and Behrend's theorem equating the weighted Euler characteristic of X and the virtual count of X by symmetric semi-perfect obstruction theories. As an application, we prove that Joyce's d-critical scheme admits a symmetric semi-perfect obstruction theory, which can be applied to the virtual Euler characteristics by Jiang—Thomas. [ABSTRACT FROM AUTHOR]

We give a direct proof for the degeneration formula of Gromov–Witten invariants including its cycle version for degenerations with smooth singular locus in the setting of stable log maps of Abramovich-Chen, Chen, Gross–Siebert. [ABSTRACT FROM AUTHOR]

We describe software for symbolic computations that we developed in order to find Hamiltonian operators for Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations, and verify their compatibility. The computation involves nonlocal (integro-differential) operators, for which specific canonical forms and algorithms have been used. [ABSTRACT FROM AUTHOR]

We propose two conjectural relationships between the equivariant Gromov–Witten invariants of the resolved conifold under diagonal and anti-diagonal actions and the Gromov–Witten invariants of P 1 and verify their validity in genus zero approximation. We also provide evidences to support the validity of these relationships in genus one and genus two. [ABSTRACT FROM AUTHOR]

We relate the quantum cohomology of minuscule flag manifolds to the tt*-Toda equations, a special case of the topological–antitopological fusion equations which were introduced by Cecotti and Vafa in their study of supersymmetric quantum field theories. To do this, we combine the Lie-theoretic treatment of the tt*-Toda equations of Guest–Ho with the Lie-theoretic description of the quantum cohomology of minuscule flag manifolds from Chaput–Manivel–Perrin and Golyshev–Manivel. [ABSTRACT FROM AUTHOR]

Given a semistable degeneration with a simple normal crossings central fiber, Abramovich et al. (Compositio Mathematica 156(10):2020–2075, 2020) proved a degeneration (or decomposition) formula that relates the moduli spaces of stable maps in smooth fibers to certain moduli spaces of log-smooth maps in the central fiber. In this paper, we study the same problem from an analytic point of view. We prove that the limiting stable maps in the central fiber satisfy specific combinatorial and analytical conditions. Furthermore, we derive a (conjectural) degeneration formula similar to ACGS’s formula, and work out an explicit example. The results are expected to hold in the symplectic category as outlined in an earlier version (Farajzadeh-Tehrani in Towards a degeneration formula for the Gromov–Witten invariants of symplectic manifolds. [math]) of this paper. [ABSTRACT FROM AUTHOR]

The space of Fredholm operators of fixed index is stratified by submanifolds according to the dimension of the kernel. Geometric considerations often lead to questions about the intersections of concrete families of elliptic operators with these submanifolds: Are the intersections nonempty? Are they smooth? What are their codimensions? The purpose of this article is to develop tools to address these questions in equivariant situations. An important motivation for this work are transversality questions for multiple covers of J-holomorphic maps. As an application, we use our framework to give a concise exposition of Wendl’s proof of the superrigidity conjecture.

We provide a construction of the moduli space of stable coherent sheaves in the world of non-archimedean geometry, where we use the notion of Berkovich non-archimedean analytic spaces. The motivation for our construction is Tony Yue Yu's non-archimedean enumerative geometry in Gromov—Witten theory. The construction of the moduli space of stable sheaves using Berkovich analytic spaces will give rise to the non-archimedean version of Donaldson—Thomas invariants. In this paper we give the moduli construction over a non-archimedean field K . We use the machinery of formal schemes, that is, we define and construct the formal moduli stack of (semi)-stable coherent sheaves over a discrete valuation ring R, and taking generic fiber we get the non-archimedean analytic moduli space of semistable coherent sheaves over the fractional non-archimedean field K . We generalize Joyce's d-critical scheme structure in [37] or Kiem—Li's virtual critical manifolds in [38] to the world of formal schemes, and Berkovich non-archimedean analytic spaces. As an application, we provide a proof for the motivic localization formula for a d-critical non-archimedean K -analytic space using global motive of vanishing cycles and motivic integration on oriented formal d-critical schemes. This generalizes Maulik's motivic localization formula for the motivic Donaldson—Thomas invariants. [ABSTRACT FROM AUTHOR]

We give a conjugacy relation on certain type of Frobenius manifold structures using the theory of flat pencils of metrics. It leads to a geometric interpretation for the inversion symmetry of solutions to Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations. [ABSTRACT FROM AUTHOR]

The orbits space of an irreducible linear representation of a finite group is a variety whose coordinate ring is the ring of invariant polynomials. Boris Dubrovin proved that the orbits space of the standard reflection representation of an irreducible finite Coxeter group W acquires a natural polynomial Frobenius manifold structure. We apply Dubrovin’s method on various orbits spaces of linear representations of finite groups. We find some of them has non or several natural Frobenius manifold structures. On the other hand, these Frobenius manifold structures include rational and trivial structures which are not known to be related to the invariant theory of finite groups. [ABSTRACT FROM AUTHOR]

In this paper, we consider double ramification cycles with orbifold targets. An explicit formula for double ramification cycles with orbifold targets, which is parallel to and generalizes the one known for the smooth case, is provided. Some applications for orbifold Gromov—Witten theory are also included. [ABSTRACT FROM AUTHOR]

Let M be a super Riemann surface with holomorphic distribution D and N a symplectic manifold with compatible almost complex structure J . We call a map Φ : M → N a super J -holomorphic curve if its differential maps the almost complex structure on D to J . Such a super J -holomorphic curve is a critical point for the superconformal action and satisfies a super differential equation of first order. Using component fields of this super differential equation and a transversality argument we construct the moduli space of super J -holomorphic curves as a smooth subsupermanifold of the space of maps M → N . [ABSTRACT FROM AUTHOR]

We derive a formula for the connected n-point functions of a tau-function of the BKP hierarchy in terms of its affine coordinates. This is a BKP-analogue of a formula for KP tau-functions proved by Zhou (Emergent geometry and mirror symmetry of a point, 2015. ). Moreover, we prove a simple relation between the KP-affine coordinates of a tau-function τ (t) of the KdV hierarchy and the BKP-affine coordinates of τ (t / 2) . As applications, we present a new algorithm to compute the free energies of the Witten–Kontsevich tau-function and the Brézin–Gross–Witten tau-function. [ABSTRACT FROM AUTHOR]

We propose an effective framework for computing the prepotential of the topological B-model on a class of local Calabi–Yau geometries related to the circle compactification of five-dimensional N = 1 super Yang–Mills theory with simple gauge group. In the simply laced case, we construct Picard–Fuchs operators from the Dubrovin connection on the Frobenius manifolds associated with the extended affine Weyl groups of type ADE . In general, we propose a purely algebraic construction of Picard–Fuchs ideals from a canonical subring of the space of regular functions on the ramification locus of the Seiberg–Witten curve, encompassing non-simply laced cases as well. We offer several precision tests of our proposal for simply laced cases by comparing with the gauge theory prepotentials obtained from the K-theoretic blow-up equations, finding perfect agreement. Whenever there is more than one candidate Seiberg-Witten curve for non-simply laced gauge groups in the literature, we employ our framework to verify which one agrees with the K-theoretic blow-up equations. [ABSTRACT FROM AUTHOR]

We prove a relationship between quantum Steenrod operations and the quantum connection. In particular, there are operations extending the quantum Steenrod power operations that, when viewed as endomorphisms of equivariant quantum cohomology, are covariantly constant. We demonstrate how this property is used in computations of examples. [ABSTRACT FROM AUTHOR]

Let (X , D) be an m-pointed compact Riemann surface of genus at least 2. For each x ∈ D , fix full flag and concentrated weight system α . Let P M ξ denote the moduli space of semi-stable parabolic vector bundles of rank r and determinant ξ over X with weight system α , where r is a prime number and ξ is a holomorphic line bundle over X of degree d which is not a multiple of r. We compute the Chen–Ruan cohomology of the orbifold for the action on P M ξ of the group of r-torsion points in Pic 0 (X) . [ABSTRACT FROM AUTHOR]