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We found an interesting application of the K-theoretic Heisenberg algebras of Weiqiang Wang to the foundations of permutation equivariant K-theoretic Gromov--Witten theory. We also found an explicit formula for the genus 0 correlators in the permutation equivariant Gromov--Witten theory of the point. In the non-equivariant limit our formula reduces to a well known formula due to Y.P. Lee., Comment: 27 pages

Let $X$ be a smooth projective variety with a semisimple quantum cohomology. It is known that the blowup $\operatorname{Bl}_{\rm pt}(X)$ of $X$ at one point also has semisimple quantum cohomology. In particular, the monodromy group of the quantum cohomology of $\operatorname{Bl}_{\rm pt}(X)$ is a reflectiongroup. We found explicit formulas for certain generators of the monodromy group of the quantum cohomology of $\operatorname{Bl}_{\rm pt}(X)$ depending only on the geometry of the exceptional divisor.

For a smooth projective variety whose anti-canonical bundle is nef, we prove confluence of the small $K$-theoretic $J$-function, i.e., after rescaling appropriately the Novikov variables, the small $K$-theoretic $J$-function has a limit when $q\to 1$, which coincides with the small cohomological $J$-function. Furthermore, in the case of a Fano toric manifold $X$ of Picard rank 2, we prove the $K$-theoretic version of an identity due to Iritani that compares the $I$-function of the toric manifold and the oscillatory integral of the toric mirror. In particular, our confluence result yields a new proof of Iritani's identity in the case of a toric manifold of Picard rank 2., Comment: 66 pages. Several typos fixed. There was a missing term in the Givental--Tonita recursion (see Theorem 3.3.4)

We construct a Hermitian matrix model for the total descendant potential of a simple singularity of type D similar to the Kontsevich matrix model for the generating function of intersection numbers on the Deligne--Mumford moduli spaces $\overline{\mathcal{M}}_{g,n}$., Comment: 45 pages

We compute the image of the Milnor lattice of an ADE singularity under a period map. We also prove that the Milnor latticecan be identified with an appropriate relative $K$-group defined through the Berglund-H\"ubsch dual of the corresponding singularity.

Fano orbifold lines are classified by the Dynkin diagrams of type $A,D,$ and $E$. It is known that the corresponding total descendant potential is a tau-function of an appropriate Kac--Wakimoto hierarchy. It is also known that in the A-case the Kac--Wakimoto hierarchies admit an extension and that the total descendant potential is a tau-function of an extended Kac--Wakimoto hierarchy. The goal of this paper is to prove that in the D-case the total descendent potential is also a tau-function of an extended Kac--Wakimoto hierarchy., Comment: 37 pages. Some typos and statements are corrected, and the title has been changed

In a companion paper to this one, we proved that the Gromov--Witten theory of a Fano orbifold line of type $D$ is governed by a system of Hirota Bilinear Equations. The goal of this paper is to prove that every solution to the Hirota Bilinear Equations determines a solution to a new integrable hierarchy of Lax equations. We suggest the name extended D-Toda hierarchy for this new system of Lax equations, because it should be viewed as the analogue of Carlet's extended bi-graded Toda hierarchy, which is known to govern the Gromov--Witten theory of Fano orbifold lines of type $A$, Comment: 73 pages

It was proved in 2010 that the principal Kac--Wakimoto hierarchy of type $D$ is a reduction of the 2-component BKP hierarchy. On the other hand, it is known that the total descendant potential of a singularity of type $D$ is a tau-function of the principal Kac--Wakimoto hierarchy. We find explicitly the point in the Grassmanian of the 2-component BKP hierarchy (in the sense of Shiota) that corresponds to the total descendant potential. We also prove that the space of tau-functions of Gaussian type is parametrized by the base of the miniversal unfolding of the simple singularity of type $D$., Comment: 36 pages

We invert the period map defined by the second structure connection of quantum cohomology of $\mathbb{P}^2$. For small quantum cohomology the inverse is given explicitly in terms of the Eisenstein series $E_4$ and $E_6$, while for big quantum cohomology the inverse is determined perturbatively as a Taylor series expansion whose coefficients are quasi-modular forms., Comment: 58 pages. Typos fixed. The previous version was extended in the following way: we proved the Laplace transform version of the $\Gamma$-conjecture for $\mathbb{P}^2$, worked out the structure of a Schwarz triangular map, and added an appendix explaining the relation to vertex algebras and W-constraints. Version accepted in Adv. in Math

These notes are based on a sequence of five lectures given to graduate students. The main goal is to prove the so-called Painleve property for semi-simple Frobenius manifolds., Comment: 61 pages

The main goal of this paper is to introduce the notion of a primitive form for a generic family of Hurwitz covers of $\mathbb{P}^1$ with a fixed ramification profile over infinity. We prove that primitive forms are in one-to-one correspondence with semi-simple Frobenius structures on the base of the family. Furthermore, we introduce the notion of a polynomial primitive form and show that the corresponding class of Frobenius manifolds contains the Hurwitz Frobenius manifolds of Dubrovin. Finally, we apply our theory to investigate the relation between the Eynard--Orantin recursion and Frobenius manifolds., Comment: 67 pages, typos corrected, references added, an invertible argument was inverted

We prove that the genus-0 invariants in K-theoretic Gromov--Witten theory are governed by an integrable hierarchy of hydrodynamic type. If the K-theoretic quantum product is semisimple, then we also prove the completeness of the hierarchy., Comment: 18 pages

We construct a global B-model for weighted homogeneous polynomials based on K. Saito's theory of primitive forms. Our main motivation is to give a rigorous statement of the so called global mirror symmetry conjecture relating Gromov-Witten invariants and Fan--Jarvis--Ruan--Witten invariants. Furthermore, our construction allows us to generalize the notion of a quasi-modular form and holomorphic anomaly equations. Finally, we prove the global mirror symmetry conjecture for the Fermat polynomials., Comment: 82 pages, in version 2: some typos corrected and references added

We derive a Bouchard--Eynard type topological recursion for the total descendant potential of $A_N$-singularity. Our argument relies on a certain twisted representation of a Heisenberg Vertex Operator Algebra (VOA) constructed via the periods of $A_N$-singularity. In particular, our approach allows us to prove that the topological recursion for the total descendant potential is equivalent to a certain generating set of $\mathcal{W}$-algebra constraints., Comment: 18 pages, main author is Todor Milanov, appendix by Danilo Lewanski

The paper \cite{BM} proposed a construction of a twisted representation of the lattice vertex algebra corresponding to the Milnor lattice of a simple singularity. The main difficulty in extending the above construction to an arbitrary isolated singularity is in the so called {\em phase factors} -- the scalar functions produced by composing two vertex operators. They are certain family of multivalued analytic functions on the space of miniversal deformations. The first result in this paper is an explicit formula for the unperturbed phase factors in terms of the classical monodromy operator and the polylogorithm functions. Our second result is that with respect to the deformation parameters the phase factors are analytic functions on the monodromy covering space., Comment: 23 pages

According to \cite{BOSS} and \cite{M1}, the ancestor correlators of any semi-simple cohomological field theory satisfy {\em local} Eynard--Orantin recursion. In this paper, we prove that for simple singularities, the local recursion can be extended to a global one. The spectral curve of the global recursion is an interesting family of Riemann surfaces defined by the invariant polynomials of the corresponding Weyl group. We also prove that for genus 0 and 1, the free energies introduced in \cite{EO} coincide up to some constant factors with respectively the genus 0 and 1 primary potentials of the simple singularity., Comment: 24 pages

We construct an integrable hierarchy in the form of Hirota quadratic equations (HQE) that governs the Gromov--Witten (GW) invariants of the Fano orbifold projective curve $\mathbb{P}^1_{a_1,a_2,a_3}$. The vertex operators in our construction are given in terms of the $K$-theory of $\mathbb{P}^1_{a_1,a_2,a_3}$ via Iritani's $\Gamma$-class modification of the Chern character map. We also identify our HQEs with an appropriate Kac--Wakimoto hierarchy of ADE type. In particular, we obtain a generalization of the famous Toda conjecture about the GW invariants of $\mathbb{P}^1$ ., Comment: v3: 62 pages, 1 figure, major revision: the correspondence under mirror symmetry between Iritani's Gamma-integral structure and the natural integral structure in LG model is proved in the case of $\mathbb{P}^1_{a_1,a_2,a_3}$ and is used to greatly improved the study of root systems; the main result is strengthened; materials in this paper are reorganized; the title of the paper is changed

In a series of papers \cite{KS,MR}, Krawitz, Milanov, Ruan, and Shen have verified the so-called Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence for simple elliptic singularities $E_N^{(1,1)}$ ($N=6,7,8$). As a byproduct it was also proved that the orbifold Gromov--Witten invariants of the orbifold projective lines $\mathbb{P}^1_{3,3,3}$, $\mathbb{P}^1_{4,4,2}$, and $\mathbb{P}^1_{6,3,2}$ are quasi-modular forms on an appropriate modular group. While the modular group for $\mathbb{P}^1_{3,3,3}$ is $\Gamma(3)$, the modular groups in the other two cases were left unknown. The goal of this paper is to prove that the modular groups in the remaining two cases are respectively $\Gamma(4)$ and $\Gamma(6)$., Comment: 28 pages. arXiv admin note: substantial text overlap with arXiv:1210.6862

We give a new reconstruction method of big quantum $K$-ring based on the $q$-difference module structure in quantum $K$-theory. The $q$-difference structure yields commuting linear operators $A_{i,\rm com}$ on the $K$-group as many as the Picard number of the target manifold. The genus-zero quantum $K$-theory can be reconstructed from the $q$-difference structure at the origin $t=0$ if the $K$-group is generated by a single element under the actions of $A_{i,\rm com}$. This method allows us to prove the convergence of the big quantum $K$-rings of certain manifolds, including the projective spaces and the complete flag manifold $\operatorname{Fl}_3$., Comment: 36 pages

K. Saito's theory of primitive forms gives a natural semi-simple Frobenius manifold structure on the space of miniversal deformations of an isolated singularity. On the other hand, Givental introduced the notion of a total ancestor potential for every semi-simple point of a Frobenius manifold and conjectured that in the settings of singularity theory his definition extends analytically to non-semisimple points as well. In this paper we prove Givental's conjecture by using the Eynard--Orantin recursion., Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv:1211.5847, arXiv:1203.3414

It was proved recently that the correlation functions of a semi-simple cohomological field theory satisfy the so called Eynard--Orantin topological recursion. We prove that in the settings of singularity theory, the relations can be expressed in terms of periods integrals and the so called phase forms. In particular, we prove that the Eynard-Orantin recursion is equivalent to $N$ copies of Virasoro constraints for the ancestor potential, which follow easily from the definition of the potential., Comment: 18 pages. arXiv admin note: text overlap with arXiv:1203.3414

A simple elliptic singularity of type $E_N^{(1,1)}$ ($N=6,7,8$) can be described in terms of a marginal deformation of an invertible polynomial $W$. In the papers \cite{KS} and \cite{MR} the authors proved a mirror symmetry statement for some particular choices of $W$ and used it to prove quasi-modularity of Gromov-Witten invariants for certain elliptic orbifold $\mathbb{P}^1$s. However, the choice of the polynomial $W$ and its marginal deformation $\phi_{\mu}$ are not unique. In this paper, we investigate the global mirror symmetry phenomenon for the one-parameter family $W+\sigma\phi_{\mu}$. In each case the mirror symmetry is governed by a certain system of hypergeometric equations. We conjecture that the Saito-Givental theory of $W+\sigma\phi_{\mu}$ at any special limit $\sigma$ is mirror to either the Gromov-Witten theory of an elliptic orbifold $\mathbb{P}^1$ or the Fan-Jarvis-Ruan-Witten theory of an invertible simple elliptic singularity with diagonal symmetries, and the limits are classified by the Milnor number of the singularity and the $j$-invariant at the special limit. We prove the conjecture when $W$ is a Fermat polynomial. We also prove that the conjecture is true at the Gepner point $\sigma=0$ in all other cases., Comment: The last section of version 1 is removed, because we added more details to it and it became a separate paper. 47 pages

In this paper, we proved generating functions of Gromov-Witten cycles of the elliptic orbifold lines with weights (3,3,3), (4,4,2), and (6,3,2) are cycle-valued quasi-modular forms. This is a generalization of Milanov and Ruan's work on cycle-valued level. First we construct a global cohomology field theory (CohFT) for simple elliptic singularities (modulo an extension problem) and prove its modularity. Then, we apply Teleman's reconstruction theorem to prove mirror theorems on cycled-valued level and match it with a CohFT from Gromov-Witten theory of a corresponding orbifold.This solves the extension property as well as inducing the modularity for a Gromov-Witten CohFT.

Simple, or Kleinian, singularities are classified by Dynkin diagrams of type ADE. Let g be the corresponding finite-dimensional Lie algebra, and W its Weyl group. The set of g-invariants in the basic representation of the affine Kac-Moody algebra g^ is known as a W-algebra and is a subalgebra of the Heisenberg vertex algebra F. Using period integrals, we construct an analytic continuation of the twisted representation of F. Our construction yields a global object, which may be called a W-twisted representation of F. Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest weight vector for the W-algebra., Comment: 61 pages, 2 figures

In this paper we prove that the GW invariants of the elliptic orbifold lines with weights (3,3,3), (4,4,2), and (6,3,2) are quasi-modular forms. Our method is based on Givental's higher genus reconstruction formalism applied to the settings of Saito's Frobenius structures for simple elliptic singularities. Our results are part of a larger project whose goal is to prove the Landau-Ginzburg/Calabi-Yau correspondence for simple elliptic singularities. The correspondence describes a relation between Gromov-Witten theory (of a certain hypersurface) and Fan-Jarvis-Ruan-Witten theory (of a certain Landau-Ginzburg potential). Roughly, the main statement is that the Saito's Frobenius manifold for simple elliptic singularities has some special points such that locally near these points the Frobenius structure governs one of the two theories. The local part of the correspondence is established in a companion article by M. Krawitz and Y. Shen, while here we describe the global picture., Comment: 49 pages

Using Picard--Lefschetz periods for the singularity of type $A_N$, we construct a projective representation of the Lie algebra of differential operators on the circle with central charge $h:=N+1$. We prove that the total descendant potential $\D_{A_N}$ of $A_N$-singularity is a highest weight vector. It is known that $\D_{A_N}$ can be interpreted as a generating function of a certain class of intersection numbers on the moduli space of $h$-spin curves. In this settings our constraints provide a complete set of recursion relations between the intersection numbers. Our methods are based entirely on the symplectic loop space formalism of A. Givental and therefore they can be applied to the mirror models of symplectic manifolds., Comment: 31 pages

Let $\CP^1_{k,m}$ be the orbifold structure on $\CP^1$ obtained via uniformizing the neighborhoods of 0 and $\infty$ respectively by $z\mapsto z^k$ and $w\mapsto w^m.$ The diagonal action of the torus on the projective line induces naturally an orbifold action on $\CP^1_{k,m}.$ In this paper we prove that if k and m are co-prime then Givental's prediction of the equivariant total descendent Gromov-Witten potential of $\CP^1_{k,m}$ satisfies certain Hirota Quadratic Equations (HQE for short). We also show that after an appropriate change of the variables, similar to Getzler's change in the equivariant Gromov-Witten theory of $\CP^1$, the HQE turn into the HQE of the 2-Toda hierarchy, i.e., the Gromov-Witten potential of $\CP^1_{k,m}$ is a tau-function of the 2-Toda hierarchy. More precisely, we obtain a sequence of tau-functions of the 2-Toda hierarchy from the descendent potential via some translations. The later condition, that all tau-functions in the sequence are obtained from a single one via translations, imposes a serious constraint on the solution of the 2-Toda hierarchy. Our theorem leads to the discovery of a new integrable hierarchy (we suggest to be called the Equivariant Bi-graded Toda Hierarchy). We conjecture that this new hierarchy governs, i.e., uniquely determines, the equivariant Gromov-Witten invariants of $\CP^1_{k,m}.$, Comment: 40 pages, 1 figure

Let $M_{k,m}$ be the space of Laurent polynomials in one variable $x^k + t_1 x^{k-1}+... t_{k+m}x^{-m},$ where $k,m\geq 1$ are fixed integers and $t_{k+m}\neq 0$. According to B. Dubrovin \cite{D}, $M_{k,m}$ can be equipped with a semi-simple Frobenius structure. In this paper we prove that the corresponding descendant and ancestor potentials of $M_{k,m}$ (defined by A. Givental) satisfy Hirota quadratic equations (HQE for short). Let $\mathcal{C}_{k,m}$ be the orbifold obtained from $\mathbb{P}^1$ by cutting small discs $D_1\simeq \{|z|\leq \epsilon\}$ and $D_2\simeq\{|z^{-1}|\leq \epsilon\}$ around $z=0$ and $z=\infty$ and gluing back the orbifolds $D_1/\mathbb{Z}_k$ and $D_2/\mathbb{Z}_m$ in the obvious way. We show that the orbifold quantum cohomology of $\mathcal{C}_{k,m}$ coincides with $M_{k,m}$ as Frobenius manifolds. Modulo some yet-to-be-clarified details, this implies that the descendant (respectively the ancestor) potential of $M_{k,m}$ is a generating function for the descendant (respectively ancestor) orbifold Gromov--Witten invariants of $\mathcal{C}_{k,m}$. There is a certain similarity between our HQE and the Lax operators of the Extended bi-graded Toda hierarchy, introduced by G. Carlet in \cite{car}. Therefore, it is plausible that our HQE characterize the tau-functions of this hierarchy and we expect that the Extended bi-graded Toda hierarchy governs the Gromov--Witten theory of $\mathcal{C}_{k,m}.$, Comment: v2: Typos and mistakes fixed, to appear in Journal fuer die reine und angewandte Mathematik (Crelle's Journal). v3: Mistakes fixed and references updated

The ancestor Gromov--Witten invariants of a compact {\Kahler} manifold $X$ can be organized in a generating function called the total ancestor potential of $X$. In this paper, we construct Hirota Quadratic Equations (HQE shortly) for the total ancestor potential of $\C P^1$. The idea is to adopt the formalism developed in \cite{G1,GM} to the mirror model of $\C P^1$. We hope that the ideas presented here can be generalized to other manifolds as well. As a corollary, using the twisted loop group formalism from \cite{G3}, we obtain a new proof of the following version of the Toda conjecture: the total descendant potential of $\C P^1$ (known also as the partition function of the $\C P^1$ topological sigma model) is a tau-function of the Extended Toda Hierarchy., Comment: 22 pages, this is the second part of an earlier version, major revision of the exposition

We construct an integrable hierarchy in terms of vertex operators and Hirota Quadratic Equations (HQE shortly) and we show that the equivariant total descendant potential of $\C P^1$ satisfies the HQE. Our prove is based on the quantization formalism developed in \cite{G1}, \cite{G2}, and on the equivariant mirror model of $\C P^1.$ The vertex operators in our construction obey certain transformation law under change of coordinates, which might be important for generalizing the HQE to other manifolds. We also show that under certain change of the variables, which is due to E. Getzler, the HQE are transformed into the HQE of the 2-Toda hierarchy. Thus we obtain a new proof of the equivariant Toda conjecture., Comment: 18 pages

The Extended Toda Hierarchy (shortly ETH) was introduce by E. Getzler \cite{Ge} and independently by Y. Zhang \cite{Z} in order to describe an integrable hierarchy which governs the Gromov--Witten invariants of $\C P^1$. The {\em Lax type} presentation of the ETH was given in \cite{CDZ}. In this paper we give a description of the ETH in terms of {\em tau-functions} and Hirota Quadratic Equations (known also as Hirota Bilinear Equations). A new feature here is that the Hirota equations are given in terms of vertex operators taking values in the algebra of differential operators on the affine line., Comment: 15 pages, pending final approval in Duke, this replacement is the first part of the old version

The paper math.AG/0108100 gives a construction of the total descendent potential corresponding to a semisimple Frobenius manifold. In math.AG/0209205, it is proved that the total descendent potential corresponding to K. Saito's Frobenius structure on the parameter space of the miniversal deformation of the A_{n-1}-singularity satisfies the modulo-n reduction of the KP-hierarchy. In this paper, we identify the hierarchy satisfied by the total descendent potential of a simple singularity of the A,D,E-type. Our description of the hierarchy is parallel to the vertex operator construction of Kac -- Wakimoto except that we give both some general integral formulas and explicit numerical values for certain coefficients which in the Kac -- Wakimoto theory are studied on a case-by-case basis and remain, generally speaking, unknown., Comment: 24 pages