BKP and projective Hurwitz numbers

We consider d-fold branched coverings of the projective plane \(\mathbb {RP}^2\) and show that the hypergeometric tau function of the BKP hierarchy of Kac and van de Leur is the generating function for weighted sums of the related Hurwitz numbers. In particular, we get the \(\mathbb {RP}^2\) analogues of the \(\mathbb {CP}^1\) generating functions proposed by Okounkov and by Goulden and Jackson. Other examples are Hurwitz numbers weighted by the Hall–Littlewood and by the Macdonald polynomials. We also consider integrals of tau functions which generate Hurwitz numbers related to base surfaces with arbitrary Euler characteristics \(\textsc {e}\), in particular projective Hurwitz numbers \(\textsc {e}=1\).     

Chiral expansion and Macdonald deformation of two‐dimensional Yang‐Mills theory

We derive the analog of the large N Gross‐Taylor holomorphic string expansion for the refinement of q‐deformed Yang‐Mills theory on a compact oriented Riemann surface. The derivation combines Schur‐Weyl duality for quantum groups with the Etingof‐Kirillov theory of generalized quantum characters which are related to Macdonald polynomials. In the unrefined limit we reproduce the chiral expansion of q‐deformed Yang‐Mills theory derived by de Haro, Ramgoolam and Torrielli. In the classical limit , the expansion defines a new β‐deformation of Hurwitz theory wherein the refined partition function is a generating function for certain parameterized Euler characters, which reduce in the unrefined limit to the orbifold Euler characteristics of Hurwitz spaces of holomorphic maps. We discuss the geometrical meaning of our expansions in relation to quantum spectral curves and β‐ensembles of matrix models arising in refined topological string theory.     

Determinantal Quintics and Mirror Symmetry of Reye Congruences

We study a certain family of determinantal quintic hypersurfaces in \({\mathbb{P}^{4}}\) whose singularities are similar to the well-studied Barth–Nieto quintic. Smooth Calabi–Yau threefolds with Hodge numbers (h ^1,1,h ^2,1) = (52, 2) are obtained by taking crepant resolutions of the singularities. It turns out that these smooth Calabi–Yau threefolds are in a two dimensional mirror family to the complete intersection Calabi–Yau threefolds in \({\mathbb{P}^{4}\times\mathbb{P}^{4}}\) which have appeared in our previous study of Reye congruences in dimension three. We compactify the two dimensional family over \({\mathbb{P}^{2}}\) and reproduce the mirror family to the Reye congruences. We also determine the monodromy of the family over \({\mathbb{P}^{2}}\) completely. Our calculation shows an example of the orbifold mirror construction with a trivial orbifold group.     

The Remodeling Conjecture and the Faber–Pandharipande Formula

In this note, we prove that the free energies F _g constructed from the Eynard–Orantin topological recursion applied to the curve mirror to ${\mathbb{C}^3}$ reproduce the Faber–Pandharipande formula for genus g Gromov–Witten invariants of ${\mathbb{C}^3}$ . This completes the proof of the remodeling conjecture for ${\mathbb{C}^3}$ .     

Quantum Curves for Hitchin Fibrations and the Eynard–Orantin Theory

We generalize the topological recursion of Eynard–Orantin (JHEP 0612:053, 2006; Commun Number Theory Phys 1:347–452, 2007) to the family of spectral curves of Hitchin fibrations. A spectral curve in the topological recursion, which is defined to be a complex plane curve, is replaced with a generic curve in the cotangent bundle T*C of an arbitrary smooth base curve C. We then prove that these spectral curves are quantizable, using the new formalism. More precisely, we construct the canonical generators of the formal ${\hbar}$ -deformation family of D modules over an arbitrary projective algebraic curve C of genus greater than 1, from the geometry of a prescribed family of smooth Hitchin spectral curves associated with the ${SL(2,\mathbb{C})}$ -character variety of the fundamental group π₁(C). We show that the semi-classical limit through the WKB approximation of these ${\hbar}$ -deformed D modules recovers the initial family of Hitchin spectral curves.     

Identification of the Givental Formula with the Spectral Curve Topological Recursion Procedure

We identify the Givental formula for the ancestor formal Gromov–Witten potential with a version of the topological recursion procedure for a collection of isolated local germs of the spectral curve. As an application we prove a conjecture of Norbury and Scott on the reconstruction of the stationary sector of the Gromov–Witten potential of ${\mathbb{C}{\rm P}^1}$ via a particular spectral curve.     

Chemical memory erasure; a photochemical model

In this paper we propose an exactly solvable model of a topological insulator defined on a spin- \(\tfrac{1}{2}\) square decorated lattice. Itinerant fermions defined in the framework of the Haldane model interact via the Kitaev interaction with spin- \(\tfrac{1}{2}\) Kitaev sublattice. The presented model, whose ground state is a non-trivial topological phase, is solved exactly. We have found out that various phase transitions without gap closing at the topological phase transition point outline the separate states with different topological numbers. We provide a detailed analysis of the model’s ground-state phase diagram and demonstrate how quantum phase transitions between topological states arise. We have found that the states with both the same and different topological numbers are all separated by the quantum phase transition without gap closing. The transition between topological phases is accompanied by a rearrangement of the spin subsystem’s spectrum from band to flat-band states.     

Equivariant GW Theory of Stacky Curves

We extend Okounkov and Pandharipande’s work on the equivariant Gromov–Witten theory of ${\mathbb{P}^1}$ to a class of stacky curves ${\mathcal{X}}$ . Our main result uses virtual localization and the orbifold ELSV formula to express the tau function ${\tau_\mathcal{X}}$ as a vacuum expectation on a Fock space. As corollaries, we prove the decomposition conjecture for these ${\mathcal{X}}$ , and prove that ${\tau_\mathcal{X}}$ satisfies a version of the 2-Toda hierarchy. Coupled with degeneration techniques, the result should lead to treatment of general orbifold curves.     

Geometric interpretation of Zhou’s explicit formula for the Witten–Kontsevich tau function

Based on the work of Itzykson and Zuber on Kontsevich’s integrals, we give a geometric interpretation and a simple proof of Zhou’s explicit formula for the Witten–Kontsevich tau function. More precisely, we show that the numbers \(A_{m,n}^\mathrm{Zhou}\) defined by Zhou coincide with the affine coordinates for the point of the Sato Grassmannian corresponding to the Witten–Kontsevich tau function. Generating functions and new recursion relations for \(A_{m,n}^\mathrm{Zhou}\) are derived. Our formulation on matrix-valued affine coordinates and on tau functions remains valid for generic Grassmannian solutions of the KdV hierarchy. A by-product of our study indicates an interesting relation between the matrix-valued affine coordinates for the Witten–Kontsevich tau function and the V-matrices associated with the R-matrix of Witten’s 3-spin structures.     

<Emphasis Type="Italic">C</Emphasis><Subscript>2</Subscript>-Cofiniteness of the 2-Cycle Permutation Orbifold Models of Minimal Virasoro Vertex Operator Algebras

In this article, we give a sufficient and necessary condition for the C ₂-cofiniteness of \({\widetilde{V} = (V\otimes V)^\sigma}\) for a C ₂-cofinite vertex operator algebra V and the 2-cycle permutation σ of \({V\otimes V}\) . As an application, we show that the 2-cycle permutation orbifold model of the simple Virasoro vertex operator algebra L(c, 0) of minimal central charge c is C ₂-cofinite.     

Lagrangian Floer Superpotentials and Crepant Resolutions for Toric Orbifolds

We investigate the relationship between the Lagrangian Floer superpotentials for a toric orbifold and its toric crepant resolutions. More specifically, we study an open string version of the crepant resolution conjecture (CRC) which states that the Lagrangian Floer superpotential of a Gorenstein toric orbifold ${\mathcal{X}}$ and that of its toric crepant resolution Y coincide after analytic continuation of quantum parameters and a change of variables. Relating this conjecture with the closed CRC, we find that the change of variable formula which appears in closed CRC can be explained by relations between open (orbifold) Gromov-Witten invariants. We also discover a geometric explanation (in terms of virtual counting of stable orbi-discs) for the specialization of quantum parameters to roots of unity which appears in Ruan’s original CRC (Gromov-Witten theory of spin curves and orbifolds, contemp math, Amer. Math. Soc., Providence, RI, pp 117–126, 2006). We prove the open CRC for the weighted projective spaces ${\mathcal{X} = \mathbb{P}(1,\ldots,1, n)}$ using an equality between open and closed orbifold Gromov-Witten invariants. Along the way, we also prove an open mirror theorem for these toric orbifolds.     

Exact Results for Topological Strings on Resolved <Emphasis Type="Italic">Y</Emphasis><Superscript><Emphasis Type="Italic"> p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Superscript> Singularities

We obtain exact results in α′ for open and closed A-model topological string amplitudes on a large class of toric Calabi-Yau threefolds by using their correspondence with five dimensional gauge theories. The toric Calabi-Yaus that we analyze are obtained as minimal resolution of cones over Y ^p,q manifolds and give rise via M-theory compactification to SU(p) gauge theories on . As an application we present a detailed study of the local case and compute open and closed genus zero Gromov-Witten invariants of the orbifold. We also display the modular structure of the topological wave function and give predictions for higher genus amplitudes. The mirror curve in this case is the spectral curve of the relativistic A ₁ Toda chain. Our results also indicate the existence of a wider class of relativistic integrable systems associated to generic Y ^p,q geometries.     

Topological complexity of certain classes of <Emphasis Type="Italic">C</Emphasis>*-algebras

We compute the topological complexity for some important classes of noncommutative C*-algebras: AF algebras, AI algebras, and even Cuntz algebras.     

The trinification model <Emphasis Type="Italic">SU</Emphasis>(3)<Superscript>3</Superscript> from orbifolds for fuzzy spheres

In this review, we consider an N = 4 supersymmetric SU(3N) gauge theory defined on the Minkowski spacetime. Then we apply an orbifold projection leading to an N = 1 supersymmetric SU(N)³ model, with a truncated particle spectrum. Then, we present the dynamical generation of (twisted) fuzzy spheres as vacuum solutions of the projected field theory, breaking the SU(N)³ spontaneously to a chiral effective theory with unbroken gauge group the trinification group, SU(3)³.     

Global Stringy Orbifold Cohomology,K-Theory and de Rham Theory

There are two approaches to constructing stringy multiplications for global quotients. The first one is given by first pulling back and then pushing forward. The second one is given by first pushing forward and then pulling back. The first approach has been used to define a global stringy extension of the functors K ₀ and K ^top by Jarvis–Kaufmann–Kimura, A* by Abramovich–Graber–Vistoli, and H* by Chen–Ruan and Fantechi–Göttsche. The second approach has been applied by the author in the case of cyclic twisted sector and in particular for singularities with symmetries and for symmetric products. The second type of construction has also been discussed in the de Rham setting for Abelian quotients by Chen–Hu. We give a rigorous formulation of de Rham theory for any global quotient from both points of view. We also show that the pull–push formalism has a solution by the push–pull equations in the setting case of cyclic twisted sectors. In the general, not necessarily cyclic case, we introduce ring extensions and treat all the stringy extension of the functors mentioned above also from the second point of view. A first extension provides formal sections and a second extension fractional Euler classes. The formal sections allow us to give a pull–push solution while fractional Euler classes give a trivialization of the co-cycles of the pull–push formalism. The main tool is the formula for the obstruction bundle of Jarvis–Kaufmann–Kimura. This trivialization can be interpreted as defining the physics notion of twist fields. We end with an outlook on applications to singularities with symmetries aka. orbifold Landau–Ginzburg models.     

Hypergeometric $${\tau}$$ -Functions,Hurwitz Numbers and Enumeration of Paths

A multiparametric family of 2D Toda \({\tau}\) -functions of hypergeometric type is shown to provide generating functions for composite, signed Hurwitz numbers that enumerate certain classes of branched coverings of the Riemann sphere and paths in the Cayley graph of S _n . The coefficients \({{F^{c_{1}, . . . , c_{l}}_{d_{1}, . . . , d_{m}}}(\mu, \nu)}\) in their series expansion over products \({P_{\mu}P^{'}_{\nu}}\) of power sum symmetric functions in the two sets of Toda flow parameters and powers of the l + m auxiliary parameters are shown to enumerate \({|\mu|=|\nu|=n}\) fold branched covers of the Riemann sphere with specified ramification profiles \({ \mu}\) and \({\nu}\) at a pair of points, and two sets of additional branch points, satisfying certain additional conditions on their ramification profile lengths. The first group consists of l branch points, with ramification profile lengths fixed to be the numbers \({(n-c_{1}, . . . , n-c_{l})}\) ; the second consists of m further groups of “coloured” branch points, of variable number, for which the sums of the complements of the ramification profile lengths within the groups are fixed to equal the numbers \({(d_{1}, . . . , d_{m})}\). The latter are counted with signs determined by the parity of the total number of such branch points. The coefficients \({{F^{c_{1}, . . . , c_{l}}_{d_{1}, . . . , d_{m}}}(\mu, \nu)}\) are also shown to enumerate paths in the Cayley graph of the symmetric group S _n generated by transpositions, starting, as in the usual double Hurwitz case, at an element in the conjugacy class of cycle type \({\mu}\) and ending in the class of type \({\nu}\), with the first l consecutive subsequences of \({(c_{1}, . . . , c_{l})}\) transpositions strictly monotonically increasing, and the subsequent subsequences of \({(d_{1}, . . . , d_{m})}\) transpositions weakly increasing.     

Topological invariants for phase transition points of one-dimensional Z<Subscript>2</Subscript> topological systems

We study topological properties of phase transition points of two topologicallynon-trivial Z₂ classes (D and DIII) in one dimension byassigning a Berry phase defined on closed circles around the gap closing points in theparameter space of momentum and a transition driving parameter. While the topologicalproperty of the Z₂ system is generally characterized by aZ₂topological invariant, we identify that it has a correspondence to the quantized Berryphase protected by the particle-hole symmetry, and then give a proper definition of Berryphase to the phase transition point. By applying our scheme to some specific models ofclass D and DIII, we demonstrate that the topological phase transition can be wellcharacterized by the Berry phase of the transition point, which reflects the change ofBerry phases of topologically different phases across the phase transition point.     

Quantum Orbifolds

This is a study of orbifold-quotients of quantum groups (quantum orbifolds \({\Theta } \rightrightarrows G_{q}\)). These structures have been studied extensively in the case of the quantum S U ₂ group. A generalized theory of quantum orbifolds over compact simple and simply connected quantum groups is developed. Associated with a quantum orbifold there is an invariant subalgebra and a crossed product algebra. For each spin quantum orbifold, there is a unitary equivalence class of Dirac spectral triples over the invariant subalgebra, and for each effective spin quantum orbifold associated with a finite group action, there is a unitary equivalence class of Dirac spectral triples over the crossed product algebra. A Hopf-equivariant Fredholm index problem is studied as an application.     

Computation of Open Gromov–Witten Invariants for Toric Calabi–Yau 3-Folds by Topological Recursion,a Proof of the BKMP Conjecture

The BKMP conjecture (2006–2008) proposed a new method to compute closed and open Gromov–Witten invariants for every toric Calabi–Yau 3-folds, through a topological recursion based on mirror symmetry. So far, this conjecture has been verified to low genus for several toric CY3folds, and proved to all genus only for \({\mathbb{C}^3}\). In this article we prove the general case. Our proof is based on the fact that both sides of the conjecture can be naturally written in terms of combinatorial sums of weighted graphs: on the A-model side this is the localization formula, and on the B-model side the graphs encode the recursive algorithm of the topological recursion.One can slightly reorganize the set of graphs obtained in the B-side, so that it coincides with the one obtained by localization in the A-model. Then it suffices to compare the weights of vertices and edges of graphs on each side, which is done in two steps: the weights coincide in the large radius limit, due to the fact that the toric graph is the tropical limit of the mirror curve. Then the derivatives with respect to Kähler radius coincide due to the special geometry property implied by the topological recursion.     

Temperature dependence of the kinetic energy in the Zr<Subscript>40</Subscript>Be<Subscript>60</Subscript> amorphous alloy

The average kinetic energy 〈E(T)〉 of the atomic nucleus for each element of the amorphous alloy Zr₄₀Be₆₀ in the temperature range 10–300 K has been measured for the first time using VESUVIO spectrometer (ISIS). The experimental values of 〈E(T)〉 have been compared to the partial ZrBe spectra refined by a recursion method based on the data obtained with thermal neutron scattering. The satisfactory agreement has been reached with the calculations using partial spectra based on thermal neutron spectra obtained with recursion method. In addition, the experimental data have been compared to the Debye model. The measurements at different temperatures (10, 200, and 300 K) will provide an opportunity to evaluate the significance of anharmonicity in the dynamics of metallic glasses.