Calabi-Yau Research Papers - Academia.edu (original) (raw)

2025

We give several applications of the thick distributional calculus. We consider homogeneous distributions, point source fields, and higher order derivatives of order 0.0.0.

2025, Forum Mathematicum

We study the geometry of the birational map between an intersection of a web of quadrics in P7 that contains a plane and the double octic branched along the discriminant of the web.

2025

The purpose of this dissertation is to study the intersection theory of the moduli spaces of stable maps of degree two from two-pointed, genus zero nodal curves to arbitrary-dimensional projective space. Toward this end, first the Betti numbers of \bar{M}_{0,2}(P^r,2) are computed using Serre polynomials and equivariant Serre polynomials. Then, specializing to the space \bar{M}_{0,2}(P^1,2), generators and relations for the Chow ring are given. Chow rings of simpler spaces are also described, and the method of localization and linear algebra is developed. Both tools are used in finding the relations. It is further demonstrated that no additional relations exist among the generators, so that a presentation for the Chow ring A^*(\bar{M}_{0,2}(P^1,2)) is obtained. As a further check of the presentation, it is applied to give a new computation of the previously known genus zero, degree two, two-pointed gravitational correlators of P^1. Portions of this work also appear in math.AG/050132...

2025, arXiv (Cornell University)

2025

In the present paper we have used the Differential forms also known as exterior calculus of E. in Pullback calculations and proving the main theorems of advanced calculus i.e. Green, Gauss and Stoke's theorems. In particular a convenient test for a transformation to be canonical is given with examples based on differential forms, which is more suitable than tests given by Goldstein et.al. [2004].

2025, Advances in Theoretical and Mathematical Physics

Large N geometric transitions and the Dijkgraaf-Vafa conjecture suggest a deep relationship between the sum over planar diagrams and Calabi-Yau threefolds. We explore this correspondence in details, explaining how to construct the Calabi-Yau for a large class of Mmatrix models, and how the geometry encodes the correlators. We engineer in particular two-matrix theories with potentials W (X, Y ) that reduce to arbitrary functions in the commutative limit. We apply the method to calculate all correlators tr X p and tr Y p in models of the form The solution of the latter example was not known, but when U is a constant we are able to solve the loop equations, finding a precise match with the geometric approach. We also discuss special geometry in multi-matrix models, and we derive an important property, the entanglement of eigenvalues, governing the expansion around classical vacua for which the matrices do not commute.

2025, arXiv (Cornell University)

2025

We develop a theory of differential equations associated to families of algebraic cycles in higher Chow groups (i.e., motivic cohomology groups). This formalism is related to inhomogenous Picard-Fuchs type differential equations. For... more

2025, Journal of Mathematics and Computer Science

The calculus of integrals is used to solve the majority of physics and engineering issues, which are frequently not immediately solved. This compels us to use approximation techniques, the selection of which is based on the class of functions that meet the necessary criteria as well as known points. Within the context of quantum calculus, we present an assessment of the error of the well-known corrected dual Euler-Simpson quadrature rule in this paper. As an auxiliary result, we create a new quantum identity. Using this identity, we prove several quantum-corrected dual Euler-Simpson type integral inequalities for functions with convex q-derivatives. We allow q to go towards 1-in order to obtain the classical inequalities. We provide a few applications to wrap up this research.

2025, Advances in High Energy Physics

Within the AdS/CFT correspondence, we review the studies of field theories with a large number of adjoint and fundamental fields, in the Veneziano limit. We concentrate in set-ups where the fundamentals are introduced by a smeared set of D-branes. We make emphasis on the general ideas and then in subsequent chapters that can be read independently and describe particular considerations in various different models. Some new material is presented along the various sections.

2025

We consider a possible approach to the Jacobian conjecture. We define the Jacobian variety of Cn of degree d, denoted by J(n, d), whose points parametrize the set of all the n-Jacobian tuples of total degree at most d normalized to map 0 ∈ Cn onto itself. We use the term ”variety” as a not necessarily irreducible algebraic set. The set Aut0,d(C) of all the polynomial automorphisms of Cn of total degree at most d that map 0 onto itself corresponds to a subset of the Jacobian variety of degree d. This subset is in fact a subvariety, i.e. it is a Zariski closed subset of the variety. This might be significant to settle the Jacobian conjecture. For instance, if the jacobian variety of degree d is irreducible, then it reduces the problem to computing the dimensions of these two varieties. The conjecture is true if and only if the two dimensions are equal. Otherwise, almost any point on the Jacobian variety will serve as a counterexample to the Jacobian conjecture. Using results of Magnus...

2025, arXiv (Cornell University)

An analytic pair of dimension n and center V is a pair (V, M) where M is a complex manifold of (complex) dimension n and V ⊂ M is a closed totally real analytic submanifold of dimension n. To an analytic pair (V, M) we associate the class U (V, M) of the functions u : M → [0, π/4[ which are plurisubharmonic in M and such that u(p) = 0 for each p ∈ V . If U (V, M) admits a maximal function u, the triple (V, M, u) is said to be a maximal plurisubharmonic model. After defining a pseudo-metric E V,M on the center V of an analytic pair (V, M) we prove (see Theorem 4.1, Theorem 5.1) that maximal plurisubharmonic models provide a natural generalization of the Monge-Ampère models introduced by Lempert and Szöke in .

2025

In this paper we generalize classical 3-set theorem related to stable partitions of arbitrary mappings due to Erdős-de Bruijn, Katětov and Kasteleyn. We consider a structural generalization of this result to partitions preserving sets of inequalities and characterize all finite sets of such inequalities which can be preserved by a "small" coloring. These results are also related to graph homomorphisms and (oriented) colorings.

2025, Oxford University Press eBooks

In this article we investigate the Duistermaat-Heckman theorem using the theory of hyperfunctions. In applications involving Hamiltonian torus actions on infinite dimensional manifolds, this more general theory seems to be necessary in order to accomodate the existence of the infinite order differential operators which arise from the isotropy representations on the tangent spaces to fixed points. We will quickly review of the theory of hyperfunctions and their Fourier transforms. We will then apply this theory to construct a hyperfunction analogue of the Duistermaat-Heckman distribution. Our main goal will be to study the Duistermaat-Heckman hyperfunction of ΩSU ( ), but in getting to this goal we will also characterize the singular locus of the moment map for the Hamiltonian action of T × S 1 on ΩG. The main goal of this paper is to present a Duistermaat-Heckman hyperfunction arising from a Hamiltonian action on an infinite dimensional manifold.

2025, SciPost Physics Core

The dualization of the scalar fields of a theory into (d-2)(d−2)-form potentials preserving all the global symmetries is one of the main problems in the construction of democratic pseudoactions containing simultaneously all the original fields and their duals. We study this problem starting with the simplest cases and we show how it can be solved for scalars parametrizing Riemannian symmetric \sigmaσ-models as in maximal and half-maximal supergravities. Then, we use this result to write democratic pseudoactions for theories in which the scalars are non-minimally coupled to (p+1)(p+1)-form potentials in any dimension. These results include a proposal of democratic pseudoaction for the generic bosonic sector of 4-dimensional maximal and half-maximal ungauged supergravities. Furthermore, we propose a democratic pseudoaction for the bosonic sector of \mathcal{N}=2B,d=10𝒩=2B,d=10 supergravity (the effective action of the type IIB superstring theory) containing two 0-, two 2-, one 4-, two...

2025, arXiv (Cornell University)

It is proved that over every countable field K there is a nil algebra R such that the algebra obtained from R by extending the field K contains noncommutative free subalgebras of arbitrarily high rank. It is also shown that over every countable field K there is an algebra R without noncommutative free subalgebras of rank two such that the algebra obtained from R by extending the field K contains a noncommutative free subalgebra of rank two. This answers a question of .

2025, Physical Review Letters

We analyze non-perturbative corrections to the superpotential of seven-brane gauge theories on type IIB and F-theory warped Calabi-Yau compactifications. We show in particular that such corrections modify the holomorphic Yukawa couplings by an exponentially suppressed contribution, generically solving the Yukawa rank-one problem of certain F-theory local models. We provide explicit expressions for the non-perturbative correction to the seven-brane superpotential, and check that it is related to a non-commutative deformation to the tree-level superpotential via a Seiberg-Witten map.

2025, Physical Review Letters

2025, Springer eBooks

We study two dimensional N = (2, 2) Landau-Ginzburg models with tensor valued superfields with the aim of constructing large central charge superconformal field theories which are solvable using large N techniques. We demonstrate the viability of such constructions and motivate the study of anisotropic tensor models. Such theories are a novel deformation of tensor models where we break the continuous symmetries while preserving the large N solvability. Specifically, we examine theories with superpotentials involving tensor contractions chosen to pick out melonic diagrams. The anisotropy is introduced by further biasing individual terms by different coefficients, all of the same order, to retain large N scaling. We carry out a detailed analysis of the resulting low energy fixed point and comment on potential applications to holography. Along the way we also examine gauged versions of the models (with partial anisotropy) and find generically that such theories have a non-compact Higgs branch of vacua.

2025, Landon Dean Cherry

The study of open and closed string worldsheets is fundamental and foundational for understanding string behavior, boundary conditions, and action formalism in modern string theory. This review presents the topological and dynamical distinctions between open and closed string configurations, including their origins, momenta, and actions that dictate the string evolution and energy profile. Central to strings evolution and energy profile is the Polyakov action, the Polyakov action offers a more holistic view of how strings evolve and interact across their worldsheet.

2025, Double stochastic processes and Generalized Chapman- Kolmogorov equations.

In this paper generalized Chapman-Kolmogorov equation is derived. 1.Introduction 2.Generalized random variables. 3.Integration over random interval. 4.Generalized Chapman-Kolmogorov equation. .

2025, arXiv (Cornell University)

We consider the spectral problem on ∂Ω in a smooth bounded domain Ω of R 2 . The factor ρ ε which appears in the first equation plays the role of a mass density and it is equal to a constant of order ε -1 in an ε-neighborhood of the boundary and to a constant of order ε in the rest of Ω. We study the asymptotic behavior of the eigenvalues λ(ε) and the eigenfunctions u ε as ε tends to zero. We obtain explicit formulas for the first and second terms of the corresponding asymptotic expansions by exploiting the solutions of certain auxiliary boundary value problems.

2025, SIAM Journal on Mathematical Analysis

2025, Homology, Homotopy and Applications

We extend the Oprea's result G1(S 2n+1 /H) = ZH to the 1 st generalized Gottlieb group G f 1 (S 2n+1 /H) for a map f : A → S 2n+1 /H. Then, we compute or estimate the groups G f m (S 2n+1 /H) and P f m (S 2n+1 /H) for some m > 1 and... more

2025

We introduce the notion of a matron M = ⊕Mn,m whose sub-modules ⊕M n,1 and ⊕M 1,m are non-Σ operads. We construct a functor from PROP to matrons and its inverse, the universal enveloping functor. We define the free matron H∞, generated by a singleton in each bidegree (m, n) = (1, 1), and define an A∞-bialgebra as an algebra over H∞. We realize H∞ as the cellular chains of polytopes KKn,m, among which KK n,0 and KK 0,m are Stasheff associahedra.

2025, arXiv (Cornell University)

Using embedding of complex curves in the complex projective plane P 2 , we develop a non planar topological 3-vertex formalism for topological strings on the family of local Calabi-Yau threefolds ∞) . The base E (t,∞) stands for the degenerate elliptic curve with Kahler parameter t; but a large complex structure µ; i.e |µ| -→ ∞. We also give first results regarding A-model topological string amplitudes on X (m,-m,0) . The 2D U (1) gauged N = 2 supersymmetric sigma models of the degenerate elliptic curve E (t,∞) as well as for the family X (m,-m,0) are studied and the role of D-and F-terms is explicitly exhibited.

2025

It is shown how some field theories in the target-space induce the splitting of the space-time into a continuous of branes, which can be p-branes or D-branes depending on what the field theory it is. The basic symmetry underlying the construction is used to build an invariant action, which is proved to be off-shell identical to the p-brane (D-brane) action. The coupling with the abelian ($p+1$)-form in this formulation it is also found. While the classical brane's embedding couple to the field strenght, the classical fields couple with its dual (in the Hodge sense), therefore providing an explicit electric-magnetic duality. Finally, the generic role of the underlying symmetry in the connection between the target-space theory and the world-volume one, is completely elucidated.

2025, The journal of fourier analysis and applications/Journal of fourier analysis & applications

In this paper we use some ideas from and consider the description of Hörmander type pseudo-differential operators on R d (d ≥ 1), including the case of the magnetic pseudo-differential operators introduced in [15, 16], with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calderón-Vaillancourt theorem and Beals' commutator criterion, and also establish local trace-class criteria.

2025, American Mathematical Society eBooks

We construct an algebraic variety by resolving singularities of a quintic Calabi-Yau threefold. The middle cohomology of the threefold is shown to contain a piece coming from a pair of elliptic surfaces. The resulting quotient is a two-dimensional Galois representation. By using the Lefschetz fixed-point theorem in étale cohomology and counting points on the variety over finite fields, this Galois representation is shown to be modular.

2025, Journal of Number Theory

BRUEN, JENSEN. AND YUI with certain Frobenius groups as Galois groups. 111.1. Preliminary results. 111.2. Realization of Frobenius groups of prime degree as Galois groups (general existence theorem). 111.3. Construction of polynomials with Galois group F,,, , bl over 0. 111.4. Construction of generic family of polynomials with Galois group F,c p I uz (p = 3 (mod 4)). 111.5. Special examples. 111.6. Remarks and problems. IV. Frobenius fields. IV.1. Frobenius fields over 0. IV.2. Frobenius fields over function fields.

2025, Expositiones Mathematicae

The proof of Serre's conjecture on Galois representations over finite fields allows us to show, using a trick due to Serre himself, that all rigid Calabi-Yau threefolds defined over Q are modular.

2025, International Journal of Robust and Nonlinear Control

In this paper we consider a nonlinear model of a biological wastewater treatment process, based on two microbial populations and two substrates. As a result of this process methane is produced. This model is known to be practically validated and reliable. Two feedback control laws are proposed (one of them is adaptive) for asymptotic stabilization of the closed-loop system towards a previously chosen operating point. Computer simulations are reported to compare the effectiveness of the proposed feedbacks.

2025, Nuclear Physics B

We obtain a closed form expression for the Action describing pure gravity, in light-cone gauge, in a four-dimensional Anti de Sitter background. We perform a perturbative expansion of this closed form result to extract the cubic... more

2025, arXiv (Cornell University)

We introduce the notion of complex G 2 manifold M C , and complexification of a G 2 manifold M ⊂ M C . As an application we show the following: If (Y, s) is a closed oriented 3-manifold with a Spin c structure, and (Y, s) ⊂ (M, ϕ) is an imbedding as an associative submanifold of some G 2 manifold (such imbedding always exists), then the isotropic associative deformations of Y in the complexified G 2 manifold M C is given by Seiberg-Witten equations.

2025

Bosonization (and fermionization) in (1+1) dimensions is viewed as two dierent gauge xings of an underlying theory containing both types of elds.

2025, Communications in Mathematical Physics

We present a complex matrix gauge model defined on an arbitrary two-dimensional orientable lattice. We rewrite the model's partition function in terms of a sum over representations of the group U (N ). The model solves the general combinatorial problem of counting branched covers of orientable Riemann surfaces with any given, fixed branch point structure. We then define an appropriate continuum limit allowing the branch points to freely float over the surface. The simplest such limit reproduces two-dimensional chiral U (N ) Yang-Mills theory and its string description due to Gross and Taylor.

2025, International Journal of Modern Physics

We study moduli space stabilization of a class of BPS configurations from the perspective of the real intrinsic Riemannian geometry. Our analysis exhibits a set of implications towards the stability of the D-term potentials, defined for a set of abelian scalar fields. In particular, we show that the nature of marginal and threshold walls of stabilities may be investigated by real geometric methods. Interestingly, we find that the leading order contributions may easily be accomplished by translations of the Fayet parameter. Specifically, we notice that the various possible linear, planar, hyper-planar and the entire moduli space stabilities may easily be reduced to certain polynomials in the Fayet parameter. For a set of finitely many real scalar fields, it may be further inferred that the intrinsic scalar curvature defines the global nature and range of vacuum correlations. Whereas, the underlying moduli space configuration corresponds to a non-interacting basis at the zeros of the scalar curvature, where the scalar fields become un-correlated. The divergences of the scalar curvature provide possible phase structures, viz., wall of stability, phase transition, if any, in the chosen moduli configuration. The present analysis opens up a new avenue towards the stabilizations of gauge and string moduli.

2025, Physical review

We evaluate the Hadamard function and the vacuum expectation value (VEV) of the current density for a charged scalar field in the region between two co-dimension one branes on the background of locally AdS spacetime with an arbitrary number of toroidally compactified spatial dimensions. Along compact dimensions periodicity conditions are considered with general values of the phases and on the branes Robin boundary conditions are imposed for the field operator. In addition, we assume the presence of a constant gauge field. The latter gives rise to Aharonov-Bohm type effect on the vacuum currents. There exists a range in the space of the Robin coefficients for separate branes where the vacuum state becomes unstable. Compared to the case of the standard AdS bulk, in models with compact dimensions the stability condition imposed on the parameters is less restrictive. The current density has nonzero components along compact dimensions only. These components are decomposed into the brane-free and brane-induced contributions. Different representations are provided for the latter well suited for the investigation of the near-brane, near-AdS boundary and near-AdS horizon asymptotics. The component along a given compact dimension is a periodic function of the gauge field flux, enclosed by that dimension, with the period of the flux quantum. An important feature, that distinguishes the current density from the expectation values of the field squared and energy-momentum tensor, is its finiteness on the branes. In particular, for Dirichlet boundary condition the current density vanishes on the branes. We show that, depending on the constants in the boundary conditions, the presence of the branes may either increase or decrease the current density compared with that for the brane-free geometry. Applications are given to the Randall-Sundrum 2-brane model with extra compact dimensions. In particular, we estimate the effects of the hidden brane on the current density on the visible brane.

2025, Selecta Mathematica

The condition of having an N = 1 spacetime supersymmetry for heterotic string leads to 4 distinct possibilities for compactifications namely compactifications down to 6,4,3 and 2 dimensions. Compactifications to 6 and 4 dimensions have been studied extensively before (corresponding to K3 and a Calabi-Yau threefold respectively). Here we complete the study of the other two cases corresponding to compactification down to 3 on a 7 dimensional manifold of G 2 holonomy and compactification down to 2 on an 8 dimensional manifold of Spin( ) holonomy. We study the extended chiral algebra and find the space of exactly marginal deformations. It turns out that the role the U (1) current plays in the N = 2 superconformal theories, is played by tri-critical Ising model in the case of G 2 and Ising model in the case of Spin(7) manifolds. Certain generalizations of mirror symmetry are found for these two cases. We also discuss the topological twisting in each case.

2025, Journal of High Energy Physics

This paper lays groundwork for the detailed study of the non-trivial renormalization group flow connecting supersymmetric fixed points in four dimensions using string theory on AdS spaces. Specifically, we consider D3-branes placed at singularities of Calabi-Yau threefolds which generalize the conifold singularity and have an ADE classification. The N = 1 superconformal theories dictating their low-energy dynamics are infrared fixed points arising from deforming the corresponding ADE N = 2 superconformal field theories by mass terms for adjoint chiral fields. We probe the geometry with a single D3-brane and discuss the near-horizon supergravity solution for a large number N of coincident D3-branes.

2025, arXiv (Cornell University)

The Bagger-Witten line bundle is a line bundle over moduli spaces of two-dimensional SCFTs, related to the Hodge line bundle of holomorphic top-forms on Calabi-Yau manifolds. It has recently been a subject of a number of conjectures, but concrete examples have proven elusive. In this paper we collect several results on this structure, including a proposal for an intrisic geometric definition over moduli spaces of Calabi-Yau manifolds and some additional concrete examples. We also conjecture a new criterion for UV completion of four-dimensional supergravity theories in terms of properties of the Bagger-Witten line bundle.

2025, arXiv (Cornell University)

This is a summary of a talk given at the Vienna homological mirror symmetry conference in June 2006. We review how both derived categories and stacks enter physics. The physical realization of each has many formal similarities. For example, in both cases, equivalences are realized via renormalization group flow: in the case of derived categories, (boundary) renormalization group flow realizes the mathematical procedure of localization on quasiisomorphisms, and in the case of stacks, worldsheet renormalization group flow realizes presentation-independence. For both, we outline current technical issues and applications.

2025, arXiv (Cornell University)

In this paper we discuss noninvertible topological operators in the context of one-form symmetries and decomposition of two-dimensional quantum field theories, focusing on twodimensional orbifolds with and without discrete torsion. As one component of our analysis, we study the ring of dimension-zero operators in two-dimensional theories exhibiting decomposition. From a commutative algebra perspective, the rings are naturally associated to a finite number of points, one point for each universe in the decomposition. Each universe is canonically associated to a representation, which defines a projector, an idempotent in the ring of dimension-zero operators. We discuss how bulk Wilson lines act as defects bridging universes, and how Wilson lines on boundaries of two-dimensional theories decompose, and compute actions of projectors. We discuss one-form symmetries of the rings, and related properties. We also give general formulas for projection operators, which previously were computed on a case-by-case basis. Finally, we propose a characterization of noninvertible higher-form symmetries in this context in terms of representations. In that characterization, non-isomorphic universes appearing in decomposition are associated with noninvertible one-form symmetries.

2025, arXiv (Cornell University)

In this short review we outline some recent developments in understanding string orbifolds. In particular, we outline the recent observation that string orbifolds do not precisely describe string propagation on quotient spaces, but rather are literally sigma models on objects called quotient stacks, which are closely related to (but not quite the same as) quotient spaces. We show how this is an immediate consequence of definitions, and also how this explains a number of features of string orbifolds, from the fact that the CFT is well-behaved to orbifold Euler characteristics. Put another way, many features of string orbifolds previously considered "stringy" are now understood as coming from the target-space geometry; one merely needs to identify the correct target-space geometry.

2025, Advances in Theoretical and Mathematical Physics

In this paper, we shall describe some correlation function computations in perturbative heterotic strings that generalize B model computations. On the (2,2) locus, correlation functions in the B model receive no quantum corrections, but off the (2,2) locus, that can change. Classically, the (0,2) analogue of the B model is equivalent to the previously discussed (0,2) analogue of the A model, but with the gauge bundle dualizedour generalization of the A model also simultaneously generalizes the B model. The A and B analogues sometimes have different regularizations, however, which distinguish them quantum-mechanically. We discuss how properties of the (2,2) B model, such as the lack of quantum corrections, are realized in (0,2) A model language. In an appendix, we also extensively discuss how the Calabi-Yau condition for the closed string B model (uncoupled to topological gravity) can be weakened slightly, a detail which does not seem to have been covered in the literature previously. That weakening also manifests in the description of the (2,2) B model as a (0,2) A model.

2025

Dirichlet series, or, series of the form D(A, s) = ∞ i=1 a i i s are central objects in complex analysis and analytic number theory. This paper seeks to generalize them by introducing & studying new types of Dirichlet series-quaternionic Dirichlet series, and lattice Dirichlet series. That is, Dirichlet series over quaternions, and, various types of Dirichlet series over lattices.

2025, The Lazenby Twist Law: Memory Spectrum and Topological Dynamics

I introduce a novel topological invariant-the Memory Spectrum-emerging from the behavior of loops in structured 3D bubble spaces. I define memory cohomology classes, construct the Memory Laplacian ∆ m , and prove formal theorems capturing how memory propagates through topological environments. This theory has implications for quantum geometry, categorical topology, and new physical invariants.

2025

While exploring Dirichlet L-functions as part of a final-year project, I stumbled upon a surprising property: a wave function ψp(x) = χ(p)e iγx , where χ is a non-trivial Dirichlet character modulo q, γ is a non-trivial zero of L(s, χ), and p is a prime, produces a discrete Fourier transform ψp(k) with a dominant peak at k ≡ p -1 mod q. This "mirror symmetry" suggests a deep arithmetic structure linking primes to their modular inverses. I formalize this observation with a quantitative conjecture, provide numerical evidence for q = 5, 13, 17, and offer a partial theoretical analysis using Gauss and Kloosterman sums. Potential applications in quantum physics and cryptography are discussed. All data and code are available on Zenodo (DOI to be inserted).