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At least 505 records · Page 28Linked to original sources
arXiv1993

Measuring Small Distances in N=2 Sigma Models

We analyze global aspects of the moduli space of Kähler forms for $N$=(2,2) conformal $σ$-models. Using algebraic methods and mirror symmetry we study extensions of the mathematical notion of length (as specified by a Kähler structure) to conformal field theory and calculate the way in which lengths change as the moduli fields are varied along distinguished paths in the moduli space. We find strong evidence supporting the notion that, in the robust setting of quantum Calabi-Yau moduli space, string theory restricts the set of possible Kähler forms by enforcing ``minimal length'' scales, provided that topology change is properly taken into account. Some lengths, however, may shrink to zero. We also compare stringy geometry to classical general relativity in this context.

hep-th↗
arXiv1995

Summing the Instantons: Quantum Cohomology and Mirror Symmetry in Toric Varieties

We use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety $V$ or a Calabi--Yau hypersurface $M \subset V$. In the linear model the instanton moduli spaces are relatively simple objects and the correlators are explicitly computable; moreover, the instantons can be summed, leading to explicit solutions for both kinds of models. In the case of smooth $V$, our results reproduce and clarify an algebraic solution of the $V$ model due to Batyrev. In addition, we find an algebraic relation determining the solution for $M$ in terms of that for $V$. Finally, we propose a modification of the linear model which computes instanton expansions about any limiting point in the moduli space. In the smooth case this leads to a (second) algebraic solution of the $M$ model. We use this description to prove some conjectures about mirror symmetry, including the previously conjectured ``monomial-divisor mirror map'' of Aspinwall, Greene, and Morrison.

hep-th↗
arXiv1994

Mirror Symmetry for Calabi-Yau Hypersurfaces in Weighted P_4 and Extensions of Landau Ginzburg Theory

Recently two groups have listed all sets of weights (k_1,...,k_5) such that the weighted projective space P_4^{(k_1,...,k_5)} admits a transverse Calabi-Yau hypersurface. It was noticed that the corresponding Calabi-Yau manifolds do not form a mirror symmetric set since some 850 of the 7555 manifolds have Hodge numbers (b_{11},b_{21}) whose mirrors do not occur in the list. By means of Batyrev's construction we have checked that each of the 7555 manifolds does indeed have a mirror. The `missing mirrors' are constructed as hypersurfaces in toric varieties. We show that many of these manifolds may be interpreted as non-transverse hypersurfaces in weighted P_4's, ie, hypersurfaces for which dp vanishes at a point other than the origin. This falls outside the usual range of Landau--Ginzburg theory. Nevertheless Batyrev's procedure provides a way of making sense of these theories.

hep-th↗
arXiv1995

Non-Abelian Monopoles on Four-Manifolds

We present a non-abelian generalization of Witten monopole equations and we analyze the associated moduli problem, which can be regarded as a generalization of Donaldson theory. The moduli space of solutions for SU(2) monopoles on Kähler manifolds is discussed. We also construct, using the Mathai-Quillen formalism, the topological quantum field theory corresponding to the new moduli problem. This theory involves the coupling of topological Yang-Mills theory to topological matter in four dimensions

hep-th↗
arXiv1995

Topological QCD

We study the twisted $N=2$ supersymmetric Yang-Mills theory coupled with the hypermultiplets (TQCD). We suggest that the family of TQCD can be served as a powerful tool for studying the quantum field theoretic properties of the underlying physical theories.

hep-th↗
arXiv1995

Mirror Symmetry and the Moduli Space for Generic Hypersurfaces in Toric Varieties

The moduli dependence of $(2,2)$ superstring compactifications based on Calabi--Yau hypersurfaces in weighted projective space has so far only been investigated for Fermat-type polynomial constraints. These correspond to Landau-Ginzburg orbifolds with $c=9$ whose potential is a sum of $A$-type singularities. Here we consider the generalization to arbitrary quasi-homogeneous singularities at $c=9$. We use mirror symmetry to derive the dependence of the models on the complexified Kähler moduli and check the expansions of some topological correlation functions against explicit genus zero and genus one instanton calculations. As an important application we give examples of how non-algebraic (``twisted'') deformations can be mapped to algebraic ones, hence allowing us to study the full moduli space. We also study how moduli spaces can be nested in each other, thus enabling a (singular) transition from one theory to another. Following the recent work of Greene, Morrison and Strominger we show that this corresponds to black hole condensation in type II string theories compactified on Calabi-Yau manifolds.

hep-th↗
arXiv1995

Higher Grading Generalisations of the Toda Systems

In the present paper we obtain some integrable generalisations of the Toda system generated by flat connection forms taking values in higher ${\bf Z}$--grading subspaces of a simple Lie algebra, and construct their general solutions. One may think of our systems as describing some new fields of the matter type coupled to the standard Toda systems. This is of special interest in nonabelian Toda theories where the latter involve black hole target space metrics. We also give a derivation of our conformal system on the base of the Hamiltonian reduction of the WZNW model; and discuss a relation between abelian and nonabelian systems generated by a gauge transformation that maps the first grading description to the second. The latter involves grades larger than one.

hep-th↗
arXiv1996

Algebras, BPS States, and Strings

We clarify the role played by BPS states in the calculation of threshold corrections of D=4, N=2 heterotic string compactifications. We evaluate these corrections for some classes of compactifications and show that they are sums of logarithmic functions over the positive roots of generalized Kac-Moody algebras. Moreover, a certain limit of the formulae suggests a reformulation of heterotic string in terms of a gauge theory based on hyperbolic algebras such as $E_{10}$. We define a generalized Kac-Moody Lie superalgebra associated to the BPS states. Finally we discuss the relation of our results with string duality.

hep-th↗
arXiv1995

BPS States, String Duality, and Nodal Curves on K3

We describe the counting of BPS states of Type II strings on K3 by relating the supersymmetric cycles of genus $g$ to the number of rational curves with $g$ double points on K3. The generating function for the number of such curves is the left-moving partition function of the bosonic string.

hep-th↗
arXiv1996

Mirror Symmetry is T-Duality

It is argued that every Calabi-Yau manifold $X$ with a mirror $Y$ admits a family of supersymmetric toroidal 3-cycles. Moreover the moduli space of such cycles together with their flat connections is precisely the space $Y$. The mirror transformation is equivalent to T-duality on the 3-cycles. The geometry of moduli space is addressed in a general framework. Several examples are discussed.

hep-th↗
arXiv1995

Instantons and affine Lie algebras

Various constructions of the affine Lie algebra action on the homology group of moduli spaces of instantons on 4-manifolds are discussed. The analogy between the local-global principle and the role of mass is also explained. The detailed proofs are given in separated papers \cite{Na-algebra,Na-Hilbert}.

alg-geom↗
arXiv1996

Algebraic cycles and topology of real Enriques surfaces

For a real Enriques surface Y we prove that every homology class in H_1(Y(R), Z/2) can be represented by a real algebraic curve if and only if all connected components of Y(R) are orientable. Furthermore, we give a characterization of real Enriques surfaces which are Galois-Maximal and/or Z-Galois-Maximal and we determine the Brauer group of any real Enriques surface.

alg-geom↗
arXiv1995

Reflection groups in hyperbolic spaces and the denominator formula for Lorentzian Kac--Moody Lie algebras

This is a continuation of our "Lecture on Kac--Moody Lie algebras of the arithmetic type" \cite{25}. We consider hyperbolic (i.e. signature $(n,1)$) integral symmetric bilinear form $S:M\times M \to {\Bbb Z}$ (i.e. hyperbolic lattice), reflection group $W\subset W(S)$, fundamental polyhedron $\Cal M$ of $W$ and an acceptable (corresponding to twisting coefficients) set $P({\Cal M})\subset M$ of vectors orthogonal to faces of $\Cal M$ (simple roots). One can construct the corresponding Lorentzian Kac--Moody Lie algebra ${\goth g}={\goth g}^{\prime\prime}(A(S,W,P({\Cal M})))$ which is graded by $M$. We show that $\goth g$ has good behavior of imaginary roots, its denominator formula is defined in a natural domain and has good automorphic properties if and only if $\goth g$ has so called {\it restricted arithmetic type}. We show that every finitely generated (i.e. $P({\Cal M})$ is finite) algebra ${\goth g}^{\prime\prime}(A(S,W_1,P({\Cal M}_1)))$ may be embedded to ${\goth g}^{\prime\prime}(A(S,W,P({\Cal M})))$ of the restricted arithmetic type. Thus, Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type is a natural class to study. Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type have the best automorphic properties for the denominator function if they have {\it a lattice Weyl vector $ρ$}. Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type with generalized lattice Weyl vector $ρ$ are called {\it elliptic}

alg-geom↗
arXiv1995

Hodge groups of abelian varieties with purely multiplicative reduction

The main result of the paper is that if $A$ is an abelian variety over a subfield $F$ of ${\bold C}$, and $A$ has purely multiplicative reduction at a discrete valuation of $F$, then the Hodge group of $A$ is semisimple. Further, we give necessary and sufficient conditions for the Hodge group to be semisimple. We obtain bounds on certain torsion subgroups for abelian varieties which do not have purely multiplicative reduction at a given discrete valuation, and therefore obtain bounds on torsion for abelian varieties, defined over number fields, whose Hodge groups are not semisimple.

alg-geom↗
arXiv1996

Secants of Abelian Varieties, Theta functions, and Soliton Equations

This paper is a survey on relations between secant identities and soliton equations and applications of soliton equations to problems of algebraic geometry, i.e., the Riemann-Schottky problem and its analogues. A short introduction into the analytic theory of theta functions is also given.

alg-geom↗
arXiv1996

The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras

We find automorphic corrections for the Lorentzian Kac--Moody algebras with the simplest generalized Cartan matrices of rank 3: A_{1,0} = 2 0 -1 0 2 -2 -1 -2 2 and A_{1,I} = 2 -2 -1 -2 2 -1 -1 -1 2 For A_{1,0} this correction is given by the Igusa Sp_4(Z)-modular form χ_{35} of weight 35, and for A_{1,I} by a Siege modular form of weight 30 with respect to a 2-congruence subgroup. We find infinite product or sum expansions for these forms. Our method of construction of χ_{35} leads to the direct construction of Siegel modular forms by infinite product expansions, whose divisors are the Humbert surfaces with fixed discriminants. Existence of these forms was proved by van der Geer in 1982 using some geometrical consideration. We announce a list of all hyperbolic symmetric generalized Cartan matrices A of rank 3 such that A has elliptic or parabolic type, A has a lattice Weyl vector, and A contains the affine submatrix \tilde{A}_1.

alg-geom↗