Vector Bundle Moduli

We give the general presciption for calculating the number of moduli of irreducible, stable U(n) holomorphic vector bundles with positive spectral covers over elliptically fibered Calabi–Yau threefolds. Explicit results are presented for Hirzebruch base surfaces B = F _r. Vector bundle moduli appear as gauge singlet scalar fields in the effective low-energy actions of heterotic superstrings and heterotic M-theory.     

Notes on Certain (0,2) Correlation Functions

In this paper we shall describe some correlation function computations in perturbative heterotic strings that, for example, in certain circumstances can lend themselves to a heterotic generalization of quantum cohomology calculations. Ordinary quantum chiral rings reflect worldsheet instanton corrections to correlation functions involving products of elements of Dolbeault cohomology groups on the target space. The heterotic generalization described here involves computing worldsheet instanton corrections to correlation functions defined by products of elements of sheaf cohomology groups. One must not only compactify moduli spaces of rational curves, but also extend a sheaf (determined by the gauge bundle) over the compactification, and linear sigma models provide natural mechanisms for doing both. Euler classes of obstruction bundles generalize to this language in an interesting way.     

Open M5-branes

We show how, in heterotic M theory, an M5-brane in the 11-dimensional bulk may end on an "M9-brane" boundary, the M5-brane boundary being a Yang-monopole 4-brane. This possibility suggests various novel 5-brane configurations of heterotic M theory, in particular, a static M5-brane suspended between the two M9-brane boundaries, for which we find the asymptotic heterotic supergravity solution.     

Compactifications of F-theory on Calabi-Yau threefolds (II)

We continue our study of compactifications of F-theory on Calabi-Yau threefolds. We gain more insight into F-theory duals of heterotic strings and provide a recipe for building F-theory duals for arbitrary heterotic compactifications on elliptically fibered manifolds. As a byproduct we find that string/string duality in six dimensions gets mapped to fiber/base exchange in F-theory. We also construct a number of new N = 1, d = 6 examples of F-theory vacua and study transitions among them. We find that some of these transition points correspond upon further compactification to 4 dimensions to transitions through analogues of Argyres-Douglas points of N = 2 moduli. A key idea in these transitions is the notion of classifying (0,4) fivebranes of heterotic strings.     

Instanton strings and hyper-Kähler geometry

We discuss two-dimensional sigma models on moduli spaces of instantons on K3 surfaces. These N = (4, 4) superconformal field theories describe the near-horizon dynamics of the D1-D5-brane system and are dual to string theory on AdS³. We derive a precise map relating the moduli of the K3 type 1113 string compactification to the moduli of these conformal field theories and the corresponding classical hyper-Kahler geometry. We conclude that in the absence of background gauge fields, the metric on the instanton moduli spaces degenerates exactly to the orbifold symmetric product of K3. Turning on a self-dual NS B-field deforms this symmetric product to a manifold that is diffeomorphic to the Hilbert scheme. We also comment on the mathematical applications of string duality to the global issues of deformations of hyper-Kähler manifolds.     

Heterotic p-branes from massive sigma models

We explicitly construct massive (0,4) supersymmetric ADHM sigma models which have heterotic p-brane solitons as their conformal fixed points. These yield the familiar gauge 5-brane and a new 1-brane solution which preserve half and a quarter of the space-time supersymmetry, respectively. We also discuss an analogous construction for the type 11 NS-NS p-branes using (4,4) supersymmetric models.     

Are (0, 2) models string miracles?

String theory exhibits a number of string miracles. These are “accidental” relations between couplings in the low-energy effective action which cannot be understood in terms of the symmetries of that action. It is shown that the existence of (0, 2) models need not be a string miracle, but can often be understood in terms of a discrete R symmetry. This understanding points to many new (0, 2) models, including some with gauge group SU(3) × SU(2) × U(1). Discrete R symmetries also protect some massless particles from acquiring a mass at special points in the moduli space of the conformal field theory. From a two dimensional point of view, this can be understood in terms of index theorems not only in the R but also in the NS sector. An important tool in the study of string compactification and conformal field theory is the perturbative expansion around a conformal field theory. We prove that the expansion of the space time superpotential around a space-time supersymmetric vacuum does not miss non-perturbative effects (as long as we do not expand around a boundary of moduli space). This shows that certain results which have been obtained to all orders in this expansion are exact. Also, it shows that the construction of the (0, 2) models in perturbation theory around (2, 2) models is exact as well.     

Supersymmetry breaking with zero vacuum energy in M-theory flux compactifications

An attractive mechanism to break supersymmetry in vacua with zero vacuum energy arose in E(8) x E(8) heterotic models with hidden sector gaugino condensate. An H flux balances the exponentially small condensate on shell and fixes the complex structure moduli. At quantum level this balancing is, however, obstructed by the quantization of the H flux. We show that the warped flux compactification background in heterotic M theory can solve this problem through a warp-factor suppression of the integer flux relative to the condensate. We discuss the suppression mechanism both in the M theory and the four-dimensional effective theory and provide a derivation of the condensate's superpotential which is free of delta-function squared ambiguities.     

Destabilizing the gaugino condensate in modular invariant supergravity

We investigate the stability of the hidden sector gaugino condensate in a SL(2,Z)-invariant supergravity model inspired by E₈E₈ heterotic string, using the chiral superfield formalism. We calculate Planck-suppressed corrections to the “truncated approximation” for the condensate value and scalar potential. A transition to a phase with zero condensate occurs near special points in moduli space and at large compactification radius. We discuss the implications for the T-modulus dependence of supersymmetry-breaking.     

Symbolic Computation of Extended Jacobian Elliptic Function Algorithm for Nonlinear Differential-Different Equations

The Jacobian elliptic function expansion method for nonlinear differential-different equations and its algorithm are presented by using some relations among ten Jacobian elliptic functions and successfully construct more new exact doubly-periodic solutions of the integrable discrete nonlinear Schr ödinger equation. When the modulous m→1 or 0, doubly-periodic solutions degenerate to solitonic solutions including bright soliton, dark soliton, new solitons as well as trigonometric function solutions.     

Non-perturbative superpotentials in string theory

The non-perturbative superpotential can be effectively calculated in M-theory compactification to three dimensions on a Calabi-Yau four-fold X. For certain X, the superpotential is identically zero, while for other X, a non-perturbative superpotential is generated. Using F-theory, these results carry over to certain Type IIB and heterotic string compactifications to four dimensions with N = 1 supersymmetry. In the heterotic string case, the non-perturbative superpotential can be interpreted as coming from space-time and world-sheet instantons; in many simple cases contributions come only from finitely many values of the instanton numbers.     

Very large intermediate scales in three-generation models

The issue of large intermediate scales is addressed in a class of three-generation models based on the best-known Calabi-Yau compactification of the heterotic string. The renormalisation group equations for this class are studied to determine the scale, μ≡M_I, at which the first “intermediate-scale” symmetry breaking occurs. Throughout most of the parameter space specifying the models, M_I is found to be extremely close to the compactification scale, M_c, as is required by phenomenology. This is due to the large number of fields which are massless below M_c. Depending on unknown normalisation factors, different directions of symmetry breaking are possible, including the one thus far preferred on phenomenological grounds.     

A Gravitational Potential with Extra-dimensions and Spin Effects in Hadronic Reactions

The impact of the KK-modes in d-brane models of gravity with large compactification radii and TeV-scale quantum gravity on the hadronic potential at small impact parameters is examined. The effects of the gravitational hadron form factors obtained from the hadron generalized parton distributions (GPDs) on the behavior of the gravitational potential and the possible spin correlation effects are also analysed.     

Discrete symmetries and the complex structure of Calabi-Yau manifolds

We show how the discrete symmetries, which may be present after Calabi-Yau compactification for specific choices of the complex structure, extend to the h_2,1 moduli — the scalar fields whose vacuum expectation values determine the complex structure. This allows us to determine much about the coupling of the moduli and hence the energetically favoured complex structure. The discrete symmetry transformation properties of the moduli are worked out in detail for a three-generation Calabi-Yau model and it is shown how minimization of the effective potential involving these fields selects the complex structure which leaves unbroken a set of discrete symmetries. The phenomenological implications of these symmetries are briefly discussed.     

On the existence of non‐supersymmetric black hole attractors for two‐parameter Calabi‐Yau's and attractor equations

We look for possible nonsupersymmetric black hole attractor solutions for type II compactification on (the mirror of) CY₃(2,128) expressed as a degree‐12 hypersurface in WCP ⁴[1,1,2,2,6]. In the process, (a) for points away from the conifold locus, we show that the existence of a non‐supersymmetric attractor along with a consistent choice of fluxes and extremum values of the complex structure moduli, could be connected to the existence of an elliptic curve fibered over C ⁸ which may also be “arithmetic” (in some cases, it is possible to interpret the extremization conditions for the black‐hole superpotential as an endomorphism involving complex multiplication of an arithmetic elliptic curve), and (b) for points near the conifold locus, we show that existence of non‐supersymmetric black‐hole attractors corresponds to a version of A₁‐singularity in the space Image( Z ⁶→ R ²/ Z ₂ (↪ R ³)) fibered over the complex structure moduli space. The (derivatives of the) effective black hole potential can be thought of as a real (integer) projection in a suitable coordinate patch of the Veronese map: CP ⁵→ CP ²⁰, fibered over the complex structure moduli space. We also discuss application of Kallosh's attractor equations (which are equivalent to the extremization of the effective black‐hole potential) for nonsupersymmetric attractors and show that (a) for points away from the conifold locus, the attractor equations demand that the attractor solutions be independent of one of the two complex structure moduli, and (b) for points near the conifold locus, the attractor equations imply switching off of one of the six components of the fluxes. Both these features are more obvious using the attractor equations than the extremization of the black hole potential.     

用推广的Jacobian椭圆函数方法解离散的mKdV格子

本文用推广的Jacobian椭圆函数方法求解了离散的mKdV格子，得到了Jacobian椭圆函数双周期解，当模取1时，可得到钟型孤波解和冲击型孤波解。     

Strength functions for fragmented doorway states

Coupling a strongly excited “doorway state” to weak “hallway states” distributes its strength into micro-resonances seen in differential cross sections taken with very good energy resolution. The distribution of strength is shown to be revealed by reduced widths of the K-matrix rather than by the imaginary part of poles of the S-matrix. Different strength functions (SF) constructed by averaging the K-matrix widths are then investigated to determine their dependences on energy and on parameters related to averages of microscopic matrix elements. A new sum rule on the integrated strength of these SF is derived and used to show that different averaging procedures actually distribute the strength differently. Finally, it is shown that the discontinuous summed strength defines spreading parameters for the doorway state only in strong coupling, where it approximates the indefinite integral of the continuous SF of MacDonald-Mekjian-Kerman-De Toledo Piza. A new method of “parametric continuation” is used to relate a discontinuous sliding box-average, or a finite sum, of discrete terms to a continuous function.