40162216
Azumaya noncommutative geometry and D-branes: An Origin of the master nature of D-branes
In this lecture I review how a matrix/Azumaya-type noncommutative geometry arises for D-branes in string theory and how such a geometry serves as an origin of the master nature of D-branes; and then highlight an abundance conjecture on D0-brane resolutions of singularities that is extracted and purified from a work of Douglas and Moore in 1996. A conjectural relation of our setting with `D-geometry' in the sense of Douglas is also given. The lecture is based on a series of works on D-branes with Shing-Tung Yau, and in part with Si Li and Ruifang Song.
14489670
Nontrivial Azumaya noncommutative schemes, morphisms therefrom, and their extension by the sheaf of algebras of differential operators: D-branes in a B-field background a la Polchinski-Grothendieck Ansatz
In this continuation of [L-Y1], [L-L-S-Y], [L-Y2], and [L-Y3] (arXiv:0709.1515 [math.AG], arXiv:0809.2121 [math.AG], arXiv:0901.0342 [math.AG], arXiv:0907.0268 [math.AG]), we study D-branes in a target-space with a fixed $B$-field background $(Y,\alpha_B)$ along the line of the Polchinski-Grothendieck Ansatz, explained in [L-Y1] and further extended in the current work. We focus first on the gauge-field-twist effect of $B$-field to the Chan-Paton module on D-branes. Basic properties of the moduli space of D-branes, as morphisms from Azumaya schemes with a twisted fundamental module to $(Y,\alpha_B)$, are given. For holomorphic D-strings, we prove a valuation-criterion property of this moduli space. The setting is then extended to take into account also the deformation-quantization-type noncommutative geometry effect of $B$-field to both the D-brane world-volume and the superstring target-space(-time) $Y$. This brings the notion of twisted ${\cal D}$-modules that are realizable as twisted locally-free coherent modules with a flat connection into the study. We use this to realize the notion of both the classical and the quantum spectral covers as morphisms from Azumaya schemes with a fundamental module (with a flat connection in the latter case) in a very special situation. The 3rd theme (subtitled "Sharp vs. Polchinski-Grothendieck") of Sec. 2.2 is to be read with the work [Sh3] (arXiv:hep-th/0102197) of Sharp while Sec. 5.2 (subtitled less appropriately "Dijkgraaf-Holland-Su{\l}kowski-Vafa vs. Polchinski-Grothendieck") is to be read with the related sections in [D-H-S-V] (arXiv:0709.4446 [hep-th]) and [D-H-S] (arXiv:0810.4157 [hep-th]) of Dijkgraaf, Hollands, Su{\l}kowski, and Vafa.
18139263
On A-twisted moduli stack for curves from Witten's gauged linear sigma models
Witten’s gauged linear sigma model [Wi1] is one of the universal frameworks or structures that lie behind stringy dualities. Its A-twisted moduli space at genus 0 case has been used in the Mirror Principle [L-L-Y] that relates Gromov-Witten invariants and mirror symmetry computations. In this paper the A-twisted moduli stack for higher genus curves is defined and systemically studied. It is proved that such a moduli stack is an Artin stack. For genus 0, it has the A-twisted moduli space of [MP] as the coarse moduli space. The detailed proof of the regularity of the collapsing morphism by Jun Li in [L-L-Y: I and II] can be viewed as a natural morphism from the moduli stack of genus 0 stable maps to the A-twisted moduli stack at genus 0. Due to the technical demand of stacks to physicists and the conceptual demand of supersymmetry to mathematicians, a brief introduction of each topic that is most relevant to the main contents of this paper is given in the beginning and the appendix respectively. Themes for further study are listed in the end.
116912536
Algebraic cobordism of filtered vector bundles on varieties: Notes on a work of Lee and Pandharipande
The construction of double point cobordism groups of vector bundles on varieties in the work [Lee-P] (arXiv:1002.1500 [math.AG]) of Yuan-Pin Lee and Rahul Pandharipande gives immediately double point cobordism groups of filtered vector bundles on varieties. We note also that among the four basic operations – direct sum, tensor product, dual, and Hom – on vector bundles on varieties, only taking dual is compatible with double point cobordisms of vector bundles on varieties in general, by a demonstration on an example of vector bundles on Calabi-Yau 3-folds. A question on refined and/or higher algebraic cobordisms of vector bundles on varieties is posed in the end.
341730
Target-space duality between simple compact Lie groups and Lie algebras under the Hamiltonian formalism: I. Remnants of duality at the classical level
AbstractIt has been suggested that a possible classical remnant of the phenomenon of target-space duality (T-duality) would be the equivalence of the classical string Hamiltonian systems. Given a simple compact Lie groupG with a bi-invariant metric and a generating function Γ suggested in the physics literature, we follow the above line of thought and work out the canonical transformation Φ generated by Γ together with an Ad-invariant metric and a B-field on the associated Lie algebra
$$\mathfrak{g}$$
 ofG so thatG and
$$\mathfrak{g}$$
 form a string target-space dual pair at the classical level under the Hamiltonian formalism. In this article, some general features of this Hamiltonian setting are discussed. We study properties of the canonical transformation Φ including a careful analysis of its domain and image. The geometry of the T-dual structure on
$$\mathfrak{g}$$
 is lightly touched. We leave the task of tracing back the Hamiltonian formalism at the quantum level to the sequel of this paper.
116949875
A Degeneration formula of Gromov-Witten invariants with respect to a curve class for degenerations from blow-ups
In two very detailed, technical, and fundamental works, Jun Li constructed a theory of Gromov-Witten invariants for a singular scheme of the gluing form Y1 ∪D Y2 that arises from a degeneration W/A 1 and a theory of relative Gromov-Witten invariants for a codimension-1 relative pair (Y,D). As a summit, he derived a degeneration formula that relates a finite summation of the usual Gromov-Witten invariants of a general smooth fiber Wt of W/A 1 to the Gromov-Witten invariants of the singular fiber W0 = Y1 ∪D Y2 via gluing the relative pairs (Y1,D) and (Y2,D). The finite sum mentioned above depends on a relative ample line bundle H on W/A 1 . His theory has already applications to string theory and mathematics alike. For other new applications of Jun Li’s theory, one needs a refined degeneration formula that depends
119180642
More on volume dependence of spectral weight function
Spectral weight functions are easily obtained from two-point correlation functions and they might be used to distinguish single-particle from multi-particle states in a finite-volume lattice calculation, a problem crucial for many lattice QCD simulations. In previous studies, it is shown that the spectral weight function for a broad resonance shares the typical volume dependence of a two-particle scattering state i.e. proportional to $1/L^3$ in a large cubic box of size $L$ while the narrow resonance case requires further investigation. In this paper, a generalized formula is found for the spectral weight function which incorporates both narrow and broad resonance cases. Within L\"uscher's formalism, it is shown that the volume dependence of the spectral weight function exhibits a single-particle behavior for a extremely narrow resonance and a two-particle behavior for a broad resonance. The corresponding formulas for both $A^+_1$ and $T^-_1$ channels are derived. The potential application of these formulas in the extraction of resonance parameters are also discussed.
119253117
SPACETIME NONCOMMUTATIVE EFFECT ON BLACK HOLE AS PARTICLE ACCELERATORS
We study the spacetime noncommutative effect on black hole as particle accelerators and, find that the particles falling from infinity with zero velocity cannot collide with unbound energy, either near the horizon or on the prograde ISCO when the noncommutative Kerr black hole is exactly extremal. Our results also show that the bigger of the spinning black hole's mass is the higher of center of mass energy that the particles obtain. For small and medium noncommutative Schwarzschild black hole, the collision energy depends on the black hole's mass.
124563066
Baryonic Isgur-Wise functions in large-N-c heavy quark effective theory
Large-N-c relations among baryonic Isgur-Wise functions appearing at the order of 1/m(Q) are analyzed. An application to Omega(b) -> Omega(c) weak decays is given.
118542460
Coherent Neutron Scattering in Polycrystalline Deuterium and its Implications for Ultracold Neutron Production
This paper presents a calculation of the neutron cross-sections in solid materials (used in practical neutron sources) with a large coherent scattering contribution. In particular, the dynamic structure function S(Q, !) of polycrystalline ortho-D2 is evaluated using a Monte-Carlo calculation that performs an average over scattering angles relative to crystal axes in random orientations. This method uses an analytical dispersion function with force constants derived from neutron scattering data of single crystal D2 in the framework of an axially symmetric force tensor. The resulting two dimensional map of S(Q, !) captures details of the phonon branches as well as the molecular rotations, that can be compared directly to data from inelastic neutron scattering on polycrystalline D2. This high resolution information is used to calculate the absolute cross-sections of production and upscattering loss of ultracold neutron (UCN). The resulting scattering cross-sections are significantly different, especially for UCN upscattering, from the previous predictions using the approach centered on the incoherent approximation.
124850278
Decay Widths of Excited Heavy Mesons from Quantum Chromodynamics Sum Rules in the Infinite Mass Limit
The widths for pionic decays of the lowest two excited doublets (0(+), 1(+)) and (1(+), 2(+)) of heavy mesons are studied with quantum Chromodynamics sum rules in the heavy quark effective theory. The interpolating currents used allow us to distinguish the two 1(+) states without contamination in the infinite mass Limit. The soft pion approximation is not used in the calculations.