Dbrane Phase Transitions and Monodromy in K-theory

Majumder and Sen have given an explicit construction of a first order phase transition in a non-supersymmetric system of Dbranes that occurs when the B field is varied. We show that the description of this transition in terms of K-theory involves a bundle of K groups of non-commutative algebras over the Kahler cone with nontrivial monodromy. Thus the study of monodromy in K groups associated with quantized algebras can be used to predict the phase structure of systems of (non-supersymmetric) Dbranes.Comment: 8 pages, RevTeX, 1 figur

Deformation Quantization in Singular Spaces

We present a method of quantizing analytic spaces $X$ immersed in an arbitrary smooth ambient manifold $M$. Remarkably our approach can be applied to singular spaces. We begin by quantizing the cotangent bundle of the manifold $M$. Using a super-manifold framework we modify the Fedosov construction in a way such that the $\star$-product of the functions lifted from the base manifold turns out to be the usual commutative product of smooth functions on $M$. This condition allows us to lift the ideals associated to the analytic spaces on the base manifold to form left (or right) ideals on (\mc{O}_{\Omega^1 M}[[\hbar]],\starl) in a way independent of the choice of generators and leading to a finite set of PDEs defining the functions in the quantum algebra associated to $X$. Some examples are included.Comment: 14 page

Universality of Fedosov's Construction for Star Products of Wick Type on Pseudo-K\"ahler Manilfolds

In this paper we construct star products on a pseudo-K\"ahler manifold $(M,\omega,I)$ using a modification of the Fedosov method based on a different fibrewise product similar to the Wick product on $\mathbb C^n$. In a first step we show that this construction is rich enough to obtain star products of every equivalence class by computing Deligne's characteristic class of these products. Among these products we uniquely characterize the ones which have the additional property to be of Wick type which means that the bidifferential operators describing the star products only differentiate with respect to holomorphic directions in the first argument and anti-holomorphic directions in the second argument. These star products are in fact strongly related to star products with separation of variables introduced and studied by Karabegov. This characterization gives rise to special conditions on the data that enter the Fedosov procedure. Moreover, we compare our results that are based on an obviously coordinate independent construction to those of Karabegov that were obtained by local considerations and give an independent proof of the fact that star products of Wick type are in bijection to formal series of closed two-forms of type $(1,1)$ on $M$. Using this result we finally succeed in showing that the given Fedosov construction is universal in the sense that it yields all star products of Wick type on a pseudo-K\"ahler manifold.Comment: terminology corrected, typos removed, appendix adde

One-dimensional Chern-Simons theory and the A^\hat{A} genus

We construct a Chern-Simons gauge theory for dg Lie and L-infinity algebras on any one-dimensional manifold and quantize this theory using the Batalin-Vilkovisky formalism and Costello's renormalization techniques. Koszul duality and derived geometry allow us to encode topological quantum mechanics, a nonlinear sigma model of maps from a 1-manifold into a cotangent bundle T*X, as such a Chern-Simons theory. Our main result is that the partition function of this theory is naturally identified with the A-genus of X. From the perspective of derived geometry, our quantization construct a volume form on the derived loop space which can be identified with the A-class.Comment: 61 pages, figures, final versio

Stability of heterogeneous parallel-bond adhesion clusters under static load

Adhesion interactions mediated by multiple bond types are relevant for many biological and soft matter systems, including the adhesion of biological cells and functionalized colloidal particles to various substrates. To elucidate advantages and disadvantages of multiple bond populations for the stability of heterogeneous adhesion clusters of receptor-ligand pairs, a theoretical model for a homogeneous parallel adhesion bond cluster under constant loading is extended to several bond types. The stability of the entire cluster can be tuned by changing densities of different bond populations as well as their extensional rigidity and binding properties. In particular, bond extensional rigidities determine the distribution of total load to be shared between different sub-populations. Under a gradual increase of the total load, the rupture of a heterogeneous adhesion cluster can be thought of as a multistep discrete process, in which one of the bond sub-populations ruptures first, followed by similar rupture steps of other sub-populations or by immediate detachment of the remaining cluster. This rupture behavior is qualitatively independent of involved bond types, such as slip and catch bonds. Interestingly, an optimal stability is generally achieved when the total cluster load is shared such that loads on distinct bond populations are equal to their individual critical rupture forces. We also show that cluster heterogeneity can drastically affect cluster lifetime.Comment: 11 pages, 8 figure

BRST quantization of quasi-symplectic manifolds and beyond

We consider a class of \textit{factorizable} Poisson brackets which includes almost all reasonable Poisson structures. A particular case of the factorizable brackets are those associated with symplectic Lie algebroids. The BRST theory is applied to describe the geometry underlying these brackets as well as to develop a deformation quantization procedure in this particular case. This can be viewed as an extension of the Fedosov deformation quantization to a wide class of \textit{irregular} Poisson structures. In a more general case, the factorizable Poisson brackets are shown to be closely connected with the notion of $n$-algebroid. A simple description is suggested for the geometry underlying the factorizable Poisson brackets basing on construction of an odd Poisson algebra bundle equipped with an abelian connection. It is shown that the zero-curvature condition for this connection generates all the structure relations for the $n$-algebroid as well as a generalization of the Yang-Baxter equation for the symplectic structure.Comment: Journal version, references and comments added, style improve

The Generalized Moyal Nahm and Continuous Moyal Toda Equations

We present in detail a class of solutions to the $4D SU(\infty)$ Moyal Anti Self Dual Yang Mills equations that are related to $reductions$ of the generalized Moyal Nahm quations using the Ivanova-Popov ansatz. The former yields solutions to the ASDYM/SDYM equations for arbitary gauge groups. A further dimensional reduction yields solutions to the Moyal Anti Self Dual Gravitational equations. The Self Dual Yang Mills /Self Dual Gravity case requires a separate study. SU(2) and $SU(\infty)$ (continuous) Moyal Toda equations are derived and solutions to the latter equations in $implicit$ form are proposed via the Lax-Brockett double commutator formalism . An explicit map taking the Moyal heavenly form (after a rotational Killing symmetry reduction) into the SU(2) Moyal Toda field is found. Finally, the generalized Moyal Nahm equations are conjectured that contain the continuous $SU(\infty)$ Moyal Toda equation after a suitable reduction. Three different embeddings of the three different types of Moyal Toda equations into the Moyal Nahm equations are discussed.Comment: Revised TEX file. 31 pages. The Legendre transform between the Moyal heavenly form and the Moyal Toda field is solve