All Stable Characteristic Classes of Homological Vector Fields

An odd vector field Q on a supermanifold M is called homological, if Q ² = 0. The operator of Lie derivative L _Q makes the algebra of smooth tensor fields on M into a differential tensor algebra. In this paper, we give a complete classification of certain invariants of homological vector fields called characteristic classes. These take values in the cohomology of the operator L _Q and are represented by Q-invariant tensors made up of the homological vector field and a symmetric connection on M by means of the algebraic tensor operations and covariant differentiation.     

A Formal Model of Berezin-Toeplitz Quantization

We give a new construction of symbols of the differential operators on the sections of a quantum line bundle L over a Kähler manifold M using the natural contravariant connection on L. These symbols are the functions on the tangent bundle TM polynomial on fibres. For high tensor powers of L, the asymptotics of the composition of these symbols leads to the star product of a deformation quantization with separation of variables on TM corresponding to some pseudo-Kähler structure on TM. Surprisingly, this star product is intimately related to the formal symplectic groupoid with separation of variables over M. We extend the star product on TM to generalized functions supported on the zero section of TM. The resulting algebra of generalized functions contains an idempotent element which can be thought of as a natural counterpart of the Bergman projection operator. Using this idempotent, we define an algebra of Toeplitz elements and show that it is naturally isomorphic to the algebra of Berezin-Toeplitz deformation quantization on M.     

A model for FL and R = FL/FT at low x and low Q2

A model for the longitudinal structure function F_L at low x and low Q² is presented, which includes the kinematical constraint F_L ～ Q⁴ as Q² → 0. It is based on the photon-gluon fusion mechanism suitably extrapolated to the region of low Q². The contribution of quarks having limited transverse momentum is treated phenomenologically assuming that it is described by the soft pomeron exchange mechanism. The ratio R = F_L/(F₂ ? F_L), with the F₂ appropriately extrapolated to the region of low Q², is also discussed.     

F L structure function and heavy quark contributions

In this paper we obtain the heavy-quark contribution to the longitudinal structure functions F _L (x, Q ²). Since F _L structure functions contains rather large heavy flavor contributions in the small x region, we need to use the massive operator matrix elements, which contribute to the heavy flavor Wilson coefficients in unpolarized deeply inelastic scattering in the region Q ²?>?>?m ². The method of QCD analysis, based on the Jacobi polynomials method, is also described. Our results for longitudinal structure function are in good agreement with the available experimental data.     

Nilpotency ofQ BRST and background field equations

Following the method of Fradkin and Vialkovsky, we study the BRST charge operator.Q _BRST , for the closed bosonic string coupled to the background of its massless states. We show that the requirement of nilpotency ofQ _BRST gives rise to the correct equations for the background fields.     

BRS cohomology in string theory and the no-ghost theorem

It is shown that the operator counting the number of non-transverse modes of the bosonic string in 26 dimensions can be expressed as the anticommutator of the BRS charge Q with another operator. As a result it is easy to exhibit the cohomology of Q and express the transverse state projection operator of Brink and Olive in terms of Q.     

A Path Integral Approach¶to the Kontsevich Quantization Formula

We give a quantum field theory interpretation of Kontsevich's deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory. Its Batalin–Vilkovisky quantization yields a superconformal field theory. The associativity of the star product, and more generally the formality conjecture can then be understood by field theory methods. As an application, we compute the center of the deformed algebra in terms of the center of the Poisson algebra. Received: 10 March 1999 / Accepted: 30 January 2000     

Tensor constructions of open string theories (I) foundations

The possible tensor constructions of open string theories are analyzed from first principles. To this end the algebraic framework of open string field theory is clarified, including the role of the homotopy associative A_∞ algebra, the odd symplectic structure, cyclicity, star conjugation, and twist. It is also shown that two string theories are off-shell equivalent if the corresponding homotopy associative algebras are homotopy equivalent in a strict sense.It is demonstrated that a homotopy associative star algebra with a compatible even bilinear form can be attached to an open string theory. If this algebra does not have a space-time interpretation, positivity and the existence of a conserved ghost number require that its cohomology is at degree zero, and that it has the structure of a direct sum of full matrix algebras. The resulting string theory is shown to be physically equivalent to a string theory with a familiar open string gauge group.     

Hypercontractivity for Semigroups of Unital Qubit Channels

Hypercontractivity is proved for products of qubit channels that belong to self-adjoint semigroups. The hypercontractive bound gives necessary and sufficient conditions for a product of the form ${e^{-t_1 H_1}\otimes \cdots \otimes e^{- t_n H_n}}$ to be a contraction from L ^p to L ^q , where L ^p is the algebra of 2 ⁿ -dimensional matrices equipped with the normalized Schatten norm, and each generator H _j is a self-adjoint positive semidefinite operator on the algebra of 2-dimensional matrices. As a particular case the result establishes the hypercontractive bound for a product of qubit depolarizing channels.     

A convergent star product for linear Poisson structures

An explicit star product ⋆ _α ^Γ on the dual of a general Lie algebra equipped with the linear Poisson bracket is constructed. An equivalence operator between this star product and the Kontsevich star product in [K₁] is given and diverse properties of the star product ⋆ _α ^Γ are studied. It is also proved that the star product ⋆ _α ^Γ provides a convergent deformation quantization in the sense of Rieffel [R₁].     

A Characterization of Vertex Operator Algebra {L(\frac{1}{2},0)\otimes L(\frac{1}{2},0)}

We study a simple, rational and C ₂-cofinite vertex operator algebra whose weight 1 subspace is zero, the dimension of weight 2 subspace is greater than or equal to 2 and with c = c? = 1. Under some additional conditions it is shown that such a vertex operator algebra is isomorphic to ${L(\frac{1}{2},0)\otimes L(\frac{1}{2},0)}We study a simple, rational and C ₂-cofinite vertex operator algebra whose weight 1 subspace is zero, the dimension of weight 2 subspace is greater than or equal to 2 and with c = c̃ = 1. Under some additional conditions it is shown that such a vertex operator algebra is isomorphic to L(\frac12,0)?L(\frac12,0){L(\frac{1}{2},0)\otimes L(\frac{1}{2},0)}.     

Generalized Homologies for the Zero Modes of the SU(2) WZNW Model

We generalize the BRS method for the (finite-dimensional) quantum gauge theory involved in the zero modes of the monodromy extended SU(2) WZNW model. The generalization consists of a nilpotent operator Q such that Q^h = 0 (h = k + 2 = 2, 3, ... being the height of the current algebra representation) acting on an extended state space. The physical subquotient is identified with the direct sum ^h-1 _n=1 Ker(Qⁿ)/Im(Q^h–n).     

Index of a family of Dirac operators on loop space

We use methods of constructive field theory to generalize index theory to an infinite-dimensional setting. We study a family of Dirac operatorsQ on loop space. These operators arise in the context of supersymmetric nonlinear quantum field models with HamiltoniansH=Q ². In these modelsQ is self-adjoint and Fredholm. A natural grading operator Γ exists such that ΓQ+QΓ=0. We studyQ ₊=P _? QP ₊, whereP _±=1/2 (1±Γ) are the orthogonal projections onto the eigenspaces of Γ. We calculate the indexi(Q ₊) for Wess-Zumino models defined by a superpotentialV(ω). HereV is a polynomial of degreen≧2. We establish thati(Q ₊)=n?1=degδV. In particular, the field theory models have unbroken supersymmetry, and (forn≧3) they have degenerate vacua. We believe that this is the first index theorem for a Dirac operator that couples infinitely many degrees of freedom.     

A Local (Perturbative) Construction of Observables in Gauge Theories: The Example of QED

Interacting fields can be constructed as formal power series in the framework of causal perturbation theory. The local field algebra is obtained without performing the adiabatic limit; the (usually bad) infrared behavior plays no role. To construct the observables in gauge theories we use the Kugo–Ojima formalism; we define the BRST-transformation as a graded derivation on the algebra of interacting fields and use the implementation of by the Kugo–Ojima operator Q _int. Since our treatment is local, the operator Q _int differs from the corresponding operator Q of the free theory. We prove that the Hilbert space structure present in the free case is stable under perturbations. All assumptions are shown to be satisfied in QED. Received: 5 August 1998 / Accepted: 20 November 1998     

String loop effect on the BRST charge

An effective BRST charge Q_BRST which incorporates the string one-loop corrections is presented for the closed basonic string in an arbitrary background. The effective σ-model action which leads to such a Q_BRST is obtained and some consequences are discussed.     

Hopf -deformation of any Lie group

We use the Neroslavsky-Vlassov (1981) method to find a star product _h on a class of exact Poisson-Lie groups such that (C^∞(G)[[h]], _h, Δ) is a Hopf algebra. We show that we can find such a nontrivial star product on every Lie group.     

Covariant quantization of string based on BRS invariance

Covariant canonical quantization of the bosonic string is performed, based on the BRS invariance principle. Defining a physical state by a modified form of Kugo-Ojima's subsidiary condition Q_B|phys〉 = 0 (Q_B = BRS charge), we prove a no-ghost theorem in a simple manner. There an interesting relation between critical dimension and BRS symmetry is noticed; namely, the conditions D = 26 and α₀ = 1 for the space-time dimension D and the zero-intercept α₀ of leading trajectory are required by the nilpotency Q_B² = 0 of the BRS charge. In addition, ghost number fractionization is observed in this system.     

Standard complex for quantum Lie algebras

For a quantum Lie algebra Γ, let Γ^ be its exterior extension (the algebra Γ^ is canonically defined). We introduce a differential on the exterior extension algebra Γ^ which provides the structure of a complex on Γ^. In the situation when Γ is a usual Lie algebra, this complex coincides with the “standard complex.” The differential is realized as a commutator with a (BRST) operator Q in a larger algebra Γ^[Ω], with extra generators canonically conjugated to the exterior generators of Γ^. A recurrent relation which uniquely defines the operator Q is given.     

Homotopy Algebras Inspired by Classical Open-Closed String Field Theory

We define a homotopy algebra associated to classical open-closed strings. We call it an open-closed homotopy algebra (OCHA). It is inspired by Zwiebach's open-closed string field theory and also is related to the situation of Kontsevich's deformation quantization. We show that it is actually a homotopy invariant notion; for instance, the minimal model theorem holds. Also, we show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures (A_∞-algebras) by closed strings (L_∞-algebras). H. K is supported by JSPS Research Fellowships for Young Scientists. J. S. is supported in part by NSF grant FRG DMS-0139799 and US-Czech Republic grant INT-0203119.