Characteristic cohomology ofp-form gauge theories

The characteristic cohomologyH ^k _char(d) for an arbitrary set of freep-form gauge fields is explicitly worked out in all form degreesk < n — 1, wheren is the spacetime dimension. It is shown that this cohomology is finite-dimensional and completely generated by the forms dual to the field strengths. The gauge invariant characteristic cohomology is also computed. The results are extended to interactingp-form gauge theories with gauge invariant interactions. Implications for the BRST cohomology are mentioned.     

Refined Chiral Slavnov–Taylor Identities: Renormalization and Local Physics

We study the quantization of chiral QED with one family of massless fermions and the Stueckelberg field in order to give mass to the Abelian gauge field in a BRST-invariant way. We show that an extended Slavnov–Taylor (ST) identity can be introduced and fulfilled to all orders in perturbation theory by a suitable choice of the local actionlike counterterms, order by order in the loopwise expansion. This ST identity incorporates the Adler–Bardeen anomaly and involves the introduction of a doublet (K, c), where K is an external source of dimension 0 and c is the ghost field. By a purely algebraic argument we show that the introduction of the source K trivializes the cohomology of the extended linearized classical ST operator S ₀ in the Fadeev–Popov (FP) charge + 1 sector.We discuss the physical content of the extended ST identity and prove that the cohomology classes associated with S ₀ are modified with respect to the ones of the classical BRST differential s in the FP neutral sector (physical observables). This provides a counterexample showing that the introduction of a doublet can modify the cohomology of the model, as a consequence of the fact that the counting operator for the doublet (K, c) does not commute with S ₀ .We explicitly check that the physical states defined by s are no more physical states of the full quantized theory by showing that the subspace of the physical states corresponding to s is not left-invariant under the application of the S matrix, as a consequence of the extended ST identity.     

Fedosov Deformation Quantization as a BRST Theory

The relationship is established between the Fedosov deformation quantization of a general symplectic manifold and the BFV-BRST quantization of constrained dynamical systems. The original symplectic manifold ℳ is presented as a second class constrained surface in the fibre bundle ?^* _ρℳ which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure. The second class system is converted into the first class one by continuation of the constraints into the extended manifold, being a direct sum of ?^* _ρℳ and the tangent bundle Tℳ. This extended manifold is equipped with a nontrivial Poisson bracket which naturally involves two basic ingredients of Fedosov geometry: the symplectic structure and the symplectic connection. The constructed first class constrained theory, being equivalent to the original symplectic manifold, is quantized through the BFV-BRST procedure. The existence theorem is proven for the quantum BRST charge and the quantum BRST invariant observables. The adjoint action of the quantum BRST charge is identified with the Abelian Fedosov connection while any observable, being proven to be a unique BRST invariant continuation for the values defined in the original symplectic manifold, is identified with the Fedosov flat section of the Weyl bundle. The Fedosov fibrewise star multiplication is thus recognized as a conventional product of the quantum BRST invariant observables. Received: 28 April 2000 / Accepted: 6 December 2000     

Quasiclassical Lian-Zuckerman Homotopy Algebras,Courant Algebroids and Gauge Theory

We define a quasiclassical limit of the Lian-Zuckerman homotopy BV algebra (quasiclassical LZ algebra) on the subcomplex, corresponding to “light modes”, i.e. the elements of zero conformal weight, of the semi-infinite (BRST) cohomology complex of the Virasoro algebra associated with vertex operator algebra (VOA) with a formal parameter. We also construct a certain deformation of the BRST differential parametrized by a constant two-component tensor, such that it leads to the deformation of the A _∞-subalgebra of the quasiclassical LZ algebra. Altogether this gives a functor the category of VOA with a formal parameter to the category of A _∞-algebras. The associated generalized Maurer-Cartan equation gives the analogue of the Yang-Mills equation for a wide class of VOAs. Applying this construction to an example of VOA generated by β - γ systems, we find a remarkable relation between the Courant algebroid and the homotopy algebra of the Yang-Mills theory.     

Reduction of Quantum Systems with Arbitrary First Class Constraints and Hecke Algebras

We propose a method for reduction of quantum systems with arbitrary first-class constraints. An appropriate mathematical setting for the problem is the homology of associative algebras. For every such algebra A and subalgebra B with augmentation ɛ there exists a cohomological complex which is a generalization of the BRST one. Its cohomology is an associative graded algebra Hk ^*(A,B) which we call the Hecke algebra of the triple (A,B,ɛ). It acts in the cohomology space H ^*(B,V) for every left A module V. In particular the zeroth graded component $Hk^{0}(A,B)$ acts in the space of B invariants of $V$ and provides the reduction of the quantum system. Received: 15 June 1998 / Accepted: 25 January 1999     

Chiral BRST Cohomology of N= 2 Strings at Arbitrary Ghost and Picture Number

We compute the BRST cohomology of the holomorphic part of the N= 1 string at arbitrary ghost and picture number. We confirm the expectation that the relative cohomology at non-zero momentum consists of a single massless state in each picture. The absolute cohomology is obtained by an independent method based on homological algebra. For vanishing momentum, the relative and absolute cohomologies both display a picture dependence – a phenomenon discovered recently also in the relative Ramond sector of N= 1 strings by Berkovits and Zwiebach [1]. Received: 5 January 1998 / Accepted: 16 November 1998     

Fock representations and BRST cohomology inSL(2) current algebra

We investigate the structure of the Fock modules overA ₁ ⁽¹⁾ introduced by Wakimoto. We show that irreducible highest weight modules arise as degree zero cohomology groups in a BRST-like complex of Fock modules. Chiral primary fields are constructed as BRST invariant operators acting on Fock modules. As a result, we obtain a free field representation of correlation functions of theSU(2) WZW model on the plane and on the torus. We also consider representations of fractional level arising in Polyakov's 2D quantum gravity. Finally, we give a geometrical, Borel-Weil-like interpretation of the Wakimoto construction.     

The Donaldson–Witten Function for Gauge Groups of Rank Larger Than One

We study correlation functions in topologically twisted , d=4 supersymmetric Yang–Mills theory for gauge groups of rank larger than one on compact four-manifolds X. We find that the topological invariance of the generator of correlation functions of BRST invariant observables is not spoiled by noncompactness of field space. We show how to express the correlators on simply connected manifolds of b _2,+(X)>0 in terms of Seiberg–Witten invariants and the classical cohomology ring of X. For manifolds X of simple type and gauge group SU(N) we give explicit expressions of the correlators as a sum over =1 vacua. We describe two applications of our expressions, one to superconformal field theory and one to large N expansions of SU(N) , d=4 supersymmetric Yang–Mills theory. Received: 30 March 1998 / Accepted: 17 April 1998     

On equivalence of Floer's and quantum cohomology

We show that the Floer cohomology and quantum cohomology rings of the almost Kähler manifoldM, both defined over the Novikov ring of the loop space M, are isomorphic. We do it using a BRST trivial deformation of the topological A-model. The relevant aspect of noncompactness of the moduli of pseudoholomorphic instantons is discussed. It is shown nonperturbatively that any BRST trivial deformation of A model which does not change the dimensions of BRST cohomology does not change the topological correlation functions either.     

BRST cohomology for certain reducible topological symmetries

In this paper, two different definitions of the BRST complex are connected. We obtain the BRST complex of topological quantum field theories (leading to equivariant cohomology) from the standard definition of the classical BRST complex (leading to Lie algebra cohomology) provided that we include ghosts for ghosts. Hereby, we use a finite dimensional model with a semi-direct product action ofH DiffM on a configuration spaceM, whereH is a compact Lie group representing the gauge symmetry in this model.     

On the classical dynamics of a pure spinor string on an AdS5×S5 background

We will discuss some properties of the pure spinor string on an AdS₅×S⁵ background. Using a classical Hamiltonian analysis we will show that the vertex operator for the massless state that is in the cohomology of the BRST charges describes on-shell fluctuations around an AdS₅×S⁵ background.     

BRST analysis of physical states for 2D gravity coupled toc≦1 matter

We consider 2D gravity coupled toc1 conformal matter in the conformal gauge. The Liouville system is represented by a free scalar field, ^L , with background charge such that the BRST operator imposing reparametrization invariance is nilpotent. We compute the cohomology of this BRST charge on the product of the Fock space of ^L with those of the ghosts and one other free scalar field, ^M representing the matter system. From this calculation the physical states of the full theory are determined. For thec<1 case the further projection from the Fock space of ^M to the irreducible representation, using Felder's resolution, reproduces the results of Lian and Zuckerman.Supported by the NSF Grant # PHY-88-04561Supported in part by the Department of Energy Contract # DE-FG03-84ER-40168 and by the USC Faculty Research and Innovation Fund     

Formality Theorem for Hochschild Cochains via Transfer

We construct a 2-colored operad Ger _∞ which, on the one hand, extends the operad Ger _∞ governing homotopy Gerstenhaber algebras and, on the other hand, extends the 2-colored operad governing open-closed homotopy algebras. We show that Tamarkin’s Ger _∞-structure on the Hochschild cochain complex C ^•(A, A) of an A _∞-algebra A extends naturally to a Ger⁺_￥{{\bf Ger}^+_{\infty}}-structure on the pair (C ^•(A, A), A). We show that a formality quasi-isomorphism for the Hochschild cochains of the polynomial algebra can be obtained via transfer of this Ger⁺_￥{{\bf Ger}^+_{\infty}}-structure to the cohomology of the pair (C ^•(A, A), A). We show that Ger⁺_￥{{\bf Ger}^+_{\infty}} is a sub DG operad of the first sheet E ¹(SC) of the homology spectral sequence for the Fulton–MacPherson version SC of Voronov’s Swiss Cheese operad. Finally, we prove that the DG operads Ger⁺_￥{{\bf Ger}^+_{\infty}} and E ¹(SC) are non-formal.     

Julia Sets in Parameter Spaces

Given a complex number λ of modulus 1, we show that the bifurcation locus of the one parameter family {f _b (z)=λz+b z ²+z ³} _{b ∈ ℂ} contains quasi-conformal copies of the quadratic Julia set J(λz+z ²). As a corollary, we show that when the Julia set J(λz+z ²) is not locally connected (for example when z↦λz+z ² has a Cremer point at 0), the bifurcation locus is not locally connected. To our knowledge, this is the first example of complex analytic parameter space of dimension 1, with connected but non-locally connected bifurcation locus. We also show that the set of complex numbers λ of modulus 1, for which at least one of the parameter rays has a non-trivial accumulation set, contains a dense G _δ subset of S ¹. Received: 22 September 2000 / Accepted: 16 January 2001     

Ue(1)-covariant Rξ gauge for the two-Higgs doublet model

A U _e (1)-covariant R _ξ gauge for the two-Higgs doublet model based on BRST (Becchi-Rouet-Stora-Tyutin) symmetry is introduced. This gauge allows one to remove a significant number of nonphysical vertices appearing in conventional linear gauges, which greatly simplifies the loop calculations, since the resultant theory satisfies QED-like Ward identities. The presence of four ghost interactions in these types of gauges and their connection with the BRST symmetry are stressed. The Feynman rules for those new vertices that arise in this gauge, as well as for those couplings already present in the linear R _ξ gauge but that are modified by this gauge-fixing procedure, are presented.     

A Riemann–Roch Theorem¶for One-Dimensional Complex Groupoids

We consider a smooth groupoid of the form Σ⋊Γ, where Σ is a Riemann surface and Γ a discrete pseudogroup acting on Σ by local conformal diffeomorphisms. After defining a K-cycle on the crossed product C ₀(Σ)⋊Γ generalising the classical Dolbeault complex, we compute its Chern character in cyclic cohomology, using the index theorem of Connes and Moscovici. This involves in particular a generalisation of the Euler class constructed from the modular automorphism group of the von Neumann algebra L ^∞(Σ)⋊Γ. Received: 1 February 2000 / Accepted: 3 December 2000     

Existence,uniqueness and cohomology of the classical BRST charge with ghosts of ghosts

A complete canonical formulation of the BRST theory of systems with redundant gauge symmetries is presented. These systems includep-form gauge fields, the superparticle, and the superstring. We first define the Koszul-Tate differential and explicitly show how the introduction of the momenta conjugate to the ghosts of ghosts makes it acyclic. The global existence of the BRST generator is then demonstrated, and the BRST charge is proved to be unique up to canonical transformations in the extended phase space, which includes the ghosts. Finally, the BRST cohomology in classical mechanics is investigated and shown to be equal to the cohomology of the exterior derivative along the gauge orbits, as in the irreducible case. This is done by re-expressing the exterior algebra along the gauge orbits as a free differential algebra containing generators of higher degree, which are identified with the ghosts of ghosts. The quantum cohomology is not dealt with.Aspirant du Fonds National de la Recherche Scientifique (Belgium)Chercheur qualifié au Fonds National de la Recherche Scientifique (Belgium)     

Solving general gauge theories on inner product spaces

By means of a generalized quartet mechanism we show in a model independent way that a BRST quantization on an inner product space leads to physical states of the form ph〉 = exp [Q, ψ]ph〉₀ where Q is the nilpotent BRST operator, ψ a hermitian fermionic gauge-fixing operator, and ph〉_o BRST invariant states determined by a hermitian set of BRST doublets in involution. ph〉₀ does not belong to an inner product space although ph〉 does. Since the BRST quartets are split into two sets of hermitian BRST doublets there are two choices for ph〉₀ and the corresponding ψ. When applied to general, both irreducible and reducible, gauge theories of arbitrary rank within the BFV formulation we find that ph〉₀ are trivial BRST invariant states which only depend on the matter variables for one set of solutions, and for the other set ph〉₀ are solutions of a Dirac quantization. This generalizes previous Lie group solutions obtained by means of a bigrading.     

Extended modes and energy dynamics in two-dimensional lattices with correlated disorder

We study the nature of the vibrational modes in a two-dimensional harmonic lattice with long-range correlated random masses, with power-law spectral density S(k)∼1/k^α. We obtain numerically the scale invariance of the fluctuations of the relative participation number and the local density of states. We find signatures of extended vibrational modes when α>α_c and α_c depends on the magnitude of disorder. In order to confirm this claim, we also study the time evolution of an initially localized perturbation of the lattice. We show that the second moment of the spatial distribution of the energy displays a ballistic regime when α>α_c, in agreement with the occurrence of extended vibrational modes.