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.     

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.     

Classes on Compactifications of the Moduli Space of Curves Through Solutions to the Quantum Master Equation

In this paper we describe a construction which produces classes in compactifications of the moduli space of curves. This construction extends a construction of Kontsevich which produces classes in the open moduli space from the initial data of a cyclic A _∞-algebra. The initial data for our construction are what we call a ‘quantum A _∞-algebra’, which arises as a type of deformation of a cyclic A _∞-algebra. The deformation theory for these structures is described explicitly. We construct a family of examples of quantum A _∞-algebras which extend a family of cyclic A _∞-algebras, introduced by Kontsevich, which are known to produce all the kappa classes using his construction.      

BFV-Complex and Higher Homotopy Structures

We present a connection between the BFV-complex (abbreviation for Batalin-Fradkin-Vilkovisky complex) and the strong homotopy Lie algebroid associated to a coisotropic submanifold of a Poisson manifold. We prove that the latter structure can be derived from the BFV-complex by means of homotopy transfer along contractions. Consequently the BFV-complex and the strong homotopy Lie algebroid structure are L _∞ quasi-isomorphic and control the same formal deformation problem. However there is a gap between the non-formal information encoded in the BFV-complex and in the strong homotopy Lie algebroid respectively. We prove that there is a one-to-one correspondence between coisotropic submanifolds given by graphs of sections and equivalence classes of normalized Maurer-Cartan elemens of the BFV-complex. This does not hold if one uses the strong homotopy Lie algebroid instead.     

The sh Lie Structure of Poisson Brackets in Field Theory

A general construction of an sh Lie algebra (L _∞-algebra) from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel'fand, Dickey and Dorfman. In this way, higher order maps are constructed which combine to form an sh Lie algebra on the graded differential algebra of horizontal forms. The same construction applies for graded brackets in field theory such as the Batalin-Fradkin-Vilkovisky bracket of the Hamiltonian BRST theory or the Batalin-Vilkovisky antibracket. Received: 5 March 1997 / Accepted: 21 May 1997     

Nonassociative Star Product Deformations for D-Brane World-Volumes in Curved Backgrounds

We investigate the deformation of D-brane world-volumes in curved backgrounds. We calculate the leading corrections to the boundary conformal field theory involving the background fields, and in particular we study the correlation functions of the resulting system. This allows us to obtain the world-volume deformation, identifying the open string metric and the noncommutative deformation parameter. The picture that unfolds is the following: when the gauge invariant combination ω=B+F is constant one obtains the standard Moyal deformation of the brane world-volume. Similarly, when dω= 0 one obtains the noncommutative Kontsevich deformation, physically corresponding to a curved brane in a flat background. When the background is curved, H=dω≠ 0, we find that the relevant algebraic structure is still based on the Kontsevich expansion, which now defines a nonassociative star product with an A _∞ homotopy associative algebraic structure. We then recover, within this formalism, some known results of Matrix theory in curved backgrounds. In particular, we show how the effective action obtained in this framework describes, as expected, the dielectric effect of D-branes. The polarized branes are interpreted as a soliton, associated to the condensation of the brane gauge field. Received: 22 March 2001 / Accepted: 13 July 2001     

Strongly Homotopy Lie Bialgebras and Lie Quasi-bialgebras

Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer–Cartan equations on corresponding governing differential graded Lie algebras using the big bracket construction of Kosmann–Schwarzbach. This approach provides a definition of an L _∞-(quasi)bialgebra (strongly homotopy Lie (quasi)bialgebra). We recover an L _∞-algebra structure as a particular case of our construction. The formal geometry interpretation leads to a definition of an L _∞ (quasi)bialgebra structure on V as a differential operator Q on V, self-commuting with respect to the big bracket. Finally, we establish an L _∞-version of a Manin (quasi) triple and get a correspondence theorem with L _∞-(quasi)bialgebras. This paper is dedicated to Jean-Louis Loday on the occasion of his 60th birthday with admiration and gratitude.     

On Hyper-Kähler Manifolds of Type A ∞ and D ∞

We shall use an infinite dimensional hyper-K?hler quotient method to obtain hyper-K?hler 4 manifolds of type A _∞ and D _∞. Hyper-K?hler manifolds of type A _∞ and D _∞ are constructed in terms of Dynkin diagrams of type A _∞ and D _∞ respectively. A hyper-K?hler manifold of type D _∞ is the minimal resolution of the quotient space of a hyper-K?hler manifold of type A _∞ by an involution. Finally we shall show that a hyper-K?hler manifold of type A _∞ can be considered as the universal cover of elliptic fibre space of type I _b . Received: 18 July 1997 / Accepted: 14 April 1998     

<Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript>-Algebras from Multisymplectic Geometry

A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n + 1. In previous work with Baez and Hoffnung, we described how the ‘higher analogs’ of the algebraic and geometric structures found in symplectic geometry should naturally arise in 2-plectic geometry. In particular, just as a symplectic manifold gives a Poisson algebra of functions, any 2-plectic manifold gives a Lie 2-algebra of 1-forms and functions. Lie n-algebras are examples of L _∞-algebras: graded vector spaces equipped with a collection of skew-symmetric multi-brackets that satisfy a generalized Jacobi identity. Here, we generalize our previous result. Given an n-plectic manifold, we explicitly construct a corresponding Lie n-algebra on a complex consisting of differential forms whose multi-brackets are specified by the n-plectic structure. We also show that any n-plectic manifold gives rise to another kind of algebraic structure known as a differential graded Leibniz algebra. We conclude by describing the similarities between these two structures within the context of an open problem in the theory of strongly homotopy algebras. We also mention a possible connection with the work of Barnich, Fulp, Lada, and Stasheff on the Gelfand–Dickey–Dorfman formalism.     

Computation of Superpotentials for D-Branes

We present a general method for the computation of tree-level superpotentials for the world-volume theory of B-type D-branes. This includes quiver gauge theories in the case that the D-brane is marginally stable. The technique involves analyzing the A_∞-structure inherent in the derived category of coherent sheaves. This effectively gives a practical method of computing correlation functions in holomorphic Chern–Simons theory. As an example, we give a more rigorous proof of previous results concerning 3-branes on certain singularities including conifolds. We also provide a new example.     

A Random Surface Wave Equation for Bosonic QCD(<Emphasis Type="Italic">SU</Emphasis>(∞))

We propose a quantum surface wave functional describing the interaction between a colored SU(N _c ) membrane and a quantized Yang-Mills field. Additionally, we deduce its associated wave equation in the t’Hooft N _c →∞ limit. We show that its reproduces the Yang-Mills Field Theory at a large rigid random surface scale.     

A Note on the Koszul Complex in Deformation Quantization

The aim of this short note is to present a proof of the existence of an A _∞-quasi-isomorphism between the A _∞-S(V ^*)-ù(V){\wedge(V)} -bimodule K, introduced in Calaque et al. (Bimodules and branes in deformation quantization, 2009), and the Koszul complex K(V) of S(V ^*), viewed as an A _∞-S(V ^*)-ù(V){\wedge(V)} -bimodule, for V a finite-dimensional (complex or real) vector space.     

Soliton equations and fundamental representations of A 2l (2)

Fundamental representations of the Euclidean Lie algebra A _2l ⁽²⁾ is constructed by decomposing the vertex representations of gI(∞). For l=1 the multiplicities of highest weights are determined. Soliton equations associated with each of these representations are also discussed.     

A simple construction of associative deformations

We propose a simple approach to formal deformations of associative algebras. It exploits the machinery of multiplicative coresolutions of an associative algebra A in the category of A-bimodules. Specifically, we show that certain first-order deformations of A extend to all orders and we derive explicit recurrent formulas determining this extension. In physical terms, this may be regarded as the deformation quantization of noncommutative Poisson structures on A.

     

Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group

We showed in Part I that the Hopf algebra ℋ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group G and that the renormalized theory is obtained from the unrenormalized one by evaluating at ɛ= 0 the holomorphic part γ₊(ɛ) of the Riemann–Hilbert decomposition γ₋(ɛ)^{− 1}γ₊(ɛ) of the loop γ(ɛ)∈G provided by dimensional regularization. We show in this paper that the group G acts naturally on the complex space X of dimensionless coupling constants of the theory. More precisely, the formula g ₀=gZ ₁ Z ₃ ^−3/2 for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra ℋ. This allows first of all to read off directly, without using the group G, the bare coupling constant and the renormalized one from the Riemann–Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter ɛ. It also allows to lift both the renormalization group and the β-function as the asymptotic scaling in the group G. This exploits the full power of the Riemann–Hilbert decomposition together with the invariance of γ₋(ɛ) under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group G for the full higher pole structure of minimal subtracted counterterms in terms of the residue. Received: 21 March 2000 / Accepted: 3 October 2000     

Enhanced Gauge Symmetry and Braid Group Actions

 Enhanced gauge symmetry appears in Type II string theory (as well as F- and M-theory) compactified on Calabi–Yau manifolds containing exceptional divisors meeting in Dynkin configurations. It is shown that in many such cases, at enhanced symmetry points in moduli a braid group acts on the derived category of sheaves of the variety. This braid group covers the Weyl group of the enhanced symmetry algebra, which itself acts on the deformation space of the variety in a compatible way. Extensions of this result are given for nontrivial B-fields on K3 surfaces, explaining physical restrictions on the B-field, as well as for elliptic fibrations. The present point of view also gives new evidence for the enhanced gauge symmetry content in the case of a local A _2n -configuration in a threefold having global ℤ/2 monodromy. Received: 28 October 2002 / Accepted: 9 December 2002 Published online: 28 May 2003 Communicated by R.H. Dijkgraaf     

On twist quantizations of <Emphasis Type="Italic">D</Emphasis> = 4 Lorentz and Poincare algebras

We use the decomposition of o(3, 1) = sl(2; ℂ)₁ ⊕sl(2; ℂ)₂ in order to describe nonstandard quantum deformation of o(3, 1) linked with Jordanian deformation of sl(2; ℂ). Using the twist quantization technique, we obtain the deformed coproducts and antipodes, which can be expressed in terms of real physical Lorentz generators. We describe the extension of the considered deformation of D = 4 Lorentz algebra to the twist deformation of D = 4 Poincare algebra with dimensionless deformation parameter. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

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     

A one-parameter family of hamiltonian structures for the KP hierarchy and a continuous deformation of the nonlinear WKP algebra

The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained from a generalized Adler map in the space of formal pseudodifferential symbols with noninteger powers. The resulting W-algebra is a one-parameter deformation of W_KP admitting a central extension for generic values of the parameter, reducing naturally to W _n for special values of the parameter, and contracting to the centrally extended W₁₊, W and further truncations. In the classical limit, all algebras in the one-parameter family are equivalent and isomorphic tow _KP. The reduction induced by setting the spin-one field to zero yields a one-parameter deformation of which contracts to a new nonlinear algebra of the W-type.Address after October 1993: Queen Mary and Westfield College, UK