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Albert Schwarz

Publications and source records attributed to Albert Schwarz.

72 records · Page 4Linked to original sources
arXiv1999

Quantum observables, Lie algebra homology and TQFT

Let us consider a Lie (super)algebra $G$ spanned by $T_α$ where $T_α$ are quantum observables in BV-formalism. It is proved that for every tensor $c^{α_1...α_k}$ that determines a homology class of the Lie algebra $G$ the expression $c^{α_1...α_k}T_{α_1}...T_{α_k}$ is again a quantum observables. This theorem is used to construct quantum observables in BV sigma-model. We apply this construction to explain Kontsevich's results about the relation between homology of the Lie algebra of Hamiltonian vector fields and topological invariants of manifolds.

hep-th↗
arXiv1998

BPS states on noncommutative tori and duality

We study gauge theories on noncommutative tori. It was proved in [5] that Morita equivalence of noncommutative tori leads to a physical equivalence (SO(d,d| Z)-duality) of the corresponding gauge theories. We calculate the energy spectrum of maximally supersymmetric BPS states in these theories and show that this spectrum agrees with the SO(d,d| Z)-duality. The relation of our results with those of recent calculations is discussed.

hep-th↗
arXiv1998

Morita equivalence of multidimensional noncommutative tori

One can describe an $n$-dimensional noncommutative torus by means of an antisymmetric $n\times n$-matrix $θ$. We construct an action of the group $SO(n,n|\bf Z)$ on the space of antisymmetric matrices and show that, generically, matrices belonging to the same orbit of this group give Morita equivalent tori. Some applications to physics are sketched.

math.QA↗
arXiv1998

Morita equivalence and duality

It was shown by Connes, Douglas, Schwarz[1] that one can compactify M(atrix) theory on noncommutative torus. We prove that compactifications on Morita equivalent tori are physically equivalent. This statement can be considered as a generalization of non-classical duality conjectured in [1] for two-dimensional tori.

hep-th↗
arXiv1998

Noncommutative Geometry and Matrix Theory: Compactification on Tori

We study toroidal compactification of Matrix theory, using ideas and results of non-commutative geometry. We generalize this to compactification on the noncommutative torus, explain the classification of these backgrounds, and argue that they correspond in supergravity to tori with constant background three-form tensor field. The paper includes an introduction for mathematicians to the IKKT formulation of Matrix theory and its relation to the BFSS Matrix theory.

hep-th↗
arXiv1997

Grassmannian and elliptic operators

The conjecture about relation between infinite-dimensional Grassmannian and string theory is based on the fact that moduli spaces of algebraic curves are embedded into Grassmannian via Krichever construction. We describe a multidimensional analog of Krichever construction, that can be used in the attempt to relate Grassmannian to membranes.

funct-an↗
arXiv1996

Grassmannian and string theory

Infinite-dimensional Grassmannian manifold contains moduli spaces of Riemann surfaces of all genera. This well known fact leads to a conjecture that non-perturbative string theory can be formulated in terms of Grassmannian. We present new facts supporting this hypothesis. In particular, it is shown that Grassmannians can be considered as generalized moduli spaces; this statement permits us to define corresponding "string amplitudes" (at least formally). One can conjecture, that it is possible to explain the relation between non-perturbative and perturbative string theory by means of localization theorems for equivariant cohomology; this conjecture is based on the characterization of moduli spaces, relevant to string theory, as sets consisting of points with large stabilizers in certain groups acting on Grassmannian. We describe an involution on the Grassmannian that could be related to S-duality in string theory.

hep-th↗
arXiv1995

Supersymmetry and localization

We study conditions under which an odd symmetry of the integrand leads to localization of the corresponding integral over a (super)manifold. We also show that in many cases these conditions guarantee exactness of the stationary phase approximation of such integrals.

hep-th↗
arXiv1995

$A_\infty$ Algebras and the Cohomology of Moduli Spaces

We introduce the notion of cyclic cohomology of an A-infinity algebra and show that the deformations of an A-infinity algebra which preserve an invariant inner product are classified by this cohomology. We use this result to construct some cycles on the moduli space of algebraic curves. The paper also contains a review of some well known notions and results about Hochschild and cyclic cohomology of associative algebras, A-infinity algebras, and deformation theory of algebras, and includes a discussion of the homology of the graph complex of metric ribbon graphs which is associated to the moduli space of Riemann surfaces with marked points.

hep-th↗
arXiv1995

Sigma-models having supermanifolds as target spaces

We study a topological sigma-model ($A$-model) in the case when the target space is an ($m_0|m_1$)-dimensional supermanifold. We prove under certain conditions that such a model is equivalent to an $A$-model having an ($m_0-m_1$)-dimensional manifold as a target space. We use this result to prove that in the case when the target space of $A$-model is a complete intersection in a toric manifold, this $A$-model is equivalent to an $A$-model having a toric supermanifold as a target space.

hep-th↗
arXiv1994

Matrix Vieta Theorem

We consider generalizations of the Vieta formula (relating the coefficients of an algebraic equation to the roots) to the case of equations whose coefficients are order-$k$ matrices. Specifically, we prove that if $X_1,\dots ,X_n$ are solutions of an algebraic matrix equation $X^n+A_1X^{n-1}+\dots +A_n=0,$ independent in the sense that they determine the coefficients $A_1,\dots ,A_n$, then the trace of $A_1$ is the sum of the traces of the $X_i$, and the determinant of $A_n$ is, up to a sign, the product of the determinants of the $X_i$. We generalize this to arbitrary rings with appropriate structures. This result is related to and motivated by some constructions in non-commutative geometry.

math.RA↗
arXiv1994

Superanalogs of symplectic and contact geometry and their applications to quantum field theory

The paper contains a short review of the theory of symplectic and contact manifolds and of the generalization of this theory to the case of supermanifolds. It is shown that this generalization can be used to obtain some important results in quantum field theory. In particular, regarding $N$-superconformal geometry as particular case of contact complex geometry, one can better understand $N=2$ superconformal field theory and its connection to topological conformal field theory. The odd symplectic geometry constitutes a mathematical basis of Batalin-Vilkovisky procedure of quantization of gauge theories. The exposition is based mostly on published papers. However, the paper contains also a review of some unpublished results (in the section devoted to the axiomatics of $N=2$ superconformal theory and topological quantum field theory). The paper will be published in Berezin memorial volume.

hep-th↗
arXiv1993

Symmetry transformations in Batalin-Vilkovisky formalism

This short note is closely related to Sen-Zwiebach paper on gauge transformations in Batalin-Vilkovisky theory (hep-th 9309027). We formulate some conditions of physical equivalence of solutions to the quantum master equation and use these conditions to give a very transparent analysis of symmetry transformations in BV-approach. We prove that in some sense every quantum observable (i.e. every even function $H$ obeying $Δ_ρ(He^S)=0$) determines a symmetry of the theory with the action functional $S$ satisfying quantum master equation $Δ_ρe^S=0$ \end

hep-th↗
arXiv1992

On Some Algebraic Structures Arising in String Theory

Lian and Zuckerman proved that the homology of a topological chiral algebra can be equipped with the structure of a BV-algebra; \ie one can introduce a multiplication, an odd bracket, and an odd operator $Δ$ having the same properties as the corresponding operations in Batalin-Vilkovisky quantization procedure. We give a simple proof of their results and discuss a generalization of these results to the non chiral case. To simplify our proofs we use the following theorem giving a characterization of a BV-algebra in terms of multiplication and an operator $Δ$: {\em If $A$ is a supercommutative, associative algebra and $Δ$ is an odd second order derivation on $A$ satisfying $Δ^2=0$, one can provide $A$ with the structure of a BV-algebra.}

hep-th↗
arXiv1992

Semiclassical approximation in Batalin-Vilkovisky formalism

The geometry of supermanifolds provided with $Q$-structure (i.e. with odd vector field $Q$ satisfying $\{ Q,Q\} =0$), $P$-structure (odd symplectic structure ) and $S$-structure (volume element) or with various combinations of these structures is studied. The results are applied to the analysis of Batalin-Vilkovisky approach to the quantization of gauge theories. In particular the semiclassical approximation in this approach is expressed in terms of Reidemeister torsion.

hep-th↗
arXiv1992

Geometry of Batalin-Vilkovisky quantization

The present paper is devoted to the study of geometry of Batalin-Vilkovisky quantization procedure. The main mathematical objects under consideration are P-manifolds and SP-manifolds (supermanifolds provided with an odd symplectic structure and, in the case of SP-manifolds, with a volume element). The Batalin-Vilkovisky procedure leads to consideration of integrals of the superharmonic functions over Lagrangian submanifolds. The choice of Lagrangian submanifold can be interpreted as a choice of gauge condition; Batalin and Vilkovisky proved that in some sense their procedure is gauge independent. We prove much more general theorem of the same kind. This theorem leads to a conjecture that one can modify the quantization procedure in such a way as to avoid the use of the notion of Lagrangian submanifold. In the next paper we will show that this is really so at least in the semiclassical approximation. Namely the physical quantities can be expressed as integrals over some set of critical points of solution S to the master equation with the integrand expressed in terms of Reidemeister torsion. This leads to a simplification of quantization procedure and to the possibility to get rigorous results also in the infinite-dimensional case. The present paper contains also a compete classification of P-manifolds and SP-manifolds. The classification is interesting by itself, but in this paper it plays also a role of an important tool in the proof of other results.

hep-th↗
arXiv1991

The Solution Space of the Unitary Matrix Model String Equation and the Sato Grassmannian

The space of all solutions to the string equation of the symmetric unitary one-matrix model is determined. It is shown that the string equation is equivalent to simple conditions on points $V_1$ and $V_2$ in the big cell $\Gr$ of the Sato Grassmannian $Gr$. This is a consequence of a well-defined continuum limit in which the string equation has the simple form $\lb \cp ,\cq_- \rb =\hbox{\rm 1}$, with $\cp$ and $\cq_-$ $2\times 2$ matrices of differential operators. These conditions on $V_1$ and $V_2$ yield a simple system of first order differential equations whose analysis determines the space of all solutions to the string equation. This geometric formulation leads directly to the Virasoro constraints $Ł_n\,(n\geq 0)$, where $Ł_n$ annihilate the two modified-KdV $\t$-functions whose product gives the partition function of the Unitary Matrix Model.

hep-th↗