Simulation of Topological Field Theories¶by Quantum Computers

Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned “topological models” having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H≡ 0. These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold: 1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”. 2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm. Received: 4 May 2001 / Accepted: 16 January 2002     

A Change of Coordinates on the Large Phase Space¶of Quantum Cohomology

The Gromov–Witten invariants of a smooth, projective variety V, when twisted by the tautological classes on the moduli space of stable maps, give rise to a family of cohomological field theories and endow the base of the family with coordinates. We prove that the potential functions associated to the tautological ψ classes (the large phase space) and the κ classes are related by a change of coordinates which generalizes a change of basis on the ring of symmetric functions. Our result is a generalization of the work of Manin–Zograf who studied the case where V is a point. We utilize this change of variables to derive the topological recursion relations associated to the κ classes from those associated to the ψ classes. Received: 2 August 1999 / Accepted: 30 September 2000     

Loop Homotopy Algebras in Closed String Field Theory

Barton Zwiebach constructed [20] “string products” on the Hilbert space of a combined conformal field theory of matter and ghosts, satisfying the “main identity”. It has been well known that the “tree level” of the theory gives an example of a strongly homotopy Lie algebra (though, as we will see later, this is not the whole truth). Strongly homotopy Lie algebras are now well-understood objects. On the one hand, strongly homotopy Lie algebra is given by a square zero coderivation on the cofree cocommutative connected coalgebra [13, 14]; on the other hand, strongly homotopy Lie algebras are algebras over the cobar dual of the operad &?om for commutative algebras [9]. As far as we know, no such characterization of the structure of string products for arbitrary genera has been available, though there are two series of papers directly pointing towards the requisite characterization. As far as the characterization in terms of (co)derivations is concerned, we need the concept of higher order (co)derivations, which has been developed, for example, in[2, 3]. These higher order derivations were used in the analysis of the ”master identity“. For our characterization we need to understand the behavior of these higher (co)derivations on (co)free (co)algebras. The necessary machinery for the operadic approach is that of modular operads, anticipated in [5] and introduced in [8]. We believe that the modular operad structure on the compactified moduli space of Riemann surfaces of arbitrary genera implies the existence of the structure we are interested in the same manner as was explained for the tree level in [11]. We also indicate how to adapt the loop homotopy structure to the case of open string field theory [19]. Received: 10 November 1999 / Accepted: 29 March 2001     

sh-Lie Algebras Induced by Gauge Transformations

Traditionally symmetries of field theories are encoded via Lie group actions, or more generally, as Lie algebra actions. A significant generalization is required when “gauge parameters” act in a field dependent way. Such symmetries appear in several field theories, most notably in a “Poisson induced” class due to Schaller and Strobl [SS94] and to Ikeda [Ike94], and employed by Cattaneo and Felder [CF99] to implement Kontsevich's deformation quantization [Kon97]. Consideration of “particles of spin > 2” led Berends, Burgers and van Dam [Bur85,BBvD84,BBvD85] to study “field dependent parameters” in a setting permitting an analysis in terms of smooth functions. Having recognized the resulting structure as that of an sh-Lie algebra (L _∞-algebra), we have now formulated such structures entirely algebraically and applied it to a more general class of theories with field dependent symmetries. Received: 14 December 2000 / Accepted: 8 February 2002?Published online: 2 October 2002     

Three-Point Functions for MN/SN Orbifolds¶with?= 4 Supersymmetry

The D1–D5 system is believed to have an “orbifold point” in its moduli space where its low energy theory is a ?=4 supersymmetric sigma model with target space M ^N /S _N , where M is T ⁴ or K3. We study correlation functions of chiral operators in CFTs arising from such a theory. We construct a basic class of chiral operators from twist fields of the symmetric group and the generators of the superconformal algebra. We find explicitly the 3-point functions for these chiral fields at large N; these expressions are “universal” in that they are independent of the choice of M. We observe that the result is a significantly simpler expression than the corresponding expression for the bosonic theory based on the same orbifold target space. Received: 29 March 2001 / Accepted: 20 January 2002     

Non-Commutative Periods and Mirror Symmetry¶in Higher Dimensions

We study an analog for higher-dimensional Calabi–Yau manifolds of the standard predictions of Mirror Symmetry. We introduce periods associated with “non-commutative” deformations of Calabi–Yau manifolds. These periods define a map on the moduli space of such deformations which is a local isomorphism. Using these non-commutative periods we introduce invariants of variations of semi-infinite generalized Hodge structures living over the moduli space ℳ. It is shown that the generating function of such invariants satisfies the system of WDVV-equations exactly as in the case of Gromov–Witten invariants. We prove that the total collection of rational Gromov–Witten invariants of complete intersection Calabi–Yau manifold can be identified with the collection of invariants of variations of generalized (semi-infinite) Hodge structures attached to the mirror variety. The basic technical tool utilized is the deformation theory. Received: 6 April 2000 / Accepted: 15 January 2002     

A Master Identity for Homotopy Gerstenhaber Algebras

We produce a master identity for a certain type of homotopy Gerstenhaber algebras, in particular suitable for the prototype, namely the Hochschild complex of an associative algebra. This algebraic master identity was inspired by the work of Getzler–Jones and Kimura–Voronov–Zuckerman in the context of topological conformal field theories. To this end, we introduce the notion of a “partitioned multilinear map” and explain the mechanics of composing such maps. In addition, many new examples of pre-Lie algebras and homotopy Gerstenhaber algebras are given. Received: 2 March 1998 / Accepted: 16 July 1999     

Automorphism Group of k((t)):¶Applications to the Bosonic String

This paper is concerned with the formulation of a non-pertubative theory of the bosonic string. We introduce a formal group G which we propose as the “universal moduli space” for such a formulation. This is motivated because G establishes a natural link between representations of the Virasoro algebra and the moduli space of curves. Received: 26 March 1999 / Accepted: 10 September 2000     

On the Extension of a TCFT to the Boundary of the Moduli Space

The purpose of this paper is to describe an analogue of a construction of Costello in the context of finite-dimensional differential graded Frobenius algebras which produces closed forms on the decorated moduli space of Riemann surfaces. We show that this construction extends to a certain natural compactification of the moduli space which is associated with the modular closure of the associative operad, due to the absence of ultra-violet divergences in the finite-dimensional case. We demonstrate that this construction is equivalent to the “dual construction” of Kontsevich.     

The descent of off-shell supersymmetry to the mass shell

It is shown that classical actions for some “physically interesting” quantum field theories can be obtained as effective actions from the single “fundamental” theory of the Chern–Simons form. The physical degrees of freedom are encoded in the space of cohomologies of a certain differential operator. This observation suggests a different perspective on some of the supersymmetric properties of these effective theories. Namely, it is possible to construct a superfield formalism which allows to find off-shell SUSY actions for the on-shell supersymmetric theories, where conventional superfield formalism does not work. This formalism contains even auxiliary variables λ^α in addition to conventional odd variables θ^α. This idea is similar to the Pure Spinor construction. This paper is a short review of papers [11, 12]. Original results discussed below were obtained in collaboration with V. Alexandrov, A. Gorodentsev, A. Losev and V. Lysov.     

On Entropy and Monotonicity for Real Cubic Maps

Consider real cubic maps of the interval onto itself, either with positive or with negative leading coefficient. This paper completes the proof of the “monotonicity conjecture”, which asserts that each locus of constant topological entropy in parameter space is a connected set. The proof makes essential use of the thesis of Christopher Heckman, and is based on the study of “bones” in the parameter triangle as defined by Tresser and R. MacKay. Received: 16 November 1998 / Accepted: 2 August 1999     

Regularisable and Minimal Orbits for Group Actions in Infinite Dimensions

We introduce a class of regularisable infinite dimensional principal fibre bundles which includes fibre bundles arising in gauge field theories like Yang-Mills and string theory and which generalise finite dimensional Riemannian principal fibre bundles induced by an isometric action. We show that the orbits of regularisable bundles have well defined, both heat-kernel and zeta function regularised volumes. We introduce a notion of μ-minimality ( ) for these orbits which extend the finite dimensional one. Our approach uses heat-kernel methods and yields both “heat-kernel” (obtained via heat-kernel regularisation) and “zeta function” (obtained via zeta function regularisation) minimality for specific values of the parameter μ. For each of these notions, we give an infinite dimensional version of Hsiang's theorem which extends the finite dimensional case, interpreting μ-minimal orbits as orbits with extremal (μ-regularised) volume. Received: 27 November 1995 / Accepted: 30 May 1997     

Grassmannian and String Theory

The 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 the 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 the Grassmannian. We describe an involution on the Grassmannian that could be related to S-duality in string theory. Received: 19 December 1996 / Accepted: 27 March 1998     

On Nonlinear σ-Models Arising in (Super-)Gravity

In a previous paper with Gibbons [1] we derived a list of three dimensional symmetric space σ-models obtained by dimensional reduction of a class of four dimensional gravity theories with abelian gauge fields and scalars. Here we give a detailed analysis of their group theoretical structure leading to an abstract parametrization in terms of “triangular” group elements. This allows for a uniform treatment of all these models. As an interesting application we give a simple derivation of a “Quadratic Mass Formula” for strictly stationary black holes. Received: 3 August 1998 / Accepted: 1 December 1999     

Einstein’s “Zur Elektrodynamik...” (1905) Revisited, With Some Consequences

Einstein, in his “Zur Elektrodynamik bewegter K?rper”, gave a physical (operational) meaning to “time” of a remote event in describing “motion” by introducing the concept of “synchronous stationary clocks located at different places”. But with regard to “place” in describing motion, he assumed without analysis the concept of a system of co-ordinates.In the present paper, we propose a way of giving physical (operational) meaning to the concepts of “place” and “co-ordinate system”, and show how the observer can define both the place and time of a remote event. Following Einstein, we consider another system “in uniform motion of translation relatively to the former”. Without assuming “the properties of homogeneity which we attribute to space and time”, we show that the definitions of space and time in the two systems are linearly related. We deduce some novel consequences of our approach regarding faster-than-light observers and particles, “one-way” and “two-way” velocities of light, symmetry, the “group property” of inertial reference frames, length contraction and time dilatation, and the “twin paradox”. Finally, we point out a flaw in Einstein’s argument in the “Electrodynamical Part” of his paper and show that the Lorentz force formula and Einstein’s formula for transformation of field quantities are mutually consistent. We show that for faster-than-light bodies, a simple modification of Planck’s formula for mass suffices. (Except for the reference to Planck’s formula, we restrict ourselves to Physics of 1905.)     

Glimmers of a Pre-geometric Perspective

Spacetime measurements and gravitational experiments are made by using objects, matter fields or particles and their mutual relationships. As a consequence, any operationally meaningful assertion about spacetime is in fact an assertion about the degrees of freedom of the matter (i.e. non gravitational) fields; those, say for definiteness, of the Standard Model of particle physics. As for any quantum theory, the dynamics of the matter fields can be described in terms of a unitary evolution of a state vector in a Hilbert space. By writing the Hilbert space as a generic tensor product of “subsystems” we analyse the evolution of a state vector on an information theoretical basis and attempt to recover the usual spacetime relations from the information exchanges between these subsystems. We consider generic interacting second quantized models with a finite number of fermionic degrees of freedom and characterize on physical grounds the tensor product structure associated with the class of “localized systems” and therefore with “position”. We find that in the case of free theories no spacetime relation is operationally definable. On the contrary, by applying the same procedure to the simple interacting model of a one-dimensional Heisenberg spin chain we recover the tensor product structure usually associated with “position”. Finally, we discuss the possible role of gravity in this framework.     

Parametric Representation of Noncommutative Field Theory

In this paper we investigate the Schwinger parametric representation for the Feynman amplitudes of the recently discovered renormalizable quantum field theory on the Moyal non commutative space. This representation involves new hyperbolic polynomials which are the non-commutative analogs of the usual “Kirchoff” or “Symanzik” polynomials of commutative field theory, but contain richer topological information. Work supported by ANR grant NT05-3-43374 “GenoPhy”.