Balanced Topological Field Theories

We describe a class of topological field theories called “balanced topological field theories”. These theories are associated to moduli problems with vanishing virtual dimension and calculate the Euler character of various moduli spaces. We show that these theories are closely related to the geometry and equivariant cohomology of “iterated superspaces” that carry two differentials. We find the most general action for these theories, which turns out to define Morse theory on field space. We illustrate the constructions with numerous examples. Finally, we relate these theories to topological sigma-models twisted using an isometry of the target space. Received: 2 September 1996 / Accepted: 3 October 1996     

Topological Field Theories and Harrison Homology

Tools and arguments developed by Kevin Costello are adapted to families of “Outer Spaces” or spaces of graphs. This allows us to prove a version of Deligne’s conjecture: the Harrison homology associated with a homotopy commutative algebra is naturally a module over a cobordism category of 3-manifolds.     

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     

Topological Conformal Field Theories from the Noncompact Coset Models

It is shown in this letter that in the bosonic coset models G_kl×G_k2/G_kl+_k2 associated with noncompact Kac-Moody algebra there exist two kinds of topological points, _k2=0 and _kl+_k2-2g = 0. At these points, the coset models may be interpreted as twisted versions of noncompact counterparts of the Kazama-Suzuki models.     

Topological Field Theories Associated with Three-Dimensional Seiberg–Witten Monopoles

Three-dimensional topological field theoriesassociated with the three-dimensional version of Abelianand non-Abelian Seiberg–Witten monopoles arepresented. These three-dimensional monopole equationsare obtained by a dimensional reduction of thefour-dimensional ones. The starting actions to beconsidered are Gaussian types with random auxiliaryfields. As the local gauge symmetries with topologicalshifts are found to be first-stage-reducible, theBatalin–Vilkovisky algorithm is suitable forquantization. Then the BRST transformation rules areautomatically obtained. Nontrivial observablesassociated with Chern classes are obtained from the geometricsector and are found to correspond to those of thetopological field theory of Bogomol'nyimonopoles.     

AKSZ–BV Formalism and Courant Algebroid-Induced Topological Field Theories

We give a detailed exposition of the Alexandrov–Kontsevich–Schwarz– Zaboronsky superfield formalism using the language of graded manifolds. As a main illustrating example, to every Courant algebroid structure we associate canonically a three-dimensional topological sigma-model. Using the AKSZ formalism, we construct the Batalin–Vilkovisky master action for the model.      

Boundary Conditions for Topological Quantum Field Theories,Anomalies and Projective Modular Functors

We study boundary conditions for extended topological quantum field theories (TQFTs) and their relation to topological anomalies. We introduce the notion of TQFTs with moduli level m, and describe extended anomalous theories as natural transformations of invertible field theories of this type. We show how in such a framework anomalous theories give rise naturally to homotopy fixed points for n-characters on ∞-groups. By using dimensional reduction on manifolds with boundaries, we show how boundary conditions for n + 1-dimensional TQFTs produce n-dimensional anomalous field theories. Finally, we analyse the case of fully extended TQFTs, and show that any fully extended anomalous theory produces a suitable boundary condition for the anomaly field theory.     

Topological Charges in Gauge Theories

Topological and geometric aspects of gauge theories are examined. The geometry of the fiber-bundle formulation of gauge theories is discussed and compared with the formalism of general relativity. The basic role played by the parallel displacement operator of this geometry is examined. With this operator a gauge independent characterization of various topological singularities and non-singular soliton configurations is carried out.     

Topological Kinks in φ2n Field Theories and Low Energy Excited States

The static kinks and their low energy excited states in scalar field theories with V[φ]= -(1/2)m²φ² + gφ²ⁿ/2n are studied in a function-series method. We give a universal formal solution, and a universal approach to find the low energy excited states as well. Excellent agreement between the FS solution and the exact answer is found. This confirms our predicted results of the energies of the excited quantum states by considering small oscillations about the kink motion.     

Exact Solutions to Pauli-Villars-Regulated Field Theories

We present a new class of quantum field theories which are exactly solvable. The theories are generated by introducing Pauli-Villars (PV) fermionic and bosonic fields with masses degenerate with the physical positive metric fields. An algorithm is given to compute the spectrum and corresponding eigensolutions. We also give the operator solution for a particular case and use it to illustrate some of the tenets of light-cone quantization. Since the solutions of the solvable theory contain ghost quanta, these theories are unphysical. However, the existence of an exact solution provides an important check on the implementation of PV-regulated discretized light-cone quantization (DLCQ). In the limit of exact mass degeneracy of the ghost and physical fields, the numerical DLCQ solutions are constrained to reduce to the explicit forms we give here. We also discuss how perturbation theory in the difference between the masses of the physical and PV particles could be developed, thus generating physical theories. The existence of explicit solutions of the solvable theory also allows one to study the relationship between the equal-time and light-cone vacua and eigensolutions.     

Two-Dimensional Field Theories

Quantum field theories in space with the dimensionality 1 + 1 are considered. Quantum electrodynamics, quantum chromodynamics, sufficiently nonlinear models (with soliton solutions) and gravitational theory are discussed from the common viewpoint. The possible correspondence of two-dimensional field models to physical reality is analyzed.     

Quantum Gravitational Contributions to Gauge Field Theories

We revisit quantum gravitational contributions to quantum gauge field theories in the gauge condition independent Vilkovisky-DeWitt formalism based on the background field method.With the advantage of Landau-DeWitt gauge,we explicitly obtain the gauge condition independent result for the quadratically divergent gravitational corrections to gauge couplings.By employing,in a general way,a scheme-independent regularization method that can preserve both gauge invariance and original divergent behavior of integrals,we show that the resulting gauge coupling is power-law running and asymptotically free.The regularization scheme dependence is clarified by comparing with results obtained by other methods.The loop regularization scheme is found to be applicable for a consistent calculation.     

Algebraic Coset Conformal Field Theories

All unitary Rational Conformal Field Theories (RCFT) are conjectured to be related to unitary coset Conformal Field Theories, i.e., gauged Wess–Zumino–Witten (WZW) models with compact gauge groups. In this paper we use subfactor theory and ideas of algebraic quantum field theory to approach coset Conformal Field Theories. Two conjectures are formulated and their consequences are discussed. Some results are presented which prove the conjectures in special cases. In particular, one of the results states that a class of representations of coset W _N (N≥ 3) algebras with critical parameters are irreducible, and under the natural compositions (Connes' fusion), they generate a finite dimensional fusion ring whose structure constants are completely determined, thus proving a long-standing conjecture about the representations of these algebras. Received: 5 November 1998 / Accepted: 18 October 1999     

Topological invariants for standard model: From semi-metal to topological insulator

We consider topological invariants describing semimetal (gapless) and insulating (gapped) states of the quantum vacuum of Standard Model and possible quantum phase transitions between these states.     

Z/NZ Conformal Field Theories

We compute the modular properties of the possible genus-one characters of some Rational Conformal Field Theories starting from their fusion rules. We show that the possible choices ofS matrices are indexed by some automorphisms of the fusion algebra. We also classify the modular invariant partition functions of these theories. This gives the complete list of modular invariant partition functions of Rational Conformal Field Theories with respect to theA _N ⁽¹⁾ level one algebra.Unité propre de Recherche du Centre National de la Recherche Scientifique, associée à l'Ècole Normale Supérieure et à l'Université de Paris-Sud     

Topological Field Theory and Computing with Instantons

It is well known that dynamical systems may be employed as computing machines. However, not all dynamical systems offer particular advantages compared to the standard paradigm of computation, in regard to efficiency and scalability. Recently, it was suggested that a new type of machines, named digital –hence scalable– memcomputing machines (DMMs), that employ non‐linear dynamical systems with memory, can solve complex Boolean problems efficiently. This result was derived using functional analysis without, however, providing a clear understanding of which physical features make DMMs such an efficient computational tool. Here, we show, using recently proposed topological field theory of dynamical systems, that the solution search by DMMs is a composite instanton. This process effectively breaks the topological supersymmetry common to all dynamical systems, including DMMs. The emergent long‐range order – a collective dynamical behavior– allows logic gates of the machines to correlate arbitrarily far away from each other, despite their non‐quantum character. We exemplify these results with the solution of prime factorization, but the conclusions generalize to DMMs applied to any other Boolean problem.     

Knotted Topological Phase Singularities of Electromagnetic Field

In this paper, knotted objects （RS vortices） in the theory of topological phase singularity in electromagnetic field have been investigated in details. By using the Duan＇s topological current theory, we rewrite the topological current form of RS vortices and use this topological current we reveal that the Hopf invariant of RS vortices is just the sum of the linking and self-linking numbers of the knotted RS vortices. Furthermore, the conservation of the Hopf invariant in the splitting, the mergence and the intersection processes of knotted RS vortices is also discussed.