Quantum strings and the functional calculus

In the present chapter we examine the connections between functional calculus and interpolation spaces. As an ‘appetiser’, in Section 6.1 we present two central ideas: a model describing the real interpolation spaces (X, D(A)) θ,p using the functional calculus and a theorem of Dore. In Section 6.2 we examine the first of these ideas, proving several representation results for the spaces (X, D(Aα)) θ,p . Then we introduce extrapolation spaces for injective operators (Section 6.3). With the help of these spaces in Section 6.4 we derive two fundamental results (Theorem 6.4.2 and Theorem 6.4.5) that lead to more characterisations of interpolation spaces by functional calculus (Section 6.5.1) and a generalisation of Dore’s theorem (Section 6.5.3). In Section 6.6 we establish all the common properties of fractional domain spaces as intermediate spaces: density, the moment inequality, reiteration. Moreover, we prove the intriguing fact that for an operator A ∈ BIP(X) the fractional domain spaces equal the complex interpolation spaces (Theorem 6.6.9). Finally we characterise growth conditions like supt≥1 ‖tθ B(t + A)−1‖ < ∞ in terms of interpolation spaces (Section 6.7).

Tent spaces of vector-valued functions were recently studied by Hytönen, van Neerven and Portal with an eye on applications to $H^{\infty }$-functional calculi. This paper extends their results to the endpoint cases $p=1$ and $p=\infty $ along the lines of earlier work by Harboure, Torrea and Viviani in the scalar-valued case. The main result of the paper is an atomic decomposition in the case $p=1$, which relies on a new geometric argument for cones. A result on the duality of these spaces is also given.

Almost all problems in applied mathematics, including the analysis of dynamical systems, deal with spaces of real-valued functions on Euclidean domains in their formulation and solution. In this paper, we describe the the tool Ariadne, which provides a rigorous calculus for working with Euclidean functions. We first introduce the Ariadne framework, which is based on a clean separation of objects as providing exact, effective, validated and approximate information. We then discuss the function calculus as implemented in Ariadne, including polynomial function models which are the fundamental class for concrete computations. We then consider solution of some core problems of functional analysis, namely solution of algebraic equations and differential equations, and briefly discuss their use for the analysis of hybrid systems. We will give examples of C++ and Python code for performing the various calculations. Finally, we will discuss progress on extensions, including improvements to the function calculus and extensions to more complicated classes of system.

String field theory from Weyl invariance

Some type-based approaches to termination use sized types: an ordinal bound for the size of a data structure is stored in its type. A recursive function over a sized type is accepted if it is visible in the type system that recursive calls occur just at a smaller size. This approach is only sound if the type of the recursive function is admissible, i.e., depends on the size index in a certain way. To explore the space of admissible functions in the presence of higher-kinded data types and impredicative polymorphism, a semantics is developed where sized types are interpreted as functions from ordinals into sets of strongly normalizing terms. It is shown that upper semi-continuity of such functions is a sufficient semantic criterion for admissibility. To provide a syntactical criterion, a calculus for semi-continuous functions is developed.

We discuss the Casimir effect in heterotic string theory. This is done by considering a [Formula: see text] twist acting on one external compact direction and three internal coordinates. The hyperplanes fixed by the orbifold generator [Formula: see text] realize the two infinite parallel plates. For the latter to behave as “conducting material,” we implement in a modular invariant way the projection [Formula: see text] on the spectrum running in the vacuum-to-vacuum amplitude at one-loop. Hence, the relevant projector to account for the Casimir effect is orthogonal to that commonly used in string orbifold models which is [Formula: see text]. We find that this setup yields the same net force acting on the plates in the context of quantum field theory and string theory. However, when supersymmetry is not present from the onset, finiteness of the resultant force in field theory is reached by adding formally infinite forces acting on either side of each plate, while in string theory, both contributions are finite. On the contrary, when supersymmetry is spontaneously broken à la Scherk–Schwarz, finiteness of each contribution is fulfilled in field and string theory.

We investigate the Hopf algebra structure in string worldsheet theory and give a unified formulation of the quantization of string and the space-time symmetry. We reformulate the path integral quantization of string as a Drinfeld twist at the worldsheet level. The coboundary relation shows that the Drinfeld twist defines a module algebra which is equivalent to operators with normal ordering. Upon applying the twist, the space-time diffeomorphism

We introduce a notion of generalized Triebel-Lizorkin spaces associated with sectorial operators in Banach function spaces. Our approach is based on holomorphic functional calculus techniques. Using the concept of \(\mathcal {R}_{s}\)-sectorial operators, which in turn is based on the notion of \(\mathcal {R}_{s}\)-bounded sets of operators introduced by Lutz Weis, we obtain a neat theory including equivalence of various norms and a precise description of real and complex interpolation spaces. Another main result of this article is that an \(\mathcal {R}_{s}\)-sectorial operator always has a bounded H∞-functional calculus in its associated generalized Triebel-Lizorkin spaces.

Functional quantum theory is defined by an isomorphism of the state space H of a conventional quantum theory into an appropriate functional state space D It is a constructive approach to quantum theory in those cases where the state spaces H of physical eigenstates cannot be calculated explicitly like in nonlinear spinor field quantum theory. For the foundation of functional quantum theory appropriate functional state spaces have to be constructed which have to be representation spaces of the corresponding invariance groups. In this paper, this problem is treated for the spinor field. Using anticommuting source operator, it is shown that the construction problem of these spaces is tightly connected with the construction of appropriate relativistic function spaces. This is discussed in detail and explicit representations of the function spaces are given. Imposing no artificial restrictions it follows that the resulting functional spaces are indefinite. Physically the indefiniteness results from the inclusion of tachyon states. It is reasonable to assume a tight connection of these tachyon states with the ghost states introduced by Heisenberg for the regularization of the nonrenormalizable spinor theory

From first to second quantized string theory (II). The dilaton and other fields

In this note we prove an identity that equates the elliptic genus partition function of a supersymmetric sigma model on the N-fold symmetric product of a manifold M to the partition function of a second quantized string theory on the space . The generating function of these elliptic genera is shown to be (almost) an automorphic form for . In the context of D-brane dynamics, this result gives a precise computation of the free energy of a gas of D-strings inside a higher-dimensional brane.

Cosmological quantum string vacua and string-string duality

Ideal fluid dynamics is studied as a relativistic field theory with particular importance on its hamiltonian structure. The Schwinger condition, whose integrated version yields the stress tensor conservation, is explicitly verified both in equal-time and light-cone coordinate systems. We also consider the hamiltonian formulation of fluids interacting with an external gauge field. The complementary roles of the canonical(Noether) stress tensor and the symmetric one obtained by metric variation are discussed. Finally, a non-relativistic reduction of the system in light-cone coordinates has been carried out which reproduces results found earlier in the literature.

Light-cone lattice approach to fermionic theories in 2D: The massive Thirring model

The Galilean covariant approach of field theory is based on a (4 + 1)-dimensional manifold using light-cone coordinates followed by a reduction to (3+1) dimensions. We use this approach to investigate Galilean covariant linear wave equations in a non-commutative phase space. After a brief review of the Galilean covariance formalism, we construct the Galilean Duffin-Kemmer-Petiau equation for the harmonic oscillator in a non-commutative phase space. The exact wave functions and their energy levels are found, and we discuss the effects of non-commutativity.