Advances in Theoretical and Mathematical Physics Articles (Project Euclid)

Equivariant cohomology of the chiral de Rham complex and the half-twisted gauged sigma model

Thu, 05 Aug 2010 15:41 EDT

Meng-Chwan Tan

Source: Adv. Theor. Math. Phys., Volume 13, Number 4, 897--946.

Abstract:
In this paper, we study the perturbative aspects of the half-twisted variant of Witten’s topological A-model coupled to a non-dynamical gauge field with Kähler target space X being a G-manifold. Our main objective is to furnish a purely physical interpretation of the equivariant cohomology of the chiral de Rham complex, recently constructed by Lian and Linshaw, called the “chiral equivariant cohomology.” In doing so, one finds that key mathematical results such as the vanishing in the chiral equivariant cohomology of positive weight classes, lend themselves to straightforward physical explanations. In addition, one can also construct topological invariants of X from the correlation functions of the relevant physical operators corresponding to the nonvanishing weight-zero classes. Via the topological invariance of these correlation functions, one can verify, from a purely physical perspective, the mathematical isomorphism between the weight-zero subspace of the chiral equivariant cohomology and the classical equivariant cohomology of X. Last but not least, one can also determine fully, the de Rham cohomology ring of X/G, from the topological chiral ring generated by the local ground operators of the physical model under study.

On the global structure of the Pomeransky–Senkov black holes

Tue, 24 Apr 2012 09:18 EDT

Piotr T. Chrúsciel, Julien Cortier, Alfonso García-Parrado Gómez-Lobo

Source: Adv. Theor. Math. Phys., Volume 14, Number 6, 1779--1856.

Abstract:
We construct analytic extensions of the Pomeransky–Senkov metrics with multiple Killing horizons and asymptotic regions. We show that, in our extensions, the singularities associated to an obstruction to differentiability of the metric lie beyond event horizons. We analyze the topology of the non-empty singular set, which turns out to be parameterdependent. The resulting global structure is somewhat reminiscent of that of Kerr space-time.

Wall-crossing of D4-branes using flow trees

Tue, 24 Apr 2012 10:48 EDT

Jan Manschot

Source: Adv. Theor. Math. Phys., Volume 15, Number 1, 1--42.

Abstract:
The moduli dependence of D4-branes on a Calabi–Yau manifold is studied using attractor flow trees, in the large volume limit of the Kähler cone. One of the moduli-dependent existence criteria of flow trees is the positivity of the flow parameters along its edges. It is shown that the sign of the flow parameters can be determined iteratively as function of the initial moduli, without explicit calculation of the flow of the moduli in the tree. Using this result, an indefinite quadratic form, which appears in the expression for the D4-D2-D0 BPS mass in the large volume limit, is proven to be positive definite for flow trees with 3 or less endpoints. The contribution of these flow trees to the BPS partition function is therefore convergent. From non-primitive wall crossing is deduced that the S-duality invariant partition function must be a generating function of the rational invariants $\bar{\Omega} (\Gamma) = \Sigma_{m|\Gamma} =\frac {\Omega(\Gamma/m)}{m^2}$ instead of the integer invariants $\Omega(\Gamma)$.

Invertible defects and isomorphisms of rational CFTs

Tue, 24 Apr 2012 10:48 EDT

Alexei Davydov, Liang Kong, Ingo Runkel

Source: Adv. Theor. Math. Phys., Volume 15, Number 1, 43--69.

Abstract:
Given two two-dimensional conformal field theories, a domain wall — or defect line—between them is called invertible if there is another defect with which it fuses to the identity defect. A defect is called topological if it is transparent to the stress tensor. A conformal isomorphism between the two CFTs is a linear isomorphism between their state spaces which preserves the stress tensor and is compatible with the operator product expansion. We show that for rational CFTs there is a one-to-one correspondence between invertible topological defects and conformal isomorphisms if both preserve the rational symmetry. This correspondence is compatible with composition.

Supersymmetric surface operators, four-manifold theory and invariants in various dimensions

Tue, 24 Apr 2012 10:48 EDT

Meng-Chwan Tan

Source: Adv. Theor. Math. Phys., Volume 15, Number 1, 71--129.

Abstract:
We continue our program initiated in "Integration over the u-plane in Donaldson theory with surface operators," J. High Energy Phys. 05 (2011), to consider supersymmetric surface operators in a topologically twisted ${\cal N}=2$ pure ${\it SU}(2)$ gauge theory, and apply them to the study of four-manifolds and related invariants. Elegant physical proofs of various seminal theorems in four-manifold theory obtained by Ozsáth and Szabó and Taubes, will be furnished. In particular, we will show that Taubes’ groundbreaking and difficult result — that the ordinary SW invariants are in fact the Gromov invariants which count pseudo-holomorphic curves embedded in a symplectic four-manifold $X$ — nonetheless lends itself to a simple and concrete physical derivation in the presence of “ordinary” surface operators. As an offshoot, we will be led to several interesting and mathematically novel identities among the Gromov and “ramified” SW invariants of $X$, which in certain cases, also involve the instanton and monopole Floer homologies of its three-submanifold. Via these identities, and a physical formulation of the “ramified” Donaldson invariants of four-manifolds with boundaries, we will uncover completely new and economical ways of deriving and understanding various important mathematical results concerning (i) knot homology groups from “ramified” instantons by Kronheimer and Mrowka; and (ii) monopole Floer homology and SW theory on symplectic four-manifolds by Kutluhan–Taubes. Supersymmetry, as well as other physical concepts such as $R$-invariance, electric–magnetic duality, spontaneous gauge symmetry breaking and localization onto supersymmetric configurations in topologically twisted quantum field theories, play a pivotal role in our story.

Non-Kähler heterotic rotations

Tue, 24 Apr 2012 10:48 EDT

Dario Martelli, James Sparks

Source: Adv. Theor. Math. Phys., Volume 15, Number 1, 131--174.

Abstract:
We study a supersymmetry-preserving solution-generating method in heterotic supergravity. In particular, we use this method to construct one-parameter non-Kähler deformations of Calabi–Yau manifolds with a $U(1)$ isometry, in which the complex structure remains invariant. We explain how to obtain corresponding solutions to heterotic string theory, up to first order in $\alpha'$, by a modified form of the standard embedding. In the course of the paper we also show that Abelian heterotic supergravity embeds into type II supergravity, and note that the solution-generating method in this context is related to a dipole-type deformation when there is a field theory dual.

Stable causality of the Pomeransky–Senkov black holes

Tue, 24 Apr 2012 10:48 EDT

Piotr T. Chrúsciel, Sebastian J. Szybka

Source: Adv. Theor. Math. Phys., Volume 15, Number 1, 175--178.

Abstract:
We show stable causality of the Pomeransky–Senkov black rings.

Two-dimensional topological field theories as taffy

Tue, 24 Apr 2012 10:48 EDT

Matt Ando, Eric Sharpe

Source: Adv. Theor. Math. Phys., Volume 15, Number 1, 179--244.

Abstract:
In this paper, we use trivial defects to define global taffy-like operations on string worldsheets, which preserve the field theory. We fold open and closed strings on a space $X$ into open strings on products of multiple copies of $X$, and perform checks that the “taffy-folded” worldsheets have the same massless spectra and other properties as the original worldsheets. Such folding tricks are a standard method in the defects community; the novelty of this paper lies in deriving mathematical identities to check that e.g., massless spectra are invariant in topological field theories. We discuss the case of the B model extensively, and also derive the same identities for string topology, where they become statements of homotopy invariance. We outline analogous results in the A model, B-twisted Landau–Ginzburg models, and physical strings. We also discuss the understanding of the closed string states as the Hochschild homology of the open string algebra, and outline possible applications to elliptic genera.

Compact supersymmetric solutions of the heterotic equations of motion in dimensions 7 and 8

Fri, 25 May 2012 09:18 EDT

Marisa Fernández, Stefan Ivanov, Luis Ugarte, Raquel Villacampa

Source: Adv. Theor. Math. Phys., Volume 15, Number 2, 245--284.

Abstract:
We construct explicit compact solutions with non-zero field strength, non-flat instanton and constant dilaton to the heterotic string equations in dimensions 7 and 8. We present a quadratic condition on the curvature, which is necessary and sufficient the heterotic supersymmetry and the anomaly cancellation to imply the heterotic equations of motion in dimensions 7 and 8. We show that some of our examples are compact supersymmetric solutions of the heterotic equations of motion in dimensions 7 and 8.

The diffeomorphism supergroup of a finite-dimensional supermanifold

Fri, 25 May 2012 09:18 EDT

C. Sachse, C. Wockel

Source: Adv. Theor. Math. Phys., Volume 15, Number 2, 285--323.

Abstract:
Using the categorical description of supergeometry we give an explicit construction of the diffeomorphism supergroup of a compact finitedimensional supermanifold. The construction provides the diffeomorphism supergroup with the structure of a Fréchet supermanifold. In addition, we derive results about the structure of diffeomorphism supergroups.

String universality in six dimensions

Fri, 25 May 2012 09:18 EDT

Vijay Kumar, Washington Taylor

Source: Adv. Theor. Math. Phys., Volume 15, Number 2, 325--353.

Abstract:
In six dimensions, cancellation of gauge, gravitational, and mixed anomalies strongly constrains the set of quantum field theories, which can be coupled consistently to gravity. We show that for some classes of six-dimensional (6D) supersymmetric gauge theories coupled to gravity, the anomaly cancellation conditions are equivalent to tadpole cancellation and other constraints on the matter content of heterotic/type I compactifications on K3. In these cases, all consistent 6D supergravity theories have a realization in string theory. We find one example that may arise from a novel string compactification, and we identify a new infinite family of models satisfying anomaly factorization. We find, however, that this infinite family of models, as well as other infinite families of models previously identified by Schwarz are pathological. We suggest that it may be feasible to demonstrate that there is a string theoretic realization of all consistent 6D supergravity theories which have Lagrangian descriptions with arbitrary gauge and matter content. We attempt to frame this hypothesis of string universality as a concrete conjecture.

Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime

Fri, 25 May 2012 09:18 EDT

Claudio Dappiaggi, Valter Moretti, Nicola Pinamonti

Source: Adv. Theor. Math. Phys., Volume 15, Number 2, 355--447.

Abstract:
The discovery of the radiation properties of black holes prompted the search for a natural candidate quantum ground state for a massless scalar field theory on Schwarzschild spacetime, here considered in the Eddington–Finkelstein representation. Among the several available proposals in the literature, an important physical role is played by the so-called Unruh state, which is supposed to be appropriate to capture the physics of a black hole formed by spherically symmetric collapsing matter. Within this respect, we shall consider a massless Klein–Gordon field and we shall rigorously and globally construct such state, that is on the algebra of Weyl observables localised in the union of the static external region, the future event horizon and the non-static black hole region. Eventually, out of a careful use of microlocal techniques, we prove that the built state fulfils, where defined, the so-called Hadamard condition; hence, it is perturbatively stable, in other words realizing the natural candidate with which one could study purely quantum phenomena such as the role of the back reaction of Hawking’s radiation. From a geometrical point of view, we shall make a profitable use of a bulk-to-boundary reconstruction technique which carefully exploits the Killing horizon structure as well as the conformal asymptotic behaviour of the underlying background. From an analytical point of view, our tools will range from Hörmander’s theorem on propagation of singularities, results on the role of passive states, and a detailed use of the recently discovered peeling behaviour of the solutions of the wave equation in Schwarzschild spacetime.

Cyclotron braid group approach to Laughlin correlations

Fri, 25 May 2012 09:18 EDT

Janusz Jacak, Ireneusz Jóźwiak, Lucjan Jacak, Konrad Wieczorek

Source: Adv. Theor. Math. Phys., Volume 15, Number 2, 449--469.

Abstract:
Homotopy braid group description including cyclotron motion of charged interacting two-dimensional (2D) particles at strong magnetic field presence is developed in order to explain, in algebraic topology terms, Laughlin correlations in fractional quantum Hall systems. There are introduced special cyclotron braid subgroups of a full braid group with 1D unitary representations suitable to satisfy Laughlin correlation requirements. In this way an implementation of composite fermions (fermions with auxiliary flux quanta attached in order to reproduce Laughlin correlations) is formulated within uniform for all 2D particles braid group approach. The fictitious fluxes — vortices attached to the composite fermions in a traditional formulation are replaced with additional cyclotron trajectory loops unavoidably occurring when ordinary cyclotron radius is too short in comparison to particle separation and does not allow for particle interchanges along single-loop cyclotron braids. Additional loops enhance the effective cyclotron radius and restore particle interchanges. A new type of 2D particles — composite anyons is also defined via unitary representations of cyclotron braid subgroups. It is demonstrated that composite fermions and composite anyons are rightful 2D particles, not auxiliary compositions with fictitious fluxes and are associated with cyclotron braid subgroups instead of the full braid group, which may open also a new opportunity for non-Abelian composite anyons for topological quantum information processing applications, due to richer representations of subgroup than of a group.

Quantization of the Hitchin moduli spaces, Liouville theory and the geometric Langlands correspondence I

Fri, 25 May 2012 09:18 EDT

J. Teschner

Source: Adv. Theor. Math. Phys., Volume 15, Number 2, 471--564.

Abstract:
We discuss the relation between Liouville theory and the Hitchin integrable system, which can be seen in two ways as a two step process involving quantization and hyperkähler rotation. The modular duality of Liouville theory and the relation between Liouville theory and the SL(2)-WZNW-model give a new perspective on the geometric Langlands correspondence and on its relation to conformal field theory.

The quantum chiral Minkowski and conformal superspaces

Fri, 25 May 2012 09:18 EDT

Dalia Cervantes, Rita Fioresi, María A. Lledó

Source: Adv. Theor. Math. Phys., Volume 15, Number 2, 565--620.

Abstract:
We give a quantum deformation of the chiral super Minkowski space in four dimensions as the big cell inside a quantum super Grassmannian. The quantization is performed in such way that the actions of the Poincar´e and conformal quantum supergroups on the quantum Minkowski and quantum conformal superspaces are preserved.

Bundle gerbes for orientifold sigma models

Sun, 10 Jun 2012 20:24 EDT

Krzysztof Gawędzki, Rafał R. Suszek, Konrad Waldorf

Source: Adv. Theor. Math. Phys., Volume 15, Number 3, 621--687.

Abstract:
Bundle gerbes with connection and their modules play an important role in the theory of two-dimensional sigma models with a background Wess–Zumino flux: their holonomy determines the contribution of the flux to the Feynman amplitudes of classical fields. We discuss additional structures on bundle gerbes and gerbe modules needed in similar constructions for orientifold sigma models describing closed and open strings.

Decoupling gravity in F-theory

Sun, 10 Jun 2012 20:24 EDT

Clay Córdova

Source: Adv. Theor. Math. Phys., Volume 15, Number 3, 689--740.

Abstract:
We study seven-brane SU(5) GUT models of string phenomenology which can be consistently analyzed in a purely local framework. The requirement that gravity can decouple constrains the form of four-dimensional physics as well as the geometry of spacetime. We rule out a large family of candidate UV completions of such models and derive a priori constraints on the local singularities of compact elliptic Calabi– Yau four-folds. These constraints are strong enough to obstruct a wide class of brane constructions from UV completion in string theory. It is demonstrated that consistent local models always have exotic Yukawa coupling structures, and hidden sectors or interesting non-perturbative superpotentials which merit further investigation.

Superstring field theory in the democratic picture

Sun, 10 Jun 2012 20:24 EDT

Michael Kroyter

Source: Adv. Theor. Math. Phys., Volume 15, Number 3, 741--781.

Abstract:
We present a new open superstring field theory, whose string fields carry an arbitrary picture number and reside in the large Hilbert space. The redundancy related to picture number is resolved by treating picture changing as a gauge transformation. A mid-point insertion is imperative for this formalism. We find that this mid-point insertion must include all multi-picture-changing operators. It is also proven that this insertion as well as all the multi-picture-changing operators are zero weight conformal primaries. This new theory solves the problems with the Ramond sector shared by other Ramond-Neveu-Schwarz (RNS) string field theories, while naturally unifying the Neveu-Schwarz (NS) and Ramond string fields. When partially gauge fixed, it reduces in the NS sector to the modified cubic superstring field theory. Hence, it shares all the good properties of this theory, e.g., it has analytical vacuum and marginal deformation solutions. Treating the redundant gauge symmetry using the Batalin-Vilkovisky (BV) formalism is straightforward and results in a cubic action with a single string field, whose quantum numbers are unconstrained. The generalization to an arbitrary brane system is simple and includes the standard Chan–Paton factors and the most general string field consistent with the brane system.

On a possible approach to general field theories with nonpolynomial interactions

Sun, 10 Jun 2012 20:24 EDT

Franco Ferrari

Source: Adv. Theor. Math. Phys., Volume 15, Number 3, 783--799.

Abstract:
In this work, a class of field theories with self-interactions described by a potential of the kind $V (\phi(x) − \phi(x_0))$ is studied. $\phi$ is a massive scalar field and $x$, $x_0$ are points in a $d$-dimensional space. Under the condition that the potential admits the Fourier representation, it is shown that such theories may be mapped into a standard field theory, in which the interaction of the new fields is a polynomial of fourth degree. With some restrictions, this mapping allows the perturbative treatment of models that are otherwise intractable with standard field theoretical methods. A nonperturbative approach to these theories is attempted. The original scalar field $\phi$ is integrated out exactly at the price of introducing auxiliary vector fields. The latter are treated in a mean field theory approximation. The singularities that arise after the elimination of the auxiliary fields are cured using the dimensional regularization. The expression of the counterterms to be subtracted is computed.

Large-spin asymptotics of Euclidean LQG flat-space wavefunctions

Sun, 10 Jun 2012 20:24 EDT

Aleksandar Miković, Marko Vojinović

Source: Adv. Theor. Math. Phys., Volume 15, Number 3, 801--847.

Abstract:
We analyze the large-spin asymptotics of a class of spin-network wavefunctions of Euclidean loop quantum gravity, which corresponds to a flat spacetime. A wavefunction from this class can be represented as a sum over the spins of an amplitude for a spin network whose graph is a composition of the wavefunction spin network graph with the dual one-complex graph and the tetrahedron graphs for a triangulation of the spatial 3-manifold. This spin-network amplitude can be represented as a product of $6j$ symbols, which is then used to find the large-spin asymptotics of the wavefunction. By using the Laplace method we show that the large-spin asymptotics is given by a sum of Gaussian functions. However, these Gaussian functions are not of the type, which gives the correct graviton propagator.

D-branes, surface operators, and ADHM quiver representations

Sun, 10 Jun 2012 20:24 EDT

Ugo Bruzzo, Wu-Yen Chuang, Duiliu-Emanuel Diaconescu, Marcos Jardim, G. Pan, Yi Zhang

Source: Adv. Theor. Math. Phys., Volume 15, Number 3, 849--911.

Abstract:
A supersymmetric quantum mechanical model is constructed for BPS states bound to surface operators in five dimensional $SU(r)$ gauge theories using D-brane engineering. This model represents the effective action of a certain D2-brane configuration, and is naturally obtained by dimensional reduction of a quiver (0, 2) gauged linear sigma model. In a special stability chamber, the resulting moduli space of quiver representations is shown to be smooth and isomorphic to a moduli space of framed quotients on the projective plane. A precise conjecture relating a K-theoretic partition function of this moduli space to refined open-string invariants of toric Lagrangian branes is formulated for conifold and local $\mathbb{P}^1 \times \mathbb{P}^1$ geometries.

Perturbative study of the transfer matrix on the string worldsheet in ${\rm AdS}^5 \times S_5$

Mon, 11 Jun 2012 14:12 EDT

Andrei Mikhailov, Sakura Schäfer-Nameki

Source: Adv. Theor. Math. Phys., Volume 15, Number 4, 913--971.

Abstract:
Quantum non-local charges are central to the quantum integrability of a sigma model. In this paper we study the quantum consistency and UV finiteness of non-local charges of string theory in ${\rm AdS}^5 \times S_5$. We use the pure spinor formalism. We develop the near-flat space expansion of the transfer matrix and calculate the one-loop divergences. We find that the logarithmic divergences cancel at the level of one loop. This gives strong support to the quantum integrability of the full string theory. We develop a calculational setup for the renormalization group analysis of Wilson line type of operators on the string worldsheet.

Classification of free actions on complete intersections of four quadrics

Mon, 11 Jun 2012 14:12 EDT

Zheng Hua

Source: Adv. Theor. Math. Phys., Volume 15, Number 4, 973--990.

Abstract:
In this paper we classify all free actions of finite groups on Calabi–Yau complete intersection of four quadrics in $\mathbb{P}^7$, up to projective equivalence. We get some examples of smooth Calabi–Yau three-folds with large nonabelian fundamental groups. We also observe the relation between some of these examples and moduli of polarized abelian surfaces.

On the final definition of the causal boundary and its relation with the conformal boundary

Mon, 11 Jun 2012 14:12 EDT

José Luis Flores, Jónatan Herrera, Miguel Sánchez

Source: Adv. Theor. Math. Phys., Volume 15, Number 4, 991--1057.

Abstract:
The notion of causal boundary $\partial M$ for a strongly causal spacetime $M$ has been a controversial topic along last decades: on one hand, some attempted definitions were not fully consistent, on the other, there were simple examples where an open conformal embedding $i : M \hookrightarrow M_0$ could be defined, but the corresponding conformal boundary $\partial_i M$ disagreed drastically with the causal one. Nevertheless, the recent progress in this topic suggests that a final option for $\partial M$ is available in most cases. Our study has two parts: (I) To give general arguments on a boundary in order to ensure that it is admissible as a causal boundary at the three natural levels, i.e., as a point set, as a chronological space and as a topological space. Then, the essential uniqueness of our choice is stressed, and the relatively few admissible alternatives are discussed. (II) To analyze the role of the conformal boundary $\partial_i M$. We show that, in general, $\partial_i M$ may present a very undesirable structure. Nevertheless, it is well behaved under certain general assumptions, and its accessible part $\partial^*_i M$ agrees with the causal boundary. This study justifies both boundaries. On one hand, the conformal boundary $\partial^*_i M$, which cannot be defined for a general spacetime but is easily computed in particular examples, appears now as a special case of the causal boundary. On the other, the new redefinition of the causal boundary not only is free of inconsistencies and applicable to any strongly causal spacetime, but also recovers the expected structure in the cases where a natural simple conformal boundary is available. The cases of globally hyperbolic spacetimes and asymptotically conformally flat ends are especially studied.

Lie crossed modules and gauge-invariant actions for 2-BF theories

Mon, 11 Jun 2012 14:12 EDT

João Martins, Aleksandar Miković

Source: Adv. Theor. Math. Phys., Volume 15, Number 4, 1059--1084.

Abstract:
We generalize the BF theory action to the case of a general Lie crossed module $(\partial : H \to G, \triangleright)$, where $G$ and $H$ are non-abelian Lie groups. Our construction requires the existence of $G$-invariant non-degenerate bilinear forms on the Lie algebras of $G$ and $H$ and we show that there are many examples of such Lie crossed modules by using the construction of crossed modules provided by short chain complexes of vector spaces. We also generalize this construction to an arbitrary chain complex of vector spaces, of finite type. We construct two gauge-invariant actions for 2-flat and fake-flat 2-connections with auxiliary fields. The first action is of the same type as the BFCG action introduced by Girelli, Pfeiffer and Popescu for a special class of Lie crossed modules, where $H$ is abelian. The second action is an extended BFCG action which contains an additional auxiliary field. However, these two actions are related by a field redefinition. We also construct a three-parameter deformation of the extended BFCG action, which we believe to be relevant for the construction of non-trivial invariants of knotted surfaces embedded in the four-sphere.

Null asymptotics of solutions of the Einstein–Maxwell equations in general relativity and gravitational radiation

Mon, 11 Jun 2012 14:12 EDT

Lydia Bieri, PoNing Chen, Shing-Tung Yau

Source: Adv. Theor. Math. Phys., Volume 15, Number 4, 1085--1113.

Abstract:
We prove that for spacetimes solving the Einstein–Maxwell (EM) equations, the electromagnetic field contributes at highest order to the nonlinear memory effect of gravitational waves. In Nonlinear nature of gravitation and gravitational-wave experiments, Christodoulou showed that gravitational waves have a nonlinear memory. He discussed how this effect can be measured as a permanent displacement of test masses in a laser interferometer gravitational-wave detector. Christodoulou derived a precise formula for this permanent displacement in the Einstein vacuum (EV) case. We prove in Theorem 2.6 that for the EM equations this permanent displacement exhibits a term coming from the electromagnetic field. This term is at the same highest order as the purely gravitational term that governs the EV situation. On the other hand, in Section 3, we show that to leading order, the presence of the electromagnetic field does not change the instantaneous displacement of the test masses. Following the method introduced by Christodoulou in Nonlinear nature of gravitation and gravitational-wave experiments, and asymptotics derived by Zipser in The global nonlinear stability of the trivial solution of the Einstein–Maxwell equations and Extensions of the stability theorem of the Minkowski space in general relativity: Solutions of the Einstein–Maxwell equations , we investigate gravitational radiation at null infinity in spacetimes solving the EM equations. We study the Bondi mass loss formula at null infinity derived in Extensions of the stability theorem of the Minkowski space in general relativity: Solutions of the Einstein–Maxwell equations . We show that the mass loss formula from Extensions of the stability theorem of the Minkowski space in general relativity: Solutions of the Einstein–Maxwell equations is compatible with the one in Bondi coordinates obtained in Gravitational waves in general relativity X: Asymptotic expansions for the Einstein–Maxwell field . And we observe that the presence of the electromagnetic field increases the total energy radiated to infinity up to leading order. Moreover, we compute the limit of the area radius at null infinity in Theorem 2.7.

Plane-symmetric spacetimes with positive cosmological constant: The case of stiff fluids

Mon, 11 Jun 2012 14:12 EDT

Philippe G. LeFloch, Sophonie B. Tchapnda

Source: Adv. Theor. Math. Phys., Volume 15, Number 4, 1115--1140.

Abstract:
We consider plane-symmetric spacetimes satisfying Einstein’s field equations with positive cosmological constant, when the matter is a fluid whose pressure is equal to its mass-energy density (i.e., a so-called stiff fluid). We study the initial-value problem for the associated Einstein equations and establish a global existence result. The late-time asymptotics of solutions is also rigorously derived, and we conclude that the spacetime approaches the de Sitter spacetime while the matter disperses asymptotically. A technical difficulty dealt with here lies in the fact that solutions may contain vacuum states as well as velocities approaching the speed of light, both possibilities leading to singular behavior in the evolution equations.

Sums over topological sectors and quantization of Fayet–Iliopoulos parameters

Mon, 11 Jun 2012 14:12 EDT

Simeon Hellerman, Eric Sharpe

Source: Adv. Theor. Math. Phys., Volume 15, Number 4, 1141--1199.

Abstract:
In this paper we discuss quantization of the Fayet–Iliopoulos parameter in supergravity theories with altered nonperturbative sectors, which were recently used to argue a fractional quantization condition. Nonlinear sigma models with altered nonperturbative sectors are the same as nonlinear sigma models on special stacks known as gerbes. After reviewing the existing results on such theories in two dimensions, we discuss examples of gerby moduli “spaces” appearing in four-dimensional field theory and string compactifications, and the effect of various dualities. We discuss global topological defects arising when a field or string theory moduli space has a gerbe structure. We also outline how to generalize the results of Bagger–Witten and more recent authors on quantization issues in supergravities from smooth moduli spaces to smooth moduli stacks, focusing particular attention on stacks that have gerbe structures.

The $n$-point functions for intersection numbers on moduli spaces of curves

Wed, 10 Oct 2012 10:25 EDT

Kefeng Liu, Hao Xu

Source: Adv. Theor. Math. Phys., Volume 15, Number 5, 1201--1236.

Abstract:
Using the celebrated Witten–Kontsevich theorem, we prove a recursive formula of the $n$-point functions for intersection numbers on moduli spaces of curves. It has been used to prove the Faber intersection number conjecture and motivated us to find some conjectural vanishing identities for Gromov–Witten invariants. The latter has been proved recently by Liu and Pandharipande. We also give a combinatorial interpretation of $n$-point functions in terms of summation over binary trees.

Model building with $F$-theory

Wed, 10 Oct 2012 10:25 EDT

Ron Donagi, Martijn Wijnholt

Source: Adv. Theor. Math. Phys., Volume 15, Number 5, 1237--1317.

Abstract:
Despite much recent progress in model building with $D$-branes, it has been problematic to find a completely convincing explanation of gauge coupling unification. We extend the class of models by considering $F$-theory compactifications, which may incorporate unification more naturally. We explain how to derive the charged chiral spectrum and Yukawa couplings in $N = 1$ compactifications of $F$-theory with $G$-flux. In a class of models which admit perturbative heterotic duals, we show that the $F$-theory and heterotic computations match.

Symmetries of massless vertex operators in $AdS_5 × S^5$

Wed, 10 Oct 2012 10:25 EDT

Andrei Mikhailov

Source: Adv. Theor. Math. Phys., Volume 15, Number 5, 1319--1372.

Abstract:
The worldsheet sigma-model of the superstring in $AdS_5 × S^5$ has a one-parameter family of flat connections parametrized by the spectral parameter. The corresponding Wilson line is not BRST invariant for an open contour, because the BRST transformation leads to boundary terms. These boundary terms define a cohomological complex associated to the endpoint of the contour. We study the cohomology of this complex for Wilson lines in some infinite-dimensional representations. We find that for these representations the cohomology is nontrivial at the ghost number 2. This implies that it is possible to define a BRST invariant open Wilson line. The central point in the construction is the existence of massless vertex operators transforming exactly covariantly under the action of the global symmetry group. In flat space massless vertices transform covariantly only up to adding BRST exact terms. But in AdS we show that it is possible to define vertices so that they transform exactly covariantly.

Division algebras and supersymmetry II

Wed, 10 Oct 2012 10:25 EDT

John C. Baez, John Huerta

Source: Adv. Theor. Math. Phys., Volume 15, Number 5, 1373--1410.

Abstract:
Starting from the four normed division algebras — the real numbers, complex numbers, quaternions and octonions — a systematic procedure gives a 3-cocycle on the Poincaré Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure gives a 4-cocycle on the Poincaré Lie superalgebra in dimensions 4, 5, 7 and 11. In general, an$ (n + 1)$-cocycle on a Lie superalgebra yields a “Lie $n$-superalgebra”: that is, roughly speaking, an $n$-term chain complex equipped with a bracket satisfying the axioms of a Lie superalgebra up to chain homotopy. We thus obtain Lie 2-superalgebras extending the Poincaré superalgebra in dimensions 3, 4, 6 and 10, and Lie 3-superalgebras extending the Poincaré superalgebra in dimensions 4, 5, 7 and 11. As shown in Sati, Schreiber and Stasheff’s work on higher gauge theory, Lie 2-superalgebra connections describe the parallel transport of strings, while Lie 3-superalgebra connections describe the parallel transport of 2-branes. Moreover, in the octonionic case, these connections concisely summarize the fields appearing in 10- and 11-dimensional supergravity.

Ricci-flow-conjugated initial data sets for Einstein equations

Wed, 10 Oct 2012 10:25 EDT

Mauro Carfora

Source: Adv. Theor. Math. Phys., Volume 15, Number 5, 1411--1484.

Abstract:
We discuss a natural form of Ricci-flow conjugation between two distinct general relativistic data sets given on a compact $n \ge 3$-dimensional manifold $\Sigma$. We establish the existence of the relevant entropy functionals for the matter and geometrical variables, their monotonicity properties, and the associated convergence in the appropriate sense. We show that in such a framework there is a natural mode expansion generated by the spectral resolution of the Ricci conjugate Hodge–DeRham operator. This mode expansion allows one to compare the two distinct data sets and gives rise to a computable heat kernel expansion of the fluctuations among the fields defining the data. In particular, this shows that Ricciflow conjugation entails a natural form of $L^2$ parabolic averaging of one data set with respect to the other with a number of desirable properties: (i) It preserves the dominant energy condition; (ii) It is localized by a heat kernel whose support sets the scale of averaging; (iii) It is characterized by a set of balance functionals, that allow the analysis of its entropic stability.

Super Yangian of superstring on $\mathrm{AdS}_5 × S^5$ revisited

Wed, 10 Oct 2012 10:25 EDT

Machiko Hatsuda, Kentaroh Yoshida

Source: Adv. Theor. Math. Phys., Volume 15, Number 5, 1485--1501.

Abstract:
We construct infinite number of conserved nonlocal charges for type IIB superstring on the $\mathrm{AdS}_5×S^5$ space in the conformal gauge without assuming any $\kappa$ gauge fixing, and show that they satisfy the super Yangian algebra. The resultant algebra is the same as our previous work, where a special gauge was assumed in such a way that the Noether current satisfies a flatness condition. However the flatness condition for the Noether current of a superstring on the AdS space is broken in general. We show that the anomalous contribution is absorbed into the current where fermionic constraints play an essential role, and a resultant conserved nonlocal charge has different expression satisfying the same super Yangian algebra.

Hermitian–Einstein connections on polystable parabolic principal Higgs bundles

Wed, 10 Oct 2012 10:25 EDT

Indranil Biswas, Matthias Stemmler

Source: Adv. Theor. Math. Phys., Volume 15, Number 5, 1503--1521.

Abstract:
Given a smooth complex projective variety $X$ and a smooth divisor $D$ on $X$, we prove the existence of Hermitian–Einstein connections, with respect to a Poincaré-type metric on $X \setminus D$, on polystable parabolic principal Higgs bundles with parabolic structure over $D$, satisfying certain conditions on their restriction to $D$.

Breaking GUT groups in $F$-theory

Wed, 12 Dec 2012 09:19 EST

Ron Donagi, Martijn Wijnholt

Source: Adv. Theor. Math. Phys., Volume 15, Number 6, 1523--1603.

Abstract:
We consider the possibility of breaking the GUT group to the Standard Model gauge group in $F$-theory compactifications by turning on certain $U(1)$ fluxes. We show that the requirement of massless hypercharge is equivalent to a topological constraint on the UV completion of the local model. The possibility of this mechanism is intrinsic to $F$-theory. We address some of the phenomenological signatures of this scenario. We show that our models predict monopoles as in conventional GUT models. We discuss in detail the leading threshold corrections to the gauge kinetic terms and their effect on unification. They turn out to be related to Ray–Singer torsion. We also discuss the issue of proton decay in $F$-theory models and explain how to engineer models which satisfy current experimental bounds.

$E_7$ groups from octonionic magic square

Wed, 12 Dec 2012 09:19 EST

Sergio L. Cacciatori, Francesco Dalla Piazza, Antonio Scotti

Source: Adv. Theor. Math. Phys., Volume 15, Number 6, 1605--1654.

Abstract:
In this paper, we continue our program, started in Euler angles for $G(2)$ , of building up explicit generalized Euler angle parameterizations for all exceptional compact Lie groups. Here we solve the problem for $E_7$, by first providing explicit matrix realizations of the Tits construction of a Magic Square product between the exceptional octonionic algebra $\mathfrak{J}$ and the quaternionic algebra $\mathbb{H}$, both in the adjoint and the 56-dimensional representations. Then, we provide the Euler parametrization of $E_7$ starting from its maximal subgroup $U = (E_6 \times U(1))/\mathbb{Z}_3$. Next, we give the constructions for all the other maximal compact subgroups.

Riemann–Hilbert approach to the time-dependent generalized sine kernel

Wed, 12 Dec 2012 09:19 EST

Karol Kajetan Kozlowski

Source: Adv. Theor. Math. Phys., Volume 15, Number 6, 1655--1743.

Abstract:
We derive the leading asymptotic behavior and build a new series representation for the Fredholm determinant of integrable integral operators appearing in the representation of the time and distance-dependent correlation functions of integrable models described by a six-vertex $R$-matrix. This series representation opens a systematic way for the computation of the long-time, long-distance asymptotic expansion for the correlation functions of the aforementioned integrable models away from their free fermion point. Our method builds on a Riemann–Hilbert based analysis.

A rigid Calabi–Yau three-fold

Wed, 12 Dec 2012 09:19 EST

Sara Angela Filippini, Alice Garbagnati

Source: Adv. Theor. Math. Phys., Volume 15, Number 6, 1745--1787.

Abstract:
The aim of this paper is to analyze some geometric properties of the rigid Calabi–Yau three-fold $\mathcal{Z}$ obtained by a quotient of $E_3$, where $E$ is a specific elliptic curve. We describe the cohomology of $\mathcal{Z}$ and give a simple formula for the trilinear form on $\mathrm{Pic}(\mathcal{Z})$. We describe some projective models of $\mathcal{Z}$ and relate these to its generalized mirror. A smoothing of a singular model is a Calabi–Yau three-fold with small Hodge numbers which was not known before.

The wave equation in a general spherically symmetric black hole geometry

Wed, 12 Dec 2012 09:19 EST

Matthew Masarik

Source: Adv. Theor. Math. Phys., Volume 15, Number 6, 1789--1815.

Abstract:
We consider the Cauchy problem for the wave equation in a general class of spherically symmetric black hole geometries. Under certain mild conditions on the far-field decay and the singularity, we show that there is a unique globally smooth solution to the Cauchy problem for the wave equation with data compactly supported away from the horizon that is compactly supported for all times and decays in $L_{loc}^\infty$ as $t$ tends to infinity . We obtain as a corollary that in the geometry of black hole solutions of the SU(2) Einstein/Yang–Mills equations, solutions to the wave equation with compactly supported initial data decay as $t$ goes to infinity.

Algebraic deformations of toric varieties II: noncommutative instantons

Wed, 12 Dec 2012 09:19 EST

Lucio Cirio, Giovanni Landi, Richard J. Szabo

Source: Adv. Theor. Math. Phys., Volume 15, Number 6, 1817--1907.

Abstract:
We continue our study of the noncommutative algebraic and differential geometry of a particular class of deformations of toric varieties, focusing on aspects pertinent to the construction and enumeration of noncommutative instantons on these varieties. We develop a noncommutative version of twistor theory, which introduces a new example of a noncommutative four-sphere. We develop a braided version of the ADHM construction and show that it parameterizes a certain moduli space of framed torsion free sheaves on a noncommutative projective plane. We use these constructions to explicitly build instanton gauge bundles with canonical connections on the noncommutative four-sphere that satisfy appropriate anti-selfduality equations. We construct projective moduli spaces for the torsion free sheaves and demonstrate that they are smooth. We define equivariant partition functions of these moduli spaces, finding that they coincide with the usual instanton partition functions for supersymmetric gauge theories on $\mathbb{C}2$.

Codes and supersymmetry in one dimension

Wed, 12 Dec 2012 09:19 EST

Charles F. Doran, Michael G. Faux, Sylvester James Gates, Tristan Hübsch, Kevin M. Iga, Gregory D. Landweber, Robert L. Miller

Source: Adv. Theor. Math. Phys., Volume 15, Number 6, 1909--1970.

Abstract:
Adinkras are diagrams that describe many useful supermultiplets in $D = 1$ dimensions. We show that the topology of the Adinkra is uniquely determined by a doubly even code. Conversely, every doubly even code produces a possible topology of an Adinkra. A computation of doubly even codes results in an enumeration of these Adinkra topologies up to $N = 28$, and for minimal supermultiplets, up to $N = 32$.

Chiral algebras of (0, 2) models

Wed, 23 Jan 2013 09:20 EST

Junya Yagi

Source: Adv. Theor. Math. Phys., Volume 16, Number 1, 1--37.

Abstract:
We explore two-dimensional sigma models with (0, 2) supersymmetry through their chiral algebras. Perturbatively, the chiral algebras of (0, 2) models have a rich infinite-dimensional structure described by the cohomology of a sheaf of chiral differential operators. Nonperturbatively, instantons can deform this structure drastically. We show that under some conditions they even annihilate the whole algebra, thereby triggering the spontaneous breaking of supersymmetry. For a certain class of Kähler manifolds, this suggests that there are no harmonic spinors on their loop spaces and gives a physical proof of the Höhn–Stolz conjecture.

Hamiltonian structure of gauge-invariant variational problems

Wed, 23 Jan 2013 09:20 EST

Marco Castrillón López, Jamie Muñoz Masqué

Source: Adv. Theor. Math. Phys., Volume 16, Number 1, 39--63.

Abstract:
Let $C \to M$ be the bundle of connections of a principal bundle on $M$. The solutions to Hamilton–Cartan equations for a gauge-invariant Lagrangian density $\Lambda$ on $C$ satisfying a weak condition of regularity, are shown to admit an affine fibre-bundle structure over the set of solutions to Euler–Lagrange equations for $\Lambda$. This structure is also studied for the Jacobi fields and for the moduli space of extremals.

A master solution of the quantum Yang–Baxter equation and classical discrete integrable equations

Wed, 23 Jan 2013 09:20 EST

Vladimir V. Bazhanov, Sergey M. Sergeev

Source: Adv. Theor. Math. Phys., Volume 16, Number 1, 65--95.

Abstract:
We obtain a new solution of the star–triangle relation with positive Boltzmann weights, which contains as special cases all continuous and discrete spin solutions of this relation, that were previously known. This new master solution defines an exactly solvable two lattice model of statistical mechanics, which involves continuous spin variables, living on a circle, and contains two temperature-like parameters. If one of the these parameters approaches a root of unity (corresponds to zero temperature), the spin variables freezes into discrete positions, equidistantly spaced on the circle. An absolute orientation of these positions on the circle slowly changes between lattice sites by overall rotations. Allowed configurations of these rotations are described by classical discrete integrable equations, closely related to the famous $Q_4$-equations by Adler, Bobenko and Suris. Fluctuations between degenerate ground states in the vicinity of zero temperature are described by a rather general integrable lattice model with discrete spin variables. In some simple special cases, the latter reduces to the Kashiwara–Miwa and chiral Potts models.

Modular realizations of hyperbolic Weyl groups

Wed, 23 Jan 2013 09:20 EST

Axel Kleinschmidt, Hermann Nicolai, Jakob Palmkvist

Source: Adv. Theor. Math. Phys., Volume 16, Number 1, 97--148.

Abstract:
We study the recently discovered isomorphisms between hyperbolic Weyl groups and modular groups over integer domains in normed division algebras. We show how to realize the group action via fractional linear transformations on generalized upper half-planes over the division algebras, focusing on the cases involving quaternions and octonions. For these we construct automorphic forms, whose explicit expressions depend crucially on the underlying arithmetic properties of the integer domains. Another main new result is the explicit octavian realization of $W+(E_{10})$, which contains as a special case a new realization of $W+(E_8)$ in terms of unit octavians and their automorphism group.

Čech cocycles for differential characteristic classes: an ∞-Lie theoretic construction

Wed, 23 Jan 2013 09:20 EST

Domenico Fiorenza, Urs Schreiber, Jim Stasheff

Source: Adv. Theor. Math. Phys., Volume 16, Number 1, 149--250.

Abstract:
What are called secondary characteristic classes in Chern–Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and from Lie groups to higher connected covers of Lie groups by smooth $\infty$-groups, i.e., by smooth groupal $A_\infty$- spaces. Namely, we realize differential characteristic classes as morphisms from $\infty$-groupoids of smooth principal $\infty$-bundles with connections to $\infty$-groupoids of higher $U(1)$-gerbes with connections. This allows us to study the homotopy fibres of the differential characteristic maps thus obtained and to show how these describe differential obstruction problems. This applies in particular to the higher twisted differential spin structures called twisted differential string structures and twisted differential fivebrane structures .

Old issues and linear sigma models

Wed, 23 Jan 2013 09:20 EST

Jock McOrist, Ilarion V. Melnikov

Source: Adv. Theor. Math. Phys., Volume 16, Number 1, 251--288.

Abstract:
Using mirror symmetry, we resolve an old puzzle in the linear sigma model description of the spacetime Higgs mechanism in a heterotic string compactification with (2,2) worldsheet supersymmetry. The resolution has a nice spacetime interpretation via the normalization of physical fields and suggests that with a little care deformations of the linear sigma model can describe heterotic Higgs branches.

Equivariant modular categories via Dijkgraaf–Witten theory

Wed, 23 Jan 2013 09:20 EST

Jennifer Maier, Thomas Nikolaus, Christoph Schweigert

Source: Adv. Theor. Math. Phys., Volume 16, Number 1, 289--358.

Abstract:
Based on a weak action of a finite group $J$ on a finite group $G$, we present a geometric construction of $J$-equivariant Dijkgraaf–Witten theory as an extended topological field theory. The construction yields an explicitly accessible class of equivariant modular tensor categories. For the action of a group $J$ on a group $G$, the category is described as the representation category of a $J$-ribbon algebra that generalizes the Drinfel’d double of the finite group $G$.

The fast Newtonian limit for perfect fluids

Wed, 23 Jan 2013 09:21 EST

Todd A. Oliynyk

Source: Adv. Theor. Math. Phys., Volume 16, Number 2, 359--391.

Abstract:
We prove the existence of a large class of dynamical solutions to the Einstein–Euler equations for which the fluid density and spatial three-velocity converge to a solution of the Poisson–Euler equations of Newtonian gravity. The results presented here generalize those of The Newtonian limit for perfect fluids to allow for a larger class of initial data. As in The Newtonian limit for perfect fluids , the proof is based on a nonlocal symmetric hyperbolic formulation of the Einstein–Euler equations, which contain a singular parameter $\epsilon = v_T /c$ with $v_T$ a characteristic speed associated to the fluid and $c$ the speed of light. Energy and dispersive estimates on weighted Sobolev spaces are the main technical tools used to analyze the solutions in the singular limit $\epsilon \searrow 0$.

An extension of Friedmann–Robertson–Walker theory beyond big bang

Wed, 23 Jan 2013 09:21 EST

Joachim Schröter

Source: Adv. Theor. Math. Phys., Volume 16, Number 2, 393--419.

Abstract:
Starting from the classic Friedmann–Robertson–Walker theory with big bang it is shown that the solutions of the field equations can be extended to negative times. Choosing a new cosmic time scale instead of proper time one achieves complete differentiability of the scale factor and of suitable thermodynamic quantities equivalent to pressure and energy density. Then, the singularity of big bang manifests itself only by the vanishing of the scale factor at time zero. Moreover, all solutions of the field equations are defined for all times from $-\infty$ to $+\infty$. In a separate section, the horizon structure of the extended theory is studied. Some weak assumptions guarantee that there are no horizons. Hence, the horizon problem in a strict sense disappears. An intensive discussion of the results is given at the end of the paper.

Higher genus BMN correlators: factorization and recursion relations

Wed, 23 Jan 2013 09:21 EST

Min–xin Huang

Source: Adv. Theor. Math. Phys., Volume 16, Number 2, 421--503.

Abstract:
We systematically study the factorization and recursion relations in higher genus correlation functions of Berenstein–Maldacena–Nastase (BMN) operators in free $\mathcal{N} = 4$ super Yang–Mills theory. These properties were found in a previous paper by the author, and were conjectured to result from the correspondence with type IIB string theory on the infinitely curved pp-wave background, where the strings become effectively infinitely long. Here we push the calculations to higher genus, provide more clarifications and verifications of the factorization and recursion relations. Our calculations provide conjectural indirect tests of the AdS/ CFT correspondence for multi-loop superstring amplitudes of stringy modes.

Generalized TKNN-equations

Wed, 23 Jan 2013 09:21 EST

Giuseppe De Nittis, Giovanni Landi

Source: Adv. Theor. Math. Phys., Volume 16, Number 2, 505--547.

Abstract:
We derive generalized TKNN-equations via bundle representations of the noncommutative torus with rational deformation parameter, the bundle coming from spectral projections in the torus algebra. These equations relate Chern numbers of dual bundles, which we interpret as Hall conductances for Dirac-like Hamiltonians describing magnetic Bloch electrons in a strong magnetic field. We also present their generalizations for irrational values of the deformation parameter.

Geometry of fractional spaces

Wed, 23 Jan 2013 09:21 EST

Gianluca Calcagni

Source: Adv. Theor. Math. Phys., Volume 16, Number 2, 549--644.

Abstract:
We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool is fractional calculus, which is cast in a way convenient for the definition of the differential structure, distances, volumes, and symmetries. By an extensive use of concepts and techniques of fractal geometry, we clarify the relation between fractional calculus and fractals, showing that fractional spaces can be regarded as fractals when the ratio of their Hausdorff and spectral dimension is greater than one. All the results are analytic and constitute the foundation for field theories living on multi-fractal spacetimes, which are presented in a companion paper.

Causal posets, loops and the construction of nets of local algebras for QFT

Wed, 23 Jan 2013 09:21 EST

Fabio Ciolli, Giuseppe Ruzzi, Ezio Vasselli

Source: Adv. Theor. Math. Phys., Volume 16, Number 2, 645--691.

Abstract:
We provide a model independent construction of a net of $C*$-algebras satisfying the Haag–Kastler axioms over any spacetime manifold. Such a net, called the net of causal loops , is constructed by selecting a suitable base $K$ encoding causal and symmetry properties of the spacetime. Considering $K$ as a partially ordered set (poset) with respect to the inclusion order relation, we define groups of closed paths (loops) formed by the elements of $K$. These groups come equipped with a causal disjointness relation and an action of the symmetry group of the spacetime. In this way, the local algebras of the net are the group $C*$-algebras of the groups of loops, quotiented by the causal disjointness relation. We also provide a geometric interpretation of a class of representations of this net in terms of causal and covariant connections of the poset K. In the case of the Minkowski spacetime, we prove the existence of Poincaré covariant representations satisfying the spectrum condition. This is obtained by virtue of a remarkable feature of our construction: any Hermitian scalar quantum field defines causal and covariant connections of $K$. Similar results hold for the chiral spacetime $S^1$ with conformal symmetry.

Persistence of gaps in the spectrum of certain almost periodic operators

Wed, 23 Jan 2013 09:21 EST

Norbert Riedel

Source: Adv. Theor. Math. Phys., Volume 16, Number 2, 693--712.

Abstract:
It is shown that for any irrational rotation number and any admissible gap labelling number the almost Mathieu operator (also known as Harper’s operator) has a gap in its spectrum with that labelling number. This answers the strong version of the so-called "Ten Martini Problem". When specialized to the particular case where the coupling constant is equal to one, it follows that the "Hofstadter butterfly" has for any quantum Hall conductance the exact number of components prescribed by the recursive scheme to build this fractal structure.

Vortex equation and reflexive sheaves

Wed, 23 Jan 2013 09:21 EST

Indranil Biswas, Matthias Stemmler

Source: Adv. Theor. Math. Phys., Volume 16, Number 2, 713--723.

Abstract:
It is known that given a stable holomorphic pair $(E,\phi)$, where $E$ is a holomorphic vector bundle on a compact Kähler manifold $X$ and $\phi$ is a holomorphic section of $E$, the vector bundle $E$ admits a Hermitian metric solving the vortex equation. We generalize this to pairs $(E,\phi)$, where $E$ is a reflexive sheaf on $X$.