arXiv subjects
Publications and source records attributed to Richard J. Szabo.
This is a mini-review about generalized instantons of noncommutative gauge theories in dimensions 4, 6 and 8, with emphasis on their realizations in type II string theory, their geometric interpretations, and their applications to the enumerative geometry of non-compact toric varieties.
We address the problem of UV/IR mixing in noncommutative quantum field theories from the perspective of braided $L_\infty$-structures and the Batalin-Vilkovisky formalism. We describe the example of braided noncommutative scalar field theory and its quantization using braided homological perturbation theory. The formalism is illustrated through one-loop calculations of the two-point functions for $ϕ^4$-theory in four dimensions and $ϕ^3$-theory in six dimensions. In both cases we find that there are no non-planar diagrams and no UV/IR mixing.
Using the recently developed formalism of braided noncommutative field theory, we construct an explicit example of braided electrodynamics, that is, a noncommutative $U(1)$ gauge theory coupled to a Dirac fermion. We construct the braided $L_\infty$-algebra of this field theory and apply the formalism to obtain the braided equations of motion, action functional and conserved matter current. The braided deformation leads to a modification of the charge conservation. Finally, the Feynman integral appearing in the one-loop contribution to the vacuum polarization diagram is calculated. There are no non-planar diagrams, but the UV/IR mixing appears nevertheless. We comment on this unexpected result.
We summarise recent perspectives on symmetries of noncommutative field theories based on homotopy algebras. We show how these viewpoints naturally lead to a new class of noncommutative field theories which possess braided gauge symmetries, and explain in detail their uses in gravity. We review how these considerations lead to a new theory of noncommutative gravity in four dimensions within the Einstein-Cartan-Palatini formalism.
We introduce T-duality invariant versions of D-branes in doubled geometry using a global covariant framework based on para-Hermitian geometry and metric algebroids. We define D-branes as conformal boundary conditions for the open string version of the Born sigma-model, where they are given by maximally isotropic vector bundles which do not generally admit the standard geometric picture in terms of submanifolds. When reduced to the conventional sigma-model description of a physical string background as the leaf space of a foliated para-Hermitian manifold, integrable branes yield D-branes as leaves of foliations which are interpreted as Dirac structures on the physical spacetime. We define a notion of generalised para-complex D-brane, which realises our D-branes as para-complex versions of topological A/B-branes. We illustrate how our formalism recovers standard D-branes in the explicit example of reductions from doubled nilmanifolds.
We give a pedagogical introduction to $L_\infty$-algebras and their uses in organising the symmetries and dynamics of classical field theories, as well as of the conventional noncommutative gauge theories that arise as low-energy effective field theories in string theory. We review recent developments which formulate field theories with braided gauge symmetries as a new means of overcoming several obstacles in the standard noncommutative theories, such as the restrictions on gauge algebras and matter fields. These theories can be constructed by using techniques from Drinfel'd twist deformation theory, which we review in some detail, and their symmetries and dynamics are controlled by a new homotopy algebraic structure called a 'braided $L_\infty$-algebra'. We expand and elaborate on several novel theoretical issues surrounding these constructions, and present three new explicit examples: the standard noncommutative scalar field theory (regarded as a braided field theory), a braided version of $BF$ theory in arbitrary dimensions (regarded as a higher gauge theory), and a new braided version of noncommutative Yang-Mills theory for arbitrary gauge algebras.
We formulate general definitions of semi-classical gauge transformations for noncommutative gauge theories in general backgrounds of string theory, and give novel explicit constructions using techniques based on symplectic embeddings of almost Poisson structures. In the absence of fluxes the gauge symmetries close a Poisson gauge algebra and their action is governed by a $P_\infty$-algebra which we construct explicitly from the symplectic embedding. In curved backgrounds they close a field dependent gauge algebra governed by an $L_\infty$-algebra which is not a $P_\infty$-algebra. Our technique produces new all orders constructions which are significantly simpler compared to previous approaches, and we illustrate its applicability in several examples of interest in noncommutative field theory and gravity. We further show that our symplectic embeddings naturally define a $P_\infty$-structure on the exterior algebra of differential forms on a generic almost Poisson manifold, which generalizes earlier constructions of differential graded Poisson algebras, and suggests a new approach to defining noncommutative gauge theories beyond the gauge sector and the semi-classical limit based on $A_\infty$-algebras.
We apply the modern Batalin-Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in the finite-dimensional case of fuzzy spaces. We further develop a generalization of this framework to theories that are equivariant under a triangular Hopf algebra symmetry, which in particular leads to quantizations of finite-dimensional analogs of the field theories proposed recently through the notion of `braided $L_\infty$-algebras'. The techniques are illustrated by computing perturbative correlation functions for scalar and Chern-Simons theories on the fuzzy $2$-sphere, as well as for braided scalar field theories on the fuzzy $2$-torus.
We define a new homotopy algebraic structure, that we call a braided $L_\infty$-algebra, and use it to systematically construct a new class of noncommutative field theories, that we call braided field theories. Braided field theories have gauge symmetries which realize a braided Lie algebra, whose Noether identities are inhomogeneous extensions of the classical identities, and which do not act on the solutions of the field equations. We use Drinfel'd twist deformation quantization techniques to generate new noncommutative deformations of classical field theories with braided gauge symmetries, which we compare to the more conventional theories with star-gauge symmetries. We apply our formalism to introduce a braided version of general relativity without matter fields in the Einstein-Cartan-Palatini formalism. In the limit of vanishing deformation parameter, the braided theory of noncommutative gravity reduces to classical gravity without any extensions.
We give a self-contained survey of some approaches aimed at a global description of the geometry underlying double field theory. After reviewing the geometry of Courant algebroids and their incarnations in the AKSZ construction, we develop the theory of metric algebroids including their graded geometry. We use metric algebroids to give a global description of doubled geometry, incorporating the section constraint, as well as an AKSZ-type construction of topological doubled sigma-models. When these notions are combined with ingredients of para-Hermitian geometry, we demonstrate how they reproduce kinematical features of double field theory from a global perspective, including solutions of the section constraint for Riemannian foliated doubled manifolds, as well as a natural notion of generalized T-duality for polarized doubled manifolds. We describe the $L_\infty$-algebras of symmetries of a doubled geometry, and briefly discuss other proposals for global doubled geometry in the literature.
We give a pedagogical introduction to spectral networks and abelianization, as well as their relevance to $\mathcal{N}=2$ supersymmetric field theories in four dimensions. Motivated by a conjecture of Nekrasov-Rosly-Shatashvili, we detail a geometric recipe for computing the effective twisted superpotential for $\mathcal{N}=2$ field theories of class $\mathcal{S}$ as a generating function of the brane of opers, with respect to the spectral coordinates found from abelianization. We present two new examples, the simplest Argyres-Douglas theory and the pure $SU(2)$ gauge theory, while we conjecture the $ε$-expansion of the effective twisted superpotential for the $E_6$ Minahan-Nemeschansky theory.
We describe discrete symmetries of two-dimensional Yang-Mills theory with gauge group $G$ associated to outer automorphisms of $G$, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted $G$-bundles, and calculate it exactly. We argue that its weak-coupling limit computes the symplectic volume of the moduli space of flat twisted $G$-bundles on a surface. Using the defect network approach to generalised orbifolds, we gauge the discrete symmetry and construct the corresponding orbifold theory, which is again two-dimensional Yang-Mills theory but with gauge group given by an extension of $G$ by outer automorphisms. With the help of the orbifold completion of the topological defect bicategory of two-dimensional Yang-Mills theory, we describe the reverse orbifold using a Wilson line defect for the discrete gauge symmetry. We present our results using two complementary approaches: in the lattice regularisation of the path integral, and in the functorial approach to area-dependent quantum field theories with defects via regularised Frobenius algebras.
We study the BPS spectrum of four-dimensional $\mathcal{N}=2$ supersymmetric Yang-Mills theory with gauge group $SU(2)$ and a massive adjoint hypermultiplet, which has an extremely intricate structure with infinite spectrum in all chambers of its Coulomb moduli space, and is not well understood. We build on previous results by employing the BPS quiver description of the spectrum, and explore the qualitative features in detail using numerical techniques. We find novel and unexpected behaviour in the form of wall-crossings involving interactions between BPS particles with negative electric-magnetic pairings, which we interpret in terms of the reverse orderings of the usual wall-crossing formulas for rank one $\mathcal{N}=2$ field theories. This identifies new a priori unrelated states in the spectrum.
We give a covariant realization of the doubled sigma-model formulation of duality-symmetric string theory within the general framework of para-Hermitian geometry. We define a notion of generalized metric on a para-Hermitian manifold and discuss its relation to Born geometry. We show that a Born geometry uniquely defines a worldsheet sigma-model with a para-Hermitian target space, and we describe its Lie algebroid gauging as a means of recovering the conventional sigma-model description of a physical string background as the leaf space of a foliated para-Hermitian manifold. Applying the Kotov-Strobl gauging leads to a generalized notion of T-duality when combined with transformations that act on Born geometries. We obtain a geometric interpretation of the self-duality constraint that halves the degrees of freedom in doubled sigma-models, and we give geometric characterizations of non-geometric string backgrounds in this setting. We illustrate our formalism with detailed worldsheet descriptions of closed string phase spaces, of doubled groups where our notion of generalized T-duality includes non-abelian T-duality, and of doubled nilmanifolds.
We study bundle gerbes on manifolds $M$ that carry an action of a connected Lie group $G$. We show that these data give rise to a smooth 2-group extension of $G$ by the smooth 2-group of hermitean line bundles on $M$. This 2-group extension classifies equivariant structures on the bundle gerbe, and its non-triviality poses an obstruction to the existence of equivariant structures. We present a new global approach to the parallel transport of a bundle gerbe with connection, and use it to give an alternative construction of this smooth 2-group extension in terms of a homotopy-coherent version of the associated bundle construction. We apply our results to give new descriptions of nonassociative magnetic translations in quantum mechanics and the Faddeev-Mickelsson-Shatashvili anomaly in quantum field theory. We also propose a definition of smooth string 2-group models within our geometric framework. Starting from a basic gerbe on a compact simply-connected Lie group $G$, we prove that the smooth 2-group extensions of $G$ arising from our construction provide new models for the string group of $G$.
We apply the $C^*$-algebraic formalism of topological T-duality due to Mathai and Rosenberg to a broad class of topological spaces that include the torus bundles appearing in string theory compactifications with duality twists, such as nilmanifolds, as well as many other examples. We develop a simple procedure in this setting for constructing the T-duals starting from a commutative $C^*$-algebra with an action of ${\mathbb R}^n$. We treat the general class of almost abelian solvmanifolds in arbitrary dimension in detail, where we provide necessary and sufficient criteria for the existence of classical T-duals in terms of purely group theoretic data, and compute them explicitly as continuous-trace algebras with non-trivial Dixmier-Douady classes. We prove that any such solvmanifold has a topological T-dual given by a $C^*$-algebra bundle of noncommutative tori, which we also compute explicitly. The monodromy of the original torus bundle becomes a Morita equivalence among the fiber algebras, so that these $C^*$-algebras rigorously describe the T-folds from non-geometric string theory.
We derive the $T\overline{T}$-perturbed version of two-dimensional $q$-deformed Yang-Mills theory on an arbitrary Riemann surface by coupling the unperturbed theory in the first order formalism to Jackiw-Teitelboim gravity. We show that the $T\overline{T}$-deformation results in a breakdown of the connection with a Chern-Simons theory on a Seifert manifold, and of the large $N$ factorization into chiral and anti-chiral sectors. For the $U(N)$ gauge theory on the sphere, we show that the large $N$ phase transition persists, and that it is of third order and induced by instantons. The effect of the $T\overline{T}$-deformation is to decrease the critical value of the 't Hooft coupling, and also to extend the class of line bundles for which the phase transition occurs. The same results are shown to hold for $(q,t)$-deformed Yang-Mills theory. We also explicitly evaluate the entanglement entropy in the large $N$ limit of Yang-Mills theory, showing that the $T\overline{T}$-deformation decreases the contribution of the Boltzmann entropy.
We give a detailed account of the cyclic $L_\infty$-algebra formulation of general relativity with cosmological constant in the Einstein-Cartan-Palatini formalism on spacetimes of arbitrary dimension and signature, which encompasses all symmetries, field equations and Noether identities of gravity without matter fields. We present a local formulation as well as a global covariant framework, and an explicit isomorphism between the two $L_\infty$-algebras in the case of parallelizable spacetimes. By duality, we show that our $L_\infty$-algebras describe the complete BV-BRST formulation of Einstein-Cartan-Palatini gravity. We give a general description of how to extend on-shell redundant symmetries in topological gauge theories to off-shell correspondences between symmetries in terms of quasi-isomorphisms of $L_\infty$-algebras. We use this to extend the on-shell equivalence between gravity and Chern-Simons theory in three dimensions to an explicit $L_\infty$-quasi-isomorphism between differential graded Lie algebras which applies off-shell and for degenerate dynamical metrics. In contrast, we show that there is no morphism between the $L_\infty$-algebra underlying gravity and the differential graded Lie algebra governing $BF$ theory in four dimensions.