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C. I. Lazaroiu

Publications and source records attributed to C. I. Lazaroiu.

At least 19 recordsLinked to original sources
arXiv2024

Strong rapid turn inflation and contact Hamilton-Jacobi equations

We consider the consistency condition for ``strong'' sustained rapid turn inflation with third order slow roll (SRRT) in two-field cosmological models with oriented scalar manifold as a geometric PDE which constrains the metric and potential of such models. When supplemented by appropriate boundary conditions, the equation determines one of these objects in terms of the other and hence selects ``fiducial'' models for strong SRRT inflation. When the scalar potential is given, the equation can be simplified by fixing the conformal class of the scalar field metric (equivalently, fixing a complex structure which makes the scalar manifold into a complex Riemann surface). Then the consistency equation becomes a contact Hamilton-Jacobi PDE which determines the scalar field metric within the given conformal class. We analyze this equation with standard methods of PDE theory, discuss its approximation near a nondegenerate critical point of the scalar potential and extract natural asymptotic conditions for its solutions at such points. We also give numerical examples of solutions to a simple Dirichlet problem. For the case of elliptic curves relevant to two-field axion cosmology, we determine the general symmetry-adapted solution of the equation for potentials with a single charge vector.

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arXiv2023

The geometry and DSZ quantization of four-dimensional supergravity

We implement the Dirac-Schwinger-Zwanziger integrality condition on four-dimensional classical ungauged supergravity and use it to obtain its duality-covariant, gauge-theoretic, differential-geometric model on an oriented four-manifold $M$ of arbitrary topology. Classical bosonic supergravity is completely determined by a submersion $π$ over $M$ equipped with a complete Ehresmann connection, a vertical euclidean metric, and a vertically-polarized flat symplectic vector bundle $Ξ$. Building on these structures, we implement the Dirac-Schwinger-Zwanziger integrality condition through the choice of an element in the degree-two sheaf cohomology group with coefficients in a locally constant sheaf $\mathcal{L}\subset Ξ$ valued in the groupoid of integral symplectic spaces. We show that this data determines a Siegel principal bundle $P_{\mathfrak{t}}$ of fixed type $\mathfrak{t}\in \mathbb{Z}^{n_v}$ whose connections provide the global geometric description of the local electromagnetic gauge potentials of the theory. Furthermore, we prove that the Maxwell gauge equations of the theory reduce to the polarized self-duality condition determined by $Ξ$ on the connections of $P_{\mathfrak{t}}$. In addition, we investigate the continuous and discrete U-duality groups of the theory, characterizing them through short exact sequences and realizing the latter through the gauge group of $P_{\mathfrak{t}}$ acting on its adjoint bundle. This elucidates the geometric origin of U-duality, which we explore in several examples, illustrating its dependence on the topology of the fiber bundles $π$ and $P_{\mathfrak{t}}$ as well as on the isomorphism type of $\mathcal{L}$.

math.DG↗
arXiv2022

Dynamical renormalization and universality in classical multifield cosmological models

We study the scaling behavior of classical multifield cosmological models with complete scalar manifold $({\cal M},{\cal G})$ and positive smooth scalar potential $Φ$, introducing a dynamical renormalization group action which relates their UV and IR limits. We show that the RG flow of such models interpolates between a modification of the geodesic flow of $({\cal M},{\cal G})$ (obtained in the UV limit) and the gradient flow of $({\cal M},{\cal G},V)$ (obtained in the IR limit), where the classical effective potential $V$ is proportional to $\sqrt{2Φ}$. Using this fact, we show that two-field models whose scalar manifold has constant Gaussian curvature equal to $-1$, $0$ or $1$ are infrared universal in the sense that they suffice to describe the first order IR approximants of cosmological orbits for all two-field models with positive smooth scalar potential.

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arXiv2021

Dirac operators on real spinor bundles of complex type

Let $(M,g)$ be a pseudo-Riemannian manifold of signature $(p,q)$. We compute the obstruction for a vector bundle $S$ over $(M,g)$ to admit a Dirac operator whose principal symbol induces on $S$ the structure of a bundle of irreducible real Clifford modules of complex type, that is, a real spinor bundle of irreducible complex type. In order to do this, we use the theory of Lipschitz structures in signature $p-q\equiv_8 3,7$ to reformulate the problem as the obstruction problem for $(M,g)$ to admit a $\mathrm{Spin}^{o}_α$ structure with $α= -1$ if $p-q \equiv_{8} 3$ or $α= +1$ if $ p-q \equiv_{8} 7$, where $\mathrm{Spin}^o_+(p,q)=\mathrm{Spin}(p,q)\cdot\mathrm{Pin}_{2,0}$ and $\mathrm{Spin}^o_-(p,q)=\mathrm{Spin}(p,q)\cdot \mathrm{Pin}_{0,2}$. This allows computing the obstruction in terms of the Karoubi Stiefel-Whitney classes of $(M,g)$ and the existence of an auxiliary $\mathrm{O}(2)$ bundle with prescribed characteristic classes. Furthermore, we explicitly show how a $\mathrm{Spin}^o_α$ structure can be used to construct $S$ and we give geometric characterizations (in terms of associated bundles) of the conditions under which the structure group of $S$ reduces to certain natural subgroups of $\mathrm{Spin}^o_α$. Finally, we prove that certain codimension two submanifolds of spin manifolds and certain products of tori with Grassmanians, which were not known to admit irreducible real spinor bundles, do admit $\mathrm{Spin}^{o}_α$ structures and therefore do admit real spinor bundles of irreducible complex type.

math.DG↗
arXiv2020

Four-dimensional geometric supergravity and electromagnetic duality: a brief guide for mathematicians

We give a gentle introduction to the global geometric formulation of the bosonic sector of four-dimensional supergravity on an oriented four-manifold $M$ of arbitrary topology, providing a geometric characterization of its U-duality group. The geometric formulation of four-dimensional supergravity is based on a choice of a vertically Riemannian submersion $π$ over $M$ equipped with a flat Ehresmann connection, which determines the non-linear section sigma model of the theory, and a choice of flat symplectic vector bundle $\mathcal{S}$ equipped with a positive complex polarization over the total space of $π$, which encodes the inverse gauge couplings and theta angles of the theory and determines its gauge sector. The classical fields of the theory consist of Lorentzian metrics on $M$, global sections of $π$ and two-forms valued in $\mathcal{S}$ that satisfy an algebraic relation which defines the notion of \emph{twisted} self-duality in four Lorentzian dimensions. We use this geometric formulation to investigate the group of electromagnetic duality transformations of supergravity, also known as the continuous classical U-duality group, which we characterize using a certain short exact sequence of automorphism groups of vector bundles. Moreover, we discuss the general structure of the Killing spinor equations of four-dimensional supergravity, providing several explicit examples and remarking on a few open mathematical problems. This presentation is aimed at mathematicians working in differential geometry.

math.DG↗
arXiv2019

Generalized Einstein-Scalar-Maxwell theories and locally geometric U-folds

We give a global formulation of the coupling of four-dimensional scalar sigma models to Abelian gauge fields for the generalized situation when the "duality structure" of the Abelian gauge theory is described by a flat symplectic vector bundle $(\mathcal{S},D,ω)$ defined over the scalar manifold $\mathcal{M}$. The construction uses a taming of $(\mathcal{S}, ω)$, which encodes globally the inverse gauge couplings and theta angles of the "twisted" Abelian gauge theory in a manner that makes no use of duality frames. We show that global solutions of the equations of motion of such models give classical locally geometric U-folds. We also describe the groups of duality transformations and scalar-electromagnetic symmetries arising in such models, which involve lifting isometries of $\mathcal{M}$ to a particular class of flat automorphisms of the bundle $\mathcal{S}$ and hence differ from expectations based on local analysis. The appropriate version of the Dirac quantization condition involves a discrete local system defined over $\mathcal{M}$ and gives rise to a smooth bundle of polarized Abelian varieties, endowed with a flat symplectic connection. This shows that a generalization of part of the mathematical structure familiar from $\mathcal{N}=2$ supergravity is already present in such purely bosonic models, without any coupling to fermions and hence without any supersymmetry.

hep-th↗
arXiv2018

$\mathcal{N}=1$ Geometric Supergravity and chiral triples on Riemann surfaces

We construct a global geometric model for the bosonic sector and Killing spinor equations of four-dimensional $\mathcal{N}=1$ supergravity coupled to a chiral non-linear sigma model and a Spin$^{c}_0$ structure. The model involves a Lorentzian metric $g$ on a four-manifold $M$, a complex chiral spinor and a map $φ\colon M\to \mathcal{M}$ from $M$ to a complex manifold $\mathcal{M}$ endowed with a novel geometric structure which we call chiral triple. Using this geometric model, we show that if $M$ is spin the Kähler-Hodge condition on a complex manifold $\mathcal{M}$ is enough to guarantee the existence of an associated $\mathcal{N}=1$ chiral geometric supergravity, positively answering a conjecture proposed by D. Z. Freedman and A. V. Proeyen. We dimensionally reduce the Killing spinor equations to a Riemann surface $X$, obtaining a novel system of partial differential equations for a harmonic map with potential $φ\colon X\to \mathcal{M}$ from $X$ into the Kähler moduli space $\mathcal{M}$ of the theory. We characterize all Riemann surfaces admitting supersymmetric solutions with vanishing superpotential, proving that they consist on holomorphic maps of Riemann surfaces into $\mathcal{M}$ satisfying certain compatibility condition with respect to the canonical bundle of $X$ and the chiral triple of the theory. Furthermore, we classify the biholomorphism type of all Riemann surfaces carrying supersymmetric solutions with complete Riemannian metric and finite-energy scalar map.

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arXiv2018

B-type Landau-Ginzburg models on Stein manifolds

We summarize the description of the open-closed TFT datum for B-type Landau-Ginzburg models with Stein manifold targets and discuss various constructions which lead to large classes of examples of such models.

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arXiv2018

Generalized two-field $α$-attractor models from geometrically finite hyperbolic surfaces

We consider four-dimensional gravity coupled to a non-linear sigma model whose scalar manifold is a non-compact geometrically finite surface $Σ$ endowed with a Riemannian metric of constant negative curvature. When the space-time is an FLRW universe, such theories produce a very wide generalization of two-field $α$-attractor models, being parameterized by a positive constant $α$, by the choice of a finitely-generated surface group $Γ\subset \mathrm{PSL}(2,\mathbb{R})$ (which is isomorphic with the fundamental group of $Σ$) and by the choice of a scalar potential defined on $Σ$. The traditional two-field $α$-attractor models arise when $Γ$ is the trivial group, in which case $Σ$ is the Poincaré disk. We give a general prescription for the study of such models through uniformization in the so-called "non-elementary" case and discuss some of their qualitative features in the gradient flow approximation, which we relate to Morse theory. We also discuss some aspects of the SRST approximation in these models, showing that it is generally not well-suited for studying dynamics near cusp ends. When $Σ$ is non-compact and the scalar potential is "well-behaved" at the ends, we show that, in the {\em naive} local one-field truncation, our generalized models have the same universal behavior as ordinary one-field $α$-attractors if inflation happens near any of the ends of $Σ$ where the extended potential has a local maximum, for trajectories which are well approximated by non-canonically parameterized geodesics near the ends, we also discuss spiral trajectories near the ends.

hep-th↗
arXiv2018

Section sigma models coupled to symplectic duality bundles on Lorentzian four-manifolds

We give the global mathematical formulation of a class of generalized four-dimensional theories of gravity coupled to scalar matter and to Abelian gauge fields. In such theories, the scalar fields are described by a section of a surjective pseudo-Riemannian submersion $π$ over space-time, whose total space carries a Lorentzian metric making the fibers into totally-geodesic connected Riemannian submanifolds. In particular, $π$ is a fiber bundle endowed with a complete Ehresmann connection whose transport acts through isometries between the fibers. In turn, the Abelian gauge fields are "twisted" by a flat symplectic vector bundle defined over the total space of $π$. This vector bundle is endowed with a vertical taming which locally encodes the gauge couplings and theta angles of the theory and gives rise to the notion of twisted self-duality, of crucial importance to construct the theory. When the Ehresmann connection of $π$ is integrable, we show that our theories are locally equivalent to ordinary Einstein-Scalar-Maxwell theories and hence provide a global non-trivial extension of the universal bosonic sector of four-dimensional supergravity. In this case, we show using a special trivializing atlas of $π$ that global solutions of such models can be interpreted as classical "locally-geometric" U-folds. In the non-integrable case, our theories differ locally from ordinary Einstein-Scalar-Maxwell theories and may provide a geometric description of classical U-folds which are "locally non-geometric".

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arXiv2018

Geometric U-folds in four dimensions

We describe a construction of geometric U-folds compatible with a non-trivial extension of the global formulation of four-dimensional extended supergravity on a differentiable spin manifold. The topology of geometric U-folds depends on certain flat fiber bundles which encode how supergravity fields are globally glued together. We show that smooth non-trivial U-folds of this type can exist only in theories where both the scalar and space-time manifolds have non-trivial fundamental group and in addition, the scalar map of the solution is homotopically non-trivial. Consistency with string theory requires smooth geometric U-folds to be glued using subgroups of the effective discrete U-duality group, implying that the fundamental group of the scalar manifold of such solutions must be a subgroup of the latter. We construct simple examples of geometric U-folds in a generalization of the axion-dilaton model of $\mathcal{N}=2$ supergravity coupled to a single vector multiplet, whose scalar manifold is a generally non-compact Riemann surface of genus at least two endowed with its uniformizing metric. We also discuss the relation between geometric U-folds and a moduli space of flat connections defined on the scalar manifold, which involves certain character varieties not studied in the literature.

hep-th↗
arXiv2017

The geometric algebra of Fierz identities in arbitrary dimensions and signatures

We use geometric algebra techniques to give a synthetic and computationally efficient approach to Fierz identities in arbitrary dimensions and signatures, thus generalizing previous work. Our approach leads to a formulation which displays the underlying real, complex or quaternionic structure in an explicit and conceptually clear manner and is amenable to implementation in various symbolic computation systems. We illustrate our methods and results with a few examples which display the basic features of the three classes of pin representations governing the structure of such identities in various dimensions and signatures.

hep-th↗
arXiv2015

Foliated backgrounds for M-theory compactifications (II)

We summarize the foliation approach to ${\cal N}=1$ compactifications of eleven-dimensional supergravity on eight-manifolds $M$ down to $\mathrm{AdS}_3$ spaces for the case when the internal part $ξ$ of the supersymmetry generator is chiral on some proper subset ${\cal W}$ of $M$. In this case, a topological no-go theorem implies that the complement $M\setminus {\cal W}$ must be a dense open subset, while $M$ admits a singular foliation ${\bar {\cal F}}$ (in the sense of Haefliger) which is defined by a closed one-form $\boldsymbolω$ and is endowed with a longitudinal $G_2$ structure. The geometry of this foliation is determined by the supersymmetry conditions. We also describe the topology of ${\bar {\cal F}}$ in the case when $\boldsymbolω$ is a Morse form.

hep-th↗
arXiv2006

Graded D-branes and skew-categories

I describe extended gradings of open topological field theories in two dimensions in terms of skew categories, proving a result which alows one to translate between the formalism of graded open 2d TFTs and equivariant cyclic categories. As an application of this formalism, I describe the open 2d TFT of graded D-branes in Landau-Ginzburg models in terms of an equivariant cyclic structure on the triangulated category of `graded matrix factorizations' introduced by Orlov. This leads to a specific conjecture for the Serre functor on the latter, which generalizes results known from the minimal and Calabi-Yau cases. I also give a description of the open 2d TFT of such models which manifestly displays the full grading induced by the vector-axial R-symmetry group.

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arXiv2006

Generating the superpotential on a D-brane category: I

I study A-infinity enhancements of algebraic Calabi-Yau triangulated categories admitting a (triangle) generator, showing that the Serre pairing on such categories determines and is determined by a cyclic pairing on an enhancement of the generator. Using this result, I construct a formal topological string field action inducing an extended D-brane superpotential for such categories. I also give a procedure for lifting certain 2d boundary topological field theories to open topological string theories generated by a single D-brane.

hep-th↗
arXiv2005

Non-commutative moduli spaces of topological D-branes

We give a general construction of extended moduli spaces of topological D-branes as non-commutative algebraic varieties. This shows that noncommutative symplectic geometry in the sense of Kontsevich arises naturally in String Theory.

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arXiv2005

On the non-commutative geometry of topological D-branes

This is a noncommutative-geometric study of the semiclassical dynamics of finite topological D-brane systems. Starting from the formulation in terms of A -infinity categories, I show that such systems can be described by the noncommutative symplectic supergeometry of Z2-graded quivers, and give a synthetic formulation of the boundary part of the generalized WDVV equations. In particular, a faithful generating function for integrated correlators on the disk can be constructed as a linear combination of quiver necklaces, i.e. a function on the noncommutative symplectic superspace defined by the quiver's path algebra. This point of view allows one to construct extended moduli spaces of topological D-brane systems as non-commutative algebraic `superschemes'. They arise by imposing further relations on a Z2-graded version of the quiver's preprojective algebra, and passing to the subalgebra preserved by a natural group of symmetries.

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