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Alexander Quintero Velez

Publications and source records attributed to Alexander Quintero Velez.

At least 19 recordsLinked to original sources
arXiv2025

A topological quantum field theory for $\mathrm{Spin}(7)$-instantons

We construct a topological quantum field theory based on the moduli space of $\mathrm{Spin}(7)$-instantons on 8-dimensional manifolds. Using the Mathai-Quillen formalism, we derive the action of the theory in purely geometric terms, which coincides with prior results in the literature. We then reformulate the theory within the AKSZ formalism, obtaining a Batalin-Vilkovisky action that, after gauge fixing, matches our Mathai-Quillen construction while making the BRST symmetry explicit and providing a natural framework for classical observables. We also show that the Batalin-Vilkovisky action can be elegantly recast as a Chern-Simons type theory.

hep-th↗
arXiv2025

One-loop $N$-point correlators in pure gravity

In this work we propose a simple algebraic recursion for the complete one-loop integrands of $N$-graviton correlators. This formula automatically yields the correct symmetry factors of individual diagrams, taking into account both the graviton and the ghost loop, and seamlessly controlling the related combinatorics.

hep-th↗
arXiv2024

Tree- and one-loop-level double copy for the (anti)self-dual sectors of Yang-Mills and gravity

By employing the perturbiner method we study the tree- and one-loop-level amplitudes in (anti)self-dual Yang-Mills, focusing on color-kinematics duality and double copy features; they arise naturally even in the fully off-shell case. In particular, we calculate the respective the Kawai-Lewellen-Tye relations for tree-level Berends-Giele currents and color-kinematics master numerators at one loop, both cases for any number of external particles.

hep-th↗
arXiv2023

Homotopy double copy and the Kawai-Lewellen-Tye relations for the non-abelian and tensor Navier-Stokes equations

Recently, a non-abelian generalisation of the Navier-Stokes equation that exhibits a manifest duality between colour and kinematics has been proposed by Cheung and Mangan. In this paper, we offer a new perspective on the double copy formulation of this equation, based on the homotopy algebraic picture suggested by Borsten, Kim, Jurčo, Macrelli, Saemann, and Wolf. In the process, we describe precisely how the double copy can be realised at the level of perturbiner expansions. Specifically, we will show that the colour-dressed Berends-Giele currents for the non-abelian version of the Navier-Stokes equation can be used to construct the Berends-Giele currents for the double copied equation by replacing the colour factors with a second copy of kinematic numerators. We will also show a Kawai-Lewellen-Tye relation stating that the full tree-level scattering amplitudes in the latter can be written as a product of tree-level colour ordered partial amplitudes in the former.

math-ph↗
arXiv2023

One-loop off-shell amplitudes from classical equations of motion

In this letter we present a recursive method for computing one-loop off-shell amplitudes in colored quantum field theories. First, we generalize the perturbiner method by recasting the multiparticle currents as generators of off-shell tree level amplitudes. After, by taking advantage of the underlying color structure, we define a consistent sewing procedure to iteratively compute the one-loop integrands. When gauge symmetries are involved, the whole procedure is extended to multiparticle solutions involving ghosts, which can then be accounted for in the full loop computation. Since the required input here is equations of motion and gauge symmetry, our framework naturally extends to one-loop computations in certain non-Lagrangian field theories.

hep-th↗
arXiv2023

Relativity

These lectures notes contain an introduction to General Relativity. They are addressed to a general mathematical audience with no specific background in physics. The goal is to motivate and explain Einstein's theory of gravity and discuss some of the fundamental examples.

gr-qc↗
arXiv2021

Manifest colour-kinematics duality and double-copy in the string-based formalism

The relation for the gravity polarisation tensor as the tensor product of two gluon polarisation vectors has been well-known for a long time, but a version of this relation for multi-particle fields is presently still not known. Here we show that in order for this to happen we first have to ensure that the multi-particle polarisations satisfy colour-kinematics duality. In previous work it has been show that this arises naturally from the Bern-Kosower formalism for one-loop gluon amplitudes, and here we show that the tensor product for multi-particle fields arise naturally in the Bern-Dunbar-Shimada formalism for one-loop gravity amplitudes. This allows us to formulate a new prescription for double-copy gravity Berends-Giele currents, and to obtain both the colour-dressed Yang-Mills Berends-Giele currents in the Bern-Carrasco-Johansson gauge and the gravitational Berends-Giele currents explicitly. An attractive feature of our formalism is that it never becomes necessary to determine gauge transformation terms. Our double-copy prescription can also be applied to other cases, and to make this point we derive the double-copy perturbiners for $α'$-deformed gravity and the bi-adjoint scalar model.

hep-th↗
arXiv2021

Color-kinematics duality from the Bern-Kosower formalism

Berends-Giele currents are fundamental building blocks for on-shell amplitudes in non-abelian gauge theory. We present a novel procedure to construct them using the Bern-Kosower formalism for one-loop gluon amplitudes. Applying the pinch procedure of that formalism to a suitable special case the currents are naturally obtained in terms of multi-particle fields and obeying colour-kinematics duality. As a feedback to the Bern-Kosower formalism we outline how the multi-particle polarisations and field-strength tensors can be used to significantly streamline the pinch procedure.

hep-th↗
arXiv2021

Chern-Weil theory for $\infty$-local systems

Let $G$ be a compact connected Lie group. We show that the category $\mathbf{Loc}_{\infty}(BG)$ of $\infty$-local systems on the classifying space of $G$, can be described infinitesimally as the category $\mathbf{InfLoc}_{\infty}(\mathfrak{g})$ of basic $\mathfrak{g}$-$L_\infty$ spaces. Moreover, we show that, given a principal bundle $π\colon P \rightarrow X$ with structure group $G$ and any connection $θ$ on $P$, there is a DG functor $$\mathcal{CW}_θ \colon \mathbf{InfLoc}_{\infty}(\mathfrak{g}) \longrightarrow \mathbf{Loc}_{\infty}(X), $$ which corresponds to the pullback functor by the classifying map of $P$. The DG functors associated to different connections are related by an $A_\infty$-natural isomorphism. This construction provides a categorification of the Chern-Weil homomorphism, which is recovered by applying the functor $\mathcal{CW}_θ$ to the endomorphisms of the constant local system.

math.AT↗
arXiv2021

The $L_{\infty}$ structure of gauge theories with matter

In this work we present an algebraic approach to the dynamics and perturbation theory at tree-level for gauge theories coupled to matter. The field theories we will consider are: Chern-Simons-Matter, Quantum Chromodynamics, and scalar Quantum Chromodynamics. Starting with the construction of the master action in the classical Batalin-Vilkovisky formalism, we will extract the $L_{\infty}$-algebra that allow us to recursively calculate the perturbiner expansion from its minimal model. The Maurer-Cartan action obtained in this procedure will then motivate a generating function for all the tree-level scattering amplitudes. There are two interesting outcomes of this construction: a generator for fully-flavoured amplitudes via a localisation on Dyck words; and closed expressions for fermion and scalar lines attached to $n$-gluons with arbitrary polarisations.

hep-th↗
arXiv2020

Singular chains on Lie groups and the Cartan relations II

Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$ and denote by $\mathrm{C}_{\bullet}(G)$ the DG Hopf algebra of smooth singular chains on $G$. In a companion paper it was shown that the category of sufficiently smooth modules over $\mathrm{C}_{\bullet}(G)$ is equivalent to the category of representations of $\mathbb{T} \mathfrak{g}$, the DG Lie algebra which is universal for the Cartan relations. In this paper we show that, if $G$ is compact, this equivalence of categories can be extended to an $\mathsf{A}_{\infty}$-quasi-equivalence of the corresponding DG categories. As an intermediate step we construct an $\mathsf{A}_{\infty}$-quasi-isomorphism between the Bott-Shulman-Stasheff DG algebra associated to $G$ and the DG algebra of Hochschild cochains on $\mathrm{C}_{\bullet}(G)$. The main ingredients in the proof are the Van Est map and Gugenheim's $\mathsf{A}_{\infty}$ version of De Rham's theorem.

math.AT↗
arXiv2019

A $\mathrm{U}(2) \times \mathrm{U}(3)$ gauge theory extension of the standard model

We consider an extension of the standard model based on the group $\mathrm{U}(2) \times \mathrm{U}(3)$, which is naturally compatible with the standard model interacting-particle representations and the spontaneous symmetry breaking of $\mathrm{U}(2) \times \mathrm{U}(3)$ to an electrostrong $\mathrm{U}(3)$. In its minimal version, the model only adds one extra $\mathrm{U}(1)$ gauge boson and it implies that the hypercharge is distributed between the factors of the hyperweak and hyperstrong forces. We show that the anomaly cancellation condition can be solved by adding exotic fermions associated with a $16$-dimensional representation of $\mathrm{U}(2) \times \mathrm{U}(3)$. A brief discussion of the mechanism of the spontaneous breakdown of $\mathrm{U}(2) \times \mathrm{U}(3)$ in the gauge boson sector is given.

hep-ph↗
arXiv2019

An $\mathsf{A}_{\infty}$ version of the Poincaré lemma

We prove a categorified version of the Poincaré lemma. The natural setting for our result is that of $\infty$-local systems. More precisely, we show that any smooth homotopy between maps $f$ and $g$ induces an $\mathsf{A}_\infty$-natural transformation between the corresponding pullback functors. This transformation is explicitly defined in terms of Chen's iterated integrals. In particular, we show that a homotopy equivalence induces a quasi-equivalence on the DG categories of $\infty$-local system.

math.DG↗
arXiv2019

Calabi-Yau completions and orbifold equivalences

Calabi-Yau algebras are particularly symmetric differential graded algebras. There is a construction called `Calabi-Yau completion' which produces a canonical Calabi-Yau algebra from any homologically smooth dg algebra. Homologically smooth dg algebras also form a 2-category to which the construction of `equivariant completion' can be applied. In this theory two objects are called `orbifold equivalent' if there is a 1-morphism $X$ with invertible quantum dimensions between them. Any such relation entails a whole family of equivalences between categories. We show that an orbifold equivalence between two homologically smooth and proper dg algebras lifts to an orbifold equivalence between their Calabi-Yau completions under certain conditions on $X$.

math.RT↗
arXiv2014

Geometric Reid's recipe for dimer models

Crepant resolutions of three-dimensional toric Gorenstein singularities are derived equivalent to noncommutative algebras arising from consistent dimer models. By choosing a special stability parameter and hence a distinguished crepant resolution $Y$, this derived equivalence generalises the Fourier-Mukai transform relating the $G$-Hilbert scheme and the skew group algebra $\CC[x,y,z]\ast G$ for a finite abelian subgroup of $\SL(3,\CC)$. We show that this equivalence sends the vertex simples to pure sheaves, except for the zero vertex which is mapped to the dualising complex of the compact exceptional locus. This generalises results of Cautis-Logvinenko and Cautis-Craw-Logvinenko to the dimer setting, though our approach is different in each case. We also describe some of these pure sheaves explicitly and compute the support of the remainder, providing a dimer model analogue of results from Logvinenko.

math.AG↗
arXiv2014

Cohomology of wheels on toric varieties

We describe explicitly the cohomology of the total complex of certain diagrams of invertible sheaves on normal toric varieties. These diagrams, called wheels, arise in the study of toric singularities associated to dimer models. Our main tool describes the generators in a family of syzygy modules associated to the wheel in terms of walks in a family of graphs.

math.AG↗
arXiv2012

Boundary coupling of Lie algebroid Poisson sigma models and representations up to homotopy

A general form for the boundary coupling of a Lie algebroid Poisson sigma model is proposed. The approach involves using the Batalin-Vilkovisky formalism in the AKSZ geometrical version, to write a BRST-invariant coupling for a representation up to homotopy of the target Lie algebroid or its subalgebroids. These considerations lead to a conjectural description of topological D-branes on generalized complex manifolds, which includes A-branes and B-branes as special cases.

math-ph↗