arXiv subjects

Brett Parker

Publications and source records attributed to Brett Parker.

At least 19 recordsLinked to original sources

arXiv2025

Gromov-Witten invariants of log Calabi-Yau 3-folds are holomorphic lagrangian correspondences

We introduce a holomorphic version of Weinstein's symplectic category, in which objects are holomorphic symplectic manifolds, and morphisms are holomorphic lagrangian correspondences. We then extend this category to log schemes, and prove that Gromov-Witten invariants of log Calabi-Yau 3-folds are naturally encoded as holomorphic lagrangian correspondences. Gromov-Witten invariants and Donaldson-Thomas invariants are then conjecturally related by a natural unitary lagrangian correspondence.

math.AG↗

arXiv2024

Natural Probability

How should we model an observer within quantum mechanics or quantum field theory? How can classical physics emerge from a quantum model, and why should classical probability be useful? How can we model a selective measurement entirely within a closed quantum system? This paper sketches a new physical theory of probability based on an attempt to model classical information within a purely quantum system. We model classical information using a version of Zurek's theory of Quantum Darwinism, with emphasis on quantum information encoded using projection operators localised in spacetime. This version of Quantum Darwinism is compatible with quantum field theory, and does not require any artificial division of a quantum system into subsystems. The main innovation is our attempt to provide a physical explanation of probability. Decoherence is the physical mechanism behind Quantum Darwinism or the `branching of quantum worlds'. Assuming a type of perfect decoherence we construct a conventional probabilistic model for classical information. This, however, is not our theory of natural probability, and does not quite demonstrate the validity of Bayesian reasoning. Instead, our theory of natural probability arises from careful consideration of errors in decoherence: roughly speaking, we don't observe low probability events because they are swamped by quantum noise.

quant-ph↗

arXiv2022

The tropological vertex

The theory of the topological vertex was originally proposed by Aganagic, Klemm, Mari\~no and Vafa as a means to calculate open Gromov-Witten invariants of toric Calabi-Yau threefolds. In this paper, we place the topological vertex within the context of relative Gromov-Witten invariants of log Calabi-Yau manifolds and describe how these invariants can be effectively computed via a gluing formula for the enumeration of tropical curves in a singular integral affine space. This richer context allows us to prove that the topological vertex possesses certain tropical symmetries. These symmetries are captured by the action of a quantum torus Lie algebra that is related to a quantisation of the Lie algebra of the tropical vertex group of Gross, Pandharipande and Siebert. Finally, we demonstrate how this algebra of symmetries leads to an explicit description of the topological vertex and related Gromov-Witten invariants.

math.AG↗

arXiv2020

De Rham theory of exploded manifolds

This paper extends de Rham theory of smooth manifolds to exploded manifolds. Included are versions of Stokes' theorem, De Rham cohomology, Poincare duality, and integration along the fiber. The resulting cohomology theory is used to define Gromov Witten invariants of exploded manifolds in a separate paper.

math.DG↗

arXiv2018

Holomorphic curves in exploded manifolds: regularity

The category of exploded manifolds is an extension of the category of smooth manifolds related to tropical geometry in which some adiabatic limits appear as smooth families. This paper studies the dbar equation on variations of a given family of curves in an exploded manifold. Roughly, we prove that the dbar equation on variations of an exploded family of curves behaves as nicely as the dbar equation on variations of a smooth family of smooth curves, even though exploded families of curves allow the development of normal crossing or log smooth singularities. The resulting regularity results are used in a series of separate papers to construct Gromov Witten invariants for exploded manifolds.

math.SG↗

arXiv2018

Holomorphic curves in exploded manifolds: virtual fundamental class

We define Gromov--Witten invariants of exploded manifolds. The technical heart of this paper is a construction of a virtual fundamental class $[\mathcal K]$ of any Kuranishi category $\mathcal K$ (which is a simplified, more general version of an embedded Kuranishi structure.) We also show how to integrate differential forms over $[\mathcal K]$ to obtain numerical invariants, and push forward differential forms from $\mathcal K$ over suitable evaluation maps. We show that such invariants are independent of any choices, and are compatible with pullbacks, products, and tropical completion of Kuranishi categories. In the case of a compact symplectic manifold, this gives an alternative construction of Gromov--Witten invariants, including gravitational descendants.

math.SG↗

arXiv2017

Exploded Manifolds

This paper provides an introduction to exploded manifolds. The category of exploded manifolds is an extension of the category of smooth manifolds with an excellent holomorphic curve theory. Each exploded manifold has a tropical part which is a union of convex polytopes glued along faces. Exploded manifolds are useful for defining and computing Gromov-Witten invariants relative to normal crossing divisors, and using tropical curve counts to compute Gromov-Witten invariants.

math.SG↗

arXiv2017

On the value of thinking tropically to understand Ionel's GW invariants relative normal crossing divisors

Ionel's GW invariants relative normal-crossing divisors appear different from Gromov-Witten invariants defined using log schemes or exploded manifolds. Appearances are, in this case, deceiving. I sketch the relationship between Ionel's invariants and their exploded cousins using the example of the moduli space of lines in the complex projective plane relative two coordinate lines. Even in this simplest of examples, 13 different types of curves appear in Ionel's compactified moduli space, but these different types of curves can be understood in a unified and intuitive fashion using tropical curves.

math.SG↗

arXiv2017

Notes on exploded manifolds and a tropical gluing formula for Gromov-WItten invariants

Notes for a short lecture series, covering exploded manifolds, the moduli stack of curves in exploded manifolds, and a tropical gluing formula for Gromov-Witten invariants: a gluing formula providing a degeneration formula for Gromov-Witten invariants in normal-crossing degenerations. I gave the original lecture series in April 2016 at the Simons Center for Geometry and Physics at Stonybrook. Video of the lectures is available on the SCGP website, http://scgp.stonybrook.edu/video_portal/video.php?id=2595

math.SG↗

arXiv2017

Gluing formula for Gromov-Witten invariants in a triple product

We present a gluing formula for Gromov-Witten invariants in the case of a triple product. This gluing formula is a simple case of a much more general gluing formula proved and stated using exploded manifolds. We present this simple case because it is relatively easy to explain without any knowledge of exploded manifolds or log schemes.

math.SG↗

arXiv2017

Three dimensional tropical correspondence formula

A tropical curve in $\mathbb R^{3}$ contributes to Gromov-Witten invariants in all genus. Nevertheless, we present a simple formula for how a given tropical curve contributes to Gromov-Witten invariants when we encode these invariants in a generating function with exponents of $λ$ recording Euler characteristic. Our main modification from the known tropical correspondence formula for rational curves is as follows: a trivalent vertex, which before contributed a factor of $n$ to the count of zero-genus holomorphic curves, contributes a factor of $2\sin(nλ/2)$. We explain how to calculate relative Gromov-Witten invariants using this tropical correspondence formula, and how to obtain the absolute Gromov-Witten and Donaldson-Thomas invariants of some $3$-dimensional toric manifolds including $\mathbb CP^{3}$. The tropical correspondence formula counting Donaldson-Thomas invariants replaces $n$ by $i^{-(1+n)}q^{n/2}+i^{1+n}q^{-n/2}$.

math.SG↗

arXiv2017

Tropical gluing formulae for Gromov-Witten invariants

We prove two tropical gluing formulae for Gromov-Witten invariants of exploded manifolds, useful for calculating Gromov-Witten invariants of a symplectic manifold using a normal-crossing degeneration. The first formula generalizes the symplectic-sum formula for Gromov-Witten invariants. The second formula is stronger, and also generalizes Kontsevich and Manin's splitting and genus-reduction axioms. Both tropical gluing formulae have versions incorporating gravitational descendants.

math.SG↗

arXiv2014

Tropical enumeration of curves in blowups of the projective plane

We describe a method for recursively calculating Gromov-Witten invariants of all blowups of the projective plane. This recursive formula is different from the recursive formulas due to G\"ottsche and Pandharipande in the zero genus case, and Caporaso and Harris in the case of no blowups. We use tropical curves and a recursive computation of Gromov-Witten invariants relative a normal crossing divisor.

math.SG↗

arXiv2014

Holomorphic curves in exploded manifolds: Compactness

This paper establishes compactness results for the moduli stack of holomorphic curves in suitable exploded manifolds. This result together with the analysis in arXiv:0902.0087 allows the definition of Gromov Witten invariants of these exploded manifolds.

math.SG↗

arXiv2013

Integral counts of pseudo-holomorphic curves

In \cite{FOinteger}, Fukaya and Ono outlined a way of counting pseudo-holomorphic curves in a general compact symplectic manifold to obtain integer valued invariants. This paper contains the details of Fukaya and Ono's suggested construction for any compact symplectic manifold and a large class of exploded manifolds.

math.SG↗

arXiv2013

Universal tropical structures for curves in exploded manifolds

For any stable curve $f$ in an exploded manifold, this paper constructs a family of curves $\hat f$ with universal tropical structure which contains $f$. Such a family has the property that any other family of curves containing $f$ is locally a small modification of a family which factors through $\hat f$. As such, families of curves with universal tropical structure play an important role in the analysis of the moduli stack of curves and the construction of Gromov-Witten invariants of exploded manifolds.

math.SG↗

arXiv2013

Holomorphic curves in exploded manifolds: Kuranishi structure

This paper constructs a Kuranishi structure for the moduli stack of holomorphic curves in exploded manifolds. To avoid some technicalities of abstract Kuranishi structures, we embed our Kuranishi structure inside an ambient moduli stack of not-necessarily-holomorphic curves. The construction also works for the moduli stack of holomorphic curves in any compact symplectic manifold.

math.SG↗

arXiv2012

Log geometry and exploded manifolds

Log Gromov-Witten invariants have recently been defined separately by Gross and Siebert and Abramovich and Chen. This paper provides a dictionary between log geometry and holomorphic exploded manifolds in order to compare Gromov-Witten invariants defined using exploded manifolds or log schemes. The gluing formula for Gromov-Witten invariants of exploded manifolds suggests an approach to proving analogous gluing formulas for log Gromov-Witten invariants.

math.SG↗