arXiv subjects

Roberto Zucchini

Publications and source records attributed to Roberto Zucchini.

At least 19 recordsLinked to original sources

arXiv2025

Calibrated hypergraph states: I calibrated hypergraph and multi qudit state monads

Hypergraph states are a special kind of multipartite states encoded by hypergraphs. They play a significant role in quantum error correction, measurement--based quantum computation, quantum non locality and entanglement. In a series of two papers, we introduce and study calibrated hypergraph states, a broad generalization of weighted hypergraph states codified by hypergraphs equipped with calibrations, an ample extension of weightings. We propose as a guiding principle that a constructive theory of hypergraph states must be based on a categorical framework for hypergraphs on one hand and multi qudit states on the other constraining hypergraph states enough to render the determination of their general structure possible. In this first paper, we introduce graded $\varOmega$ monads, concrete Pro categories isomorphic to the Pro category $\varOmega$ of finite von Neumann ordinals and equipped with an associative and unital graded multiplication, and their morphisms, maps of $\varOmega$ monads compatible with their monadic structure. We then show that both calibrated hypergraphs and multi qudit states naturally organize in graded $\varOmega$ monads. In this way, we lay the foundation for the construction of calibrated hypergraph state map as a special morphism of these $\varOmega$ monads in the companion paper.

quant-ph↗

arXiv2025

Calibrated hypergraph states: II calibrated hypergraph state construction and applications

Hypergraph states are a special kind of multipartite states encoded by hypergraphs relevant in quantum error correction, measurement--based quantum computation, quantum non locality and entanglement. In a series of two papers, we introduce and investigate calibrated hypergraph states, an extension of weighted hypergraph states codified by hypergraphs equipped with calibrations, a broad generalization of weightings. The guiding principle informing our approach is that a constructive theory of hypergraph states must be based on a categorical framework for both hypergraphs and multi qudit states constraining hypergraph states enough to render the determination of their general structure possible. In this second paper, we build upon the graded $\varOmega$ monadic framework worked out in the companion paper, focusing on qudits over a generic Galois ring. We explicitly construct a calibrated hypergraph state map as a special morphism of the calibrated hypergraph and multi qudit state $\varOmega$ monads. We further prove that the calibrated hypergraph states so yielded are locally maximally entangleable stabilizer states, elucidate their relationship to weighted hypergraph states, show that they reduce to the weighted ones in the familiar qubit case and prove through examples that this is no longer the case for higher qudits.

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arXiv2023

A new quantum computational set-up for algebraic topology via simplicial sets

In this paper, a quantum computational framework for algebraic topology based on simplicial set theory is presented. This extends previous work, which was limited to simplicial complexes and aimed mostly to topological data analysis. The proposed set--up applies to any parafinite simplicial set and proceeds by associating with it a finite dimensional simplicial Hilbert space, whose simplicial operator structure is studied in some depth. It is shown in particular how the problem of determining the simplicial set's homology can be solved within the simplicial Hilbert framework. Further, the conditions under which simplicial set theoretic algorithms can be implemented in a quantum computational setting with finite resources are examined. Finally a quantum algorithmic scheme capable to compute the simplicial homology spaces and Betti numbers of a simplicial set combining a number of basic quantum algorithms is outlined.

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arXiv2022

Quantum field theoretic representation of Wilson surfaces: II higher topological coadjoint orbit model

This is the second of a series of two papers devoted to the partition function realization of Wilson surfaces in strict higher gauge theory. A higher 2--dimensional counterpart of the topological coadjoint orbit quantum mechanical model computing Wilson lines is presented based on the derived geometric framework, which has shown its usefulness in 4--dimensional higher Chern--Simons theory. Its symmetries are described. Its quantization is analyzed in the functional integral framework. Strong evidence is provided that the model does indeed underlie the partition function realization of Wilson surfaces. The emergence of the vanishing fake curvature condition is explained and homotopy invariance for a flat higher gauge field is shown. The model's Hamiltonian formulation is further furnished highlighting the model's close relationship to the derived Kirillov-Kostant-Souriau theory developed in the companion paper.

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arXiv2022

Quantum field theoretic representation of Wilson surfaces: I higher coadjoint orbit theory

This is the first of a series of two papers devoted to the partition function realization of Wilson surfaces in strict higher gauge theory. A higher version of the Kirillov-Kostant-Souriau theory of coadjoint orbits is presented based on the derived geometric framework, which has shown its usefulness in 4--dimensional higher Chern--Simons theory. An original notion of derived coadjoint orbit is put forward. A theory of derived unitary line bundles and Poisson structures on regular derived orbits is constructed. The proper derived counterpart of the Bohr--Sommerfeld quantization condition is then identified. A version of derived prequantization is proposed. The difficulties hindering a full quantization, shared with other approaches to higher quantization, are pinpointed and a possible way--out is suggested. The theory we elaborate provide the geometric underpinning for the field theoretic constructions of the companion paper.

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arXiv2021

4-d Chern-Simons Theory: Higher Gauge Symmetry and Holographic Aspects

We present and study a 4d Chern-Simons (CS) model whose gauge symmetry is encoded in a balanced Lie group crossed module. Using the derived formal set-up recently found, the model can be formulated in a way that in many respects closely parallels that of the familiar 3d CS one. In spite of these formal resemblance, the gauge invariance properties of the 4d CS model differ considerably. The 4d CS action is fully gauge invariant if the underlying base 4fold has no boundary. When it does, the action is gauge variant, the gauge variation being a boundary term. If certain boundary conditions are imposed on the gauge fields and gauge transformations, level quantization can then occur. In the canonical formulation of the theory, it is found that, depending again on boundary conditions, the 4d CS model is characterized by surface charges obeying a non trivial Poisson bracket algebra. This is a higher counterpart of the familiar WZNW current algebra arising in the 3d model. 4d CS theory thus exhibits rich holographic properties. The covariant Schroedinger quantization of the 4d CS model is performed. A preliminary analysis of 4d CS edge field theory is also provided. The toric and Abelian projected models are described in some detail.

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arXiv2019

Operational total space theory of principal 2-bundles II: 2-connections and 1- and 2--gauge transformations

The geometry of the total space of a principal bundle with regard to the action of the bundle's structure group is elegantly described by the bundle's operation, a collection of derivations consisting of the de Rham differential and the contraction and Lie derivatives of all vertical vector fields and satisfying the six Cartan relations. Connections and gauge transformations are defined by the way they behave under the action of the operation's derivations. In the first paper of a series of two extending the ordinary theory, we constructed an operational total space theory of strict principal 2--bundles with reference to the action of the structure strict 2--group. Expressing this latter through a crossed module $(\mathsans{E},\mathsans{G})$, the operation is based on the derived Lie group $\mathfrak{e}[1]\rtimes\mathsans{G}$. In this paper, the second of the series, an original formulation of the theory of $2$--connections and $1$-- and $2$--gauge transformations of principal $2$--bundles based on the operational framework is provided.

math-ph↗

arXiv2019

Operational total space theory of principal 2--bundles I: operational geometric framework

It is a classic result that the geometry of the total space of a principal bundle with reference to the action of the bundle's structure group is codified in the bundle's operation, a collection of derivations comprising the de Rham differential and the contraction and Lie derivatives of all vertical vector fields and obeying the six Cartan relations. In particular, connections and gauge transformations can be defined through the way they are acted upon by the operation's derivations. In this paper, the first of a series of two extending the ordinary theory, we construct an operational total space theory of strict principal 2--bundles with regard to the action of the structure strict 2--group. Expressing this latter via a crossed module $(\mathsans{E},\mathsans{G})$, the operation is based on the derived Lie group $\mathfrak{e}[1]\rtimes\mathsans{G}$. In the second paper, an original formulation of the theory of $2$--connections and $1$-- and $2$--gauge transformations based on the operational framework worked out here will be provided.

math-ph↗

arXiv2019

Wilson Surfaces for Surface Knots

Holonomy invariants in strict higher gauge theory have been studied in depth, aiming to applications to higher Chern-Simons theory. For a flat 2-connection, the holonomy of surface knots of arbitrary genus has been defined and its covariance properties under 1-gauge transformation and change of base data have been determined. Using quandle theory, a definition of trace over a crossed module has been given that yields surface knot invariants upon application to 2-holonomies.

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arXiv2018

Exact renormalization group in Batalin--Vilkovisky theory

In this paper, inspired by the Costello's seminal work, we present a general formulation of exact renormalization group (RG) within the Batalin-Vilkovisky (BV) quantization scheme. In the spirit of effective field theory, the BV bracket and Laplacian structure as well as the BV effective action (EA) depend on an effective energy scale. The BV EA at a certain scale satisfies the BV quantum master equation at that scale. The RG flow of the EA is implemented by BV canonical maps intertwining the BV structures at different scales. Infinitesimally, this generates the BV exact renormalization group equation (RGE). We show that BV RG theory can be extended by augmenting the scale parameter space R to its shifted tangent bundle T[1]R. The extra odd direction in scale space allows for a BV RG supersymmetry that constrains the structure of the BV RGE bringing it to Polchinski's form. We investigate the implications of BV RG supersymmetry in perturbation theory. Finally, we illustrate our findings by constructing free models of BV RG flow and EA exhibiting RG supersymmetry in the degree -1 symplectic framework and studying the perturbation theory thereof. We find in particular that the odd partner of effective action describes perturbatively the deviation of the interacting RG flow from its free counterpart.

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arXiv2017

Exact renormalization group and effective action: a Batalin--Vilkovisky algebraic formulation

In the present paper, which is a mathematical follow--up of [16] taking inspiration from [11], we present an abstract formulation of exact renormalization group (RG) in the framework of Batalin--Vilkovisky (BV) algebra theory. In the first part, we work out a general algebraic and geometrical theory of BV algebras, canonical maps, flows and flow stabilizers. In the second part, relying on this formalism, we build a BV algebraic theory of the RG. In line with the graded geometric outlook of our approach, we adjoin the RG scale with an odd parameter and analyse in depth the implications of the resulting RG supersymmetry and find that the RG equation (RGE) takes Polchinski's form [3]. Finally, we study abstract purely algebraic odd symplectic free models of RG flow and effective action (EA) and the perturbation theory thereof to illustrate and exemplify the general theory.

math-ph↗

arXiv2017

Algebraic formulation of higher gauge theory

In this paper, we present a purely algebraic formulation of higher gauge theory and gauged sigma models based on the abstract theory of graded commutative algebras and their morphisms. The formulation incorporates naturally BRST symmetry and is also suitable for AKSZ type constructions. It is also shown that for a full--fledged BV formulation including ghost degrees of freedom, higher gauge and gauged sigma model fields must be viewed as internal smooth functions on the shifted tangent bundle of a space time manifold valued in a shifted L_infty-algebroid encoding symmetry. The relationship to other formulations where the L_infty-algebroid arises from a higher Lie groupoid by Lie differentiation is highlighted.

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arXiv2016

A Lie based 4-dimensional higher Chern-Simons theory

We present and study a model of 4-dimensional higher Chern-Simons theory, special Chern-Simons (SCS) theory, instances of which have appeared in the string literature, whose symmetry is encoded in a skeletal semistrict Lie 2-algebra constructed from a compact Lie group with non discrete center. The field content of SCS theory consists of a Lie valued 2-connection coupled to a background closed 3-form. SCS theory enjoys a large gauge and gauge for gauge symmetry organized in an infinite dimensional strict Lie 2-group. The partition function of SCS theory is simply related to that of a topological gauge theory localizing on flat connections with degree 3 second characteristic class determined by the background 3-form. Finally, SCS theory is related to a 3-dimensional special gauge theory whose 2-connection space has a natural symplectic structure with respect to which the 1-gauge transformation action is Hamiltonian, the 2-curvature map acting as moment map.

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arXiv2015

On higher holonomy invariants in higher gauge theory I

This is the first of a series of two technical papers devoted to the analysis of holonomy invariants in strict higher gauge theory with end applications in higher Chern--Simons theory. For a flat 2--connection, we define the 2-holonomy of surface knots of arbitrary genus and determine its covariance properties under 1--gauge transformation and change of base data.

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arXiv2015

On higher holonomy invariants in higher gauge theory II

This is the second of a series of two technical papers devoted to the analysis of holonomy invariants in strict higher gauge theory with end applications in higher Chern--Simons theory. We provide a definition of trace over a crossed module such to yield surface knot invariants upon application to 2-holonomies. We show further that the properties of the trace are best described using the theory quandle crossed modules.

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arXiv2014

A new formulation of higher parallel transport in higher gauge theory

In this technical paper, we present a new formulation of higher parallel transport in strict higher gauge theory required for the rigorous construction of Wilson lines and surfaces. Our approach is based on an original notion of Lie crossed module cocycle and cocycle 1- and 2-gauge transformation with a non standard double category theoretic interpretation. We show its equivalence to earlier formulations.

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arXiv2014

4-d semistrict higher Chern-Simons theory I

We formulate a 4-dimensional higher gauge theoretic Chern-Simons theory. Its symmetry is encoded in a semistrict Lie 2-algebra equipped with an invariant non singular bilinear form. We analyze the gauge invariance of the theory and show that action is invariant under a higher gauge transformation up to a higher winding number. We find that the theory admits two seemingly inequivalent canonical quantizations. The first is manifestly topological, it does not require a choice of any additional structure on the spacial 3-fold. The second, more akin to that of ordinary Chern-Simons theory, involves fixing a CR structure on the latter. Correspondingly, we obtain two sets of semistrict higher WZW Ward identities and we find the explicit expressions of two higher versions of the WZW action. We speculate that the model could be used to define 2-knot invariants of 4-folds.

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arXiv2012

AKSZ models of semistrict higher gauge theory

In the first part of this paper, we work out a perturbative Lagrangian formulation of semistrict higher gauge theory, that avoids the subtleties of the relationship between Lie 2-groups and algebras by relying exclusively on the structure semistrict Lie 2-algebra v and its automorphism 2-group Aut(v). Gauge transformations are defined and shown to form a strict 2-group depending on v. Fields are v-valued and their global behaviour is controlled by appropriate gauge transformation gluing data organized as a strict 2-groupoid. In the second part, using the BV quantization method in the AKSZ geometrical version, we write down a 3-dimensional semistrict higher BF gauge theory generalizing ordinary BF theory, carry out its gauge fixing and obtain as end result a semistrict higher topological gauge field theory of the Witten type. We also introduce a related 4-dimensional semistrict higher Chern-Simons gauge theory.

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