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Alexander A. Voronov

Publications and source records attributed to Alexander A. Voronov.

At least 19 recordsLinked to original sources
arXiv2025

Toward the Universal Mumford form on Sato Grassmannians

We construct a local universal Mumford form on a product of Sato Grassmannians using the flow of the Virasoro algebra. The existence of this universal Mumford form furthers the proposal that the Sato Grassmannian provides a universal moduli space with applications to string theory. Our approach using the Virasoro flow is an alternative to using the KP flow, which in particular allows for a bosonic universal Mumford form to be constructed. Applying the same method, we construct a local universal super Mumford form on a product of super Sato Grassmannians using the flow of the Neveu-Schwarz algebra.

math.AG↗
arXiv2025

The Neveu-Schwarz group and Schwarz's extended super Mumford form

In 1987, Albert Schwarz suggested a formula which extends the super Mumford form from the moduli space of super Riemann surfaces into the super Sato Grassmannian. His formula is a remarkably simple combination of super tau functions. We compute the Neveu-Schwarz action on super tau functions, and show that Schwarz's extended Mumford form is invariant under the the super Heisenberg-Neveu-Schwarz action, which strengthens Schwarz's proposal that a locus within the Grassmannian can serve as a universal moduli space with applications to superstring theory. Along the way, we construct the Neveu-Schwarz, super Witt, and super Heisenberg formal groups.

math.AG↗
arXiv2025

Rational Homotopy Theory

This is a survey of Rational Homotopy Theory, intended for a Mathematical Physics readership.

math.AT↗
arXiv2024

Homotopy G-algebras and moduli space operad

This paper emphasizes the ubiquitous role of moduli spaces of algebraic curves in associative algebra and algebraic topology. The main results are: (1) the space of an operad with multiplication is a homotopy Gerstenhaber (i.e., homotopy graded Poisson) algebra; (2) the singular cochain complex is naturally an operad; (3) the operad of decorated moduli spaces acts naturally on the de Rham complex $Ω^\bullet X$ of a Kähler manifold $X$, thereby yielding the most general type of homotopy G-algebra structure on $Ω^\bullet X$.

hep-th↗
arXiv2024

Mysterious Triality and the Exceptional Symmetry of Loop Spaces

In previous work, we introduced Mysterious Triality, extending the Mysterious Duality of Iqbal, Neitzke, and Vafa between physics and algebraic geometry to include algebraic topology in the form of rational homotopy theory. Starting with the rational Sullivan minimal model of the 4-sphere $S^4$, capturing the dynamics of M-theory via Hypothesis H, this progresses to the dimensional reduction of M-theory on torus $T^k$, $k \ge 1$, with its dynamics described via the iterated cyclic loop space $\mathcal{L}_c^k S^4$ of the 4-sphere. From this, we also extracted data corresponding to the maximal torus/Cartan subalgebra and the Weyl group of the exceptional Lie group/algebra of type $E_k$. In this paper, we discover much richer symmetry by extending the data of the Cartan subalgebra to a maximal parabolic subalgebra $\mathfrak{p}_k^{k(k)}$ of the split real form $\mathfrak{e}_{k(k)}$ of the exceptional Lie algebra of type $E_k$ by exhibiting an action, in rational homotopy category, of $\mathfrak{p}_k^{k(k)}$ on the slightly more symmetric than $\mathcal{L}_c^k S^4$ toroidification $\mathcal{T}^k S^4$. This action universally represents symmetries of the equations of motion of supergravity in the reduction of M-theory to $11-k$ dimensions. Along the way, we identify the minimal model of the toroidification $\mathcal{T}^k S^4$, generalizing the results of Vigué-Poirrier, Sullivan, and Burghelea, and establish an algebraic toroidification/totalization adjunction.

math.AT↗
arXiv2024

Tree-Level Superstring Amplitudes: The Neveu-Schwarz Sector

We present a complete computation of superstring scattering amplitudes at tree level, for the case of Neveu-Schwarz insertions. Mathematically, this is to say that we determine explicitly the superstring measure on the moduli space $\mathcal{M}_{0,n,0}$ of super Riemann surfaces of genus zero with $n \ge 3$ Neveu-Schwarz punctures. While, of course, an expression for the measure was previously known, we do this from first principles, using the canonically defined super Mumford isomorphism. We thus determine the scattering amplitudes, explicitly in the global coordinates on $\mathcal{M}_{0,n,0}$, without the need for picture changing operators or ghosts, and are also able to determine canonically the value of the coupling constant. Our computation should be viewed as a step towards performing similar analysis on $\mathcal{M}_{0,0,n}$, to derive explicit tree-level scattering amplitudes with Ramond insertions.

hep-th↗
arXiv2024

The supermoduli space of genus zero SUSY curves with Ramond punctures

We give an explicit quotient construction of the supermoduli space $\mathfrak{M}_{0, n_R}$ of genus zero super Riemann surfaces with $n_R \ge 4$ Ramond punctures and prove it is a Deligne-Mumford superstack. We also make an explicit quotient construction of the moduli space of genus zero supercurves without a SUSY structure and thereby prove it is an algebraic superstack.

math.AG↗
arXiv2023

Mysterious Triality and Rational Homotopy Theory

Mysterious Duality was discovered by Iqbal, Neitzke, and Vafa in 2001 as a convincing, yet mysterious correspondence between certain symmetry patterns in toroidal compactifications of M-theory and del Pezzo surfaces, both governed by the root system series $E_k$. It turns out that the sequence of del Pezzo surfaces is not the only sequence of objects in mathematics that gives rise to the same $E_k$ symmetry pattern. We present a sequence of topological spaces, starting with the four-sphere $S^4$, and then forming its iterated cyclic loop spaces $\mathcal{L}_c^k S^4$, within which we discover the $E_k$ symmetry pattern via rational homotopy theory. For this sequence of spaces, the correspondence between its $E_k$ symmetry pattern and that of toroidal compactifications of M-theory is no longer a mystery, as each space $\mathcal{L}_c^k S^4$ is naturally related to the compactification of M-theory on the $k$-torus via identification of the equations of motion of $(11-k)$-dimensional supergravity as the defining equations of the Sullivan minimal model of $\mathcal{L}_c^k S^4$. This gives an explicit duality between algebraic topology and physics. Thereby, we extend Iqbal-Neitzke-Vafa's Mysterious Duality between algebraic geometry and physics into a triality, also involving algebraic topology. Via this triality, duality between physics and mathematics is demystified, and the mystery is transferred to the mathematical realm as duality between algebraic geometry and algebraic topology. Now the question is: Is there an explicit relation between the del Pezzo surfaces $\mathbb{B}_k$ and iterated cyclic loop spaces of $S^4$ which would explain the common $E_k$ symmetry pattern?

hep-th↗
arXiv2023

Mysterious Triality and M-Theory

In a previous paper, we introduced Mysterious Triality as an extension, via algebraic topology in the form of rational homotopy theory, of Mysterious Duality by Iqbal, Neitzke, and Vafa, which provides connections between physics, in the form of dimensional reduction of M-theory, and algebraic geometry, in the form of intersection theory on del Pezzo surfaces. The starting point for that connection to rational homotopy theory is the description of M-theory dynamics using the 4-sphere, via Hypothesis H. This progresses to dimensional reduction of M-theory on tori $T^k$ with its dynamics described via cyclic loop spaces of the 4-sphere $\mathcal{L}_c^k S^4$, producing a series of data analogous to that given by the del Pezzo surfaces $\mathbb{B}_k$, for $k=0, \dots, 8$. With the mathematical constructions established in the previous paper, in this companion physics paper we present novel connections to M-theory that enhance the triality, including those strengthening the duality. This uncovers interesting ties between algebraic geometry, algebraic topology, and M-theory and provides tantalizing links. We further expand on the extension of the duality and triality to the Kac-Moody setting.

hep-th↗
arXiv2019

On the BV structure on the cohomology of moduli space

The question of vanishing of the BV operator on the cohomology of the moduli space of Riemann surfaces is investigated. The BV structure, which comprises a BV operator and an antibracket, is identified, vanishing theorems are proven, and a counterexample is provided.

math.AG↗
arXiv2018

Quantizing deformation theory II

A quantization of classical deformation theory, based on the Maurer-Cartan Equation $dS + \frac{1}{2}[S,S] = 0$ in dg-Lie algebras, a theory based on the Quantum Master Equation $dS + \hbar ΔS + \frac{1}{2} \{S,S\} = 0$ in dg-BV-algebras, is proposed. Representability theorems for solutions of the Quantum Master Equation are proven. Examples of "quantum" deformations are presented.

math.QA↗
arXiv2017

On homotopy Lie bialgebroids

A well-known result of A. Vaintrob characterizes Lie algebroids and their morphisms in terms of homological vector fields on supermanifolds. We give an interpretation of Lie bialgebroids and their morphisms in terms of odd symplectic dg-manifolds, building on the approach of D. Roytenberg. This extends naturally to the homotopy Lie case and leads to the notion of $L_\infty$-bialgebroids and $L_\infty$-morphisms between them.

math.QA↗
arXiv2017

The MV formalism for ${\rm IBL}_\infty$- and ${\rm BV}_\infty$-algebras

We develop a new formalism for the Quantum Master Equation $Δe^{S/\hbar} = 0$ and the category of ${\rm IBL}_\infty$-algebras and simplify some homotopical algebra arising in the context of oriented surfaces with boundary. We introduce and study a category of MV-algebras, which, on the one hand, contains such important categories as those of ${\rm IBL}_\infty$-algebras and ${\rm L}_\infty$-algebras, and on the other hand, is homotopically trivial, in particular allowing for a simple solution of the quantum master equation. We also present geometric interpretation of our results.

math.QA↗
arXiv2016

Homotopy Gerstenhaber algebras

The goal of this paper is to complete Getzler-Jones' proof of Deligne's Conjecture, thereby establishing an explicit relationship between the geometry of configurations of points in the plane and the Hochschild complex of an associative algebra. More concretely, it is shown that the $B_\infty$-operad, which is generated by multilinear operations known to act on the Hochschild complex, is a quotient of a certain operad associated to the compactified configuration spaces. Different notions of homotopy Gerstenhaber algebras are discussed: one of them is a $B_\infty$-algebra, another, called a homotopy G-algebra, is a particular case of a $B_\infty$-algebra, the others, a $G_\infty$-algebra, an $\bar E^1$-algebra, and a weak $G_\infty$-algebra, arise from the geometry of configuration spaces. Corrections to the paper arXiv:q-alg/9602009 of Kimura, Zuckerman, and the author related to the use of a nonextant notion of a homotopy Gerstenhaber algebra are made.

math.QA↗
arXiv2016

Categorification of Dijkgraaf-Witten Theory

The goal of the paper is to categorify Dijkgraaf-Witten theory, aiming at providing foundation for a direct construction of Dijkgraaf-Witten theory as an Extended Topological Quantum Field Theory. The main tool is cohomology with coefficients in a Picard groupoid, namely the Picard groupoid of hermitian lines.

math.AT↗
arXiv2016

The BV formalism for L$_\infty$-algebras

Functorial properties of the correspondence between commutative BV$_\infty$-algebras and L$_\infty$-algebras are investigated. The category of L$_\infty$-algebras with L$_\infty$-morphisms is characterized as a certain category of pure BV$_\infty$-algebras with pure BV$_\infty$-morphisms. The functor assigning to a commutative BV$_\infty$-algebra the L$_\infty$-algebra given by higher derived brackets is also shown to have a left adjoint. Cieliebak-Fukaya-Latschev's machinery of IBL$_\infty$- and BV$_\infty$-morphisms is further developed with introducing the logarithm of a map.

math.QA↗
arXiv2011

A higher category of cobordisms and topological quantum field theory

The goal of this work is to describe a categorical formalism for (Extended) Topological Quantum Field Theories (TQFTs) and present them as functors from a suitable category of cobordisms with corners to a linear category, generalizing 2d open-closed TQFTs to higher dimensions. The approach is based on the notion of an n-fold category by C. Ehresmann, weakened in the spirit of monoidal categories (associators, interchangers, Mac Lane's pentagons and hexagons), in contrast with the simplicial (weak Kan and complete Segal) approach of Jacob Lurie. We show how different Topological Quantum Field Theories, such as gauge, Chern-Simons, Yang-Mills, WZW, Seiberg-Witten, Rozansky-Witten, and AKSZ theories, as well as sigma model, may be described as functors from the pseudo n-fold category of cobordisms to a pseudo n-fold category of sets.

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