arXiv subjects

Mauricio Romo

Publications and source records attributed to Mauricio Romo.

At least 19 recordsLinked to original sources

arXiv2026

On the length conjecture for twist knots

Using the newly developed $3$d quantum trace map, we compile more evidence for the length conjecture, a refinement of the all-order volume conjecture that incorporates insertions of additional links in the skein module of the knot complement. In particular, we prove that the length conjecture holds up to first order for an infinite family of twist knots. Along the way, we form a conjecture that the $3$d quantum trace map behaves naturally with respect to Dehn filling.

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arXiv2026

Monodromy of Calabi-Yau threefold flops via grade restriction rule and their quantum Kahler moduli

We present exact expressions, based on the grade restriction rule and window categories, for monodromies associated to certain Calabi-Yau threefold flops. We show a general formula for the monodromy action on the lattice of B-brane charges, based on the hemisphere partition function for abelian and nonabelian gauged linear sigma models. We exploit the explicit form of the discriminant in the quantum K\"ahler moduli to further refine the form of the monodromies, in several examples, using their relation to the fundamental group of nested torus links.

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arXiv2025

Non-commutative resolutions and pre-quotients of Calabi-Yau double covers

Following an earlier proposal arXiv:2307.02038 to apply the GLSM formalism to understand the so-called non-commutative resolution, this paper takes one important step further to extend this formalism to a much larger class of non-commutative resolutions. The proposal was initially motivated by the discovery of a new class of mirror pairs singular Calabi-Yau varieties arXiv:2003.07148, given by certain branched double covers over toric varieties of MPCP type. The overarching problem was to understand these mirror pairs from the viewpoint of homological mirror symmetry arXiv:alg-geom/9411018. In the present paper, we propose two main results along this line. First, one new insight is that the `gauge-fixing' condition on the branching locus of the double cover used in arXiv:2003.07148 can be relaxed in an interesting way. This turns out to produce GLSMs that describe a much larger class of non-commutative resolutions, leading to $A$-periods for a larger class of non-commutative resolutions, as well as the GKZ systems for their $A$-periods. Second, we show that the $A$-periods can also be realized as $A$-periods of a certain smooth CICY family in a toric variety of MPCP type, such that a suitable finite quotient of this family recovers the double cover CY we have started with. We call this CICY family the `pre-quotient' of the double cover CY. This realization strongly suggests that pre-quotient may provide an important approach for understanding homological mirror symmetry for singular double cover CY varieties and non-commutative resolutions.

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arXiv2025

B-brane transport in nonabelian GLSMs for $K_{Gr(2,N)}$

We study the properties of B-branes in a class of nonabelian GLSMs realizing the canonical line bundle $K_{Gr(2,N)}$ in their geometric phase. By analysing the hemisphere partition function, i.e. B-brane central charge, we propose a grade restriction rule and the corresponding window categories for a specific class of paths between phases. We find very striking differences between the cases of N even and N odd. In particular, for the case of N even, we suggest that more than one window category can be possible, for a fixed path. A detailed computation of the open Witten index and some monodromies provides evidence for our proposal for window categories. In addition, we make some remarks about B-branes on the the strongly coupled phase, for the case $N=4$, based on our window proposal.

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arXiv2024

Exponential Networks for Linear Partitions

Previous work has given proof and evidence that BPS states in local Calabi-Yau 3-folds can be described and counted by exponential networks on the punctured plane, with the help of a suitable non-abelianization map to the mirror curve. This provides an appealing elementary depiction of moduli of special Lagrangian submanifolds, but so far only a handful of examples have been successfully worked out in detail. In this note, we exhibit an explicit correspondence between torus fixed points of the Hilbert scheme of points on $\mathbb C^2\subset\mathbb C^3$ and anomaly free exponential networks attached to the quadratically framed pair of pants. This description realizes an interesting, and seemingly novel, "age decomposition" of linear partitions. We also provide further details about the networks' perspective on the full D-brane moduli space.

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arXiv2024

B-brane Transport and Grade Restriction Rule for Determinantal Varieties

We study autoequivalences of $D^{b}Coh(X)$ associated to B-brane transport around loops in the stringy K\"ahler moduli of $X$. We consider the case of $X$ being certain resolutions of determinantal varieties embedded in $\mathbb{P}^{d}\times G(k,n)$. Such resolutions have been modeled, in general, by nonabelian gauged linear sigma models (GLSM). We use the GLSM construction to determine the window categories associated with B-brane transport between different geometric phases using the machinery of grade restriction rule and the hemisphere partition function. In the family of examples analyzed the monodromies around phase boundaries enjoy the interpretation as loop inside link complements. We exploit this interpretation to find a decomposition of autoequivalences into simpler spherical functors and we illustrate this in two examples of Calabi-Yau 3-folds $X$, modeled by an abelian and nonabelian GLSM respectively. In additon we also determine explicitly the action of the autoequivalences on the Grothendieck group $K(X)$ (or equivalently, B-brane charges).

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arXiv2023

Modelling $A$-branes with foliations

A certain class of $A$-branes in mirrors of toric Calabi-Yau threefolds can be described through the framework of foliations. This allows to develop an explicit description of their moduli spaces based on a cell decomposition, with strata of various dimensions glued together in a way that is dictated by partial degenerations of the underlying special Lagrangian. Examples of $A$-branes associated with `wild' BPS states are considered in detail. The torus fixed points in their moduli spaces provide a decomposition of $m$-herds spectral networks into a number $|Ω|$ of basic connected objects, where $Ω$ is the the corresponding rank-zero Donaldson-Thomas (DT) invariant. A relation between the surgery parameters of the special Lagrangian and the baryonic semi-invariants of the representation theory of $m$-Kronecker quivers is also discussed, providing a local map between moduli spaces of branes related by homological mirror symmetry.

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arXiv2023

Non-commutative resolutions as mirrors of singular Calabi--Yau varieties

It has been conjectured that the hemisphere partition function arXiv:1308.2217, arXiv:1308.2438 in a gauged linear sigma model (GLSM) computes the central charge arXiv:math/0212237 of an object in the bounded derived category of coherent sheaves for Calabi--Yau (CY) manifolds. There is also evidence in arXiv:alg-geom/ 9511001, arXiv:hep-th/0007071. On the other hand, non-commutative resolutions of singular CY varieties have been studied in the context of abelian GLSMs arXiv:0709.3855. In this paper, we study an analogous construction of abelian GLSMs for non-commutative resolutions and propose they can be used to study a class of recently discovered mirror pairs of singular CY varieties. Our main result shows that the hemisphere partition functions (a.k.a. $A$-periods) in the new GLSM are in fact period integrals (a.k.a. $B$-periods) of the singular CY varieties. We conjecture that the two are completely equivalent: $B$-periods are the same as $A$-periods. We give some examples to support this conjecture and formulate some expected homological mirror symmetry (HMS) relation between the GLSM theory and the CY. As shown in arXiv:2003.07148, the $B$-periods in this case are precisely given by a certain fractional version of the $B$-series of arXiv:alg-geom/9511001. Since a hemisphere partition function is defined as a contour integral in a cone in the complexified secondary fan (or FI-theta parameter space) arXiv:1308.2438, it can be reduced to a sum of residues (by theorems of Passare--Tsikh--Zhdanov and Tsikh--Zhdanov). Our conjecture shows that this residue sum may now be amenable to computations in terms of the $B$-series.

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arXiv2022

Hybrid models for homological projective duals and noncommutative resolutions

We study hybrid models arising as homological projective duals (HPD) of certain projective embeddings $f:X\rightarrow\mathbb{P}(V)$ of Fano manifolds $X$. More precisely, the category of B-branes of such hybrid models corresponds to the HPD category of the embedding $f$. B-branes on these hybrid models can be seen as global matrix factorizations over some compact space $B$ or, equivalently, as the derived category of the sheaf of $\mathcal{A}$-modules on $B$, where $\mathcal{A}$ is an $A_{\infty}$ algebra. This latter interpretation corresponds to a noncommutative resolution of $B$. We compute explicitly the algebra $\mathcal{A}$ by several methods, for some specific class of hybrid models, and find that in general it takes the form of a smash product of an $A_{\infty}$ algebra with a cyclic group. Then we apply our results to the HPD of $f$ corresponding to a Veronese embedding of projective space and the projective embedding of Fano complete intersections in $\mathbb{P}^{n}$.

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arXiv2022

A GLSM view on Homological Projective Duality

Given a gauged linear sigma model (GLSM) $\mathcal{T}_{X}$ realizing a projective variety $X$ in one of its phases, i.e. its quantum Kähler moduli has a maximally unipotent point, we propose an \emph{extended} GLSM $\mathcal{T}_{\mathcal{X}}$ realizing the homological projective dual category $\mathcal{C}$ to $D^{b}Coh(X)$ as the category of B-branes of the Higgs branch of one of its phases. In most of the cases, the models $\mathcal{T}_{X}$ and $\mathcal{T}_{\mathcal{X}}$ are anomalous and the analysis of their Coulomb and mixed Coulomb-Higgs branches gives information on the semiorthogonal/Lefschetz decompositions of $\mathcal{C}$ and $D^{b}Coh(X)$. We also study the models $\mathcal{T}_{X_{L}}$ and $\mathcal{T}_{\mathcal{X}_{L}}$ that correspond to homological projective duality of linear sections $X_{L}$ of $X$. This explains why, in many cases, two phases of a GLSM are related by homological projective duality. We study mostly abelian examples: linear and Veronese embeddings of $\mathbb{P}^{n}$ and Fano complete intersections in $\mathbb{P}^{n}$. In such cases, we are able to reproduce known results as well as produce some new conjectures. In addition, we comment on the construction of the HPD to a nonabelian GLSM for the Plücker embedding of the Grassmannian $G(k,N)$.

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arXiv2022

Quantum trace map for 3-manifolds and a 'length conjecture'

We introduce a quantum trace map for an ideally triangulated hyperbolic knot complement $S^3\backslash \mathcal{K}$. The map assigns a quantum operator to each element of Kauffmann Skein module of the 3-manifold. The quantum operator lives in a module generated by products of quantized edge parameters of the ideal triangulation modulo some equivalence relations determined by gluing equations. Combining the quantum map with a state-integral model of $SL(2,\mathbb{C})$ Chern-Simons theory, one can define perturbative invariants of knot $K$ in the knot complement whose leading part is determined by its complex hyperbolic length. We then conjecture that the perturbative invariants determine an asymptotic expansion of the Jones polynomial for a link composed of $\mathcal{K}$ and $K$. We propose the explicit quantum trace map for figure-eight knot complement and confirm the length conjecture up to the second order in the asymptotic expansion both numerically and analytically.

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arXiv2022

A-branes, foliations and localization

This paper studies a notion of enumerative invariants for stable $A$-branes, and discusses its relation to invariants defined by spectral and exponential networks. A natural definition of stable $A$-branes and their counts is provided by the string theoretic origin of the topological $A$-model. This is the Witten index of the supersymmetric quantum mechanics of a single $D3$ brane supported on a special Lagrangian in a Calabi-Yau threefold. Geometrically, this is closely related to the Euler characteristic of the $A$-brane moduli space. Using the natural torus action on this moduli space, we reduce the computation of its Euler characteristic to a count of fixed points via equivariant localization. Studying the $A$-branes that correspond to fixed points, we make contact with definitions of spectral and exponential networks. We find agreement between the counts defined via the Witten index, and the BPS invariants defined by networks. By extension, our definition also matches with Donaldson-Thomas invariants of $B$-branes related by homological mirror symmetry.

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arXiv2021

D-brane central charge and Landau-Ginzburg orbifolds

We propose a formula for the exact central charge of a B-type D-brane that is expected to hold in all regions of the Kahler moduli space of a Calabi-Yau. For Landau-Ginzburg orbifolds we propose explicit expressions for the mathematical objects that enter into the central charge formula. We show that our results are consistent with results in FJRW theory and the hemisphere partition function of the gauged linear sigma model.

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arXiv2020

Exponential BPS graphs and D-brane counting on toric Calabi-Yau threefolds: Part II

We study BPS states of 5d $\mathcal{N}=1$ $SU(2)$ Yang-Mills theory on $S^1\times \mathbb{R}^4$. Geometric engineering relates these to enumerative invariants for the local Hirzebruch surface $\mathbb{F}_0$. We illustrate computations of Vafa-Witten invariants via exponential networks, verifying fiber-base symmetry of the spectrum at certain points in moduli space, and matching with mirror descriptions based on quivers and exceptional collections. Albeit infinite, parts of the spectrum organize in families described by simple algebraic equations. Varying the radius of the M-theory circle interpolates smoothly with the spectrum of 4d $\mathcal{N}=2$ Seiberg-Witten theory, recovering spectral networks in the limit.

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arXiv2019

Exponential BPS graphs and D brane counting on toric Calabi-Yau threefolds: Part I

We study BPS spectra of D-branes on local Calabi-Yau threefolds $\mathcal{O}(-p)\oplus\mathcal{O}(p-2)\to \mathbb{P}^1$ with $p=0,1$, corresponding to $\mathbb{C}^3/\mathbb{Z}_{2}$ and the resolved conifold. Nonabelianization for exponential networks is applied to compute directly unframed BPS indices counting states with D2 and D0 brane charges. Known results on these BPS spectra are correctly reproduced by computing new types of BPS invariants of 3d-5d BPS states, encoded by nonabelianization, through their wall-crossing. We also develop the notion of exponential BPS graphs for the simplest toric examples, and show that they encode both the quiver and the potential associated to the Calabi-Yau via geometric engineering.

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arXiv2018

B-brane transport in anomalous (2,2) models and localization

We study how B-branes in two-dimensional N=(2,2) anomalous models behave as we vary the energy scale and bulk parameters in the quantum K\"ahler moduli space. We focus on (2,2) theories defined by abelian gauged linear sigma models (GLSM). Guided by the hemisphere partition function we find how B-branes split in arbitrary phases into components on the Higgs branch and other branches: this generalizes the band restriction rules of Herbst-Hori-Page to (abelian) anomalous models. Secondly, we address divergences in non-compact models, through the central example of GLSMs for Hirzebruch-Jung resolutions of cyclic surface singularities. For a brane with compact support we explain how to regularize and compute the hemisphere partition function and extract its Higgs branch component, which we match in the zero-instanton sector to the geometric central charge of the brane. To this aim, we clarify the definition of zero-instanton geometric central charge for objects in the derived category of a non-compact toric orbifold.

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arXiv2018

Exploring 5d BPS Spectra with Exponential Networks

We develop geometric techniques for counting BPS states in five-dimensional gauge theories engineered by M theory on a toric Calabi-Yau threefold. The problem is approached by studying framed 3d-5d wall-crossing in presence of a single M5 brane wrapping a special Lagrangian submanifold $L$. The spectrum of 3d-5d BPS states is encoded by the geometry of the manifold of vacua of the 3d-5d system, which further coincides with the mirror curve describing moduli of the Lagrangian brane. Information about the BPS spectrum is extracted from the geometry of the mirror curve by construction of a nonabelianization map for exponential networks. For the simplest Calabi-Yau, $\mathbb{C}^3$ we reproduce the count of 5d BPS states encoded by the Mac Mahon function in the context of topological strings, and match predictions of 3d $tt^*$ geometry for the count of 3d-5d BPS states. We comment on applications of our construction to the study of enumerative invariants of toric Calabi-Yau threefolds.

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arXiv2018

All-Order Volume Conjecture for Closed 3-Manifolds from Complex Chern-Simons Theory

We propose an extension of the recently-proposed volume conjecture for closed hyperbolic 3-manifolds, to all orders in perturbative expansion. We first derive formulas for the perturbative expansion of the partition function of complex Chern-Simons theory around a hyperbolic flat connection, which produces infinitely-many perturbative invariants of the closed oriented 3-manifold. The conjecture is that this expansion coincides with the perturbative expansion of the Witten-Reshetikhin-Turaev invariants at roots of unity $q=e^{2 πi/r}$ with $r$ odd, in the limit $r \to \infty$. We provide numerical evidence for our conjecture.

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