arXiv subjects

Pietro Longhi

Publications and source records attributed to Pietro Longhi.

At least 37 records · Page 2Linked to original sources

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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arXiv2020

Localization of 4d $\mathcal{N}=1$ theories on $\mathbb{D}^2\times \mathbb{T}^2$

We consider 4d $\mathcal{N}=1$ gauge theories with R-symmetry on a hemisphere times a torus. We apply localization techniques to evaluate the exact partition function through a cohomological reformulation of the supersymmetry transformations. Our results represent the natural elliptic lifts of the lower dimensional analogs as well as a field theoretic derivation of the conjectured 4d holomorphic blocks, from which partition functions of compact spaces with diverse topology can be recovered through gluing. We also analyze the different boundary conditions which can naturally be imposed on the chiral multiplets, which turn out to be either Dirichlet or Robin-like. We show that different boundary conditions are related to each other by coupling the bulk to 3d $\mathcal{N}=1$ degrees of freedom on the boundary three-torus, for which we derive explicit 1-loop determinants.

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arXiv2020

Physics and geometry of knots-quivers correspondence

The recently conjectured knots-quivers correspondence relates gauge theoretic invariants of a knot $K$ in the 3-sphere to representation theory of a quiver $Q_{K}$ associated to the knot. In this paper we provide geometric and physical contexts for this conjecture within the framework of the large $N$ duality of Ooguri and Vafa, that relates knot invariants to counts of holomorphic curves with boundary on $L_{K}$, the conormal Lagrangian of the knot in the resolved conifold, and corresponding M-theory considerations. From the physics side, we show that the quiver encodes a 3d ${\mathcal N}=2$ theory $T[Q_{K}]$ whose low energy dynamics arises on the worldvolume of an M5 brane wrapping the knot conormal and we match the (K-theoretic) vortex partition function of this theory with the motivic generating series of $Q_{K}$. From the geometry side, we argue that the spectrum of (generalized) holomorphic curves on $L_{K}$ is generated by a finite set of basic disks. These disks correspond to the nodes of the quiver $Q_{K}$ and the linking of their boundaries to the quiver arrows. We extend this basic dictionary further and propose a detailed map between quiver data and topological and geometric properties of the basic disks that again leads to matching partition functions. We also study generalizations of A-polynomials associated to $Q_{K}$ and (doubly) refined version of LMOV invariants.

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arXiv2020

Multi-cover skeins, quivers, and 3d $\mathcal{N}=2$ dualities

The relation between open topological strings and representation theory of symmetric quivers is explored beyond the original setting of the knot-quiver correspondence. Multiple cover generalizations of the skein relation for boundaries of holomorphic disks on a Lagrangian brane are observed to generate dual quiver descriptions of the geometry. Embedding into M-theory, a large class of dualities of 3d $\mathcal{N}=2$ theories associated to quivers is obtained. The multi-cover skein relation admits a compact formulation in terms of quantum torus algebras associated to the quiver and in this language the relations are similar to wall-crossing identities of Kontsevich and Soibelman.

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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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arXiv2019

An infrared bootstrap of the Schur index with surface defects

The infrared formula relates the Schur index of a 4d $\mathcal{N}=2$ theory to its wall-crossing invariant, a.k.a. BPS monodromy. A further extension of this formula, proposed by Córdova, Gaiotto and Shao, includes contributions by various types of line and surface defects. We study BPS monodromies in the presence of vortex surface defects of arbitrary vorticity for general class $\mathcal{S}$ theories of type $A_1$ engineered by UV curves with at least one regular puncture. The trace of these defect BPS monodromies is shown to coincide with the action of certain $q$-difference operators acting on the trace of the (pure) 4d BPS monodromy. We use these operators to develop a "bootstrap" (of traces) of BPS monodromies, relying only on their infrared properties, thereby reproducing the very general ultraviolet characterization of the Schur index.

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arXiv2019

Gluing two affine Yangians of $\mathfrak{gl}_1$

We construct a four-parameter family of affine Yangian algebras by gluing two copies of the affine Yangian of $\mathfrak{gl}_1$. Our construction allows for gluing operators with arbitrary (integer or half integer) conformal dimension and arbitrary (bosonic or fermionic) statistics, which is related to the relative framing. The resulting family of algebras is a two-parameter generalization of the $\mathcal{N}=2$ affine Yangian, which is isomorphic to the universal enveloping algebra of $\mathfrak{u}(1)\oplus \mathcal{W}^{\mathcal{N}=2}_{\infty}[λ]$. All algebras that we construct have natural representations in terms of "twin plane partitions", a pair of plane partitions appropriately joined along one common leg. We observe that the geometry of twin plane partitions, which determines the algebra, bears striking similarities to the geometry of certain toric Calabi-Yau threefolds.

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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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arXiv2017

Rigid limit for hypermultiplets and five-dimensional gauge theories

We study the rigid limit of a class of hypermultiplet moduli spaces appearing in Calabi-Yau compactifications of type IIB string theory, which is induced by a local limit of the Calabi-Yau. We show that the resulting hyperkahler manifold is obtained by performing a hyperkahler quotient of the Swann bundle over the moduli space, along the isometries arising in the limit. Physically, this manifold appears as the target space of the non-linear sigma model obtained by compactification of a five-dimensional gauge theory on a torus. This allows to compute dyonic and stringy instantons of the gauge theory from the known results on D-instantons in string theory. Besides, we formulate a simple condition on the existence of a non-trivial local limit in terms of intersection numbers of the Calabi-Yau, and find an explicit form for the hypermultiplet metric including corrections from all mutually non-local D-instantons, which can be of independent interest.

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arXiv2017

S duality and Framed BPS States via BPS Graphs

We study a realization of S dualities of four-dimensional $\mathcal{N}=2$ class $\mathcal{S}$ theories based on BPS graphs. S duality transformations of the UV curve are explicitly expressed as a sequence of topological transitions of the graph, and translated into cluster transformations of the algebra associated to the dual BPS quiver. Our construction applies to generic class $\mathcal{S}$ theories, including those with non-maximal flavor symmetry, generalizing previous results based on higher triangulations. We study the the action of S duality on UV line operators, and show that it matches precisely with the mapping class group, by a careful analysis of framed wall-crossing. We comment on the implications of our results for the computation of three-manifold invariants via cluster partition functions.

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arXiv2017

Wall-Crossing Invariants from Spectral Networks

A new construction of BPS monodromies for 4d ${\mathcal N}=2$ theories of class S is introduced. A novel feature of this construction is its manifest invariance under Kontsevich-Soibelman wall crossing, in the sense that no information on the 4d BPS spectrum is employed. The BPS monodromy is encoded by topological data of a finite graph, embedded into the UV curve $C$ of the theory. The graph arises from a degenerate limit of spectral networks, constructed at maximal intersections of walls of marginal stability in the Coulomb branch of the gauge theory. The topology of the graph, together with a notion of framing, encode equations that determine the monodromy. We develop an algorithmic technique for solving the equations, and compute the monodromy in several examples. The graph manifestly encodes the symmetries of the monodromy, providing some support for conjectural relations to specializations of the superconformal index. For $A_1$-type theories, the graphs encoding the monodromy are "dessins d'enfants" on $C$, the corresponding Strebel differentials coincide with the quadratic differentials that characterize the Seiberg-Witten curve.

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arXiv2017

BPS Graphs: From Spectral Networks to BPS Quivers

We define "BPS graphs" on punctured Riemann surfaces associated with $A_{N-1}$ theories of class $\mathcal{S}$. BPS graphs provide a bridge between two powerful frameworks for studying the spectrum of BPS states: spectral networks and BPS quivers. They arise from degenerate spectral networks at maximal intersections of walls of marginal stability on the Coulomb branch. While the BPS spectrum is ill-defined at such intersections, a BPS graph captures a useful basis of elementary BPS states. The topology of a BPS graph encodes a BPS quiver, even for higher-rank theories and for theories with certain partial punctures. BPS graphs lead to a geometric realization of the combinatorics of Fock-Goncharov $N$-triangulations and generalize them in several ways.

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arXiv2017

The Vector-like Twin Higgs

We present a version of the twin Higgs mechanism with vector-like top partners. In this setup all gauge anomalies automatically cancel, even without twin leptons. The matter content of the most minimal twin sector is therefore just two twin tops and one twin bottom. The LHC phenomenology, illustrated with two example models, is dominated by twin glueball decays, possibly in association with Higgs bosons. We further construct an explicit four-dimensional UV completion and discuss a variety of UV completions relevant for both vector-like and fraternal twin Higgs models.

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arXiv2017

ADE Spectral Networks and Decoupling Limits of Surface Defects

We study vacua and BPS spectra of canonical surface defects of class $\mathcal{S}$ theories in different decoupling limits using ADE spectral networks. In some regions of the IR moduli spaces of these 2d-4d systems, the mixing between 2d and 4d BPS states is suppressed, and the spectrum of 2d-4d BPS states becomes that of a 2d $\mathcal{N}=(2,2)$ theory. For some decoupling limits, we identify the 2d theories describing the surface defects with nonlinear sigma models and coset models that have been previously studied. We also study certain cases where the decoupling limit of a surface defect exhibits a set of vacua and a BPS spectrum that appear to be entirely new. A detailed analysis of these spectra and their wall-crossing behavior is performed.

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arXiv2016

ADE Spectral Networks

We introduce a new perspective and a generalization of spectral networks for 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ associated to Lie algebras $\mathfrak{g} = \textrm{A}_n$, $\textrm{D}_n$, $\textrm{E}_{6}$, and $\textrm{E}_{7}$. Spectral networks directly compute the BPS spectra of 2d theories on surface defects coupled to the 4d theories. A Lie algebraic interpretation of these spectra emerges naturally from our construction, leading to a new description of 2d-4d wall-crossing phenomena. Our construction also provides an efficient framework for the study of BPS spectra of the 4d theories. In addition, we consider novel types of surface defects associated with minuscule representations of $\mathfrak{g}$.

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arXiv2015

The Orbifold Higgs

We introduce and systematically study an expansive class of "orbifold Higgs" theories in which the weak scale is protected by accidental symmetries arising from the orbifold reduction of continuous symmetries. The protection mechanism eliminates quadratic sensitivity of the Higgs mass to higher scales at one loop (or more) and does not involve any new states charged under the Standard Model. The structures of the Higgs and top sectors are universal and determined exclusively by group theoretical considerations. The twin Higgs model fits within our framework as the simplest example of an orbifold Higgs. Our models admit UV completions as geometric orbifolds in higher dimensions, and fit naturally within frameworks of low scale gauge coupling unification.

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arXiv2014

Neutral Naturalness from the Orbifold Higgs

We present a general class of natural theories in which the Higgs is a pseudo-goldstone boson in an orbifolded gauge theory. The symmetry protecting the Higgs at low energies is an accidental global symmetry of the quadratic action, rather than a full continuous symmetry. The lightest degrees of freedom protecting the weak scale carry no Standard Model (SM) quantum numbers and interact with visible matter principally through the Higgs portal. We find that the twin Higgs is the simplest example of an orbifold Higgs. This opens the door to the systematic study of "neutral naturalness": natural theories with SM-neutral states that are as yet untested by the LHC.

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arXiv2014

Spectral Networks with Spin

The BPS spectrum of d=4 N=2 field theories in general contains not only hyper- and vector-multipelts but also short multiplets of particles with arbitrarily high spin. This paper extends the method of spectral networks to give an algorithm for computing the spin content of the BPS spectrum of d=4 N=2 field theories of class S. The key new ingredient is an identification of the spin of states with the writhe of paths on the Seiberg-Witten curve. Connections to quiver representation theory and to Chern-Simons theory are briefly discussed.

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