Pietro Longhi

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Longhi

First Name

Pietro

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0000-0002-5558-0386

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Publications 1 - 10 of 13

Multi-cover skeins, quivers, and 3d N = 2 dualities
Ekholm, Tobias; Kucharski, Piotr; Longhi, Pietro (2020)
Journal of High Energy Physics
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 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.
Gluing two affine Yangians of gl(1)
Li, Wei; Longhi, Pietro (2019)
Journal of High Energy Physics
We construct a four-parameter family of affine Yangian algebras by gluing two copies of the affine Yangian of 𝔤�𝔩�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 N = 2 affine Yangian, which is isomorphic to the universal enveloping algebra of 𝔲� (1)⊕ 𝒲�N=2∞[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.
From Quantum Curves to Topological String Partition Functions II
Coman, Ioana; Longhi, Pietro; Teschner, Jörg (2025)
Annales Henri Poincaré
We propose a geometric characterisation of the topological string partition functions associated with the local Calabi–Yau (CY) manifolds used in the geometric engineering of d = 4, N = 2 supersymmetric field theories of class S. A quantisation of these CY manifolds defines differential operators called quantum curves. The partition functions are extracted from the isomonodromic tau-functions associated with the quantum curves by expansions of generalised theta series type. It turns out that the partition functions are in one-to-one correspondence with preferred coordinates on the moduli spaces of quantum curves defined using the Exact WKB method. The coordinates defined in this way jump across certain loci in the moduli space. The changes of normalization of the tau-functions associated with these jumps define a natural line bundle playing a key role in the geometric characterisation of the topological string partition functions proposed here.
Exploring 5d BPS Spectra with Exponential Networks
Banerjee, Sibasish; Longhi, Pietro; Romo, Mauricio (2019)
Annales Henri Poincaré
Quiver Symmetries and Wall-Crossing Invariance
Del Monte, Fabrizio; Longhi, Pietro (2023)
Communications in Mathematical Physics
We study the BPS particle spectrum of five-dimensional superconformal field theories on R4×S1 with one-dimensional Coulomb branch, by means of their associated BPS quivers. By viewing these theories as arising from the geometric engineering within M-theory, the quivers are naturally associated to the corresponding local Calabi–Yau threefold. We show that the symmetries of the quiver, descending from the symmetries of the Calabi–Yau geometry, together with the affine root lattice structure of the flavor charges, provide equations for the Kontsevich–Soibelman wall-crossing invariant. We solve these equations iteratively: the pattern arising from the solution is naturally extended to an exact conjectural expression, that we provide for the local Hirzebruch F0, and local del Pezzo dP3 and dP5 geometries. Remarkably, the BPS spectrum consists of two copies of suitable 4d N=2 spectra, augmented by Kaluza-Klein towers.
Exponential BPS Graphs and D Brane Counting on Toric Calabi-Yau Threefolds: Part I
Banerjee, Sibasish; Longhi, Pietro; Romo, Mauricio (2021)
Communications in Mathematical Physics
We study BPS spectra of D-branes on local Calabi-Yau threefolds O(-p)circle plus O(p-2) -> P-1 with p = 0,1, corresponding to C-3/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.
An infrared bootstrap of the Schur index with surface defects
Fluder, Martin; Longhi, Pietro (2019)
Journal of High Energy Physics
The infrared formula relates the Schur index of a 4d 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 S theories of type A1 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.
Instanton Particles and Monopole Strings in 5D SU(2) Supersymmetric Yang-Mills Theory
Longhi, Pietro (2021)
Physical Review Letters
We provide a closed-form expression for the motivic Kontsevich-Soibelman invariant for M theory in the background of the toric Calabi-Yau threefold KF0. This encodes the refined Bogomol'nyi-Prasad-Sommerfield spectrum of SU(2) 5D N=1 Yang-Mills theory on S1×R4, corresponding to rank-zero Donaldson-Thomas invariants for KF0, anywhere on the Coulomb branch.
Exponential BPS Graphs and D-Brane Counting on Toric Calabi-Yau Threefolds: Part II
Banerjee, Sibasish; Longhi, Pietro; Romo, Mauricio (2025)
Communications in Mathematical Physics
We study BPS states of 5d N = 1 SU(2) Yang-Mills theory on S¹ x R⁴. Geometric engineering relates these to enumerative invariants for the local Hirzebruch surface F₀. 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 N = 2 Seiberg–Witten theory, recovering spectral networks in the limit.
A-branes, Foliations and Localization
Banerjee, Sibasish; Longhi, Pietro; Romo, Mauricio (2023)
Annales Henri Poincaré
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.

Publications 1 - 10 of 13

Pietro Longhi

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