47 research outputs found
BPS Spectra, Barcodes and Walls
BPS spectra give important insights into the non-perturbative regimes of
supersymmetric theories. Often from the study of BPS states one can infer
properties of the geometrical or algebraic structures underlying such theories.
In this paper we approach this problem from the perspective of persistent
homology. Persistent homology is at the base of topological data analysis,
which aims at extracting topological features out of a set of points. We use
these techniques to investigate the topological properties which characterize
the spectra of several supersymmetric models in field and string theory. We
discuss how such features change upon crossing walls of marginal stability in a
few examples. Then we look at the topological properties of the distributions
of BPS invariants in string compactifications on compact threefolds, used to
engineer black hole microstates. Finally we discuss the interplay between
persistent homology and modularity by considering certain number theoretical
functions used to count dyons in string compactifications and by studying
equivariant elliptic genera in the context of the Mathieu moonshine
Persistent Homology and String Vacua
We use methods from topological data analysis to study the topological
features of certain distributions of string vacua. Topological data analysis is
a multi-scale approach used to analyze the topological features of a dataset by
identifying which homological characteristics persist over a long range of
scales. We apply these techniques in several contexts. We analyze N=2 vacua by
focusing on certain distributions of Calabi-Yau varieties and Landau-Ginzburg
models. We then turn to flux compactifications and discuss how we can use
topological data analysis to extract physical informations. Finally we apply
these techniques to certain phenomenologically realistic heterotic models. We
discuss the possibility of characterizing string vacua using the topological
properties of their distributions.Comment: 32 pages, 12 pdf figure
Curve counting, instantons and McKay correspondences
We survey some features of equivariant instanton partition functions of
topological gauge theories on four and six dimensional toric Kahler varieties,
and their geometric and algebraic counterparts in the enumerative problem of
counting holomorphic curves. We discuss the relations of instanton counting to
representations of affine Lie algebras in the four-dimensional case, and to
Donaldson-Thomas theory for ideal sheaves on Calabi-Yau threefolds. For
resolutions of toric singularities, an algebraic structure induced by a quiver
determines the instanton moduli space through the McKay correspondence and its
generalizations. The correspondence elucidates the realization of gauge theory
partition functions as quasi-modular forms, and reformulates the computation of
noncommutative Donaldson-Thomas invariants in terms of the enumeration of
generalized instantons. New results include a general presentation of the
partition functions on ALE spaces as affine characters, a rigorous treatment of
equivariant partition functions on Hirzebruch surfaces, and a putative
connection between the special McKay correspondence and instanton counting on
Hirzebruch-Jung spaces.Comment: 79 pages, 3 figures; v2: typos corrected, reference added, new
summary section included; Final version to appear in Journal of Geometry and
Physic
Hodge-Elliptic Genera, K3 Surfaces and Enumerative Geometry
K3 surfaces play a prominent role in string theory and algebraic geometry. The properties of their enumerative invariants have important consequences in black hole physics and in number theory. To a K3 surface, string theory associates an Elliptic genus, a certain partition function directly related to the theory of Jacobi modular forms. A multiplicative lift of the Elliptic genus produces another modular object, an Igusa cusp form, which is the generating function of BPS invariants of K3×E. In this note, we will discuss a refinement of this chain of ideas. The Elliptic genus can be generalized to the so-called Hodge-Elliptic genus which is then related to the counting of refined BPS states of K3×E. We show how such BPS invariants can be computed explicitly in terms of different versions of the Hodge-Elliptic genus, sometimes in closed form, and discuss some generalizations
Line defects and (framed) BPS quivers
The BPS spectrum of certain N=2 supersymmetric field theories can be
determined algebraically by studying the representation theory of BPS quivers.
We introduce methods based on BPS quivers to study line defects. The presence
of a line defect opens up a new BPS sector: framed BPS states can be bound to
the defect. The defect can be geometrically described in terms of laminations
on a curve. To a lamination we associate certain elements of the Leavitt path
algebra of the BPS quiver and use them to compute the framed BPS spectrum. We
also provide an alternative characterization of line defects by introducing
framed BPS quivers. Using the theory of (quantum) cluster algebras, we derive
an algorithm to compute the framed BPS spectra of new defects from known ones.
Line defects are generated from a framed BPS quiver by applying certain
sequences of mutation operations. Framed BPS quivers also behave nicely under a
set of "cut and join" rules, which can be used to study how N=2 systems with
defects couple to produce more complicated ones. We illustrate our formalism
with several examples.Comment: 80 pages, 16 figures; v2: references added, note added, minor
corrections, final version to be published in JHE
On the M2-Brane Index on Noncommutative Crepant Resolutions
On certain M-theory backgrounds which are a circle fibration over a smooth Calabi-Yau the quantum theory of M2 branes can be studied in terms of the K-theoretic Donaldson-Thomas theory on the threefold. We extend this relation to noncommutative crepant resolutions. In this case the threefold develops a singularity and classical smooth geometry is replaced by the algebra of paths of a certain quiver. K-theoretic quantities on the quiver representation moduli space can be computed via toric localization and result in certain rational functions of the toric parameters. We discuss in particular the case of the conifold and certain orbifold singularities
Indefinite theta functions for counting attractor backgrounds
In this note, we employ indefinite theta functions to regularize canonical
partition functions for single-center dyonic BPS black holes. These partition
functions count dyonic degeneracies in the Hilbert space of four-dimensional
toroidally compactified heterotic string theory, graded by electric and
magnetic charges. The regularization is achieved by viewing the weighted sums
of degeneracies as sums over charge excitations in the near-horizon attractor
geometry of an arbitrarily chosen black hole background, and eliminating the
unstable modes. This enables us to rewrite these sums in terms of indefinite
theta functions. Background independence is then implemented by using the
transformation property of indefinite theta functions under elliptic
transformations, while modular transformations are used to make contact with
semi-classical results in supergravity.Comment: 24 pages, LaTe
Indefinite theta functions and black hole partition functions
We explore various aspects of supersymmetric black hole partition functions
in four-dimensional toroidally compactified heterotic string theory. These
functions suffer from divergences owing to the hyperbolic nature of the charge
lattice in this theory, which prevents them from having well-defined modular
transformation properties. In order to rectify this, we regularize these
functions by converting the divergent series into indefinite theta functions,
thereby obtaining fully regulated single-centered black hole partitions
functions.Comment: 35 pages; v2: various comments added; v3: a few typos correcte