Stable pairs and BPS invariants

R. Pandharipande; R. P. Thomas

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Stable pairs and BPS invariants

R. Pandharipande, R. P. Thomas

Algebraic Geometry · Submitted 25 Nov 2007

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that for irreducible classes the stable pairs generating function satisfies the strong BPS rationality conjectures.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.54 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.42 · compares with prior results 0.74 · mentions code or data availability 0.02.

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arXiv listing abstract v4. Results and the main claim are underlined; show every role.

We define the BPS invariants of Gopakumar-Vafa in the case of irreducible curve classes on Calabi-Yau 3-folds. The main tools are the theory of stable pairs in the derived category and Behrend's constructible function approach to the virtual class. We prove that for irreducible classes the stable pairs generating function satisfies the strong BPS rationality conjectures. We define the contribution of each curve to the BPS invariants. A curve $C$ only contributes to the BPS invariants in genera lying between the geometric genus and arithmetic genus of $C$. Complete formulae are derived for nonsingular and nodal curves. A discussion of primitive classes on K3 surfaces from the point of view of stable pairs is given in the Appendix via calculations of Kawai-Yoshioka. A proof of the Yau-Zaslow formula for rational curve counts is obtained. A connection is made to the Katz-Klemm-Vafa formula for BPS counts in all genera.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We define the BPS invariants of Gopakumar-Vafa in the case of irreducible curve classes on Calabi-Yau 3-folds.	contribution 1.00	0.94	0.10	asserted 0.99
s1	The main tools are the theory of stable pairs in the derived category and Behrend's constructible function approach to the virtual class.	method 1.00	0.27	0.04	asserted 0.99
s2	We prove that for irreducible classes the stable pairs generating function satisfies the strong BPS rationality conjectures.	result 0.98	0.95	0.76	asserted 1.00
s3	We define the contribution of each curve to the BPS invariants.	contribution 0.99	0.94	0.06	asserted 1.00
s4	A curve $C$ only contributes to the BPS invariants in genera lying between the geometric genus and arithmetic genus of $C$.	result 0.99	0.82	0.10	asserted 1.00
s5	Complete formulae are derived for nonsingular and nodal curves.	result 0.84	0.92	0.07	asserted 1.00
s6	A discussion of primitive classes on K3 surfaces from the point of view of stable pairs is given in the Appendix via calculations of Kawai-Yoshioka.	contribution 0.78	0.30	0.02	asserted 0.97
s7	A proof of the Yau-Zaslow formula for rational curve counts is obtained.	result 0.98	0.91	0.10	asserted 1.00
s8	A connection is made to the Katz-Klemm-Vafa formula for BPS counts in all genera.	interpretation 0.37	0.74	0.07	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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The paper

Original abstract

We define the BPS invariants of Gopakumar-Vafa in the case of irreducible curve classes on Calabi-Yau 3-folds. The main tools are the theory of stable pairs in the derived category and Behrend's constructible function approach to the virtual class. We prove that for irreducible classes the stable pairs generating function satisfies the strong BPS rationality conjectures. We define the contribution of each curve to the BPS invariants. A curve $C$ only contributes to the BPS invariants in genera lying between the geometric genus and arithmetic genus of $C$. Complete formulae are derived for nonsingular and nodal curves. A discussion of primitive classes on K3 surfaces from the point of view of stable pairs is given in the Appendix via calculations of Kawai-Yoshioka. A proof of the Yau-Zaslow formula for rational curve counts is obtained. A connection is made to the Katz-Klemm-Vafa formula for BPS counts in all genera.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Topological String on elliptic CY 3-folds and the ring of Jacobi forms
hep-th 2015-01 conditional novelty 7.0 of 10

The all-genus topological string partition function on an elliptic Calabi-Yau threefold is a ratio of weak Jacobi forms with a fixed product denominator, and it matches curve-counting invariants.
The refined BPS index from stable pair invariants
hep-th 2012-10 conditional novelty 7.0 of 10

A torus-localisation definition of refined stable-pair invariants reproduces, in low degree on local P2 and local P1 x P1, the refined BPS spectra obtained independently from the holomorphic anomaly.
Refined BPS invariants of 6d SCFTs from anomalies and modularity
hep-th 2017-01 conditional novelty 6.0 of 10

The refined topological string partition function on elliptically fibered local Calabi-Yau manifolds is shown to be a meromorphic Jacobi form whose numerator is fixed by vanishing conditions on 5d BPS numbers, yieldin...
Refined stable pair invariants for E-, M- and [p,q]-strings
hep-th 2013-08 conditional novelty 6.0 of 10

Refined stable-pair invariants for local del Pezzo and half K3 geometries are computed via mirror symmetry and the refined holomorphic anomaly equation, and are shown to fall into E8 Weyl orbits.

How the literature uses it

29 arXiv papers cite this in 71 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 49 sentences without a confident label

Builds on it 1 paper

The following result lifts [ref] from Euler characteristics to $$; the proof is essentially the same.

Refined curve counting on complex surfaces · Refined invariants · 0.99

Uses its method or tool 4 papers

Then the sequence [equation] passes by $_C(·,_C)$ into [equation] This induces an isomorphism of $C$ with the moduli space of pairs [ref] [equation] The isomorphism matches length $l$ schemes to sheaves of holomorphic Euler characteristic $l+1-δ$.

The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link · The genus expansion · 0.85
We use the method of [ref] , which can be made rigorous using Donaldson-Thomas-type invariants [ref] (see also [ref] ).

Topological Strings on Non-Commutative Resolutions · The octic double solid · 1.00
For unit D4 charge $p=1$, the wall-crossing formula [ref] gives [ref] [ Instead of DT, the wall-crossing analysis of [ref] considers stable-pair invariants [ref] , also known as Pandharipande-Thomas (PT) invariants, whose generating function is related to that of DT by stripping of a MacMahon factor $M(-q)^χ$.

Blackhole/blackring transition · Black ring microstates with unit M5 charge · 0.92

Compares with it 2 papers

We observe that these relations agree with [ref], [ref], after identifying [ref] [equation] We shall now apply these relations to determine the GV invariants $(Q)$ and $(Q)-1$ for degree $Q=κd$ with $d∈ℕ$ for the 13 hypergeometric CY threefolds.

Quantum geometry, stability and modularity · GV invariants at maximal and submaximal genus · 0.97
This description of stable pairs in terms of Hilbert schemes parameterising either the subschemes $Z_i$ [ref] or the dual subschemes $D_i$ [ref] is related to, but different from, the description [ref] of stable pairs on surfaces in terms of relative Hilbert schemes.

Stable pairs with descendents on local surfaces I: the vertical component · The dual description · 0.99

Cites it as support 2 papers

By the result mentioned above, for any $n ≥0$ there is a one-to-one correspondence [equation] which is in fact an isomorphism of schemes ( [ref] ).

Hilbert schemes and stable pairs: GIT and derived category wall crossings · GIT wall crossing · 0.97
In fact it is proved in [ref] that $P_n(S,β)$ is a relative Hilbert scheme of points on the fibres of the universal curve over $_β(S)$.

Reduced classes and curve counting on surfaces I: theory · Det and div · 0.96

Cites it as background 10 papers

We recall from [ref] that $P_i(S,β)$ is the relative Hilbert scheme of $i-1+h$ points on the fibres of the universal curve over $_β(S)$.

Reduced classes and curve counting on surfaces I: theory · Severi degrees as reduced stable pair invariants · 0.97
We begin by reviewing [ref] and [ref] , presenting them now in the Grothendieck ring of varieties.

The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link · Rationality and Symmetry · 0.95
For stable pairs on $S$ in class $β$ (or on $X:=K_S$ in class $ι_*β$) with holomorphic Euler characteristic [equation] we refer to [ref] .

Reduced classes and curve counting on surfaces II: calculations · The moduli space as a zero locus · 0.92

What it cites · 15 sentences without a confident label

Compares with 2 papers

In fact, the wall-crossing is much simpler than the wall-crossing conjectured in [ref] to equate the invariants $P_n,β$ to the reduced DT invariants of [ref] .

Curve counting via stable pairs in the derived category · Wall-crossing · 0.91
Other papers on the subject [ref] have defined BPS invariants following the original perspective of [ref] using $(𝔰𝔩_2×𝔰𝔩_2)$-actions on sophisticated cohomology theories, but have been unable to incorporate the virtual class.

Birational Calabi-Yau 3-folds and BPS state counting · Remarks · 0.86

Cites as background 3 papers

Invariants counting curves in $X$ via stable pairs have been defined in [ref] .

Curve counting via stable pairs in the derived category · Introduction · 0.97
The Yau-Zaslow formula was proven in the primitive[A proof of the Yau-Zaslow formula for all curve classes on $K3$ surfaces has been recently obtained in [ref] .] case by Bryan-Leung [ref] via Gromov-Witten theory.

Noether-Lefschetz theory and the Yau-Zaslow conjecture · Introduction · 0.80
The corresponding calculations in Gromov-Witten theory have not yet been completed.[A discussion of the Gromov-Witten side can be found in [ref] .]

Gromov-Witten theory and Noether-Lefschetz theory · Introduction · 0.92

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