Quantum cohomology and enumerative geometry

2006 ◽  
pp. 185-196
Author(s):  
Sheldon Katz
Keyword(s):  
Quantum Cohomology ◽  
Enumerative Geometry

ON GROMOV–WITTEN INVARIANTS OF DEL PEZZO SURFACES

2013 ◽  
Vol 24 (07) ◽  
pp. 1350054 ◽  
Author(s):  
MENDY SHOVAL ◽  
EUGENII SHUSTIN
Keyword(s):  
Quantum Cohomology ◽  
Enumerative Geometry ◽  
Del Pezzo Surfaces ◽  
Manuscripta Math ◽  
Moduli Space Of Curves ◽  
Divisor Class ◽  
Genus Zero ◽  
Type Formula ◽  
Del Pezzo ◽  
Generic Points

We compute Gromov–Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 2. The genus zero invariants have been computed a long ago [P. Di Francesco and C. Itzykson, Quantum intersection rings, in The Moduli Space of Curves, eds. R. Dijkgraaf et al., Progress in Mathematics, Vol. 129 (Birkhäuser, Boston, 1995), pp. 81–148; L. Göttsche and R. Pandharipande, The quantum cohomology of blow-ups of ℙ2 and enumerative geometry, J. Differential Geom.48(1) (1998) 61–90], Gromov–Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 3 have been found by Vakil [Counting curves on rational surfaces, Manuscripta Math.102 (2000) 53–84]. We solve the problem in two steps: (1) we consider curves on [Formula: see text], the plane blown up at one point, which have given degree, genus, and prescribed multiplicities at fixed generic points on a conic that avoids the blown-up point; then we obtain a Caporaso–Harris type formula counting such curves subject to arbitrary additional tangency conditions with respect to the chosen conic; as a result we count curves of any given divisor class and genus on a surface of type [Formula: see text], the plane blown up at six points on a given conic and at one more point outside the conic; (2) in the next step, we express the Gromov–Witten invariants of [Formula: see text] via enumerative invariants of [Formula: see text], using Vakil's extension of the Abramovich–Bertram formula.

scholarly journals Gromov-Witten classes, quantum cohomology, and enumerative geometry

1994 ◽  
Vol 164 (3) ◽  
pp. 525-562 ◽  
Author(s):  
M. Kontsevich ◽  
Yu. Manin
Keyword(s):  
Quantum Cohomology ◽  
Enumerative Geometry

scholarly journals The quantum cohomology of blow-ups of ${\bf P}\sp 2$ and enumerative geometry

1998 ◽  
Vol 48 (1) ◽  
pp. 61-90 ◽  
Author(s):  
L. Göttsche ◽  
R. Pandharipande
Keyword(s):  
Quantum Cohomology ◽  
Enumerative Geometry

scholarly journals ON THE HAMILTONIAN REDUCTION OF THE ASSOCIATIVITY EQUATIONS IN THE CASE OF FOUR PRIMARY FIELDS

2019 ◽  
Vol 47 (1) ◽  
pp. 118-122
Author(s):  
N.A. Strizhova
Keyword(s):  
Quantum Cohomology ◽  
Mathematical Physics ◽  
Fundamental Principle ◽  
Stationary Points ◽  
Hamiltonian Reduction ◽  
Enumerative Geometry ◽  
Associativity Equations ◽  
Topological Field ◽  
Hamiltonian Property ◽  
Arbitrary Flow

The associativity equations arose in the papers of Witten (Witten, 1990) and Dijkgraaf, Verlinde, (Dijkgraaf et al., 1991) on two-dimensional topological field theories and subsequently they became to play a key role in many other important domains of mathematics and mathematical physics: in quantum cohomology, Gromov–Witten invariants, enumerative geometry, theory of submanifolds and so on. In Mokhov’s papers (Mokhov, 1984), (Mokhov, 1987) a general fundamental principle stating a canonical Hamiltonian property for the restriction of an arbitrary flow on the set of stationary points of its nondegenerate integral was proposed and proved. In this paper the Hamiltonians of the reductions of the associativity equations with antidiagonal matrix ηij in the case of four primary fields according to Mokhov`s construction is found in an explicit form.

scholarly journals Tangency quantum cohomology and characteristic numbers

2001 ◽  
Vol 73 (3) ◽  
pp. 319-326
Author(s):  
JOACHIM KOCK
Keyword(s):  
2D Gravity ◽  
Quantum Cohomology ◽  
Rational Curves ◽  
The Other ◽  
Enumerative Geometry ◽  
Poincaré Metric ◽  
Characteristic Numbers ◽  
Systems Of Differential Equations ◽  
Topological Recursion ◽  
Homogeneous Variety

This work establishes a connection between gravitational quantum cohomology and enumerative geometry of rational curves (in a projective homogeneous variety) subject to conditions of infinitesimal nature like, for example, tangency. The key concept is that of modified psi classes, which are well suited for enumerative purposes and substitute the tautological psi classes of 2D gravity. The main results are two systems of differential equations for the generating function of certain top products of such classes. One is topological recursion while the other is Witten-Dijkgraaf-Verlinde-Verlinde. In both cases, however, the background metric is not the usual Poincaré metric but a certain deformation of it, which surprisingly encodes all the combinatorics of the peculiar way modified psi classes restrict to the boundary. This machinery is applied to various enumerative problems, among which characteristic numbers in any projective homogeneous variety, characteristic numbers for curves with cusp, prescribed triple contact, or double points.

scholarly journals Wilson loop algebras and quantum K-theory for Grassmannians

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Hans Jockers ◽  
Peter Mayr ◽  
Urmi Ninad ◽  
Alexander Tabler
Keyword(s):  
Wilson Loop ◽  
Gauge Theories ◽  
Wilson Line ◽  
Quantum Cohomology ◽  
Three Dimensional ◽  
Loop Algebra ◽  
Loop Algebras ◽  
Verlinde Algebra ◽  
K Theory ◽  
Chern Simons

Abstract We study the algebra of Wilson line operators in three-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric U(M ) gauge theories with a Higgs phase related to a complex Grassmannian Gr(M, N ), and its connection to K-theoretic Gromov-Witten invariants for Gr(M, N ). For different Chern-Simons levels, the Wilson loop algebra realizes either the quantum cohomology of Gr(M, N ), isomorphic to the Verlinde algebra for U(M ), or the quantum K-theoretic ring of Schubert structure sheaves studied by mathematicians, or closely related algebras.

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