Gopakumar–Vafa duality is a duality in

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

, hence a correspondence between two different theories, in this case between

Chern–Simons theory The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and James Harris Simons, who intr ...

and Gromov–Witten theory. The latter is known as the mathematical equivalent of string theory in mathematics and counts pseudoholomorphic curves on a

symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...

, similar to Gopakumar–Vafa invariants and Pandharipande–Thomas invariants. Gopakumar–Vafa duality is named after Rajesh Gopakumar and

Cumrun Vafa Cumrun Vafa (, ; born 1 August 1960) is an Iranian-American theoretical physicist and the Hollis Professor of Mathematicks and Natural Philosophy at Harvard University. Early life and education Cumrun Vafa was born in Tehran, Iran on 1 August 1 ...

, who first described it in 1998.

Formulation

Gopakumar–Vafa duality describes a correspondence between

on the

cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...

T^*S^3

over the three-dimensional

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

S^3

and Gromov–Witten theory on the

Whitney sum In mathematics, a vector bundle is a topological construction that makes precise the idea of a Family of sets, family of vector spaces parameterized by another space (mathematics), space X (for example X could be a topological space, a manifold, ...

\mathcal(-2)
=\mathcal(-1)\oplus\mathcal(-1)

of the

tautological bundle In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k- dimensional subspaces of V, given a point in the Grassmannian corresponding to a k-dimensional vector s ...

over the two-dimensional sphere

S^2\cong\mathbbP^1

. One has a canonical inclusion

S^3\hookrightarrow\mathbb^4

, which induces an inclusion

T^*S^3\hookrightarrow T^*\mathbb^4\cong\mathbb^4\times\mathbb^4\cong\mathbb^4\cong\operatorname_2(\mathbb)

. With a suitable endomorphism

\mathbb^4\rightarrow\mathbb^4

in between, it reduces to a

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...

T^*S^3\rightarrow\operatorname_2(\mathbb)

to the

special linear group In mathematics, the special linear group \operatorname(n,R) of degree n over a commutative ring R is the set of n\times n Matrix (mathematics), matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix ...

and through composition with the

zero section In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...

0\colon S^3\rightarrow T^*S^3

further to a diffeomorphism

S^3\rightarrow\operatorname(2)

to the

special unitary group In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 ...

.Auckly & Koshkin 2007, p. 210 One also has: :

\mathcal(-1)
=\left\,

with

-1

denoting the first

Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches ...

of the complex line bundle. By descreasing the determinant to vanish completely,

T^*S^3\cong\operatorname_2(\mathbb)

can be shrunken down to a

conifold In mathematics and string theory, a conifold is a generalization of a manifold. Unlike manifolds, conifolds can contain conical singularities, i.e. points whose neighbourhoods look like cones over a certain base. In physics, in particular in flux ...

, which can be obtained as a resolution from

\mathcal(-2)
=\mathcal(-1)\oplus\mathcal(-1)

. From the perspective of

surgery theory In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while An ...

, this corresponds to the surgery

S^3\times\mathbb^3\rightsquigarrow S^2\times\mathbb^4

. A obvious generalization of the sphere

S^3

is additionally considering the

cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...

\mathbb_p

to act on it, which leads to Lense spaces

L(p,q)

. Gopakumar–Vafa duality can only be carried over to the Lense space

L(p,1)

.Brini, Griguolo, Seminara & Tanzini 2008, Claim 1

Literature

* * * * {{cite journal , last=Brini , first=Andrea , last2=Griguolo , first2=Luca , date=2008-09-09 , title=Chern-Simons theory on L(p,q) lens spaces and Gopakumar-Vafa duality , journal=

Journal of Geometry and Physics The ''Journal of Geometry and Physics'' is a scientific journal in mathematical physics. Its scope is to stimulate the interaction between geometry and physics by publishing primary research and review articles which are of common interest to pract ...

, volume=60 , pages=417–429 , arxiv=0809.1610 , last3=Seminara , first3=Domenico , last4=Tanzini , first4=Alessandro

References

External links

* Gopakumar–Vafa duality on ''n''Lab String theory