Gopakumar–Vafa duality is a duality in string theory, hence a correspondence between two different theories, in this case between Chern–Simons theory 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, similar to Gopakumar–Vafa invariants and Pandharipande–Thomas invariants. Gopakumar–Vafa duality is named after Rajesh Gopakumar and Cumrun Vafa, who first described it in 1998.
Formulation Gopakumar–Vafa duality describes a correspondence between Chern–Simons theory on the cotangent bundle T ∗ S 3 {\displaystyle T^{*}S^{3}} over the three-dimensional sphere S 3 {\displaystyle S^{3}} and Gromov–Witten theory on the Whitney sum O ( − 2 ) = O ( − 1 ) ⊕ O ( − 1 ) {\displaystyle {\mathcal {O}}(-2)={\mathcal {O}}(-1)\oplus {\mathcal {O}}(-1)} of the tautological bundle over the two-dimensional sphere S 2 ≅ C P 1 {\displaystyle S^{2}\cong \mathbb {C} P^{1}} . One has a canonical inclusion S 3 ↪ R 4 {\displaystyle S^{3}\hookrightarrow \mathbb {R} ^{4}} , which induces an inclusion T ∗ S 3 ↪ T ∗ R 4 ≅ R 4 × R 4 ≅ C 4 ≅ Mat 2 ( C ) {\displaystyle T^{*}S^{3}\hookrightarrow T^{*}\mathbb {R} ^{4}\cong \mathbb {R} ^{4}\times \mathbb {R} ^{4}\cong \mathbb {C} ^{4}\cong \operatorname {Mat} _{2}(\mathbb {C} )} . With a suitable endomorphism C 4 → C 4 {\displaystyle \mathbb {C} ^{4}\rightarrow \mathbb {C} ^{4}} in between, it reduces to a diffeomorphism T ∗ S 3 → SL 2 ( C ) {\displaystyle T^{*}S^{3}\rightarrow \operatorname {SL} _{2}(\mathbb {C} )} to the special linear group and through composition with the zero section 0 : S 3 → T ∗ S 3 {\displaystyle 0\colon S^{3}\rightarrow T^{*}S^{3}} further to a diffeomorphism S 3 → SU ( 2 ) {\displaystyle S^{3}\rightarrow \operatorname {SU} (2)} to the special unitary group. One also has:
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