Tetrahedron Instantons
Elli Pomoni, Wenbin Yan, Xinyu Zhang
Abstract
Abstract We introduce and study tetrahedron instantons, which can be realized in string theory by $$\hbox {D}1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>D</mml:mtext> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> -branes probing a configuration of intersecting $$\hbox {D}7$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>D</mml:mtext> <mml:mn>7</mml:mn> </mml:mrow> </mml:math> -branes in flat spacetime with a proper constant B -field. Physically they capture instantons on $$\mathbb {C}^{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> in the presence of the most general intersecting real codimension-two supersymmetric defects. Moreover, we construct the tetrahedron instantons as particular solutions of general instanton equations in noncommutative field theory. We analyze the moduli space of tetrahedron instantons and discuss the geometric interpretations. We compute the instanton partition function both via the equivariant localization on the moduli space of tetrahedron instantons and via the elliptic genus of the worldvolume theory on the $$\hbox {D}1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>D</mml:mtext> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> -branes probing the intersecting $$\hbox {D}7$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>D</mml:mtext> <mml:mn>7</mml:mn> </mml:mrow> </mml:math> -branes, obtaining the same result. The instanton partition function of the tetrahedron instantons lies between the higher-rank Donaldson–Thomas invariants on $$\mathbb {C}^{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> and the partition function of the magnificent four model, which is conjectured to be the mother of all instanton partition functions. Finally, we show that the instanton partition function admits a free field representation, suggesting the existence of a novel kind of symmetry which acts on the cohomology of the moduli spaces of tetrahedron instantons.