Giant graviton expansion of Schur index and quasimodular forms

Abstract

A bstract The flavored superconformal Schur index of $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 U( N ) SYM has finite N corrections encoded in its giant graviton expansion in terms of D3 branes wrapped in AdS 5 × S 5 . The key element of this decomposition is the non-trivial index of the theory living on the wrapped brane system. A remarkable feature of the Schur limit is that the brane index is an analytic continuation of the flavored index of $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 U( n ) SYM, where n is the total brane wrapping number. We exploit recent exact results about the Schur index of $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 U( N ) SYM to evaluate the closed form of the brane indices appearing in the giant graviton expansion. Away from the unflavored limit, they are characterized by quasimodular forms providing exact information at all orders in the index universal fugacity. As an application of these results, we present novel exact expressions for the giant graviton expansion of the unflavored Schur index in a class of four dimensional $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 theories with equal central charges a = c , i.e. the non-Lagrangian theories $$ \hat{\Gamma}\left(\textrm{SU}(N)\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>Γ</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mfenced> <mml:mrow> <mml:mi>SU</mml:mi> <mml:mfenced> <mml:mi>N</mml:mi> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:math> with Γ = E 6 , E 7 , E 8 .

Locations

• Journal of High Energy Physics - View - PDF