Branched SL(2,ℤ) duality

Abstract

We investigate how SL(2,ℤ) duality is realized in nonrelativistic type IIB superstring theory, which is a self-contained corner of relativistic string theory. Within this corner, we realize manifestly SL(2,ℤ)-invariant (p, q)-string actions. The construction of these actions imposes a branching between strings of opposite charges associated with the two-form fields. The branch point is determined by these charges and the axion background field. Both branches must be incorporated in order to realize the full SL(2,ℤ) group. Besides these string actions, we also construct D-instanton and D3-brane actions that manifestly realize the branched SL(2,ℤ) symmetry.

Anisotropic compactification of nonrelativistic M-theory

“…We will then use this formalism to provide the M-theory interpretation of the SL(2 , Z) duality in nonrelativistic IIB string theory. This procedure will naturally lead to the polynomial realization of SL(2 , Z) discovered in [31], which we will review in section 3.3.…”

“…However, after taking the membrane limit later in section 3.2, the metric formalism becomes invalid, and the anisotropic toroidal geometry is only accessible via the vielbein fields. In this latter case of anisotropic compactification, the branching factor becomes physical and is responsible for the branched SL(2 , Z) duality in nonrelativistic IIB superstring theory [31]. As a simple example for illustrating how the above zweibein formalism generates the SL(2 , Z) transformations of all the background fields in type IIB superstring theory, we consider the gauge theory to arise from the system with open M2-branes ending on M5branes.…”

“…It is famously known that the S-duality in type IIB superstring theory extends to the SL(2 , Z) duality [27][28][29], which acquires a simple geometric interpretation in M-theory as the isometry group of the two-torus over which M-theory compactifies. Shown in [30,31], the SL(2 , Z) duality can also be realized in nonrelativistic IIB string theory. To be more precise, we define the SL(2 , Z) transformation matrix to be…”

“…We refer to this realization of the global SL(2 , Z) symmetry in nonrelativistic IIB string theory as a polynomial realization [31]. This novel mathematical structure bears an intriguing connection to the classical invariant theory of binary forms (homogeneous polynomials in two variables) in algebra and provides a useful formalism for classifying SL(2 , Z) invariants in nonrelativistic IIB string theory.…”

“…This novel mathematical structure bears an intriguing connection to the classical invariant theory of binary forms (homogeneous polynomials in two variables) in algebra and provides a useful formalism for classifying SL(2 , Z) invariants in nonrelativistic IIB string theory. For example, in [31], the complete set of field strengths describing nonrelativistic IIB supergravity, together with their SL(2 , Z) transformation rules, are encoded by a quadratic and quartic binary form. Our study of anisotropic compactification of nonrelativistic M-theory in this paper will provide a natural geometric interpretation of the polynomial realization of SL(2 , Z) duality.…”