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Universality and exactness of Schrodinger geometries in string and M-theory

Per Kraus, Eric Perlmutter

TL;DR

This work establishes a universal framework in which Schrödinger-invariant geometries in string/M-theory arise as nonlinear completions of linear KK vector and graviton fluctuations around any AdS×M vacuum, with nonlinearities constrained to quadratic order by AdS symmetry. The dynamical exponent z is fixed by the KK mass via (m_L)^2 = z(z+d-2), enabling infinite families Sch_{d+1}(z) for each AdS background and allowing mixed KK modes to couple only at quadratic order. The authors develop bulk perturbation theory and boundary conformal perturbation theory to justify truncation at quadratic order, derive nonrenormalization theorems for z when deforming KK modes lie in short AdS supermultiplets, and prove exactness in several cases. They present three new explicit Schrödinger constructions (Sch_7 from M5, Sch_3 from D1-D5/F1-NS5, and Sch_4 from wrapped M5) and discuss implications for consistent truncations, higher-derivative corrections, and nonrelativistic holography, including potential RG flows and condensed-matter applications.

Abstract

We propose an organizing principle for classifying and constructing Schrodinger-invariant solutions within string theory and M-theory, based on the idea that such solutions represent nonlinear completions of linearized vector and graviton Kaluza-Klein excitations of AdS compactifications. A crucial simplification, derived from the symmetry of AdS, is that the nonlinearities appear only quadratically. Accordingly, every AdS vacuum admits infinite families of Schrodinger deformations parameterized by the dynamical exponent z. We exhibit the ease of finding these solutions by presenting three new constructions: two from M5 branes, both wrapped and extended, and one from the D1-D5 (and S-dual F1-NS5) system. From the boundary perspective, perturbing a CFT by a null vector operator can lead to nonzero beta-functions for spin-2 operators; however, symmetry restricts them to be at most quadratic in couplings. This point of view also allows us to easily prove nonrenormalization theorems: for any Sch(z) solution of two-derivative supergravity constructed in the above manner, z is uncorrected to all orders in higher derivative corrections if the deforming KK mode lies in a short multiplet of an AdS supergroup. Furthermore, we find infinite classes of 1/4 BPS solutions with 4-,5- and 7-dimensional Schrodinger symmetry that are exact.

Universality and exactness of Schrodinger geometries in string and M-theory

TL;DR

This work establishes a universal framework in which Schrödinger-invariant geometries in string/M-theory arise as nonlinear completions of linear KK vector and graviton fluctuations around any AdS×M vacuum, with nonlinearities constrained to quadratic order by AdS symmetry. The dynamical exponent z is fixed by the KK mass via (m_L)^2 = z(z+d-2), enabling infinite families Sch_{d+1}(z) for each AdS background and allowing mixed KK modes to couple only at quadratic order. The authors develop bulk perturbation theory and boundary conformal perturbation theory to justify truncation at quadratic order, derive nonrenormalization theorems for z when deforming KK modes lie in short AdS supermultiplets, and prove exactness in several cases. They present three new explicit Schrödinger constructions (Sch_7 from M5, Sch_3 from D1-D5/F1-NS5, and Sch_4 from wrapped M5) and discuss implications for consistent truncations, higher-derivative corrections, and nonrelativistic holography, including potential RG flows and condensed-matter applications.

Abstract

We propose an organizing principle for classifying and constructing Schrodinger-invariant solutions within string theory and M-theory, based on the idea that such solutions represent nonlinear completions of linearized vector and graviton Kaluza-Klein excitations of AdS compactifications. A crucial simplification, derived from the symmetry of AdS, is that the nonlinearities appear only quadratically. Accordingly, every AdS vacuum admits infinite families of Schrodinger deformations parameterized by the dynamical exponent z. We exhibit the ease of finding these solutions by presenting three new constructions: two from M5 branes, both wrapped and extended, and one from the D1-D5 (and S-dual F1-NS5) system. From the boundary perspective, perturbing a CFT by a null vector operator can lead to nonzero beta-functions for spin-2 operators; however, symmetry restricts them to be at most quadratic in couplings. This point of view also allows us to easily prove nonrenormalization theorems: for any Sch(z) solution of two-derivative supergravity constructed in the above manner, z is uncorrected to all orders in higher derivative corrections if the deforming KK mode lies in a short multiplet of an AdS supergroup. Furthermore, we find infinite classes of 1/4 BPS solutions with 4-,5- and 7-dimensional Schrodinger symmetry that are exact.

Paper Structure

This paper contains 31 sections, 125 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Infinite families of Schrödinger solutions from every AdS
  3. Review and motivation
  4. In the bulk: AdS perturbation theory
  5. On the boundary: conformal perturbation theory
  6. A corollary: consistent truncations with massive KK modes
  7. Quantum/string corrections
  8. Uncorrected Schrödinger solutions from short multiplets
  9. Exact solutions
  10. At $O(\alpha'^3)$, explicit nonrenormalization of Sch$_5$
  11. Sch$_5$ solutions from AdS$_5 \times S^5$: $\sigma=0$
  12. Sch$_5$ solutions from AdS$_5 \times S^5$: $\sigma \neq 0$
  13. New Schrödinger solutions
  14. Sch$_7$ from M5 branes
  15. Sch$_3$ from D1-D5 and F1-NS5 branes
  16. ...and 16 more sections

Figures (1)

  • Figure 1: The lower states of the spectrum of Sch$_5(z)$ solutions formed from Kaluza-Klein vector deformations of AdS$_5 \times S^5$. This is an adaptation of Figure 1 in van nieu which depicts the vector mass spectrum, here converted via the massive vector relation $(mL)^2_{KK}=z(z+2)$. The $SO(6)$ representation of each vector, listed above each point, partly determines the degeneracy of solutions. At right are the $D=10$ origin of fields, where $\mu$ lies along the noncompact directions and $(a,b,c)$ lie on $S^5$. The middle branch of solutions, formed from diagonal vectors descending from the complex two-form, also requires nonzero Kaluza-Klein gravitons not pictured. The lower branch of mixed vectors, while present in the spectrum, gives solutions with negative values of $z$; hence, it appears dotted and unfilled. For details, see the main text.