Exploring Pintopia: Reflection Branes, Bordisms, and U-Dualities

Abstract

The U-dualities of maximally supersymmetric non-chiral supergravity (SUGRA) theories lead to strong constraints on the non-perturbative structure of quantum gravity. In this paper we determine Spin- and Pin-lifts of these dualities, which extend this action to fermionic degrees of freedom. Among other things, this allows us to access non-supersymmetric sectors of these low energy effective field theories in which bosonic and fermionic degrees of freedom are treated differently. We use this refinement of the duality groups, in tandem with the Swampland Cobordism Conjecture, to predict new codimension-two branes. These are a natural generalization of the recently discovered R7-branes of type II string theories. The first bordism groups for Spin-twisted duality bundles follow directly from the Abelianization of the duality groups. Viewing the SUGRA theory as the low energy limit of a toroidal compactification of M-theory, winding around these codimension-two defects enacts a reflection around one of the torus directions, which in the effective field theory appears as a charge conjugation symmetry. We establish some basic properties of such branes, including determining BPS objects which can end on it, as well as braiding rules and bound states realized by multiple reflection branes.

Figures (6)

• Figure 1: Bounding of every one-dimensional manifold with Spin-$\widetilde{G_U}$ structure (the duality bundle is depicted via transition functions on codimension-one sub-manifolds).
• Figure 2: Depicition of a pair of BPS D$p$-branes near a IIA R7-brane. Passing one D$p$-brane through the branch cut of the R7-brane (red) becomes a $\overline{\textrm{D}p}$-brane when the $(p+1)$-form potential it is coupled to picks up a minus sign. Under toroidal compactification, similar considerations hold for all reflection branes and probes by BPS objects.
• Figure 3: Top down view of the cylindrical configuration of the type IIA/IIB wall with topology $S^1 \times \mathbb{R}^{D-1} \times T^d$ with a $(-1)^{F_L}$ monodromy cut. IIB is on the inside of the wall while IIA is on the outside. The configuration collapses due to the tension of the wall, and the endpoint of the collapse is the reflection brane. This configuration lifts to a M- / F-theory wall.
• Figure 4: Type IIA $(-1)^{F_L}$ R7-brane in the presence of BPS brane probes. (i) Two D$p$-branes (black lines) are joined by passing through the branch cut of the reflection brane (red). The branes combine at a junction (black square) and extend to infinity. (ii) If two D$p$-branes can be lassoed in this way, then this implies that any number of the brane (not just an even number) can terminate on the R7-brane. Wrapping these branes on directions of an internal $T^{d}$ results in similar configurations for reflection branes probed by BPS objects of the $D$-dimensional effective field theory.
• Figure 5: Steenrod structure of twisted Spin structure for the Spin-lift of perfect U-duality groups (left) and associated Adams chart (right).

Theorems & Definitions (22)

• Proposition D.3
• proof
• Proposition D.11
• Lemma D.14
• proof
• proof : Proof of \ref{['spin_lift_H1']}
• Proposition D.24
• proof
• Theorem E.1
• Theorem E.3