Open-Closed Correspondence of K-theory and Cobordism

Abstract

Non-trivial K-theory groups and non-trivial cobordism groups can lead to global symmetries which are conjectured to be absent in quantum gravity. Inspired by open-closed string duality, we propose a correspondence between the two groups, which can be considered as the physical manifestation of a generalisation of the classic Conner-Floyd isomorphism. The picture is exemplified by the relations between KO-groups and Spincobordisms and between K-groups and Spin c -cobordisms. We mainly focus on the case in which the global symmetries are eventually gauged and, by combining K-theory and cobordism, we recover tadpole cancellation conditions in type I string theory and F-theory from a purely bottom-up perspective. In the spirit of the swampland program, we provide a prescription to compute the relative normalisation between cobordism and K-theory charges in these tadpoles, without relying on string theory input.

At the end of the world: Local Dynamical Cobordism

“…log y , (A. 13) where we have set an integration constant to zero without loss of generality. We recover the equivalent to (2.18), albeit with a and δ kept independent.…”

“…1 Such end-of-the-world configuration may correspond to a boundary (such as the 10d Horava-Witten boundary of 11d M-theory [2,3]), a bubble of nothing in which some compactification space shrinks to zero size [4] (see [5][6][7][8] for some recent works), or more exotic possibilities, and may possibly be dressed with charged objects, such as branes, orientifold planes or generalizations (dubbed I-folds in [9]). The cobordism conjecture, already at this topological level, has produced interesting results, see [6,[9][10][11][12][13] for some references. 2 An exploration of the Cobordism Conjecture beyond the topological level was undertaken in [21,22] via the study of spacetime varying solutions to the equations of motion in theories with dynamical tadpoles, namely, potentials which do not have a minimum and thus do not admit maximally symmetric solutions (see [23][24][25][26] for early work and [27][28][29][30] for related recent developments, and [31,32] for a complementary approach to cobordism solutions).…”

Looking for structure in the cobordism conjecture

“…When M P → ∞, gravity is decoupled and the constraint becomes either trivially satisfied, empty of content, or violated. 10 This is consistent since swampland criteria are not meant to constrain field theories without gravity. In this section, we discuss the gravity decoupling limit in the case of the cobordism conjecture (2.14).…”

“…Therefore, the existence of a process connecting any two quantum gravity backgrounds is far from obvious too. Indeed, even if the conjecture is quite recent, already a number of works appeared studying some of its aspects and consequences, including [5][6][7][8][9][10][11]. Bordisms are relevant in several areas in mathematics and physics, a nonexhaustive list of related works is [12][13][14][15][16] for anomalies, [17,18] in relation to bubbles of nothing, [19] for the conjecture in a holographic context.…”

At the End of the World: Local Dynamical Cobordism

“…Such end-of-the-world configuration may correspond to a boundary (such as the 10d Horava-Witten boundary of 11d M-theory [2,3]), a bubble of nothing in which some compactification space shrinks to zero size [4] (see [5][6][7][8] for some recent works), or more exotic possibilities, and may possibly be dressed with charged objects, such as branes, orientifold planes or generalizations (dubbed Ifolds in [9]). The cobordism conjecture, already at this topological level, has produced interesting results, see [6,[9][10][11][12][13] for some references 2 .…”