IIB flux non-commutativity and the global structure of field theories

Abstract

We discuss the origin of the choice of global structure for six dimensional (2, 0) theories and their compactifications in terms of their realization from IIB string theory on ALE spaces. We find that the ambiguity in the choice of global structure on the field theory side can be traced back to a subtle effect that needs to be taken into account when specifying boundary conditions at infinity in the IIB orbifold, namely the known non-commutativity of RR fluxes in spaces with torsion. As an example, we show how the classification of N = 4 theories by Aharony, Seiberg and Tachikawa can be understood in terms of choices of boundary conditions for RR fields in IIB. Along the way we encounter a formula for the fractional instanton number of N = 4 ADE theories in terms of the torsional linking pairing for rational homology spheres. We also consider six-dimensional (1, 0) theories, clarifying the rules for determining commutators of flux operators for discrete 2-form symmetries. Finally, we analyze the issue of global structure for four dimensional theories in the presence of duality defects. arXiv:1908.08027v2 [hep-th] 9 Oct 20191 The separation into background and intrinsic data is sometimes arbitrary: if we restrict ourselves to fourdimensional Yang-Mills theories with constant coupling τ we could view τ as part of the data defining T . However, if we wish to allow for the possibility that τ varies across M then we must include it as part of the background data to be specified for each manifold. The second interpretation will be more natural from the point of view in this paper, and such configurations will play an interesting role below.2 In this paper we will take M6 to be closed, Spin and orientable, and furthermore we will assume that the cohomology groups of M6 are freely generated, so there is no torsion. 3 The free, or "abelian", (2, 0) theory can be obtained by replacing C 2 /Γ by a single-centered Taub-NUT space.5 The two groups are related by the short exact sequence 0 → W 4 → H 4 (M6 × S 3 /Zn; U (1)) → Tor(H 5 (M6 × S 3 /ZN ; Z)) → 0 , with W 4 the group of topologically trivial C4 Wilson lines on M6 × S 3 /Zn.

Higher form symmetries of Argyres-Douglas theories

“…As there are no known Lagrangian descriptions of most T [g, g ] SCFTs (with the exception of some cases; we will come back to these momentarily) it might seem hard to find the 1-form symmetries for these theories using purely field theoretical tools. However, as we explain below -extending previous results [22,[31][32][33] to the four dimensional setting -there is a way of rephrasing the results from the IIB analysis in purely field theoretical terms. We find that the role played by the unscreened part of the center of the gauge group 7 in the analysis in [34] is played in the non-Lagrangian setting in this paper by Tor(coker Q), with Q the BPS quiver for the theory [49,50].…”

“…The local degrees of freedom of T [g, g ] are in this way fully determined by the choice of geometry, but in presence of a nontrivial defect group at the horizon of the IIB compactification, extra information is required to fully specify the theory and its partition function on M 4 [31] -see also [22,32,33]. The most well known example of this fact is the case of the N = 4 theory with simple ADE algebra g corresponding to IIB on T 2 × C 2 /Γ g .…”

“…The structure of this paper is as follows. In section 2 we review the results of [22,31] that are needed for our analysis, as well as some aspects of the geometric engineering of…”

“…In this section we review how to compute the spectrum of higher form symmetries for geometrically engineered N = 2 theories. We refer the reader to [31] for a more indepth discussion, here we will just highlight the modifications necessary when dealing with Calabi-Yau threefolds.…”

Lens space index and global properties for 4d $$ \mathcal{N} $$ = 2 models

“…It should be interesting apply the results of [17][18][19], like the relation between the twisted character of the chiral algebra and the Lens space index, in order to reproduce the lattices and the S-duality structure obtained in [20][21][22][23][24][25][26][27][28]. Also the results of [29], investigating the geometric origin of the global properties from the 6d N = (1, 0) perspective, may be useful for the study of N = 2 models. of pair of paths connecting each pair on nodes with opposite orientation.…”