Theories of class F and anomalies

Abstract

We consider the 6d (2, 0) theory on a fibration by genus g curves, and dimensionally reduce along the fiber to 4d theories with duality defects. This generalizes class S theories, for which the fibration is trivial. The non-trivial fibration in the present setup implies that the gauge couplings of the 4d theory, which are encoded in the complex structures of the curve, vary and can undergo S-duality transformations. These monodromies occur around 2d loci in spacetime, the duality defects, above which the fiber is singular. The key role that the fibration plays here motivates refering to this setup as theories of class F. In the simplest instance this gives rise to 4d N = 4 Super-Yang-Mills with space-time dependent coupling that undergoes SL(2, Z) monodromies. We determine the anomaly polynomial for these theories by pushing forward the anomaly polynomial of the 6d (2, 0) theory along the fiber. This gives rise to corrections to the anomaly polynomials of 4d N = 4 SYM and theories of class S. For the torus case, this analysis is complemented with a field theoretic derivation of a U (1) anomaly in 4d N = 4 SYM. The corresponding anomaly polynomial is tested against known expressions of anomalies for wrapped D3-branes with varying coupling, which are known field theoretically and from holography. Extensions of the construction to 4d N = 0 and 1, and 2d theories with varying coupling, are also discussed.

Anomalies in the space of coupling constants and their dynamical applications II

“…We start with a pure gauge SppNq theory. 9 The theory has a Z 2 one-form symmetry. We want to construct a SppNq{Z 2 bundle with second SW class B.…”

“…The anomaly becomes trivial when N " 0 mod 4 (on spin manifolds it is trivial when N " 0 mod 2). 9 We use the notation SppN q " U Spp2N q. Specifically Spp1q " SU p2q and Spp2q " Spinp5q.…”

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

“…Bringing the four insertions together we obtain the commutator s( w 2 ⊗ x, w 2 ⊗ y) which is a c-number, and can be taken out of the path integral. The c(M 4 ) factor in (3.28) is associated to the change in the partition function with no flux, so it is natural to conjecture that it is associated with the value of 25 Recall that we are taking M4 to be a Spin manifold, so 1 2 M 4 w2 w2 is an integer. 26 In comparing with the results of [18], it might be useful to recall that in the case at hand one can define the Pontryagin square of x ∈ H 2 (M4, Z2) by P(x) = x 2 mod 4, where x ∈ H 2 (M4) is an uplift of x.…”

“…The answer is well known in the context of F-theory (and before that, from the analysis of the Seiberg-Witten solution of N = 2 SU (2) with four flavours [74,75]): the six zeroes of ∆ split into four mutually local degenerations (the D7 branes, in F-theory) and two mutually non-local degenerations (the O7 − plane). 31 See [19][20][21][22][23][24][25][26] for studies of such backgrounds. 32 We refer the reader interested in reading more about F-theory to the excellent reviews [70][71][72][73].…”

“…Along the way we encounter a simple geometric reinterpretation of the fractional instanton number in N = 4 theories with simply-connected gauge group, which we expect to generalize to less supersymmetric cases. In §4 we explore these ideas in less familiar backgrounds: we will discuss global aspects of 4d theories in the presence of duality defects (as studied in [19][20][21][22][23][24][25][26], for instance) and subtleties having to do with modular invariance in the context of 4d/2d dualities that arise when the four dimensional manifold has two-cycles. We point out an interesting relation between the Vafa-Witten partition function of self-dual su(p) theories on K3 and Hecke operators acting on the partition function of chiral bosons, and briefly discuss a (speculative, but suggestive) connection between these partition functions and the j invariant.…”