Abstract
We investigate the role of gauge and gravitational instantons in the context of the Swampland program. Our focus is on the global symmetry breaking they induce, especially in the presence of fermions. We first recall and make more precise the description of the dilute instanton gas through a 3-form gauge theory. In this language, the familiar suppression of instanton effects by light fermions can be understood as the decoupling of the 3-form. Even if all fermions remain massive, such decoupling may occur on the basis of an explicitly unbroken but anomalous global symmetry in the fermionic sector. This should be forbidden by quantum gravity, which leads us to conjecture a related, cutoff-dependent lower bound on the induced axion potential. Finally, we note that the gravitational counterpart of the above are K3 instantons. These are small fluctuations of Euclidean spacetime with K3 topology, which induce fermionic operators analogous to the ’t Hooft vertex in gauge theories. Although Planck-suppressed, they may be phenomenologically relevant if accompanied by other higher-dimension fermion operators or if the K3 carries appropriate gauge fluxes.
Highlights
If exact, such global symmetries should be in the swampland and we propose a lower bound on axion masses to quantify the minimal strength of symmetry breaking
To do so we focus on the calculable case of a weakly coupled Higgsed YM theory such that we can employ the dilute instanton gas approximation in our computations
Whenever a global symmetry of the full UV theory forbids an effective axion potential, the effective 3-form description must decouple as Λ2 = 0 is required for a vanishing potential
Summary
We go on to study Higgsed YM theories which are coupled to an axion. Gauge instantons generically induce an axion potential ∝ e−S where S denotes the instanton action. This is achieved by shift-symmetry-preserving Yukawa interactions, which provide effective mass operators in addition to the hard mass terms Such a structure arises in the Standard Model if an axion coupling to trSU(2)F Fis introduced [18]. Once the hard masses are taken to zero, the axion potential vanishes due to the emergence of a global symmetry This shows that the problematic feature of the theory is not massless fermions but rather the presence of a global symmetry. There are many counterexample in string theory: Calabi-Yau compactifications of type II strings lead to 4d N = 2 supergravity models with perfectly flat moduli spaces, including axionic directions In this case we expect that the global axionic symmetries are broken by higherdimension operators involving fermions. This constraint does not make a claim about the absolute size of axion potentials
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