Weak Gravity Conjecture for the Effective Field Theories with N Species

We conjecture an intrinsic UV cutoff for the validity of the effective field theory with a large number of species coupled to gravity. In four dimensions such a UV cutoff takes the form Λ = √ λ/NMp for N scalar fields with the same potential λφ4i , i = 1, ..., N . This conjecture implies that the assisted chaotic inflation or N-flation might be in the swampland, not in the landscape. Similarly a UV cutoff Λ = gMp/ √ N is conjectured for the U(1) gauge theory with N species. String theory is proposed as a candidate of quantum gravity. The consistent perturbative superstring theory can only live in ten-dimensional spacetime. A key problem in string theory is how it eventually connect with experiments. To understand the effective low-energy physics in four dimensions, string theory must be compactified on some six dimensional manifold. Different choice of the extra dimensional manifold leads to different low-energy effective theory. Recent development on the flux compactification [1–3] can help us to stabilize some moduli. However there are so many choices of flux. Supposing that each flux can take of order 10 values, we find of order 10 distinct solutions, where K is the number of distinct topological types of flux. Calabi-Yau manifolds typically have K ∼ O(102). It is quite unclear for us to get some testable predictions for string theory because there are so many different low-energy effective field theories. In [4] Vafa suggested that the vast series of the metastable vacua is only semi-classically self-consistent, not really self-consistent. We say that they are in the swampland. The really self-consistent landscape is surrounded by the swampland. Even though we can not pick out a unique compactification or a unique low-energy effective field theory in four dimensions, we still can ask which low-energy effective field theory can/cannot be compatible with gravity. Usually we believe that the effective field theory breaks down at the Planck scale because the gravity becomes important and the full theory of quantum gravity is called for. However this estimation seems too naively. The gauge force strength is characterized by g and the gravitational strength is roughly GΛ = Λ/M p at energy scale Λ. Gravity becomes dominant when the energy scale goes to gMp which is much lower than Planck scale in the perturbative region. Arkani-Hamed et al. [5] conjecture that a U(1) gauge theory with gauge coupling g has an intrinsic UV cutoff Λ ≤ gMp (1) for the validity. In cosmology scalar field plays a crucial role. Both Inflaton and quintessence are scalar fields. As a simple example, we consider the λφ theory coupled to gravity. The interaction strength of scale field is the dimensionless scalar coupling λ, and the gravitational strength is still GΛ. Similarly in [6] we proposed a weak gravity conjecture for the λφ theory as follows Λ ≤ λMp. (2) This conjecture infers that the single-field chaotic inflation cannot be achieved in string landscape. Recently the authors constructed a D-term chaotic inflation model in supergravity [7]. The potential of the scalar field takes the form λφ which comes from the D-term of a U(1) gauge field in the same supermultiplet. The supersymmetry implies a