Timo Weigand

As a refinement of the Swampland Distance Conjecture, we propose that a quantum gravitational theory in an infinite distance limit of its moduli space either decompactifies, or reduces to an asymptotically tensionless, weakly coupled string theory. We support our claim by classifying, as special cases, the behaviour of M-Theory and Type IIA string theory compactifications on Calabi-Yau three-folds at infinite distances in Kähler moduli space.The analysis comprises three parts: we first classify the possible infinite distance limits in the classical Kähler moduli space of a Calabi-Yau three-fold. Each such limit at finite volume is characterized by a universal fibration structure, for which the generic fiber shrinking in the limit is either an elliptic curve, a K3 surface, or an Abelian surface.In the second part we focus on M-Theory and investigate the nature of the towers of asymptotically massless states that arise from branes wrapped on the shrinking fibers. Depending on which of the three classes of fibrations are considered, we obtain decompactification to F-Theory, or a theory with a unique asymptotically tensionless, weakly coupled heterotic or Type II string, respectively. The latter probes a dual D-manifold which is in general non-geometric. In addition to the intrinsic string excitations, towers of states from M2-branes along non-contractible curves become light and correspond to further wrapping and winding modes of the tensionless heterotic or Type II string.In the third part of the analysis, we consider Type IIA string theory on Calabi-Yau three-folds and show that quantum effects obstruct taking finite volume infinite distance limits in the Kähler moduli space. The only possible infinite distance limit which is not a decompactification limit involves K3-fibrations with string scale fiber volume and gives rise to an emergent tensionless heterotic string.

As a refinement of the Swampland Distance Conjecture, we propose that a quantum gravitational theory in an infinite distance limit of its moduli space either decompactifies, or reduces to an asymptotically tensionless, weakly coupled string theory. We support our claim by classifying, as special cases, the behaviour of M-Theory and Type IIA string theory compactifications on Calabi-Yau three-folds at infinite distances in Kahler moduli space. The analysis comprises three parts: We first classify the possible infinite distance limits in the classical Kahler moduli space of a Calabi-Yau three-fold. Each such limit at finite volume is characterized by a universal fibration structure, for which the generic fiber shrinking in the limit is either an elliptic curve, a K3 surface, or an Abelian surface. In the second part we focus on M-Theory and investigate the nature of the towers of asymptotically massless states that arise from branes wrapped on the shrinking fibers. Depending on which of the three classes of fibrations are considered, we obtain decompactification to F-Theory, or a theory with a unique asymptotically tensionless, weakly coupled heterotic or Type II string, respectively. The latter probes a dual D-manifold which is in general non-geometric. In addition to the intrinsic string excitations, towers of states from M2-branes along non-contractible curves become light and correspond to further wrapping and winding modes of the tensionless heterotic or Type II string. In the third part of the analysis, we consider Type IIA string theory on Calabi-Yau three-folds and show that quantum effects obstruct taking finite volume infinite distance limits in the Kahler moduli space. The only possible infinite distance limit which is not a decompactification limit involves K3-fibrations with string scale fiber volume and gives rise to an emergent tensionless heterotic string.

We systematically analyse weak coupling limits for 2-form tensor fields in the presence of gravity. Such limits are significant for testing various versions of the Weak Gravity and Swampland Distance Conjectures, and more broadly, the phenomenon of emergence. The weak coupling limits for 2-forms correspond to certain infinite-distance limits in the moduli space of string compactifications, where asymptotically tensionless, solitonic strings arise. These strings are identified as weakly coupled fundamental strings in a dual frame, which makes the idea of emergence manifest. Concretely we first consider weakly coupled tensor fields in six-dimensional compactifications of F-theory, where the arising tensionless strings play the role of dual weakly coupled heterotic strings. As the main part of this work, we consider certain infinite distance limits of Type IIB strings on K3 surfaces, for which we show that the asymptotically tensionless strings describe dual fundamental Type IIB strings, again on K3 surfaces. By contrast the analogous weak coupling limits of M-theory compactifications are found to correspond to an F-theory limit where an extra dimension emerges rather than tensionless strings. We comment on extensions of our findings to four-dimensional compactifications.

We investigate the interplay between the enumerative geometry of Calabi-Yau fourfolds with fluxes and the modularity of elliptic genera in four-dimensional string theories. We argue that certain contributions to the elliptic genus are given by derivatives of modular or quasi-modular forms, which encode BPS invariants of Calabi-Yau or non-Calabi-Yau threefolds that are embedded in the given fourfold. As a result, the elliptic genus is only a quasi-Jacobi form, rather than a modular or quasi-modular one in the usual sense. This manifests itself as a holomorphic anomaly of the spectral flow symmetry, and in an elliptic holomorphic anomaly equation that maps between different flux sectors. We support our general considerations by a detailed study of examples, including non-critical strings in four dimensions. For the critical heterotic string, we explain how anomaly cancellation is restored due to the properties of the derivative sector. Essentially, while the modular sector of the elliptic genus takes care of anomaly cancellation involving the universal B-field, the quasi-Jacobi one accounts for additional B-fields that can be present. Thus once again, diverse mathematical ingredients, namely here the algebraic geometry of fourfolds, relative Gromow-Witten theory pertaining to flux backgrounds, and the modular properties of (quasi-)Jacobi forms, conspire in an intriguing manner precisely as required by stringy consistency.

We investigate the interplay between the enumerative geometry of Calabi-Yau fourfolds with fluxes and the modularity of elliptic genera in four-dimensional string theories. We argue that certain contributions to the elliptic genus are given by derivatives of modular or quasi-modular forms, which may encode BPS invariants of Calabi-Yau or non-Calabi-Yau threefolds that are embedded in the given fourfold. As a result, the elliptic genus is only a quasi-Jacobi form, rather than a modular or quasi-modular one in the usual sense. This manifests itself as a holomorphic anomaly of the spectral flow symmetry, and in an elliptic holomorphic anomaly equation that maps between different flux sectors. We support our general considerations by a detailed study of examples, including non-critical strings in four dimensions.For the critical heterotic string, we explain how anomaly cancellation is restored due to the properties of the derivative sector. Essentially, while the modular sector of the elliptic genus takes care of anomaly cancellation involving the universal B-field, the quasi-Jacobi one accounts for additional B-fields that can be present.Thus once again, diverse mathematical ingredients, namely here the algebraic geometry of fourfolds, relative Gromow-Witten theory pertaining to flux backgrounds, and the modular properties of (quasi-)Jacobi forms, conspire in an intriguing manner precisely as required by stringy consistency.

We analyse the Weak Gravity Conjecture for chiral four-dimensional F-theory compactifications withN= 1 supersymmetry. Extending our previous work on nearly tensionless heterotic strings in six dimensions, we show that under certain assumptions a tower of asymptotically massless states arises in the limit of vanishing coupling of a U(1) gauge symmetry coupled to gravity. This tower contains super-extremal states whose charge-to-mass ratios are larger than those of certain extremal dilatonic Reissner-Nordström black holes, precisely as required by the Weak Gravity Conjecture. Unlike in six dimensions, the tower of super-extremal states does not always populate a charge sub-lattice.The main tool for our analysis is the elliptic genus of the emergent heterotic string in the chiralN= 1 supersymmetric effective theories. This also governs situations where the heterotic string is non-perturbative. We show how it can be computed in terms of BPS invariants on elliptic four-folds, by making use of various dualities and mirror symmetry. Compared to six dimensions, the geometry of the relevant elliptically fibered four-folds is substantially richer than that of the three-folds, and we classify the possibilities for obtaining critical, nearly tensionless heterotic strings. We find that the (quasi-)modular properties of the elliptic genus crucially depend on the choice of flux background. Our general results are illustrated in a detailed example.

We analyse the Weak Gravity Conjecture for chiral four-dimensional F-theory compactifications with N=1 supersymmetry. Extending our previous work on nearly tensionless heterotic strings in six dimensions, we show that under certain assumptions a tower of asymptotically massless states arises in the limit of vanishing coupling of a U(1) gauge symmetry coupled to gravity. This tower contains super-extremal states whose charge-to-mass ratios are larger than those of certain extremal dilatonic Reissner-Nordstrom black holes, precisely as required by the Weak Gravity Conjecture. Unlike in six dimensions, the tower of super-extremal states does not always populate a charge sub-lattice. The main tool for our analysis is the elliptic genus of the emergent heterotic string in the chiral N=1 supersymmetric effective theories. This also governs situations where the heterotic string is non-perturbative. We show how it can be computed in terms of BPS invariants on elliptic four-folds, by making use of various dualities and mirror symmetry. Compared to six dimensions, the geometry of the relevant elliptically fibered four-folds is substantially richer than that of the three-folds, and we classify the possibilities for obtaining critical, nearly tensionless heterotic strings. We find that the (quasi-)modular properties of the elliptic genus crucially depend on the choice of flux background. Our general results are illustrated in a detailed example.

We study the behaviour of abelian gauge symmetries in six-dimensional N = (1,0) theories upon decoupling gravity and investigate abelian flavour symmetries in the context of 6d N = (1, 0) SCFTs. From a supergravity perspective, the anomaly cancellation mechanism implies that abelian gauge symmetries can only survive as global symmetries as gravity is decoupled. The flavour symmetries obtained in this way are shown to be free of ABJ anomalies, and their ’t Hooft anomaly polynomial in the decoupling limit is obtained explicitly. In an F-theory realisation the decoupling of abelian gauge symmetries implies that a mathematical object known as the height pairing of a rational section is not contractible as a curve on the base of an elliptic Calabi-Yau threefold. We prove this prediction from supergravity by making use of the properties of the Mordell-Weil group of rational sections. In the second part of this paper we study the appearance of abelian flavour symmetries in 6d N = (1, 0) SCFTs. We elucidate both the geometric origin of such flavour symmetries in F-theory and their field theoretic interpretation in terms of suitable linear combinations of geometrically massive U(1)s. Our general results are illustrated in various explicit examples.