Zhihao Duan, David Jaramillo Duque, Amir-Kian Kashani-Poor
备注：
57 pages; v2 64 pages: discussion of specialization of Gromov-Witten invariants enhanced and moved to body of paper, with additional tables in appendix
Using topological string techniques, we compute BPS counting functions of 5d gauge theories which descend from 6d superconformal field theories upon circle compactification. Such theories are naturally organized in terms of nodes of Higgsing trees. We demonstrate that the specialization of the partition function as we move from the crown to the root of a tree is determined by homomorphisms between rings of Weyl invariant Jacobi forms. Our computations are made feasible by the fact that symmetry enhancements of the gauge theory which are manifest on the massless spectrum are inherited by the entire tower of BPS particles. In some cases, these symmetry enhancements have a nice relation to the 1-form symmetry of the associated gauge theory.
We investigate the Jacobi forms for the root system $E_8$ invariant under the Weyl group. This type of Jacobi forms has significance in Frobenius manifolds, Gromov--Witten theory and string theory. In 1992, Wirthmüller proved that the space of Jacobi forms for any irreducible root system not of type $E_8$ is a polynomial algebra. But very little has been known about the case of $E_8$. In this paper we show that the bigraded ring of Weyl invariant $E_8$ Jacobi forms is not a polynomial algebra and prove that every such Jacobi form can be expressed uniquely as a polynomial in nine algebraically …
We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with their fibers realized as hypersurfaces in the toric varieties associated to the 16 reflexive 2D polyhedra. We present a base-independent analysis of the codimension one, two and three singularities of these fibrations. We use these geometric results to determine the gauge groups, matter representations, 6D matter multiplicities and 4D Yukawa couplings of the corresponding effective theories. All these theories have a non-trivial gauge group and matter content. We explore the network of Higgsings relating thes…
In 1992 Wirthmüller showed that for any irreducible root system not of type $E_8$ the ring of weak Jacobi forms invariant under Weyl group is a polynomial algebra. However, it has recently been proved that for $E_8$ the ring is not a polynomial algebra. Weyl invariant $E_8$ Jacobi forms have many applications in string theory and it is an open problem to describe such forms. The scaled refined free energies of $E$-strings with certain $η$-function factors are conjectured to be Weyl invariant $E_8$ quasi holomorphic Jacobi forms. It is further observed that the scaled refined free energies up t…
We study the ring of Weyl invariant $E_8$ weak Jacobi forms. Wang recently proved that the ring is not a polynomial algebra. We consider a polynomial algebra which contains the ring as a subset and clarify the necessary and sufficient condition for an element of the polynomial algebra to be a Weyl invariant $E_8$ weak Jacobi form. This serves as a new algorithm for constructing all the Jacobi forms of given weight and index. The algorithm is purely algebraic and does not require Fourier expansion. Using this algorithm we determine the generators of the free module of Weyl invariant $E_8$ weak …
The weak Jacobi forms of integral weight and integral index associated to an even positive definite lattice form a bigraded algebra. In this paper we prove a criterion for this type of algebra being free. As an application, we give an automorphic proof of K. Wirthmüller's theorem which asserts that the bigraded algebra of weak Jacobi forms invariant under the Weyl group is a polynomial algebra for any irreducible root system not of type $E_8$. This approach is also applicable to $E_8$. Even if the algebra of $E_8$ Jacobi forms is known to be non-free, we still derive a new structure result.
We associate a Jacobi form over a rank s lattice to N=2, D=4 heterotic string compactifications which have s Wilson lines at a generic point in the vector multiplet moduli space. Jacobi forms of index m=1 and m=2 have appeared earlier in the context of threshold corrections to heterotic string couplings. We emphasize that higher index Jacobi forms as well as Jacobi forms of several variables over more generic even lattices also appear and construct models in which they arise. In particular, we construct an orbifold model which can be connected to models that give index m=3, 4 or 5 Jacobi forms…
The computation of anomalies in quantum field theory may be carried out by evaluating path integral Jacobians, as first shown by Fujikawa. The evaluation of these Jacobians can be cast in the form of a quantum mechanical problem, whose solution has a path integral representation. For the case of Weyl anomalies, also called trace anomalies, one is immediately led to study the path integral for a particle moving in curved spaces. We analyze the latter in a manifestly covariant way and by making use of ghost fields. The introduction of the ghost fields allows us to represent the path integral mea…
We discuss Jacobi forms that are invariant under the action of the Weyl group of type E_n (n=6,7,8). For n=6,7 we explicitly construct a full set of generators of the algebra of E_n weak Jacobi forms. We first construct n+1 independent E_n Jacobi forms in terms of Jacobi theta functions and modular forms. By using them we obtain Seiberg-Witten curves of type E_6 and E_7 for the E-string theory. The coefficients of each curve are E_n weak Jacobi forms of particular weights and indices specified by the root system, realizing the generators whose existence was shown some time ago by Wirthmüller.
We show that the Pauli-Villars regularized action for a scalar field in a gravitational background in 1+1 dimensions has, for any value of the cutoff M, a symmetry which involves non-local transformations of the regulator field plus (local) Weyl transformations of the metric tensor. These transformations, an extension to the regularized action of the usual Weyl symmetry transformations of the classical action, lead to a new interpretation of the conformal anomaly in terms of the (non-anomalous) Jacobian for this symmetry. Moreover, the Jacobian is automatically regularized, and yields the corr…
Conventional quantization of two-dimensional diffeomorphism and Weyl invariant theories sacrifices the latter symmetry to anomalies, while maintaining the former. When alternatively Weyl invariance is preserved by abandoning diffeomorphism invariance, we find that some invariance against coordinate redefinition remains: one can still perform at will transformations possessing a constant Jacobian. The alternate approach enjoys as much ``gauge'' symmetry as the conventional formulation.
The Hamilton--Jacobi formalism generalized to 2--dimensional field theories according to Lepage's canonical framework is applied to several covariant real scalar fields, e.g. massless and massive Klein--Gordon, Sine--Gordon, Liouville and $ϕ^4$ theories. For simplicity we use the Hamilton--Jacobi equation of DeDonder and Weyl. Unlike mechanics we have to impose certain integrability conditions on the velocity fields to guarantee the transversality relations between Hamilton--Jacobi wave fronts and the corresponding families of extremals embedded therein. Bäcklund Transformations play a crucial…
The Hamilton-Jacobi formalism generalized to 2-dimensional field theories according to Lepage's canonical framework is applied to several relativistic real scalar fields, e.g. massless and massive Klein-Gordon, Sinh and Sine-Gordon, Liouville and $ϕ^4$ theories. The relations between the Euler-Lagrange and the Hamilton-Jacobi equations are discussed in DeDonder and Weyl's and the corresponding wave fronts are calculated in Carathéodory's formulation. Unlike mechanics we have to impose certain integrability conditions on the velocity fields to guarantee the transversality relations and especial…
For a Calabi-Yau threefold admitting both a $K3$ fibration and an elliptic fibration (with some extra conditions) we discuss candidate asymptotic expressions of the genus 0 and 1 Gromov-Witten potentials in the limit (possibly corresponding to the perturbative regime of a heterotic string) where the area of the base of the $K3$ fibration is very large. The expressions are constructed by lifting procedures using nearly holomorphic Weyl-invariant Jacobi forms. The method we use is similar to the one introduced by Borcherds for the constructions of automorphic forms on type IV domains as infinite…
We apply the modular approach to computing the topological string partition function on non-compact elliptically fibered Calabi-Yau 3-folds with higher Kodaira singularities in the fiber. The approach consists in making an ansatz for the partition function at given base degree, exact in all fiber classes to arbitrary order and to all genus, in terms of a rational function of weak Jacobi forms. Our results yield, at given base degree, the elliptic genus of the corresponding non-critical 6d string, and thus the associated BPS invariants of the 6d theory. The required elliptic indices are determi…
A path integral evaluation of the Green's function for the hydrogen atom initiated by Duru and Kleinert is studied by recognizing it as a special case of the general treatment of the separable Hamiltonian of Liouville-type. The basic dynamical principle involved is identified as the Jacobi's principle of least action for given energy which is reparametrization invariant, and thus the appearance of a gauge freedom is naturally understood. The separation of variables in operator formalism corresponds to a choice of gauge in path integral, and the Green's function is shown to be gauge independent…
A family of covariant non-linear projections from the space of SO(10) Weyl spinors onto the space of pure SO(10) Weyl spinors is presented. The Jacobian matrices of these projections are related to a linear projector which was previously discussed in pure spinor string literature and which maps the antighost to its gauge invariant part. Only one representative of the family leads to a Hermitian Jacobian matrix and can itself be derived from a scalar potential. Comments on the SO(1,9) case are given as well as on the non-covariant version of the projection map. The insight is applied to the gho…
The fake monster Lie algebra is determined by the Borcherds function Phi_{12} which is the reflective modular form of the minimal possible weight with respect to O(II_{2,26}). We prove that the first non-zero Fourier-Jacobi coefficient of Phi_{12} in any of 23 Niemeier cusps is equal to the Weyl-Kac denominator function of the affine Lie algebra of the root system of the corresponding Niemeier lattice. This is an automorphic answer (in the case of the fake monster Lie algebra) on the old question of I. Frenkel and A. Feingold (1983) about possible relations between hyperbolic Kac-Moody algebra…
The structure of the divergences for transverse theories of gravity is studied to one-loop order. These theories are invariant only under those diffeomorphisms that enjoy unit Jacobian determinant (TDiff), so that the determinant of the metric transforms as a true scalar instead of a density. Generically, the models include an additional scalar degree of freedom contained in the metric besides the usual spin two component. When the cosmological constant is fine tuned to zero, there are only two theories which are on shell finite, namely the one in which the symmetry is enhanced to the full gro…
Polymomentum canonical theories, which are manifestly covariant multi-parameter generalizations of the Hamiltonian formalism to field theory, are considered as a possible basis of quantization. We arrive at a multi-parameter hypercomplex generalization of quantum mechanics to field theory in which the algebra of complex numbers and a time parameter are replaced by the space-time Clifford algebra and space-time variables treated in a manifestly covariant fashion. The corresponding covariant generalization of the Schroedinger equation is shown to be consistent with several aspects of the corresp…
In this paper we consider axisymmetric black holes in supergravity and address the general issue of defining a first order description for them. The natural setting where to formulate the problem is the De Donder-Weyl-Hamilton-Jacobi theory associated with the effective two-dimensional sigma-model action describing the axisymmetric solutions. We write the general form of the two functions S_m defining the first-order equations for the fields. It is invariant under the global symmetry group G_(3) of the sigma-model. We also discuss the general properties of the solutions with respect to these g…
The role of the contribution from the fermion mass term in the axial vector Ward identity in generating the U(1) axial anomaly, both local and global, is elucidated. Gauge invariance requires the fermion to decouple from the gauge field if it is very heavy. This identifies the Adler-Bell-Jackiw (ABJ) anomaly with the asymptotic limit of the sign reversed mass term. In an instanton background, the chiral limit $(m = 0)$ of the mass term does not vanish but consists of contributions from fermion zero modes. Space time integral of these zero mode contributions exactly cancels, thanks to the Atiya…
