In the context of type IIB orientifold compactification with the presence of (non-)geometric fluxes, we conjecture a modular completed version of the generalized flux-orbits of various NS-NS and RR fluxes. Subsequently, considering two explicit examples with frozen complex structure moduli, we illustrate the utility of these new flux orbits in a very compact rearrangement of the four dimensional effective scalar potential.
Utilizing a setup of type IIB superstring theory compactified on an orientifold of T^6/(Z2xZ2), we propose a modular invariant dimensional oxidation of the four-dimensional scalar potential. In the oxidized ten-dimensional supergravity action, the standard NS-NS and RR three form fluxes (H-, F-) as well as the non-geometric fluxes (Q-, P-) are found to nicely rearrange themselves to form generalized flux-combinations. As an application towards moduli stabilization, using the same S-duality invariant scalar potential, we examine the recently proposed No-Go theorem (in arXiv:1409.7075) about cre…
We extend our previous work arXiv:1012.5721 [hep-th] on the non-compact N=2 SCFT_2 defined as the supersymmetric SL(2,R)/U(1)-gauged WZW model. Starting from path-integral calculations of torus partition functions of both the axial-type (`cigar') and the vector-type (`trumpet') models, we study general models of the Z_M-orbifolds and M-fold covers with an arbitrary integer M. We then extract contributions of the degenerate representations (`discrete characters') in such a way that good modular properties are preserved. The `modular completion' of the extended discrete characters introduced in …
False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular completions of indefinite theta functions of any signature and thereby develop a structure parallel to the recently developed theory of higher depth mock modular forms. We then demonstrate this theoretical base on a number of examples up to depth three coming from characters of modules for the vertex algebra $W^0(p)_{A_n}$, $1 \leq n \leq 3$, and from $\hat{Z}$-invari…
We study modular symmetry anomalies in four-dimensional low-energy effective field theory, which is derived from six-dimensional supersymmetric $U(N)$ Yang-Mills theory by magnetic flux compactification. The gauge symmetry $U(N)$ is broken to $U(N_a) \times U(N_b)$ by magnetic fluxes. It is found that Abelian subgroup of the modular symmetry corresponding to discrete part of $U(1)$ can be anomalous, but other elements independent of $U(1)$ in the modular symmetry are always anomaly-free.
False theta functions closely resemble ordinary theta functions, however they do not have the modular transformation properties that theta functions have. In this paper, we find modular completions for false theta functions, which among other things gives an efficient way to compute their obstruction to modularity. This has potential applications for a variety of contexts where false and partial theta series appear. To exemplify the utility of this derivation, we discuss the details of its use on two cases. First, we derive a convergent Rademacher-type exact formula for the number of unimodal …
We classify discrete modular symmetries in the effective action of Type IIB string on toroidal orientifolds with three-form fluxes, emphasizing on $T^6/\mathbb{Z}_2$ and $T^6/(\mathbb{Z}_2\times \mathbb{Z}_2^\prime)$ orientifold backgrounds. On the three-form flux background, the modular group is spontaneously broken down to its congruence subgroup whose pattern is severely constrained by a quantization of fluxes and tadpole cancellation conditions. We explicitly demonstrate that the congruence subgroups appearing in the effective action arise on magnetized D-branes wrapping certain cycles of …
Flux compactification of IIB string theory associates special points in Calabi-Yau moduli space to choices of (pairs of) integral three-form fluxes. In this paper, we propose that supersymmetric flux vacua are modular. That is, to a supersymmetric flux vacuum arising in a variety defined over $\mathbb{Q}$, we associate a two-dimensional Galois representation that we conjecture to be modular. We provide numerical evidence for our conjecture by examining flux vacua arising on the octic hypersurface in $\mathbb{P}^{4}(1,1,2,2,2)$.
In the appropriate limit, a type IIB string theory setup involving D3 branes, wrapped D5 branes, and fluxes on a conifold generally leads to a supergravity background involving a warped version of the conifold with fluxes. We study the supergravity dual of the baryonic branch of the Klebanov Strassler theory by writing down a very general conifold metric--the non-Kähler resolved warped-deformed conifold--and a general set of fluxes that satisfy the supergravity equations of motion, and derive the necessary constraints that allow the geometry to be dual to an $\mathcal{N}=1$ supersymmetric gaug…
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 dimen…
Iterated integrals for generalized Appell functions
Chennai Mathematical Institute（金奈数学研究所）；School of Mathematics, Trinity College（都柏林圣三一学院数学学院）；Hamilton Mathematical Institute, Trinity College（都柏林圣三一学院哈密顿数学研究所）
Gravity duals for a class of UV complete minimally supersymmetric non-conformal gauge theories require deformed conifolds with fluxes. However these manifolds do not allow for the standard Kahler or conformally Kahler metrics on them, instead the metrics are fully non-Kahler. We take a generic such configuration of a non-Kahler deformed conifold with fluxes and ask what constraints do supersymmetry impose in the Baryonic branch. We study the supersymmetry conditions and show that for the correct choices of the vielbeins and the complex structure all the equations may be consistently solved. Th…
We discuss the period geometry and the topological string amplitudes on elliptically fibered Calabi-Yau fourfolds in toric ambient spaces. In particular, we describe a general procedure to fix integral periods. Using some elementary facts from homological mirror symmetry we then obtain Bridgelands involution and its monodromy action on the integral basis for non-singular elliptically fibered fourfolds. The full monodromy group contains a subgroup that acts as PSL(2,Z) on the Kähler modulus of the fiber and we analyze the consequences of this modularity for the genus zero and genus one amplitud…
We discuss the modular symmetry and zeros of zero-mode wave functions on two-dimensional torus $T^2$ and toroidal orbifolds $T^2/\mathbb{Z}_N$ ($N=2,3,4,6$) with a background homogeneous magnetic field. As is well-known, magnetic flux contributes to the index in the Atiyah-Singer index theorem. The zeros in magnetic compactifications therefore play an important role, as investigated in a series of recent papers. Focusing on the zeros and their positions, we study what type of boundary conditions must be satisfied by the zero modes after the modular transformation. The consideration in this pap…
We study the modular symmetry in $T^2$ and orbifold comfactifications with magnetic fluxes. There are $|M|$ zero-modes on $T^2$ with the magnetic flux $M$. Their wavefunctions as well as massive modes behave as modular forms of weight $1/2$ and represent the double covering group of $Γ\equiv SL(2,\mathbb{Z})$, $\widetildeΓ \equiv \widetilde{SL}(2,\mathbb{Z})$. Each wavefunction on $T^2$ with the magnetic flux $M$ transforms under $\widetildeΓ(2|M|)$, which is the normal subgroup of $\widetilde{SL}(2,\mathbb{Z})$. Then, $|M|$ zero-modes are representations of the quotient group $\widetildeΓ'_{2…
We study the modular symmetry in magnetized D-brane models on $T^2$. Non-Abelian flavor symmetry $D_4$ in the model with magnetic flux $M=2$ (in a certain unit) is a subgroup of the modular symmetry. We also study the modular symmetry in heterotic orbifold models. The $T^2/Z_4$ orbifold model has the same modular symmetry as the magnetized brane model with $M=2$, and its flavor symmetry $D_4$ is a subgroup of the modular symmetry.
We explore the geometrical origin of CP and the spontaneous CP violation in Calabi-Yau compactifications. We find that the CP symmetry is identified with an outer automorphism of the symplectic modular group in the large complex structure regime of Calabi-Yau threefolds, thereby enlarging the symplectic modular group to their semidirect product group. The spontaneous CP violation is realized by the introduction of fluxes, whose effective action is invariant under CP as well as the discrete $\mathbb{Z}_2$ symmetry or $\mathbb{Z}_4$ R-symmetry. We explicitly demonstrate the spontaneous CP violat…
We systematically investigate possibilities of realizing the spontaneous CP violation in Type IIB flux compactifications on toroidal orientifolds. Our detailed analysis leads to the presence of flat directions at degenerate CP-breaking and -conserving vacua for a generic choice of three-form fluxes, indicating that flux compactifications are not sufficient to realize the spontaneous CP violation. Furthermore, the four-dimensional CP can be embedded into the duality symmetries, namely modular symmetries for a particular choice of fluxes.
As the effective field theory of the superstring theory, ten-dimensional ${\cal N}=1$ supersymmetric Yang-Mills theory is induced. We consider the ten-dimensional space-time ${\cal M}_{10}$ as direct products of our four-dimensional space-time ${\cal M}_{4}$ and three two-dimensional compact spaces $X_i\ (i=1,2,3)$, i.e. ${\cal M}_{10} = {\cal M}_4 \times X_2 \times X_2 \times X_3$. In particular, we consider $X_i$ as a torus, the torus orbifold, and also the blow-up manifold of the torus orbifold with background magnetic fluxes. We discuss the modular symmetry in the magnetized torus and its …
We revisit the flavor symmetries arising from compactifications on tori with magnetic background fluxes. Using Euler's Theorem, we derive closed form analytic expressions for the Yukawa couplings that are valid for arbitrary flux parameters. We discuss the modular transformations for even and odd units of magnetic flux, M, and show that they give rise to finite metaplectic groups the order of which is determined by the least common multiple of the number of zero-mode flavors involved. Unlike in models in which modular flavor symmetries are postulated, in this approach they derive from an under…
Theta series for lattices with indefinite signature $(n_+,n_-)$ arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case ($n_+=1$), but have remained obscure when $n_+\geq 2$. Using a higher-dimensional generalization of the usual (complementary) error function, discovered in an independent physics project, we construct the modular completion of a class of `conformal' holomorphic theta series ($n_+=2$). As an application, we determine the modular properties of a generalized Appell-…
