Points In Moduli Space Research Articles

Saturating unitarity bounds at U-duality symmetric points

It has recently been shown that the leading Wilson coefficient in type II string theory can take (almost) all values allowed by unitarity, crossing symmetry and maximal supersymmetry in D = 10 and D = 9 dimensions. This suggests that string theory might define the unique consistent quantum theory of gravity with maximal supersymmetry. We study the minima of the leading Wilson coefficient in D = 6, 7 and 8 dimensions and find the global minimum at the point in moduli space with maximal symmetry. The minimum value turns out to always be negative for D ≤ 7.

Tadpoles and gauge symmetries

The tadpole conjecture proposes that complex structure moduli stabilisation by fluxes that have low tadpole charge can be realised only at special points in moduli space, leading generically to (large) gauge symmetries. Here we provide an exhaustive survey of the gauge symmetries arising in F-theory flux compactifications on products of attractive K3 surfaces, with complex structure moduli fully stabilised. We compute the minimal rank of the left-over non-abelian gauge group for all flux configurations within the tadpole bound, finding that it is always non-zero. It decreases in a roughly linear fashion with the tadpole charge, reaching zero at charge 30. By working out possible gauge algebras for different values of the tadpole, we find that all simple ADE Lie algebras of rank ≤ 18 appear.

Bounds on Species Scale and the Distance Conjecture

AbstractThe species scale serves as a UV cutoff in the gravitational sector of an EFT and can depend on the moduli of the theory as the spectrum of the theory varies. It is argued that the dependence of the species scale on massless (or light) modes satisfies . This bound is true at all points in moduli space including also its interior. The argument is based on the idea that the short distance contribution of massless modes to gravitational terms in the EFT cannot dramatically affect the black hole entropy. Based on string theory arguments the constant in this bound is expected to be equal to as the boundary of the moduli space is approached. However, it turns out that along trajectories going from interior points to the boundaries of moduli space the slope of the species scale can approach its asymptotic value from above, thereby implying that the constant in the bound must be larger than . The bound on the variation of the species scale also implies that the mass of towers of light modes cannot go to zero faster than exponential in field distance in accordance with the Distance Conjecture.

Arakelov–Milnor inequalities and maximal variations of Hodge structure

In this paper we study the $\mathbb {C}^*$-fixed points in moduli spaces of Higgs bundles over a compact Riemann surface for a complex semisimple Lie group and its real forms. These fixed points are called Hodge bundles and correspond to complex variations of Hodge structure. We introduce a topological invariant for Hodge bundles that generalizes the Toledo invariant appearing for Hermitian Lie groups. An important result of this paper is a bound on this invariant which generalizes the Milnor–Wood inequality for a Hodge bundle in the Hermitian case, and is analogous to the Arakelov inequalities of classical variations of Hodge structure. When the generalized Toledo invariant is maximal, we establish rigidity results for the associated variations of Hodge structure which generalize known rigidity results for maximal Higgs bundles and their associated maximal representations in the Hermitian case.

Universal lifting in the D1-D5 CFT

We consider D1-D5-P states in the untwisted sector of the D1-D5 orbifold CFT where one copy of the seed CFT has been excited with a left-moving superconformal primary. Despite being BPS at the orbifold point, such states can ‘lift’ as the theory is deformed away from this point in moduli space. We compute this lifting at second order in the deformation parameter for arbitrary left-moving dimension h of this class of states. This result displays an interesting universality since the lifting does not depend on the details of the superconformal primary; it depends only on the dimension. In the large-dimension limit the lift scales as sqrt{h} ; it is observed that such scaling appears to be a universal property of the lift of D1-D5-P states.

Ising-like and Fibonacci anyons from KZ-equations

In this work we present solutions to Knizhnik-Zamolodchikov (KZ) equations corresponding to conformal block wavefunctions of non-Abelian Ising-like and Fibonacci Anyons. We solve these equations around regular singular points in configuration space in terms of hypergeometric functions and derive explicit monodromy representations of the braid group action. This confirms the correct non-Abelian statistics of the solutions. One novelty of our approach is that we explicitly keep track of spin basis states and identify conformal blocks uniquely with such states at relevant points in moduli space.

Discretely shrinking targets in moduli space

We consider the discrete shrinking target problem for Teichmüller geodesic flow on the moduli space of abelian or quadratic differentials and prove that the discrete geodesic trajectory of almost every differential will hit a shrinking family of targets infinitely often provided the measures of the targets are not summable. This result applies to any ergodic \(\mathrm {SL}(2,\mathbb {R})\)–invariant measure and any nested family of spherical targets. Under stronger conditions on the targets, we moreover prove that almost every differential will eventually always hit the targets. As an application, we obtain a logarithm law describing the rate at which generic discrete trajectories accumulate on a given point in moduli space. These results build on work of Kelmer (Geom Funct Anal 27:1257–1287, 2017) and generalize theorems of Aimino, Nicol, and Todd (Ann Inst Henri Poincaré Probab Stat 53:1371–1401, 2017).

The tadpole problem

We examine the mechanism of moduli stabilization by fluxes in the limit of a large number of moduli. We conjecture that one cannot stabilize all complex-structure moduli in F-theory at a generic point in moduli space (away from singularities) by fluxes that satisfy the bound imposed by the tadpole cancellation condition. More precisely, while the tadpole bound in the limit of a large number of complex-structure moduli goes like 1/4 of the number of moduli, we conjecture that the amount of charge induced by fluxes stabilizing all moduli grows faster than this, and is therefore larger than the allowed amount. Our conjecture is supported by two examples: K3 × K3 compactifications, where by using evolutionary algorithms we find that moduli stabilization needs fluxes whose induced charge is 44% of the number of moduli, and Type IIB compactifications on mathbbm{CP} 3, where the induced charge of the fluxes needed to stabilize the D7-brane moduli is also 44% of the number of these moduli. Proving our conjecture would rule out de Sitter vacua obtained via antibrane uplift in long warped throats with a hierarchically small supersymmetry breaking scale, which require a large tadpole.

On rational points in CFT moduli spaces

Motivated by the search for rational points in moduli spaces of two-dimensional conformal field theories, we investigate how points with enhanced symmetry algebras are distributed there. We first study the bosonic sigma-model with S1 target space in detail and uncover hitherto unknown features. We find for instance that the vanishing of the twist gap, though true for the S1 example, does not automatically follow from enhanced symmetry points being dense in the moduli space. We then explore the supersymmetric sigma-model on K3 by perturbing away from the torus orbifold locus. Though we do not reach a definite conclusion on the distribution of enhanced symmetry points in the K3 moduli space, we make several observations on how chiral currents can emerge and disappear under conformal perturbation theory.

Hurwitz theory of elliptic orbifolds, I

An elliptic orbifold is the quotient of an elliptic curve by a finite group. Eskin and Okounkov proved that generating functions for the number of branched covers of an elliptic curve with specified ramification are quasimodular forms for the full modular group $SL_2(\mathbb{Z})$. They later generalized this theorem to the enumeration of branched covers of a pillowcase, i.e. the quotient of an elliptic curve by the elliptic involution, proving quasi-modularity for $\Gamma_1(2)$. We generalize their work to the quotient of an elliptic curve by cyclic groups of orders $N=3$, $4$, $6$, proving quasi-modularity for level $\Gamma_1(N)$. One corollary is that certain generating functions of hexagon, square, and triangle tilings of compact surfaces are quasi-modular. These tilings enumerate lattice points in moduli spaces of flat surfaces. We analyze the asymptotic behavior as the number of tiles goes to infinity, theoretically giving an algorithm to compute the Masur-Veech volumes of moduli spaces of cubic, quartic, and sextic differentials. We also deduce that the volume is polynomial in $\pi$.

Exploring the landscape of heterotic strings on Td

Compactifications of the heterotic string on Td are the simplest, yet rich enough playgrounds to uncover swampland ideas: the U(1)d+16 left-moving gauge symmetry gets enhanced at special points in moduli space only to certain groups. We state criteria, based on lattice embedding techniques, to establish whether a gauge group is realized or not. For generic d, we further show how to obtain the moduli that lead to a given gauge group by modifying the method of deleting nodes in the extended Dynkin diagram of the Narain lattice II1,17. More general algorithms to explore the moduli space are also developed. For d = 1 and 2 we list all the maximally enhanced gauge groups, moduli, and other relevant information about the embedding in IId,d+16. In agreement with the duality between heterotic on T2 and F-theory on K3, all possible gauge groups on T2 match all possible ADE types of singular fibers of elliptic K3 surfaces. We also present a simple method to transform the moduli under the duality group, and we build the map that relates the charge lattices and moduli of the compactification of the E8 × E8 and Spin(32)/ℤ2 heterotic theories.

Numerical metrics, curvature expansions and Calabi-Yau manifolds

We discuss the extent to which numerical techniques for computing approximations to Ricci-flat metrics can be used to investigate hierarchies of curvature scales on Calabi-Yau manifolds. Control of such hierarchies is integral to the validity of curvature expansions in string effective theories. Nevertheless, for seemingly generic points in moduli space it can be difficult to analytically determine if there might be a highly curved region localized somewhere on the Calabi-Yau manifold. We show that numerical techniques are rather efficient at deciding this issue.

Lifting of level-1 states in the D1D5 CFT

The D1D5 CFT has a large set of states that are supersymmetric at the ‘free’ orbifold point in moduli space. When we perturb away from this point, some of these states join into long multiplets and lift in energy, while others remain supersymmetric. The count of unlifted states can be bounded below by an index, but the index does not yield the pattern of lifting; i.e., which states join into a long multiplet and how much this multiplet lifts. In this paper we consider the simple case of the D1D5 CFT where the orbifold CFT is a sigma model with targets space (T4)2/S2 and consider states at energy level 1. There are 2688 states at this level. The lifted states form a triplet of long multiplets, and we compute their lift at second order in perturbation theory. Half the members of the long multiplet are in the untwisted sector and half are in the twisted sector. This and other similar studies should help in the understanding of fuzzball states that describe extremal holes, since CFT sectors with low twist describe shallow throats in the dual gravity solution while sectors with high twist describe deep throats.

Transplanckian censorship and the local swampland distance conjecture

The swampland distance conjecture (SDC) addresses the ability of effective field theory to describe distant points in moduli space. It is natural to ask whether there is a local version of the SDC: is it possible to construct local excitations in an EFT that sample extreme regions of moduli space? In many cases such excitations exhibit horizons or instabilities, suggesting that there are bounds on the size and structure of field excitations that can be achieved in EFT. Static bubbles in ordinary Kaluza-Klein theory provide a simple class of examples: the KK radius goes to zero on a smooth surface, locally probing an in- finite distance point, and the bubbles are classically unstable against radial perturbations. However, it is also possible to stabilize KK bubbles at the classical level by adding flux. We study the impact of imposing the Weak Gravity Conjecture (WGC) on these solutions, finding that a rapid pair production instability arises in the presence of charged matter with q/m ≳ 1. We also analyze 4d electrically charged dilatonic black holes. Small curvature at the horizon imposes a bound log (MBH) ,≳ |∆\U0001d719|, independent of the WGC, and the bound can be strengthened if the particle satisfying the WGC is sufficiently light. We conjecture that quantum gravity in asymptotically flat space requires a general bound on large localized moduli space excursions of the form |∆\U0001d719| ≲ | log(RΛ)|, where R is the size of the minimal region enclosing the excitation and Λ−1 is the short-distance cutoff on local EFT. The bound is qualitatively saturated by the dilatonic black holes and Kaluza-Klein monopoles.

Jumping spectra and vanishing couplings in heterotic Line Bundle Standard Models

We study two aspects of the physics of heterotic Line Bundle Standard Models on smooth Calabi-Yau threefolds. First, we investigate to what degree modern moduli stabilization scenarios can affect the standard model spectrum in such compactifications. Specifically, we look at the case where some of the complex structure moduli are fixed by a choice of hidden sector bundle. In this context, we study the frequency with which the system tends to be forced to a point in moduli space where the cohomology groups determining the spectrum in the standard model sector jump in dimension. Second, we investigate to what degree couplings, that are permitted by all of the obvious symmetries of the theory, actually vanish due to certain topological constraints associated to their higher dimensional origins. We find that both effects are prevalent within the data set of heterotic Line Bundle Standard Models studied.

Heterotic/type II duality and non-geometric compactifications

We present a new class of dualities relating non-geometric Calabi-Yau com- pactifications of type II string theory to T-fold compactifications of the heterotic string, both preserving four-dimensional mathcal{N} = 2 supersymmetry. The non-geometric Calabi-Yau space is a K 3 fibration over T2 with non-geometric monodromies in the duality group O(Γ4,20); this is dual to a heterotic reduction on a T4 fibration over T2 with the O(Γ4,20) monodromies now viewed as heterotic T-dualities. At a point in moduli space which is a minimum of the scalar potential, the type II compactification becomes an asymmetric Gepner model and the monodromies become automorphisms involving mirror symmetries, while the heterotic dual is an asymmetric toroidal orbifold. We generalise previous constructions to ones in which the automorphisms are not of prime order. The type II construction is perturbatively consistent, but the naive heterotic dual is not modular invariant. Modular invariance on the heterotic side is achieved by including twists in the circles dual to the winding numbers round the T2, and this in turn introduces non-perturbative phases depending on NS5-brane charge in the type II construction.

Microstate geometries at a generic point in moduli space

We systematically study all supersymmetric solutions of six-dimensional (2,0) supergravity with a null isometry. In particular, every such solution with at least four real supersymmetries is also a supersymmetric solution of a (1,0) theory preserving the same absolute amount of supersymmetry. This implies that no genuinely new solutions of this type can be found in this framework. The microstate geometries associated to supersymmetric black holes within Mathur's proposal are generically supersymmetric solutions of six-dimensional supergravity. A direct consequence of our result is that supersymmetric microstate geometries of single centre supersymmetric black holes should carry only one compact 3-cycle.

A Solvable Deformation of Quantum Mechanics

The conventional Hamiltonian $H= p^2+ V_N(x)$, where the potential $V_N(x)$ is a polynomial of degree $N$, has been studied intensively since the birth of quantum mechanics. In some cases, its spectrum can be determined by combining the WKB method with resummation techniques. In this paper we point out that the deformed Hamiltonian $H=2 \cosh(p)+ V_N(x)$ is exactly solvable for any potential: a conjectural exact quantization condition, involving well-defined functions, can be written down in closed form, and determines the spectrum of bound states and resonances. In particular, no resummation techniques are needed. This Hamiltonian is obtained by quantizing the Seiberg-Witten curve of $\mathcal{N}=2$ Yang-Mills theory, and the exact quantization condition follows from the correspondence between spectral theory and topological strings, after taking a suitable four-dimensional limit. In this formulation, conventional quantum mechanics emerges in a scaling limit near the Argyres-Douglas superconformal point in moduli space. Although our deformed version of quantum mechanics is in many respects similar to the conventional version, it also displays new phenomena, like spontaneous parity symmetry breaking.

Symmetry enhancement interpolation, non-commutativity and Double Field Theory

We present a moduli dependent target space effective field theory action for (truncated) heterotic string toroidal compactifications. When moving continuously along moduli space, the stringy gauge symmetry enhancement-breaking effects, which occur at particular points of moduli space, are reproduced.Besides the expected fields, originating in the ten dimensional low energy effective theory, new vector and scalar fields are included. These fields depend on “double periodic coordinates” as usually introduced in Double Field Theory. Their mode expansion encodes information about string states, carrying winding and KK momenta, associated to gauge symmetry enhancements. It is found that a non-commutative product, which introduces an intrinsic non-commutativity on the compact target space, is required in order to make contact with string theory amplitude results.

Towards exotic matter and discrete non-abelian symmetries in F-theory

We present a prescription in F-theory for realizing matter in “exotic” representations of product gauge groups. For 6D vacua, bifundamental hypermultiplets are engineered by starting at a singular point in moduli space which includes 6D superconformal field theories coupled to gravity. A deformation in Higgs branch moduli space takes us to a weakly coupled gauge theory description. In the corresponding elliptically fibered Calabi-Yau threefold, the minimal Weierstrass model parameters (f, g, Δ) vanish at collisions of the discriminant at least to order (4, 6, 12), but with sufficiently high order of tangency to ensure the existence of T-brane deformations to a weakly coupled gauge theory with exotic bifundamentals. We present explicit examples including bifundamental hypermultiplets of {mathfrak{e}}_7times mathfrak{s}{mathfrak{u}}_2 and {mathfrak{e}}_6times mathfrak{s}{mathfrak{u}}_3 , each of which have dual heterotic orbifold descriptions. Geometrically, these matter fields are delocalized across multiple points of an F-theory geometry. Symmetry breaking with such representations can be used to produce high dimension representations of simple gauge groups such as the four-index symmetric representation of mathfrak{s}{mathfrak{u}}_2 and the three-index symmetric representation of mathfrak{s}{mathfrak{u}}_3 , and after further higgsing can yield discrete non-abelian symmetries.