arXiv subjects
Publications and source records attributed to Jock McOrist.
We compute the heterotic moduli-space metric through $α'^2$ for backgrounds admitting a smooth $α'\to0$ limit. The Kaehler potential is unchanged when written in an appropriate field basis suggested by the ten-dimensional supersymmetry equations, while the metric receives $α'^2$ corrections. We discuss this and clarify the roles of the extension bundle, F-terms, the D-terms and the string-derived moduli space metric.
A family of heterotic compactifications carries more structure than a collection of solutions parametrised by moduli. Once the compactification data are fibred over moduli space, deformations become components of universal curvatures. This note reviews that organisation and explains how it incorporates the $α'^2$ supersymmetry corrections.
We analyse the $α'^2$ corrections to the supersymmetry transformations constructed by Bergshoeff--de Roo for heterotic compactifications on SU(3) manifolds. The internal geometry remains complex and conformally balanced. The graviton equation of motion receives an explicit $α'^2$ correction because the tangent-bundle connection appearing in the action is the composite Hull connection. We show that supersymmetry and the Bianchi identity imply all the equations of motion, including this correction. No instanton condition is required.
In this article we dimensionally reduce a heterotic supergravity on a $G_2$ background with Minkowski spacetime using a certain cohomology as a basis for the Kaluza-Klein expansion, up to and including first order in $α'$. We construct the moduli space heterotic $G_2$ compactifications. The $α'$-correction induces a curvature correction to the Weyl-Peterson metric. In the limit in which the $G_2$ manifold reduces to $SU(3)$, we recover known results.
In previous works, an operator was developed for heterotic compactifications on $\mathbb{R}^{2,1}\times G_2$ and $AdS_3 \times G_2$, which preserves $N=1$ $d=3$ supersymmetry and whose kernel is related to the moduli of the compactification. The operator is described in terms of non-physical spurious degrees of freedom, specifically, deformations of a connection on the tangent bundle. In this paper, we eliminate these spurious degrees of freedom by linking deformations of the spin connection to the moduli of the $G_2$ manifold $Y$. This results in an operator $\check D$ that captures the physical moduli space of the $G_2$ heterotic string theory. When $Y=X\times S^1$, with $X$ an $SU(3)$ manifold, we show $\check D$ produces results that align with existing literature. This allows us to propose a $G_2$ moduli space metric. We check that this metric reduces to the $SU(3)$ moduli metric constructed in the literature. We then define an adjoint operator ${\check D}^†$. We show the $G_2$ moduli correspond to the intersection of the kernels of $\check D$ and ${\check D}^†$. These kernels reduce to the $SU(3)$ F-terms and D-terms respectively on $X\times S^1$. This gives two non-trivial consistency checks of our proposed moduli space metric. Working perturbatively in $α'$, we also demonstrate that the heterotic $G_2$ moduli problem can be characterised in terms of a double extension of ordinary bundles, just like in the $SU(3)$ case.
We study the moduli of the universal geometry of $d=4$ $N=1$ heterotic vacua. Universal geometry refers to a family of heterotic vacua fibered over the moduli space. The universal geometry mimics aspects of the original heterotic vacua, in particular holomorphic data such as F-terms, as well as the Green-Schwarz Bianchi identity. Here we study first order deformations of the universal geometry and find this provides a shortcut to computing second order deformations of the original problem. The equations governing the moduli of the universal geometry are remarkably similar to the equations of the underlying heterotic theory and we find a fascinating double extension structure that mirrors the original heterotic problem. As an application we find first order universal deformations determine second order deformations of the original heterotic theory. This gives a shortcut to determining results that are otherwise algebraically unwieldy. The role of the D-terms is closely related to the existence of flat connections on the moduli space. Finally, we re-derive some of these results by direct differentiation - this direct approach requires significantly more calculation.
We study the cohomology of an elliptic differential complex arising from the infinitesimal moduli of heterotic string theory. We compute these cohomology groups at the standard embedding, and show that they decompose into a direct sum of cohomologies. While this is often assumed in the literature, it had not been explicitly demonstrated. Given a stable gauge bundle over a complex threefold with trivial canonical bundle and no holomorphic vector fields, we also show that the Euler characteristic of this differential complex is zero. This points towards a perfect obstruction theory for the heterotic moduli problem, at least for the most physically relevant compactifications.
We consider N=1, d=4 vacua of heterotic theories in the large radius limit in which alpha' << 1. We construct a real differential operator $\mathcal{D}= D+\bar{D}$ on an extension bundle $(Q, \mathcal{D})$ with underlying topology $Q=(T^{1,0}X)^* \oplus {\rm End} \, E \oplus T^{1,0} X$ whose curvature is holomorphic and Hermitian-Yang-Mills with respect to the complex structure and metric on the underlying non-Kahler complex 3-fold X if and only if the heterotic supersymmetry equations and Bianchi identity are satisfied. This is suggestive of an analogue of the Donaldson--Uhlenbeck--Yau correspondence for heterotic vacua of this type.
We reexamine the massless spectrum of a heterotic string vacuum at large radius and present two results. The first result is to construct a vector bundle $\mathcal{Q}$ and operator $\overline{\mathcal{D}}$ whose kernel amounts to deformations solving `F-term' type equations. This resolves a dilemma in previous works in which the spin connection is treated as an independent degree of freedom, something that is not the case in string theory. The second result is to utilise the moduli space metric, constructed in previous work, to define an adjoint operator $\overline{\mathcal{D}}^†$. The kernel of $\overline{\mathcal{D}}^†$ amounts to deformations solving `D-term' type equations. Put together, we show there is a vector bundle $\mathcal{Q}$ with a metric, a $\overline{\mathcal{D}}$ operator and a gauge fixing (holomorphic gauge) in which the massless spectrum are harmonic representatives of $\overline{\mathcal{D}}$. This is remarkable as previous work indicated the Hodge decomposition of massless deformations were complicated and in particular not harmonic except at the standard embedding.
The first part of this paper describes in detail the action of small gauge transformations in heterotic supergravity. We show a convenient gauge fixing is `holomorphic gauge' together with a condition on the holomorphic top form. This gauge fixing, combined with supersymmetry and the Bianchi identity, allows us to determine a set of non-linear PDEs for the terms in the Hodge decomposition. Although solving these in general is highly non-trivial, we give a prescription for their solution perturbatively in alpha' and apply this to the moduli space metric. The second part of this paper relates small gauge transformations to a choice of connection on the moduli space. We show holomorphic gauge is related to a~choice of holomorphic structure and Lee form on a `universal bundle'. Connections on the moduli space have field strengths that appear in the second order deformation theory and we point out it is generically the case that higher order deformations do not commute.
We consider a family of perturbative heterotic string backgrounds. These are complex threefolds X with c_1 = 0, each with a gauge field solving the Hermitian Yang-Mill's equations and compatible B and H fields that satisfy the anomaly cancellation conditions. Our perspective is to consider a geometry in which these backgrounds are fibred over a parameter space. If the manifold X has coordinates x, and parameters are denoted by y, then it is natural to consider coordinate transformations x \to \tilde{x}(x,y) and y \to \tilde{y}(y). Similarly, gauge transformations of the gauge field and B field also depend on both x and y. In the process of defining deformations of the background fields that are suitably covariant under these transformations, it turns out to be natural to extend the gauge field A to a gauge field \IA on the extended (x,y)-space. Similarly, the B, H, and other fields are also extended. The total space of the fibration of the heterotic structures is the Universal Geometry of the title. The extension of gauge fields has been studied in relation to Donaldson theory and monopole moduli spaces. String vacua furnish a richer application of these ideas. One advantage of this point of view is that previously disparate results are unified into a simple tensor formulation. In a previous paper, by three of the present authors, the metric on the moduli space of heterotic theories was derived, correct through order α', and it was shown how this was related to a simple Kahler potential. With the present formalism, we are able to rederive the results of this previously long and involved calculation, in less than a page.
Heterotic vacua of string theory are realised, at large radius, by a compact threefold with vanishing first Chern class together with a choice of stable holomorphic vector bundle. These form a wide class of potentially realistic four-dimensional vacua of string theory. Despite all their phenomenological promise, there is little understanding of the metric on the moduli space of these. What is sought is the analogue of special geometry for these vacua. The metric on the moduli space is important in phenomenology as it normalises D-terms and Yukawa couplings. It is also of interest in mathematics, since it generalises the metric, first found by Kobayashi, on the space of gauge field connections, to a more general context. Here we construct this metric, correct to first order in alpha', in two ways: first by postulating a metric that is invariant under background gauge transformations of the gauge field, and also by dimensionally reducing heterotic supergravity. These methods agree and the resulting metric is Kahler, as is required by supersymmetry. Checking that the metric is in fact Kahler is quite intricate and uses the anomaly cancellation equation for the H-field, in an essential way. The Kahler potential nevertheless takes a remarkably simple form: it is Kahler potential for special geometry with the Kahler form replaced by the alpha'-corrected hermitian form.
The effective field theory of heterotic vacua that realise $\mathbb{R}^{3,1}$ preserving $\mathcal{N} =1$ supersymmetry are studied. The vacua in question admit large radius limits taking the form $\mathbb{R}^{3,1}\times {X}$ , with ${X}$ a smooth three-fold with vanishing first Chern class and a stable holomorphic gauge bundle $\mathcal{E}$. In a previous paper we calculated the kinetic terms for moduli, deducing the moduli metric and Kahler potential. In this paper, we compute the remaining couplings in the effective field theory, correct to first order in alpha prime. In particular, we compute the contribution of the matter sector to the Kahler potential, derive the Yukawa couplings and other quadratic fermionic couplings. From this we write down a Kahler potential $\mathcal{K}$ and superpotential $\mathcal{W}$ .
We prove that N=2 theories that arise by taking n free hypermultiplets and gauging a subgroup of Sp(n), the non-R global symmetry of the free theory, have a remaining global symmetry which is a direct sum of unitary, symplectic, and special orthogonal factors. This implies that theories that have SU(N) but not U(N) global symmetries, such as Gaiotto's T_N theories, are not likely to arise as IR fixed points of RG flows from weakly coupled N=2 gauge theories.
Type IIB toroidal orientifolds are among the earliest examples of flux vacua. By applying T-duality, we construct the first examples of massive IIA flux vacua with Minkowski space-times, along with new examples of type IIA flux vacua. The backgrounds are surprisingly simple with no four-form flux at all. They serve as illustrations of the ingredients needed to build type IIA and massive IIA solutions with scale separation. To check that these backgrounds are actually solutions, we formulate the complete set of type II supergravity equations of motion in a very useful form that treats the R-R fields democratically.
We consider compactifications of M-theory and type IIA string theory to four dimensions. For Minkowski space-time, a supergravity no-go theorem forbids flux supported in the internal space. We show how to evade this no-go theorem by exhibiting new sources of brane charge: in string theory, the basic physical phenomenon is the generation of new brane charges from D-branes in transverse fluxes. In M-theory, there is a new source of M5-brane charge from novel higher derivative couplings that involve fluxes as well as curvatures. We present some explicit orientifold examples with both N=1 and N=2 space-time supersymmetry. Finally, we explain the status of massive type IIA flux compactifications.
We study monopole-instantons in M2-brane theories, focussing on the ABJM class of Chern-Simons gauge theories coupled to matter. We calculate calculate explicitly the 8-fermion term in the effective action induced by these monopole-instantons, and discuss their role in resolving a classical singularity in the moduli space. The results are compared with monopole-instantons in N=8 3d SYM and D-brane theories, as well the dual supergravity description as a membrane scattering process.
In a previous paper we used T-duality to construct a new type of 1/4-BPS solution describing a pair of NS5-branes intersecting in 1+3 dimensions and localised in all other directions except for a single transverse circle. This led to an explicit solution to a sourced Monge--Ampere equation, of which there are few known examples. In this paper we refine this formalism and apply it to two important generalisations: the resolved and deformed conifolds. In doing so we construct two new solutions describing, respectively, a pair of NS5-branes separated in a transverse direction and a pair of NS5-branes with smooth `diamond' profile. We show how the parameter of the resolved conifold (size of the S^2) maps to a transverse separation of the NS5-branes, while the modulus of the deformed conifold (size of the S^3) maps to the deformation parameter of the diamond web.