We derive topological string amplitudes on local Calabi-Yau manifolds in terms of polynomials in finitely many generators of special functions. These objects are defined globally in the moduli space and lead to a description of mirror symmetry at any point in the moduli space. Holomorphic ambiguities of the anomaly equations are fixed by global information obtained from boundary conditions at few special divisors in the moduli space. As an illustration we compute higher genus orbifold Gromov-Witten invariants for C^3/Z_3 and C^3/Z_4.
We use the remodeling approach to the B-model topological string in terms of recursion relations to study open string amplitudes at orbifold points. To this end, we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest. We exemplify this procedure for the C^3/Z_3 orbifold point of local P^2, where we present results for topological string amplitudes for genus zero and up to three holes, and for the one-holed torus. These amplitudes can be understood as generating functions for either open orbifol…
In this expository paper written for physicists and geometers we introduce the notions of TQFT and of orbifold. Then we survey the construction of TQFT's originating from orbifolds such as Chen-Ruan theory and Orbifold String Topology.
We investigate asymmetric orbifold models constructed from non-supersymmetric heterotic strings. We systematically classify the asymmetric orbifold models with standard embeddings and present a list of asymmetric orbifolds which are geometrically interpreted as toroidal compactifications of non-supersymmetric heterotic strings. By studying non-standard embedding models, we also construct examples of the {\em supersymmetric} asymmetric orbifold models based on non-supersymmetric heterotic strings.
We study symmetries between untwisted and twisted strings on asymmetric orbifolds. We present a list of asymmetric orbifold models to possess intertwining currents which convert untwisted string states to twisted ones, and vice versa. We also present a list of heterotic strings on asymmetric orbifolds with supersymmetry between untwisted and twisted string states. Some of properties inherent in asymmetric orbifolds, which are not shared by symmetric orbifolds, are pointed out.
I construct a complete 1-loop partition function of a bosonic closed string on orbifolds. Furthermore, I derive sufficient conditions for the modular invariance of the partition function.
In this short review we outline some recent developments in understanding string orbifolds. In particular, we outline the recent observation that string orbifolds do not precisely describe string propagation on quotient spaces, but rather are literally sigma models on objects called quotient stacks, which are closely related to (but not quite the same as) quotient spaces. We show how this is an immediate consequence of definitions, and also how this explains a number of features of string orbifolds, from the fact that the CFT is well-behaved to orbifold Euler characteristics. Put another way, …
Including world-sheet orientation-reversing automorphisms in the orbifold program, we recently reported the twisted operator algebra and twisted KZ equations in each open-string sector of the general WZW orientation orbifold. In this paper we work out the corresponding classical description of these sectors, including the {\it WZW orientation-orbifold action} -- which is naturally defined on the solid half cylinder -- and its associated WZW orientation-orbifold branes. As a generalization, we also obtain the {\it sigma-model orientation-orbifold action}, which describes a much larger class of …
We investigate the perturbative and non-perturbative correspondence of a class of four dimensional dual string constructions with N=4 and N=2 supersymmetry, obtained as Z_2 or Z_2 x Z_2 orbifolds of the type II, heterotic and type I string. In particular, we discuss the heterotic and type I dual of all the symmetric Z_2 x Z_2 orbifolds of the type II string, classified in hep-th/9901123. .
Including {\it world-sheet orientation-reversing automorphisms} $\hat{h}_σ \in H_-$ in the orbifold program, we construct the operator algebras and twisted KZ systems of the general WZW {\it orientation orbifold} $A_g (H_-) /H_-$. We find that the orientation-orbifold sectors corresponding to each $\hat{h}_σ \in H_-$ are {\it twisted open} WZW strings, whose properties are quite distinct from conventional open-string orientifold sectors. As simple illustrations, we also discuss the classical (high-level) limit of our construction and free-boson examples on abelian $g$.
We investigate the following three consistency conditions for constructing string theories on orbifolds: i) the invariance of the energy-momentum tensors under twist operators, ii) the duality of amplitudes and iii) modular invariance of partition functions. It is shown that this investigation makes it possible to obtain the general class of consistent orbifold models, which includes a new class of orbifold models.
We give various examples of asymmetric orbifold models to possess intertwining currents which convert untwisted string states to twisted ones, and vice versa, and see that such asymmetric orbifold models are severely restricted. The existence of the intertwining currents leads to the enhancement of symmetries in asymmetric orbifold models.
We consider several orbifold compactifications of M-theory and their corresponding type II duals in two space-time dimensions. In particular, we show that while the orbifold compactification of M-theory on $T^9/J_9$ is dual to the orbifold compactification of type IIB string theory on $T^8/I_8$, the same orbifold $T^8/I_8$ of type IIA string theory is dual to M-theory compactified on a smooth product manifold $K3 \times T^5$. Similarly, while the orbifold compactification of M-theory on $(K3 \times T^5)/σ... J_5$ is dual to the orbifold compactification of type IIB string theory on $(K3 \times…
Target space modular symmetries relevant to string loop threshold corrections are studied for $Z_N$ orbifold compactified string theories containing Wilson line background fields.
We study branes and open strings in a large class of orbifolds of a curved background using microscopic techniques of boundary conformal field theory. In particular, we obtain factorizing operator product expansions of open string vertex operators for such branes. Applications include branes in Z2 orbifolds of the SU(2) WZW model and in the D-series of unitary minimal models considered previously by Runkel.
Recently, there is an explosive growth of activities to understand stringy properties of orbifolds. In this article, we survey some of recent developments.
In this short note we review the main features of open-string orbifolds with a quantised flux for the NS-NS antisymmetric tensor in the context of the open descendants of non-supersymmetric asymmetric orbifolds with a vanishing cosmological constant.
We study toroidal orbifold models with topologically invariant terms in the path integral formalism and give physical interpretations of the terms from an operator formalism point of view. We briefly discuss a possibility of a new class of modular invariant orbifold models.
In this short note, we examine the relationship between the string coupling constant, $\gstring{}$, and the grand unified gauge coupling constant, $\alphagut{}$, in a highly successful class of models based on anisotropic orbifold compactifications of the weakly coupled heterotic string. These models represent a stringy embedding of SU(6) gauge-Higgs unification in a five dimensional orbifold GUT. We find that the requirement that the theory be perturbative provides a non-trivial constraint on these models. Interestingly, there is a correlation between the proton decay rate (due to dimension s…
We study interacting features of $S_N$ Orbifold CFTs. Concentrating on characters (associated with $S_N$ Orbifold primaries) we first formulate a novel procedure for evaluating them through $GL(\infty)_+$ tracing. The result is a polynomial formula which we show gives results equivalent to those found by Bantay. From this we deduce a hierarchy of commuting Hamiltonians featuring locality in the induced space, and nonlinear string-type interactions.
In this paper we consider a class of exactly solvable closed string flux backgrounds that exhibit non-commutativity in the closed string coordinates. They are realized in terms of freely-acting asymmetric Z_N-orbifolds, which are themselves close relatives of twisted torus fibrations with elliptic Z_N-monodromy (elliptic T-folds). We explicitly construct the modular invariant partition function of the models and derive the non-commutative algebra in the string coordinates, which is exact to all orders in α'. Finally, we relate these asymmetric orbifold spaces to inherently stringy Scherk-Schwa…
We obtain the spectrum of heterotic strings compactified on orbifolds, focusing on its algebraic structure. Affine Lie algebra provides its current algebra and representations. In particular the twisted spectrum and the Abelian charge are understood. The twisted version of algebra is used in the homomorphism from the orbifold action to the group action. The relation between the conformal weight and the mass gives a handy rule.
We consider worldsheet string theory on $Z_N$ orbifolds of $AdS_3$ associated with conical singularities. If the orbifold action includes a similar twist of $S^3$, supersymmetry is preserved, and there is a moduli space of vacua arising from blowup modes of the orbifold singularity. We exhibit the spectrum, including the properties of twisted sectors and states obtained by fractional spectral flow. A subalgebra of the spacetime superconformal symmetry remains intact after the $Z_N$ quotient, and serves as the spacetime symmetry algebra of the orbifold.
I consider the construction of heterotic orbifold models based on a toroidal orbifold with non Abelian point group. I construct an explicit model based on the point group $S_3$ and calculate the spectrum and remnant symmetries. This model provides a simple example of rank reduction of the Yang--Mills gauge group directly in the string theory rather than in the effective field theory.
We discuss the partition function of a single M5-brane on a circle with transverse orbifold of ADE type and show that the modes captured by the partition function are those of the tensor multiplet and the thee form field. We show that the bound states of M-strings corresponding to pair of simple roots appear, for all ADE, only when the momentum on the circle is turned on.
We present several classes of new 6d string theories which arise via branes at orbifold singularities. They have compact moduli spaces, associated with tensor multiplets, given by Weyl alcoves of non-Abelian groups. We discuss T-duality and Matrix model applications upon compactification.
We continue and extend our earlier investigation ``Strings in a Time-Dependent Orbifold'' (hep-th/0204168). We formulate conditions for an orbifold to be amenable to perturbative string analysis and classify the low dimensional orbifolds satisfying these conditions. We analyze the tree and torus amplitudes of some of these orbifolds. The tree amplitudes exhibit a new kind of infrared divergences which are a result of some ultraviolet effects. These UV enhanced IR divergences can be interpreted as due to back reaction of the geometry. We argue that for this reason the three dimensional paraboli…
In quantum field theory, an orbifold is a way to obtain a new theory from an old one by gauging a finite global symmetry. This definition of orbifold does not make sense for quantum gravity theories, that admit (conjecturally) no global symmetries. In string theory, the orbifold procedure involves the gauging of a global symmetry on the world-sheet theory describing the fundamental string. Alternatively, it is a way to obtain a new string background from an old one by quotienting some isometry. We propose a new formulation of string orbifolds in terms of the group of gauge symmetries of a give…
It is shown that many of the conjectured dualities involving orbifold compactification of M-theory follow from the known dualities involving M-theory and string theory in ten dimensions, and the ansatz that orbifolding procedure commutes with the duality transformation. This ansatz also leads to a new duality conjecture, namely that M-theory compactified on $T^8/Z_2$ is dual to type I string theory on $T^7$. In this case the `twisted sector states' in M-theory live on sixteen membranes transverse to the internal manifold.
We study string theory on orbifolds in the presence of an antisymmetric constant background field and discuss some of new aspects of the theory. It is shown that the term with the antisymmetric field has a topological nature like a Chern-Simons term or a Wess-Zumino term. Due to this property, the theory exhibits various anomalous behavior: Zero mode variables obey nontrivial quantization conditions. Coordinate transformations which define orbifolds are modified at the quantum level. Wavefunctions of twisted strings acquire phase factors when they move around non-contractible loops on orbifold…
In this fourth paper of the series, I clarify the somewhat mysterious relation between the large class of {\it orientation orbifolds} (with twisted open-string CFT's at $\hat c=52$) and {\it orientifolds} (with untwisted open strings at $c=26$), both of which have been associated to division by world-sheet orientation-reversing automorphisms. In particular -- following a spectral clue in the previous paper -- I show that, even as an {\it interacting string system}, a certain half-integer-moded orientation orbifold-string system is in fact equivalent to the archetypal orientifold. The subtitle …
We discuss the calculation of threshold corrections to gauge coupling constants for the, only, non-decomposable class of abelian (2, 2) symmetric N=1 four dimensional heterotic orbifold models, where the internal twist is realized as a generalized Coxeter automorphism. The latter orbifold was singled out in earlier work as the only N=1 heterotic $Z_N$ orbifold that satisfy the phenomenological criteria of correct minimal gauge coupling unification and cancellation of target space modular anomalies.
In these lectures, we review the physics of time-dependent orbifolds of string theory, with particular attention to orbifolds of three-dimensional Minkowski space. We discuss the propagation of free particles in the orbifold geometries, together with their interactions. We address the issue of stability of these string vacua and the difficulties in defining a consistent perturbation theory, pointing to possible solutions. In particular, it is shown that resumming part of the perturbative expansion gives finite amplitudes. Finally we discuss the duality of some orbifold models with the physics …
Orbifolds in field theory are potentially singular objects for at their fixed points the curvature becomes infinite, therefore one may wonder whether field theory calculations near orbifold singularities can be trusted. String theory is perfectly well defined on orbifolds and can therefore be taken as a UV completion of field theory on orbifolds. We investigate the properties of field and string theory near orbifold singularities by reviewing the computation of a one loop gauge field tadpole. We find that in string theory the twisted states give contributions that have a spread of a couple of …
We summarize the structure of N=2 multi-instanton and string loop corrections to the Kähler potential and the gauge kinetic function in type I toroidal orbifold models. For that aim, we exploit perturbative calculations of the physical gauge couplings on the heterotic dual, whenever such a perturbative description is available.
We discuss type II and type I branes at general ADE type orbifold singularities. We show that there are new phases of type I or heterotic string theory in six dimensions, involving extra tensor multiplets, which arise when small instantons sit on orbifold singularities. The theories with extra tensor multiplets are explicitly constructed via orientifolds. The world-volume theories in type IIB or type I five-branes at orbifold singularities lead to the existence of several infinite classes of six dimensional, interacting, renormalization group fixed point theories.
In this note we observe that, contrary to the usual lore, string orbifolds do not describe strings on quotient spaces, but rather seem to describe strings on objects called quotient stacks, a result that follows from simply unraveling definitions, and is further justified by a number of results. Quotient stacks are very closely related to quotient spaces; for example, when the orbifold group acts freely, the quotient space and the quotient stack are homeomorphic. We explain how sigma models on quotient stacks naturally have twisted sectors, and why a sigma model on a quotient stack would be a …
We investigate some aspects of the relationship between matrix string theory and light-cone string field theory by analysing the correspondence between the two-loop thermal partition function of DLCQ strings in flat space and the integrated two-point correlator of twist fields in a symmetric product orbifold conformal field theory at one-loop order. This is carried out by deriving combinatorial expressions for generic twist field correlation functions in permutation orbifolds using the covering surface method, by deriving the one-loop modification of the twist field interaction vertex, and by …
We study cocycle properties of vertex operators and present an operator representation of cocycle operators, which are attached to vertex operators to ensure the duality of amplitudes. It is shown that this analysis makes it possible to obtain the general class of consistent string theories on orbifolds.
We discuss tachyon-free examples of (Type IIB on) non-compact non-supersymmetric orbifolds. Tachyons are projected out by discrete torsion between orbifold twists, while supersymmetry is broken by a Scherk-Schwarz phase (+1/-1 when acting on space-time bosons/fermions) accompanying some even order twists. The absence of tachyons is encouraging for constructing non-supersymmetric D3-brane gauge theories with stable infrared fixed points. The D3-brane gauge theories in our orbifold backgrounds have chiral N = 1 supersymmetric spectra, but non-supersymmetric interactions.
We summarize the results obtained in the last few years about permutation orbifolds in two-dimensional conformal field theories, their application to string theory and their use in the construction of four-dimensional heterotic string models.
We present several examples of T^8/P orbifolds with $P \subset SU(4)$. We compute their Hodge numbers and consider turning on discrete torsion. We then study supersymmetric compactifications of type II, heterotic, and type I strings on these orbifolds. Heterotic compactifications to D=2 have a B-field tadpole with coefficient given by that of the anomaly polynomial. In the SO(32) heterotic with standard embedding the tadpole is absent provided the internal space has a precise value of the Euler number. Guided by their relation to type I, we find tadpole-free SO(32) heterotic orbifolds with non…
We construct more dual pairs of type II-heterotic strings in four dimensions with $N=2,1$ spacetime supersymmetry. On the type II side the construction utilizes the various possible choices of K3 automorphisms with fixed points which transform the holomorphic two-form nontrivially, and rotation plus translation on $T^2$. The Calabi-Yau orbifolds so obtained have non-zero Euler numbers, so quantum corrections exist on the type IIA strings. The heterotic string (asymmetric) orbifold duals are found which depend on going to the enhanced symmetry points. Some aspects of the construction are discus…
We consider Z2, freely acting orbifolds of the type IIB string with 16 parallel D5-branes. When the string is compactified on T2 X T4 and the D5-branes are wrapped on T2, these systems possess N=2 supersymmetry, originating from the spontaneous, partial breaking of N=4. Extended supersymmetry allows us to investigate the duality with certain heterotic and type I constructions, and to obtain informations about their non-perturbative regime.
We study closed string tachyon condensation on general non-supersymmetric orbifolds of C^2. Extending previous analyses on Abelian cases, we present the classification of quotients by discrete finite subgroups of GL(2; C) as well as the generalised Hirzebruch-Jung continued fractions associated with the resolution data. Furthermore, we discuss the intimate connexions with certain generalised versions of the McKay Correspondence.
We investigate the D-brane contents of asymmetric orbifolds. Using T-duality we find that the consistent description of open strings in asymmetric orbifolds requires to turn on background gauge fields on the D-branes. We derive the corresponding noncommutative geometry arising on such D-branes with mixed Neumann-Dirichlet boundary conditions directly by applying an asymmetric rotation to open strings with pure Dirichlet or Neumann boundary conditions. As a concrete application of our results we construct asymmetric type I vacua requiring open strings with mixed boundary conditions for tadpole …
We consider closed-string tachyon condensation in the twisted sectors on the C/Z_{2n+1} \times R^{7,1} orbifold. We calculate the localized energy density in the fixed plane on the orbifold at the one-loop level, and we obtain the decay rate per unit volume of the fixed plane to leading order. We show that the decay rate increases monotonically as a function of n.
We investigate the natural occurence of exponentially small couplings in effective field theories deduced from higher dimensional models. We calculate the coupling between twisted fields of the Z_3 Abelian orbifold compactification of the heterotic string. Due to the propagation of massive Kaluza-Klein modes between the fixed points of the orbifold, the massless twisted fields located at these singular points become weakly coupled. The resulting small couplings have an exponential dependence on the mass of the intermediate states and the distance between the fixed points.
This thesis is concerned with the geometry of toroidal orbifolds and their applications in string theory. By resolving the orbifold singularities via blow-ups, one arrives at a smooth Calabi-Yau manifold. The systematic method to do so is explained in detail. Also the transition to the Orientifold quotient is explained. In the second part of this thesis, applications in string phenomenology are discussed. The applications belong to the framework of compactifications with fluxes in type IIB string theory. The first example belongs to the category of model building, flux-induced soft supersymmet…
We study closed N=2 strings on orbifolds of the form T^4/Z_2 and C^2/Z_2. We compute the torus partition function and prove its modular invariance. We analyse the BRST cohomology of the theory, construct the vertex operators, and compute three and four point amplitudes of twisted and untwisted states. We introduce a background of D-branes, and compute twist states correlators.
We study BPS bound states of little strings in a limit where they realise monopole strings in five dimensional gauge theories. The latter have gauge group $U(M)^N$ and arise from compactification of $(1,0)$ little string theories of type $A_{M-1} \times A_{N-1}$. We find evidence that the partition function of a certain subclass of monopole strings of charge $(k,\ldots,k)$ ($k\geq 1$) is expressible as the partition function of a symmetric orbifold sigma model, whose target space is precisely the symmetric product of the moduli space of monopoles with charge $(1, \ldots, 1)$.
Closed string theories on orbifolds contain both untwisted and twisted states. The latter are normally assumed to live exactly at the orbifold fixed points. We perform a calculation of a gauge field tadpole amplitude and show that off-shell both the twisted and untwisted states give rise to non-trivial momentum profiles over the orbifold C^3/Z_3. These profiles take the form of Gaussian distributions integrated over the fundamental domain of the modular parameter of the torus. The propagators of the internal coordinate fields on the torus world sheet determine the width of the Gaussian profile…
We propose correspondences between 4d quantum field theories with N=2,1,0 (super)conformal invariance and Type IIB string theory on various orbifolds. We argue using the spacetime string theory, and check using the beta functions (exactly for N=2,1 and so far at 1-loop for the gauge couplings in the N=0 case), that these theories have conformal fixed lines. The latter case potentially gives well-defined non-supersymmetric vacua of string theory, with a mechanism for making the curvature and cosmological constant small at nontrivial string coupling. We suggest a correspondence between nonsupers…
We study the construction of the minimal supersymmetric standard model from the $Z_8$ orbifold models. We use a target-space duality anomaly cancellation and a unification of gauge couplings as constraints. It is shown that some models obtained through a systematical search realize the unification of SU(3) and SU(2) coupling constants.
It has long been known that strings wound around incontractible cycles can play a vital role in cosmology. In particular, in a spacetime with toroidal spatial hypersurfaces, the dynamics of the winding modes may help yield three large spatial dimensions. However, toroidal compactifications are phenomenologically unrealistic. In this paper we therefore take a first step toward extending these cosmological considerations to $D$-dimensional toroidal orbifolds. We use numerical simulation to study the timescales over which "pseudo-wound" strings unwind on these orbifolds with trivial fundamental g…
The "orbifold covariance principle", or OCP for short, is presented to support a conjecture of Pradisi, Sagnotti and Stanev on the expression of the Klein- bottle amplitude.
We calculate the moduli dependent part of string one-loop threshold corrections to gauge couplings for the heterotic string theory compactified on abelian toroidal orbifolds, allowing for arbitrary discrete Wilson lines. We show that the knowledge of threshold corrections for any such compactification is equivalent to solving a class of integrals. We solve a sub-class of these integrals and show how any model can be mapped onto this class by fractional linear transformations of its fixed plane moduli. Modular symmetries of the final expression are discussed.
The moduli dependence of string loop threshold corrections to gauge coupling constants is investigated for those ${\bf Z}_N$ Coxeter orbifolds with the property that some twisted sectors have fixed planes for which the six-torus ${\bf T}_6$ can not be decomposed into a direct sum ${\bf T}_4 \bigoplus{\bf T}_2$ with the fixed plane lying in ${\bf T}_2.$
A three-generation Pati-Salam model is constructed by compactifying the heterotic string on a particular T^6/Z_6 Abelian symmetric orbifold with two discrete Wilson lines. The compactified space is taken to be the Lie algebra lattice G_2+SU(3)+SO(4). When one dimension of the SO(4) lattice is large compared to the string scale, this model reproduces many features of a 5d SO(10) grand unified theory compactified on an S^1/Z_2 orbifold. (Of course, with two large extra dimensions we can obtain a 6d SO(10) grand unified theory.) We identify the orbifold parities and other ingredients of the orbif…
