We study B-brane superpotentials depending on several closed- and open- moduli on Calabi-Yau hypersurfaces and complete intersections. By blowing up the ambient space along a curve wrapped by B-branes in a Calabi-Yau manifold, we obtain a blow-up new manifold and the period integral satisfying the GKZ-system. Via mirror symmetry to A-model, we calculate the superpotentials and extract Ooguri-Vafa invariants for concrete examples of several open-closed moduli in Calabi-Yau manifolds.
The integrality of Ooguri-Vafa disk invariants is verified using discrete symmetries of the superpotential of the mirror Landau-Ginzburg theory to calculate quantum corrections to the boundary variables. We show that these quantum corrections are completely determined if we assume that the discrete symmetry of the superpotential also holds in terms of the quantum corrected variables. We discuss the case of local P^2 blown up at three points and local F_2 blown up at two points in detail.
We show how to compute terms in an expansion of the world-volume superpotential for fairly general D-branes on the quintic Calabi-Yau using linear sigma model techniques, and show in examples that this superpotential captures the geometry and obstruction theory of bundles and sheaves on this Calabi-Yau.
The D-brane superpotential is very important in the low energy effective theory. As the generating function of all disk instantons from the worldsheet point of view, it plays a crucial role in deriving some important properties of the compact Calabi-Yau manifolds. By using the GKZ-generalized hypergeometric system, we will calculate the B-brane superpotentials of two non-fermat type compact Calabi-Yau hypersurfaces in toric varieties,respectively. Then according to the mirror symmetry, we obtain the A-model superpotentials and the Ooguri-Vafa invariants for the mirror Calabi-Yau manifolds.
According to the generalized GKZ system for CY manifold with brane and the open-closed Mirror map, We calculate the off-shell D-Brane/F-Theory effective superpotentials and Ooguri-Vafa invariants of several compact Calabi-Yau manifolds .
By blowing up the ambient space along the curve wrapped by B-branes, we study the brane superpotentials and Ooguri-Vafa invariants on complete intersections Calabi-Yau threefolds. On the topological B-model side, B-brane superpotentials are expressed in terms of the period integral of the blow-up manifolds. By mirror maps, the superpotentials are generating functions of Ooguri-Vafa invariants counting holomorphic disks on the topological A-model side.
For compact Calabi-Yau geometries with D5-branes we study N=1 effective superpotentials depending on both open- and closed-string fields. We develop methods to derive the open/closed Picard-Fuchs differential equations, which control D5-brane deformations as well as complex structure deformations of the compact Calabi-Yau space. Their solutions encode the flat open/closed coordinates and the effective superpotential. For two explicit examples of compact D5-brane Calabi-Yau hypersurface geometries we apply our techniques and express the calculated superpotentials in terms of flat open/closed co…
We calculate the D-brane superpotentials for two non-Fermat type compact Calabi-Yau manifolds which are the hypersurfaces of the weighed projective spaces in type II string theory. By constructing the open-closed mirror maps, we also compute the Ooguri-Vafa invariants, which are related to the open Gromov-Witten invariants.
We develop computational tools for the tree-level superpotential of B-branes in Calabi-Yau orientifolds. Our method is based on a systematic implementation of the orientifold projection in the geometric approach of Aspinwall and Katz. In the process we lay down some ground rules for orientifold projections in the derived category.
Compactification of M theory in the presence of G-fluxes yields N=2 five-dimensional gauged supergravity with a potential that lifts all supersymmetric vacua. We derive the effective superpotential directly from the Kaluza-Klein reduction of the eleven-dimensional action on a Calabi-Yau three-fold and compare it with the superpotential obtained by means of calibrations. We discuss an explicit domain wall solution, which represents five-branes wrapped over holomorphic cycles. This solution has a ``running volume'' and we comment on the possibility that quantum corrections provide a lower bound …
We study Landau Ginzburg (LG) theories mirror to 2D N=2 gauged linear sigma models on toric Calabi-Yau manifolds. We derive and solve new constraint equations for Landau Ginzburg elliptic Calabi-Yau superpotentials, depending on the physical data of dual linear sigma models. In Calabi-Yau threefolds case, we consider two examples. First, we give the mirror symmetry of the canonical line bundle over the Hirzebruch surfaces $\bf F_n$. Second, we find a special geometry with the affine so(8) Lie algebra toric data extending the geometry of elliptically fibered K3. This geometry leads to a pure N=…
In this paper we explore contributions to non-perturbative superpotentials arising from instantons wrapping effective divisors in smooth Calabi-Yau four-folds. We concentrate on the case of manifolds constructed as complete intersections in products of projective spaces (CICYs) or generalizations thereof (gCICYs). We systematically investigate the structure of the cone of effective (algebraic) divisors in the four-fold geometries and employ the same tools recently developed in arXiv:1507.03235 to construct more general instanton geometries than have previously been considered in the literature…
The disk partition function of the open topological string computes the spacetime superpotential for D-branes wrapping cycles of a compact Calabi-Yau threefold. We use string duality to show that when appropriately formulated, the problem admits a natural geometrization in terms of a non-compact Calabi-Yau fourfold without D-branes. The duality relates the D-brane superpotential to a flux superpotential on the fourfold. This sheds light on several features of superpotential computations appearing in the literature, in particular on the observation that Calabi-Yau fourfold geometry enters the p…
Under heterotic/F-theory duality it was argued that a wide class of heterotic five-branes is mapped into the geometry of an F-theory compactification manifold. In four-dimensional compactifications this identifies a five-brane wrapped on a curve in the base of an elliptically fibered Calabi-Yau threefold with a specific F-theory Calabi-Yau fourfold containing the blow-up of the five-brane curve. We argue that this duality can be reformulated by first constructing a non-Calabi-Yau heterotic threefold by blowing up the curve of the five-brane into a divisor with five-brane flux. Employing hetero…
We present an explicit supersymmetric deformation of supergravity backgrounds describing D3-branes on Calabi-Yau cones. From the geometrical point of view, it corresponds to blowing up a 4-cycle in the Calabi-Yau and can be done universally. In the field theory, we identify this deformation with motion on non-mesonic directions in the full moduli space of vacua. For the case of a Z_2 orbifold of the conifold, we discuss an explicit gravity solution with two deformation parameters: one corresponding to blowing up a 2-cycle and one corresponding to blowing up a 4-cycle. The generic case where th…
We study Pfaffians that appear in non-perturbative superpotential terms arising from worldsheet instantons in heterotic theories. A result by Beasley and Witten shows that these instanton contributions cancel among curves within a given homology class for Calabi-Yau manifolds that can be described as hypersurfaces or complete intersections in projective or toric ambient spaces. We provide a prescription that identifies all $\mathbb{P}^1$ curves in certain homology classes of complete intersection Calabi-Yau manifolds in products of projective spaces (CICYs) and cross-check our results by a com…
For D-brane system with three D-branes on compact Calabi-Yau threefolds, the dual F-theory fourfolds for parallel/complete coincident/part coincident D-brane system is constructed by the type II/F-theory duality. Complete coincident means that the three D-branes coincide and part coincident represents the coincident of two of the three D branes. The low energy effective superpotentials are calculated by mirror symmetry, GKZ-system method and the type II/F-theory duality on the B-model side, respectively. Using the mirror symmetry, A-model superpotentials and the Ooguri-Vafa invariants are obta…
We analyse the (rigid) special geometry of a class of local Calabi-Yau manifolds given by hypersurfaces in C^4 as W'(x)^2+f_0(x)+v^2+w^2+z^2=0, that arise in the study of the large N duals of four-dimensional N=1 supersymmetric SU(N) Yang-Mills theories with adjoint field Φand superpotential W(Φ). The special geometry relations are deduced from the planar limit of the corresponding holomorphic matrix model. The set of cycles is split into a bulk sector, for which we obtain the standard rigid special geometry relations, and a set of relative cycles, that come from the non-compactness of the man…
We perform the Kaluza-Klein reduction of M-theory on warped Calabi-Yau fourfolds with non-trivial four-form flux turned on. The resulting scalar- and superpotential is computed and compared with the superpotential obtained by Gukov, Vafa and Witten using different methods.
We briefly report on some recent progress in the computation of B-brane superpotentials for Type II strings compactified on Calabi-Yau manifolds, obtained by using a parametrization of tubular neighborhoods of complex submanifolds, also known as local spaces. In particular, we propose a closed expression for the superpotential of a brane on a genus-g curve in a Calabi-Yau threefold in the case in which there exists a holomorphic projection from the local space around the curve to the curve itself.
We study issues related to F-theory on Calabi-Yau fourfolds and its duality to heterotic theory for Calabi-Yau threefolds. We discuss principally fourfolds that are described by reflexive polyhedra and show how to read off some of the data for the heterotic theory from the polyhedron. We give a procedure for constructing examples with given gauge groups and describe some of these examples in detail. Interesting features arise when the local pieces are fitted into a global manifold. An important issue is how to compute the superpotential explicitly. Witten has shown that the condition for a div…
In four-dimensional F-theory compactifications with N=1 supersymmetry the fields describing the dynamics of space-time filling 7-branes are part of the complex structure moduli space of the internal Calabi-Yau fourfold. We explicitly compute the flux superpotential in F-theory depending on all complex structure moduli, including the 7-brane deformations and the field corresponding to the dilaton-axion. Since fluxes on the 7-branes induce 5-brane charge, a local limit allows to effectively match the F-theory results to a D5-brane in a non-compact Calabi-Yau threefold with threeform fluxes. We a…
We study aspects of Calabi-Yau four-folds as compactification manifolds of F-theory, using mirror symmetry of toric hypersurfaces. Correlation functions of the topological field theory are determined directly in terms of a natural ring structure of divisors and the period integrals, and subsequently used to extract invariants of moduli spaces of rational curves subject to certain conditions. We then turn to the discussion of physical properties of the space-time theories, for a number of examples which are dual to $E_8\times E_8$ heterotic N=1 theories. Non-critical strings of various kinds, w…
In this work we compute relative periods for B-branes, realized in terms of divisors in a compact Calabi-Yau hypersurface, by means of direct integration. Although we exemplify the method of direct integration with a particular Calabi-Yau geometry, the recipe automatically generalizes for divisors in other Calabi-Yau geometries as well. From the calculated relative periods we extract double-logarithmic periods. These periods qualify to describe disk instanton generated N=1 superpotentials of the corresponding compact mirror Calabi-Yau geometry in the large volume regime. Finally we extract the…
We calculate the D-brane superpotentials for two Calabi-Yau manifolds with three deformations by the generalized hypergeometric GKZ systems, which give rise to the flux superpotentials $\mathcal{W}_{GVW}$ of the dual F-theory compactification on the relevant Calabi-Yau fourfolds in the weak decoupling limit. We also compute the Ooguri-Vafa invariants from A-model expansion with mirror symmetry, which are related to the open Gromov-Witten invariants.
The N=1 effective action for generic type IIA Calabi-Yau orientifolds in the presence of background fluxes is computed from a Kaluza-Klein reduction. The Kahler potential, the gauge kinetic functions and the flux-induced superpotential are determined in terms of geometrical data of the Calabi-Yau orientifold and the background fluxes. The moduli space is found to be a Kahler subspace of the N=2 moduli space and shown to coincide with the moduli space arising in compactification of M-theory on a specific class of G_2 manifolds. The superpotential depends on all geometrical moduli and vanishes a…
We present a detailed study of D-brane superpotentials depending on several open and closed-string deformations. The relative cohomology group associated with the brane defines a generalized hypergeometric GKZ system which determines the off-shell superpotential and its analytic properties under deformation. Explicit expressions for the N=1 superpotential for families of type II/F-theory compactifications are obtained for a list of multi-parameter examples. Using the Hodge theoretic approach to open-string mirror symmetry, we obtain new predictions for integral disc invariants in the A model i…
In this paper we study several issues related to the generation of superpotential induced by background Ramond-Ramond fluxes in compactification of Type IIA string theory on Calabi-Yau four-folds. Identifying BPS solitons with D-branes wrapped over calibrated submanifolds in a Calabi-Yau space, we propose a general formula for the superpotential and justify it comparing the supersymmetry conditions in D=2 and D=10 supergravity theories. We also suggest a geometric interpretation to the supersymmetric index in the two-dimensional effective theory in terms of topological invariants of the Calabi…
We consider orientifolds of Calabi-Yau 3-folds in the context of Type IIA and Type IIB superstrings. We show how mirror symmetry can be used to sum up worldsheet instanton contributions to the superpotential for Type IIA superstrings. The relevant worldsheets have the topology of the disc and ${\bf RP^2}$.
We present a concrete and consistent procedure to generate one kind of non-perturbative superpotential, including the gaugino condensation corrections and poly-instanton corrections, in type IIB orientifold compactification with four Kahler Moduli. Then we use this kind of superpotential as well as the alphaprime-corrections to Kahler potential to fix all of the four Kahler moduli on a general Calabi-Yau manifold with typical K3-fibred volume form. In our construction, the considered Calabi-Yau threefolds are K3-fibred and admit at least one del Pezzo surface and one W-surface. Searching throu…
We consider splitting type phase transitions between Calabi-Yau fourfolds. These transitions generalize previously known types of conifold transitions between threefolds. Similar to conifold configurations the singular varieties mediating the transitions between fourfolds connect moduli spaces of different dimensions, describing ground states in M- and F-theory with different numbers of massless modes as well as different numbers of cycles to wrap various p-branes around. The web of Calabi-Yau fourfolds obtained in this way contains the class of all complete intersection manifolds embedded in …
This article first reviews the calculation of the N = 1 effective action for generic type IIA and type IIB Calabi-Yau orientifolds in the presence of background fluxes by using a Kaluza-Klein reduction. The Kahler potential, the gauge kinetic functions and the flux-induced superpotential are determined in terms of geometrical data of the Calabi-Yau orientifold and the background fluxes. As a new result, it is shown that the chiral description directly relates to Hitchin's generalized geometry encoded by special odd and even forms on a threefold, whereas a dual formulation with several linear m…
We investigate the instanton effect due to D3 branes wrapping a four-cycle in a Calabi-Yau orientifold with D7 branes. We study the condition for the nonzero superpotentials from the D3 instantons. For that matter we work out the zero mode structures of D3 branes wrapping a four-cycle both in the presence of the fluxes and in the absence of the fluxes. In the presence of the fluxes, the condition for the nonzero superpotential could be different from that without the fluxes. We explicitly work out a simple example of the orientifold of $K3 \times T^2/Z_2$ with a suitable flux to show such beha…
We investigate the dynamics of space-time filling five-branes wrapped on curves in heterotic and orientifold Calabi-Yau compactifications. We first study the leading N=1 scalar potential on the infinite deformation space of the brane-curve around a supersymmetric configuration. The higher order potential is also determined by a brane superpotential which we compute for a subset of light deformations. We argue that these deformations map to new complex structure deformations of a non-Calabi-Yau manifold which is obtained by blowing up the brane-curve into a four-cycle and by replacing the brane…
From the worldsheet perspective, the superpotential on a D-brane wrapping internal cycles of a Calabi-Yau manifold is given as a generating functional for disk correlation functions. On the other hand, from the geometric point of view, D-brane superpotentials are captured by certain chain integrals. In this work, we explicitly show for branes wrapping internal 2-cycles how these two different approaches are related. More specifically, from the worldsheet point of view, D-branes at the Landau-Ginzburg point have a convenient description in terms of matrix factorizations. We use a formula derive…
We use F. Ferrari's methods relating matrix models to Calabi-Yau spaces in order to explain Intriligator and Wecht's ADE classification of $\N=1$ superconformal theories which arise as RG fixed points of $\N = 1$ SQCD theories with adjoints. The connection between matrix models and $\N = 1$ gauge theories can be seen as evidence for the Dijkgraaf--Vafa conjecture. We find that ADE superpotentials in the Intriligator--Wecht classification exactly match matrix model superpotentials obtained from Calabi-Yau's with corresponding ADE singularities. Moreover, in the additional $\Hat{O}, \Hat{A}, \Ha…
We provide a proof of the equivalence of N=1 dynamics obtained by deforming N=2 supersymmetric gauge theories by addition of certain superpotential terms, with that of type IIB superstring on Calabi-Yau threefold geometries with fluxes. In particular we show that minimization of the superpotential involving gaugino fields is equivalent to finding loci where Seiberg-Witten curve has certain factorization property. Moreover, by considering the limit of turning off of the superpotential we obtain the full low energy dynamics of N=2 gauge systems from Calabi-Yau geometries with fluxes.
We study superpotentials from worldsheet instantons in heterotic Calabi-Yau compactifications for vector bundles constructed from line bundle sums, monads and extensions. Within a certain class of manifolds and for certain second homology classes, we derive simple necessary conditions for a non-vanishing instanton superpotential. These show that non-vanishing instanton superpotentials are rare and require a specific pattern for the bundle construction. For the class of monad and extension bundles with this pattern, we derive a sufficient criterion for non-vanishing instanton superpotentials ba…
In this paper, we extend the GKZ-system method to the more general case: compact Complete Intersection Calabi-Yau manifolds (CICY). For several one-deformation modulus compact CICYs with D-branes, the on-shell superpotentials in this paper from the extended GKZ-system method are exactly consistent with published results obtained from other methods. We further compute the off-shell superpotentials of these models. Then we obtain both the on-shell and off-shell superpotentials for sev- eral two-deformation moduli compact CICYs with D-branes by using the extended GKZ-system method. The discrete s…
The presence of RR and NS three-form fluxes in type IIB string compactification on a Calabi-Yau orientifold gives rise to a nontrivial superpotential W for the dilaton and complex structure moduli. This superpotential is computable in terms of the period integrals of the Calabi-Yau manifold. In this paper, we present explicit examples of both supersymmetric and nonsupersymmetric solutions to the resulting 4d N=1 supersymmetric no-scale supergravity, including some nonsupersymmetric solutions with relatively small values of W. Our examples arise on orientifolds of the hypersurfaces in $WP^{4}_{…
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…
Using recent results on eleven-dimensional superspace, we compute the contribution of fivebrane instantons with four fermionic zeromodes in M-theory compactifications on Calabi-Yau fourfolds with flux. We find that no superpotential is generated in this case. This result is compatible with a certain flux-dependent modification of the arithmetic genus criterion.
We determine the N=1 low energy effective action for compactifications of type IIB string theory on compact Calabi-Yau orientifolds in the presence of background fluxes from a Kaluza-Klein reduction. The analysis is performed for Calabi-Yau threefolds which admit an isometric and holomorphic involution. We explicitly compute the Kahler potential, the superpotential and the gauge kinetic functions and check the consistency with N=1 supergravity. We find a new class of no-scale Kahler potentials and show that their structure can be best understood in terms of a dual formulation where some of the…
We discuss the vacuum structure of type IIA/B Calabi-Yau string compactifications to four dimensions in the presence of n-form H-fluxes. These will lift the vacuum degeneracy in the Calabi-Yau moduli space, and for generic points in the moduli space, N=2 supersymmetry will be broken. However, for certain `aligned' choices of the H-flux vector, supersymmetric ground states are possible at the degeneration points of the Calabi-Yau geometry. We will investigate in detail the H-flux induced superpotential and the corresponding scalar potential at several degeneration points, such as the Calabi-Yau…
A formalism for calculating the open supermembrane contribution to the non-perturbative superpotential of moduli in heterotic M-theory is presented. This is explicitly applied to the Calabi-Yau (1,1)-moduli and the separation modulus of the end-of-the-world BPS three-branes, whose non-perturbative superpotential is computed. The role of gauge bundles on the boundaries of the open supermembranes is discussed in detail, and a topological criterion presented for the associated superpotential to be non-vanishing.
We consider issues of moduli stabilization and "area codes" for type II flux compactifications, and the "Inverse Problem" and "Fake Superpotentials" for extremal (non)supersymmetric black holes in type II compactifications on (orientifold of) a compact two-parameter Calabi-Yau expressed as a degree-18 hypersurface in WCP^4[1,1,1,6,9] which has multiple singular loci in its moduli space. We argue the existence of extended "area codes" [1] wherein for the same set of large NS-NS and RR fluxes, one can stabilize all the complex structure moduli and the axion-dilaton modulus (to different sets of …
Each smooth elliptic Calabi-Yau 4-fold determines both a three-dimensional physical theory (a compactification of ``M-theory'') and a four-dimensional physical theory (using the ``F-theory'' construction). A key issue in both theories is the calculation of the ``superpotential''of the theory. We propose a systematic approach to identify these divisors, and derive some criteria to determine whether a given divisor indeed contributes. We then apply our techniques in explicit examples, in particular, when the base B of the elliptic fibration is a toric variety or a Fano 3-fold. When B is Fano, we…
I study A-infinity enhancements of algebraic Calabi-Yau triangulated categories admitting a (triangle) generator, showing that the Serre pairing on such categories determines and is determined by a cyclic pairing on an enhancement of the generator. Using this result, I construct a formal topological string field action inducing an extended D-brane superpotential for such categories. I also give a procedure for lifting certain 2d boundary topological field theories to open topological string theories generated by a single D-brane.
We study the D3-brane moduli dependence of the non-perturbative superpotential generated by the D7-brane gaugino condensation in a type IIB compactification on an orientifold of the elliptic Calabi-Yau threefold $\Bbb{P}_{[1,1,1,6,9]}[18].$ Building on the work of Ganor, in the weak coupling limit, we compute the D3-brane dependent one-loop pfaffian in the flat coordinates and find that the one-loop pfaffian is written as the Jacobi theta function. We also comment on the complex structure dependence of the D3-brane dependent one-loop pfaffian.
We present a method for explicitly computing the non-perturbative superpotentials associated with the vector bundle moduli in heterotic superstrings and M-theory. This method is applicable to any stable, holomorphic vector bundle over an elliptically fibered Calabi-Yau threefold. For specificity, the vector bundle moduli superpotential, for a vector bundle with structure group G=SU(3), generated by a heterotic superstring wrapped once over an isolated curve in a Calabi-Yau threefold with base B=F1, is explicitly calculated. Its locus of critical points is discussed. Superpotentials of vector b…
The four-dimensional effective action for D5-branes in generic compact Calabi-Yau orientifolds is computed by performing a Kaluza-Klein reduction. The N=1 Kaehler potential, the superpotential, the gauge-kinetic coupling function and the D-terms are derived in terms of the geometric data of the internal space and of the two-cycle wrapped by the D5-brane. In particular, we obtain the D5-brane and flux superpotential by integrating out four-dimensional three-forms which couple via the Chern-Simons action. Also the infinitesimal complex structure deformations of the two-cycle induced by the defor…
In this short note we apply methods of Aspinwall-Katz to compute superpotentials of D-branes wrapped on more general obstructed rational curves in Calabi-Yau threefolds. We find an a priori unexpected match between superpotentials from certain such curves and the superpotentials of Landau-Ginzburg models corresponding to minimal models.
We study type IIA string theory compactified on Calabi-Yau fourfolds and heterotic string theory compactified on Calabi-Yau threefolds times a two-torus. We derive the resulting effective theories which have two space-time dimensions and preserve four supercharges. The duality between such vacua is established at the level of the effective theory. For type IIA vacua with non-trivial Ramond-Ramond background fluxes a superpotential is generated. We show that for a specific choice of background fluxes and a fourfold which has the structure of a threefold fibred over a sphere the superpotential c…
Three dimensional N=2 gauge theories with arbitrary gauge group and fundamental flavors are engineered from degenerations of Calabi-Yau four-folds. We show how Coulomb and Higgs branches emerge in the geometric picture. The analysis of instanton generated superpotentials unravels interesting aspects of the five-brane effective action in M theory.
We investigate the one-parameter Calabi-Yau models and identify families of D5-branes which are associated to lines embedded in these manifolds. The moduli spaces are given by sets of Riemann curves, which form a web whose intersection points are described by permutation branes. We arrive at a geometric interpretation for bulk-boundary correlators as holomorphic differentials on the moduli space and use this to compute effective open-closed superpotentials to all orders in the open string couplings. The fixed points of D5-brane moduli under bulk deformations are determined.
We study Euclidean D3-branes wrapping divisors $D$ in Calabi-Yau orientifold compactifications of type IIB string theory. Witten's counting of fermion zero modes in terms of the cohomology of the structure sheaf $\mathcal{O}_D$ applies when $D$ is smooth, but we argue that effective divisors of Calabi-Yau threefolds typically have singularities along rational curves. We generalize the counting of fermion zero modes to such singular divisors, in terms of the cohomology of the structure sheaf $\mathcal{O}_{\overline{D}}$ of the normalization $\overline{D}$ of $D$. We establish this by detailing …
We discuss D3-branes on cohomogeneity-three resolved Calabi-Yau cones over L^{abc} spaces, for which a 2-cycle or 4-cycle has been blown up. In terms of the dual quiver gauge theory, this corresponds to motion along the non-mesonic, or baryonic, directions in the moduli space of vacua. In particular, a dimension-two and/or dimension-six scalar operator gets a vacuum expectation value. These resolved cones support various harmonic (2,1)-forms which reduce the ranks of some of the gauge groups either by a Seiberg duality cascade or by Higgsing. We also discuss higher-dimensional resolved Calabi-…
We study the F-terms in N=1 supersymmetric, d=4 gauge theories arising from D(p+3)-branes wrapping supersymmetric p-cycles in a Calabi-Yau threefold. If p is even the spectrum and superpotential for a single brane are determined by purely classical ($α^\prime \to 0$) considerations. If p=3, superpotentials for massless modes are forbidden to all orders in $α^\prime$ and may only be generated by open string instantons. For this latter case we find that such instanton effects are generically present. Mirror symmetry relates even and odd p and thus perturbative and nonperturbative superpotentials…
We investigate nonperturbative effects in M-theory compactifications arising from wrapped membranes. In particular, we show that in $d=4, \mathcal{N}=1$ compactifications along manifolds of $G_2$ holonomy, membranes wrapped on rigid supersymmetric 3-cycles induce nonzero corrections to the superpotential. Thus, membrane instantons destabilize many M-theory compactifications. Our computation shows that the low energy description of membrane physics is usefully described in terms of three-dimensional topological field theories, and the superpotential is expressed in terms of topological invarian…
