We reveal non-manifest gauge and SO(1,5) Lorentz symmetries in the Lagrangian description of a six-dimensional free chiral field derived from the Bagger-Lambert-Gustavsson model in arXiv:0804.3629 and make this formulation covariant with the use of a triplet of auxiliary scalar fields. We consider the coupling of this self-dual construction to gravity and its supersymmetrization. In the case of the non-linear model of arXiv:0805.2898 we solve the equations of motion of the gauge field, prove that its non-linear field strength is self-dual and find a gauge-covariant form of the non-linear action. Issues of the relation of this model to the known formulations of the M5-brane worldvolume theory are discussed.
We study small fluctuations around a BPS three-sphere vacuum solution of mass deformed BLG theory. We realize the BLG theory by a Nambu bracket and find a maximally supersymmetric lagrangian for the fluctuation fields corresponding to a single M5 brane on $\mb{R}^{1,2}\times S^3$.
This paper is concerned with the problem of coupling the N=8 superconformal Bagger-Lambert-Gustavsson (BLG) theory to N=8 conformal supergravity in three dimensions. We start by constructing the on-shell N=8 conformal supergravity in three dimensions consisting of a Chern-Simons type term for each of the gauge fields: the spin connection, the SO(8) R-symmetry gauge field and the spin 3/2 Rarita-Schwinger (gravitino) field. We then proceed to couple this theory to the BLG theory. The final theory should have the same physical content, i.e., degrees of freedom, as the ordinary BLG theory. We dis…
Working to lowest non-trivial order in fermions, we consider the four-derivative order corrected Lagrangian and supersymmetry transformations of the Euclidean Bagger-Lambert-Gustavsson theory. By demonstrating supersymmetric invariance of the Lagrangian we determine all numerical coefficients in the system. In addition, the supersymmetry algebra is shown to close on the scalar and gauge fields. We also comment on the extension to Lorentzian and other non-Euclidean $\mathcal{N}=8$ 3-algebra theories.
Concise summary of the recent progress in the search for the world-volume action for multiple M2 branes. After a recent discovery of simplified version of BLG action, which is based on the ordinary Lie-algebra structure, does not have coupling constants and extra dynamical fields, attention should be switched to the study of M2 brane dynamics. A viable brane analogue of Polyakov formalism and Belavin-Knizhnik theorem for strings can probably be provided by Palatini formalism for 3d (super)gravity.
In this paper we will analyse a system of multiple M2-branes in between two M5-branes. This will be analysed by studying Bagger-Lambert-Gustavsson (BLG) theory on a manifold with two boundaries. The original BLG theory will be modified to make it gauge and supersymmetric invariant in presence of the boundaries. However, this modified theory will only preserve half the supersymmetry of the original theory. We will also analyse the deformation of this theory caused by noncommutativity between Grassman coordinates and spacetime coordinates. Finally, we will analyse the Higgsing of this theory to …
Recently it was proposed that the Bagger-Lambert-Gustavsson theory with Nambu-Poisson structure describes an M5-brane in a three-form flux background. In this paper we investigate the superalgebra associated with this theory. We derive the central charges corresponding to M5-brane solitons in 3-form backgrounds. We also show that double dimensional reduction of the superalgebra gives rise to the Poisson bracket terms of a non-commutative D4-brane superalgebra. We provide interpretations of the D4-brane charges in terms of spacetime intersections.
In this paper we begin an investigation into the $l_p^3$ corrections to the supersymmetry transformations of the Bagger-Lambert-Gustavsson (BLG) theory. We begin with a review of the dNS duality transformation which allows a non-abelian gauge field to be dualised to a scalar field in 2+1 dimensions. Applying this duality to $α'^2$ terms of the non-abelian D2-brane theory gives rise to the $l_p^3$ corrections of the Lorentzian BLG theory. We then apply this duality transformation to the $α'^2$ corrections of the D2-brane supersymmetry transformations. For the `abelian' BLG theory we are able to…
By using the recently proposed prescription arXiv:0804.3629 for obtaining the $M5$ brane action from multiple $M2$ branes action in BLG theory, we examine such transition when 11 Dimensional background antisymmetric fluxes couple to the $M2$ brane world volume. Such couplings was suggested in arXiv:0805.3427 where it was used the fact that various fields in BLG theory are valued in a Lie 3-algebra. We argue that this action and promoting it by Nambu-Poisson bracket gives the expected coupling of fluxes with $M5$ brane at least at weak coupling limit. We also study some other aspects of the act…
Starting from the three dimensional N=8 BLG and N=6 BL theories with CS level k=1, a dimensional reduction is performed on the M2-branes worldvolume. The IR limit of these 2D EFTs should represent the action of F1 strings in type IIA. The CS term reduces to a particular Phi*F term and the scalars and fermions get "mass" terms. The reduced actions still rely on three-algebra, which we did not require to define precisely.
We show that the Bagger-Lambert-Gustavsson (BLG) theory with two pairs of negative norm generators is derived from the scaling limit of an orbifolded Aharony-Bergman-Jafferis-Maldacena (ABJM) theory. The BLG theory with many Lorentzian pairs is known to be reduced to the Dp-brane theory via Higgs mechanism, so our scaling procedure can be used to derive Dp-branes directly from M2-branes in the field theory language. In this paper, we focus on the D3-brane case and investigate the scaling limits of various quiver Chern-Simons theories obtained from different orbifolding actions. Remarkably, in …
We obtain a general class of polynomials for which the Schrodinger operator has a discrete spectrum. This class includes all the scalar potentials in membrane, 5-brane, p-branes, multiple M2 branes, BLG and ABJM theories. We provide a proof of the discreteness of the spectrum of the associated Schrodinger operators. This a a first step in order to analyze BLG and ABJM supersymmetric theories from a non-perturbative point of view.
We investigate conformal symmetries of the Aharony-Bergman-Jafferis-Maldacena (ABJM) theory for multiple M2 branes and the Lorentzian Bagger-Lambert-Gustavsson (L-BLG) theory which can be obtained by taking a scaling limit k (>>N) -> \infty of the ABJM theory. The conformal symmetry is maintained in the L-BLG by considering general space-time varying solutions to the constraint equations. The dual geometry is reduced to d=10 AdS4 x CP3 in the scaling limit and has the same conformal symmetry. The curvature radius R satisfies l_{11p} << l_{10p} << R << l_s (l_{dp} and l_s are the d-dimensional …
The radial quantization of N=8 theory in three dimensions is considered i.e. we study the N=8 BLG theory on RxS^2. We present the explicit from of the Lagrangian and the corresponding supersymmetry transformations and supersymmetry algebra. We study spectrum of this theory and some of its BPS configurations.
The purpose of this thesis is to explore the properties of multiple coincident M2- and M5-branes. We begin with a review of the BLG and ABJM models of multiple M2-branes and our focus will be on their formulation in terms of 3-algebras. We then examine the coupling of multiple M2-branes to the background 3-form and 6-form gauge fields of eleven-dimensional supergravity. In particular we show in detail how a natural generalisation of the Myers flux-terms, along with the resulting curvature of the background metric, leads to mass terms in the effective field theory. Working to lowest nontrivial …
We present the light-cone gauge fixed Lagrangian for the M5-brane; it has a residual `exotic' gauge invariance with the group of 5-volume preserving diffeomorphisms, SDiff(5), as gauge group. For an M5-brane of topology R2 x M3, for closed 3-manifold M3, we find an infinite tension limit that yields an SO(8)-invariant (1+2)-dimensional field theory with `exotic' SDiff(3) gauge invariance. We show that this field theory is the Carrollian limit of the Nambu bracket realization of the `BLG' model for multiple M2-branes.
One of the most important questions in the Bagger-Lambert-Gustavsson (BLG) theory of multiple M2-branes is the choice of the Lie 3-algebra. The Lie 3-algebra should be chosen such that the corresponding BLG model is unitary and admits fuzzy 3-sphere as a solution. In this paper we propose another new condition: the Lie 3-algebras of use must be connected to the semisimple Lie algebras describing the gauge symmetry of D-branes via a certain reduction condition. We show that this reduction condition leads to a natural generalization of the Cartan-Weyl 3-algebras introduced in arXiv:1004.1397. Si…
We present an ansatz for black 2-branes in five dimensional N=2 supergravity theory that carry magnetic charge with respect to general hypermultiplet scalars. We find explicit solutions in certain special cases and examine the constraints on the general case. These branes may be thought of as arising from M-branes by wrapping eleven dimensional supergravity over special Lagrangian calibrated cycles of a Calabi-Yau 3-fold.
We investigate the Bagger-Lambert-Gustavsson model associated with the Nambu-Poisson algebra as a theory describing a single M5-brane. We argue that the model is a gauge theory associated with the volume-preserving diffeomorphism in the three-dimenisonal internal space. We derive gauge transformations, actions, supersymmetry transformations, and equations of motions in terms of six-dimensional fields. The equations of motions are written in gauge-covariant form, and the equations for tensor fields have manifest self-dual structure. We demonstrate that the double dimensional reduction of the mo…
We give a brief review of Bagger-Lambert-Gustavsson (BLG) model, with emphasis on its version invariant under the volume preserving diffeomorphisms (SDiff3) symmetry. We describe the on-shell superfield formulation of this SDiff3 BLG model in standard N=8, d=3 superspace, as well as its superfield action in the pure spinor N=8 superspace. We also briefly address the Aharony-Bergman-Jafferis-Maldacena (ABJM/ABJ) model invariant under $SU(M)_{k}\times SU(N')_{-k}$ gauge symmetry, and discuss the possible form of their N=6 and, for the case of Chern-Simons level k=1,2, N=8 superfield equations.
We present a nonabelian Lagrangian that appears to have $(2,0)$ superconformal symmetry and that can be coupled to a supergravity background. But for our construction to work, we have to break this superconformal symmetry by imposing as a constraint on top of the Lagrangian that the fields have vanishing Lie derivatives along a Killing direction.
The BLG model for multiple M2-branes motivates an M5-brane theory with a novel gauge symmetry defined by the Nambu-Poisson structure. This Nambu-Poisson gauge symmetry for an M5-brane in large C-field background can be matched, on double dimension reduction, with the Poisson limit of the noncommutative gauge symmetry for a D4-brane in B-field background. Naively, one expects that there should exist a certain deformation of the Nambu-Poisson structure to match with the full noncommutative gauge symmetry including higher order terms. However, We prove the no-go theorem that there is no way to de…
This article seeks to relate a recent proposal for the association of a covariant Field Theory with a string or brane Lagrangian to the Hamilton-Jacobi formalism for strings and branes. It turns out that since in this special case, the Hamiltonian depends only upon the momenta of the Jacobi fields and not the fields themselves, it is the same as a Lagrangian, subject to a constancy constraint. We find that the associated Lagrangians for strings or branes have a covariant description in terms of the square root of the same Lagrangian. If the Hamilton-Jacobi function is zero, rather than a const…
We propose expressions for refined open topological string partition function on certain non-compact Calabi Yau 3-folds with topological branes wrapped on the special lagrangian submanifolds. The corresponding web diagrams are partially compact and a lagrangian brane is inserted on one of the external legs. Partial compactification introduces a mass deformation in the corresponding gauge theory. We propose conjectures that equate these open topological string partition functions with the generating function of equivaraint indices on certain quiver moduli spaces. To obtain these conjectures we …
We characterize the geometry produced by M5-branes wrapping a Special Lagrangian 3-cycle in a Calabi-Yau 3-fold. The presence of the brane replaces the the Calabi-Yau by a real manifold with an almost complex structure. We show that, in this classification, a distinguished (1,1) form as well as a globally defined (3,0) form play an important role. The requirements of supersymmetry preservation impose constraints on these structures which can be used to classify the background.
In this work, we study Lagrangian formulations for self-dual gauge theories, also known as chiral $n$-form gauge theories, for $n = 2p$ in $D = 4p+2$ dimensional spacetime. Motivated by a recent formulation of M5-branes derived from the BLG model, we generalize the earlier Lagrangian formulation based on a decomposition of spacetime into $(D-1)$ dimensions plus a special dimension, to construct Lagrangian formulations based on a generic decomposition of spacetime into $D'$ and $D" = D - D'$ dimensions. Although the Lorentz symmetry is not manifest, we prove that the action is invariant under m…
This paper presents a general covariant lagrangian framework for the dynamics of a system of closed n-branes and dual (D-n-4)-branes in D dimensions, interacting with a dynamical (n+1)-form gauge potential. The framework proves sufficiently general to include also a coupling of the branes to (the bosonic sector of) a dynamical supergravity theory. We provide a manifestly Lorentz-invariant and S-duality symmetric Lagrangian, involving the (n+1)-form gauge potential and its dual (D-n-3)-form gauge potential in a symmetric way. The corresponding action depends on generalized Dirac-strings. The re…
In this article we use 5-brane junctions to study the 5D T_N SCFTs corresponding to the 5D N=1 uplift of the 4D N=2 strongly coupled gauge theories, which are obtained by compactifying N M5 branes on a sphere with three full punctures. Even though these theories have no Lagrangian description, by using the 5-brane junctions proposed by Benini, Benvenuti and Tachikawa, we are able to derive their Seiberg-Witten curves and Nekrasov partition functions. We cross-check our results with the 5D superconformal index proposed by Kim, Kim and Lee. Through the AGTW correspondence, we discuss the relatio…
Recently, Maldacena, Moore and Seiberg introduced non-maximally symmetric boundary states on group manifold using T-duality. In the work presented here we suggest simple description of these branes in terms of group elements. We show that T-dualization actually reduces to multiplication of conjugacy classes by the corresponding U(1) subgroups. Using this description we find the two-form trivializing the WZW three-form on the branes. SU(2) and $SL(2,R)$ examples are considered in details.
We extend a recent analysis of gravitational perturbations on Dirac-Nambu-Goto strings, membranes and higher dimensional branes. In an arbitrary gauge, it is shown that the relevant first order equations governing the displacement vector of the worldsheet and metric perturbation are obtainable from a variational principle whose Lagrangian is constructed as a second order perturbation of the standard Dirac-Goto-Nambu action density. A symplectic current functional is obtained as a by-product that is potentially useful for the derivation of conservation laws in particular circumstances.
We integrate the nonpropagating B_μ gauge field for the non-linear BF Lagrangian describing N M2-branes which includes terms with even number of the totally antisymmetric tensor M^{IJK} in arXiv:0808.2473 and for the two-types of non-linear BF Lagrangians which include terms with odd number of M^{IJK} as well in arXiv:0809:0985. For the former Lagrangian we derive directly the DBI-type Lagrangian expressed by the SU(N) dynamical A_μ gauge field with a spacetime dependent coupling constant, while for the low-energy expansions of the latter Lagrangians the B_μ integration is iteratively performe…
Constructing M2-brane and its boundary conditions from D2-brane and the related boundary conditions and vice versa has been possible in our recent work by using 3-Lie bialgebra for BLG model with N = 8 supersymmetry. This could be generalized for BL model with N = 6 by the concept of the 3-Leibniz bialgebra. The 3-Lie bialgebra is an especial case of 3-Leibniz bialgebra, then more comprehensive information will be obtained in this work. Consequently, according to the correspondence of these 3-Leibniz bialgebras with Lie bialgebras, we reduce to D2-brane such that with some restrictions on the …
A generic Lagrangian, in arbitrary spacetime dimension, describing the interaction of a graviton, a dilaton and two antisymmetric tensors is considered. An isotropic p-brane solution consisting of three blocks and depending on four parameters in the Lagrangian and two arbitrary harmonic functions is obtained. For specific values of parameters in the Lagrangian the solution may be identified with previously known superstring solutions.
We present a manifestly SO(8) invariant non-linear Lagrangian for describing the non-abelian dynamics of the bosonic degrees of freedom of N coinciding M2 branes in flat spacetime. The theory exhibits a gauge symmetry structure of the BF type (semidirect product of SU(N) and translations) and at low energies it reduces exactly to the bosonic part of the Lorentzian Bagger-Lambert Lagrangian for group SU(N). There are eight scalar fields satisfying a free-scalar equation. When one of them takes a large expectation value, the non-linear Lagrangian gets simplified and the theory can be connected t…
In this paper we will study tachyon effective action for Dp-brane in bosonic string theory in the linear dilaton background. We obtain the tachyon effective Lagrangian from boundary state coeficient of Dp-brane in the linear dilaton background and compare it with tachyon effective Lagrangians that were proposed in previous papers.
We propose a natural generalisation of the BLG multiple M2-brane action to membranes in curved plane wave backgrounds, and verify in two different ways that the action correctly captures the non-trivial space-time geometry. We show that the M2 to D2 reduction of the theory along a non-trivial direction in field space is equivalent to the D2-brane world-volume Yang-Mills theory with a non-trivial (null-time dependent) dilaton in the corresponding IIA background geometry. As another consistency check of this proposal we show that the properties of metric 3-algebras ensure the equivalence of the …
We derive the Lagrangian and the transformation laws of N=4 gauged supergravity coupled to matter multiplets whose sigma-model of the scalars is SU(1,1)/U(1)x SO(6,6+n)/SO(6)xSO(6+n) and which corresponds to the effective Lagrangian of the Type IIB string compactified on the T^6/Z_2 orientifold with fluxes turned on and in presence of n D3-branes. The gauge group is T^12 x G where G is the gauge group on the brane and T^12 is the gauge group on the bulk corresponding to the gauged translations of the R-R scalars coming from the R-R four--form. The N=4 bulk sector of this theory can be obtained…
We obtain the Hamiltonian form of the worldvolume action for the M5-brane in a general D=11 supergravity background. We use this result to obtain a new version of the covariant M5-brane Lagrangian in which the tension appears as a dynamical variable, although this Lagrangian has some unsatisfactory features which we trace to peculiarities of the null limit. We also show that the M5-brane action is invariant under all (super)isometries of the background.
In the BRST-BFV scheme for noncommutative D-branes with constant NS $B$-field, introducing ghost degrees of freedom we construct the gauge fixed Hamiltonian and corresponding effective Lagrangian invariant under nilpotent BRST charge. It is also shown that the presence of auxiliary variables introduced via the improved Dirac formalism plays a crucial role in the construction of the BRST invariant Lagrangian.
We complete the investigation of N=(2,2) supersymmetric nonlinear sigma-models in the presence of a boundary. We study the full bihermitian geometry parameterized by chiral, twisted chiral and semi-chiral superfields and identify the D-brane configurations preserving an N=2 worldsheet supersymmetry. Combining twisted with semi-chiral superfields leads to a clearly defined notion of lagrangian and coisotropic branes generalizing lagrangian and coisotropic A-branes on Kahler manifolds to manifolds which are not necessarily Kahler (but still bihermitian). Adding chiral fields complicates the pict…
We construct a class of supersymmetric boundary interactions in N=2 field theories on the half-space, which depend on parameters that are not at all renormalized or not renormalized in perturbation theory beyond one-loop. This can be used to study D-branes wrapped on a certain class of Lagrangian submanifolds as well as holomorphic cycles. The construction of holomorphic D-branes is in close relationship with the background independent open string field theory approach to brane/anti-brane systems. As an application, mirror pairs of Lagrangian and holomorphic D-branes are identified. The mirror…
It was argued that there are two different descriptions of the effective Lagrangian of gauge fields on D-branes by non-commutative gauge theory and by ordinary gauge theory in the presence of a constant B field background. In the case of bosonic string theory, however, it was found in the previous works that the two descriptions are incompatible under the field redefinition which relates the non-commutative gauge field to the ordinary one found by Seiberg and Witten. In this paper we resolve this puzzle to observe the necessity of gauge-invariant but B-dependent correction terms involving metr…
We study M-theory fivebranes wrapped on Special Lagrangian submanifolds ($§_n$) in Calabi-Yau three- and fourfolds. When the M5 wraps a four-cycle, the resulting theory is a two-dimensional domain wall embedded in three-dimensional bulk with four supercharges. The theory on the wall is specified in terms of the geometry of the CY manifold and the cycle $§_4$. It is chiral and anomalous, however the presence of a three-dimensional gravitational Chern-Simons terms with a coefficient that jumps when crossing the wall allows to cancel the anomaly by inflow. Kahler manifolds of special type, where …
Due to the difficulties of finding superconformal Lagrangian theories for multiple M2-branes, we will in this paper instead focus on the field equations. By relaxing the requirement of a Lagrangian formulation we can explore the possibility of having structure constants $f^{ABC}{}_D$ satisfying the fundamental identity but which are not totally antisymmetric. We exemplify this discussion by making use of an explicit choice of a non-antisymmetric $f^{ABC}{}_D$ constructed from the Lie algebra structure constants $f^{ab}{}_c$ of an arbitrary gauge group. Although this choice of $f^{ABC}{}_D$ doe…
We explain microscopically why split attractor flows, known to underlie certain stationary BPS solutions of four dimensional N=2 supergravity, are the relevant data to describe wrapped D-branes in Calabi-Yau compactifications of type II string theory. We work entirely in the context of the classical geometry of A-branes, i.e. special Lagrangian submanifolds, avoiding both the use of homological algebra and explicit constructions of special Lagrangians. Our results provide a way to disassemble and assemble arbitrary special Lagrangians to and from more simple building blocks, giving a concrete …
We introduce a systematic approach to constructing $\mathcal{N}=1$ Lagrangians for a class of interacting $\mathcal{N}=2$ SCFTs. We analyse in detail the simplest case of the construction, arising from placing branes at an orientifolded $\mathbb{C}^2/\mathbb{Z}_2$ singularity. In this way we obtain Lagrangian descriptions for all the $R_{2,k}$ theories. The rank one theories in this class are the $E_6$ Minahan-Nemeschansky theory and the $C_2\times U(1)$ Argyres-Wittig theory. The Lagrangians that arise from our brane construction manifestly exhibit either the entire expected flavour symmetry …
The purpose of this article is to initiate a study of a class of Lorentz invariant, yet tractable, Lagrangian Field Theories which may be viewed as an extension of the Klein-Gordon Lagrangian to many scalar fields in a novel manner. These Lagrangians are quadratic in the Jacobians of the participating fields with respect to the base space co-ordinates. In the case of two fields, real valued solutions of the equations of motion are found and a phenomenon reminiscent of instanton behaviour is uncovered; an ansatz for a subsidiary equation which implies a solution of the full equations yields rea…
We construct new type of non-relativistic D-branes which are defined with the help of T-duality along null direction. We find Lagrangian and Hamiltonian formulation of these D-branes and study their properties under T-duality transformations.
Phenomenological Lagrangians for physical systems with spontaneously broken symmetries are reformulated in terms of gauge field theory. Description of the Dirac $p$-branes in terms of the Yang-Mills-Cartan gauge multiplets interacting with gravity, is proved to be equivalent to their description as a closed dynamical system with the symmetry $ISO(1,D-1)$ spontaneously broken to $ISO(1,p)\times SO(D-p-1)$.
The idea of a companion Lagrangian associated with $p$-Branes is extended to include the presence of U(1) fields. The Brane Lagrangians are constructed with $F_{ij}$ represented in terms of Lagrange Brackets, which make manifest the reparametrisation invariance of the theory; these are replaced by Poisson Brackets in the companion Lagrangian, which is now covariant under field redefinition. The ensuing Lagrangians possess a similar formal structure to those in the absence of an anti-symmetric field tensor.
We study normal functions capturing D-brane superpotentials on several one- and two-parameter Calabi-Yau hypersurfaces and complete intersections in weighted projective space. We calculate in the B-model and interpret the results using mirror symmetry in the large volume regime, albeit without identifying the precise A-model geometry in all cases. We identify new classes of extensions of Picard-Fuchs equations, as well as a novel type of topology changing phase transition involving quantum D-branes. A 4-d domain wall which is obtained in one region of closed string moduli space from wrapping a…
Path and path deviation equations for neutral, charged, spinning and spinning charged test particles, using a modified Bazanski Lagrangian, are derived. We extend this approach to strings and branes. We show how the Bazanski Lagrangian for charged point particles and charged branes arises `a la Kaluza-Klein from the Bazanski Lagrangian in 5-dimensions.
We describe the higher-form and non-invertible symmetries of 4d $\mathcal N= 3$ S-folds using the brane dynamics of their holographic duals. In cases with enhancement to $\mathcal N=4$ supersymmetry, our analysis reproduces the known field theory results of Aharony, Seiberg and Tachikawa, and is compatible with the effective action recently given by Bergman and Hirano. Likewise, for two specific $\mathcal N=3$ theories for which Zafrir has conjectured $\mathcal N=1$ Lagrangians our results agree with those implied by the Lagrangian description. In all other cases, our results imply novel predi…
We study complex structure deformations of special Lagrangian cycles associated to fractional D-branes at $\mathbb{Z}_2$ singularities in Type II/$Ω\mathcal{R}$ orientifold models. By means of solving hypersurface constraints, we show how to compute the volumes of the most simple D-brane configurations. These volumes are given as a function of the deformation parameters depending on the D-brane position relative to the smoothed out singularity. We observe which cycles keep the special Lagrangian property in various deformation scenarios and what orientifold involutions are allowed. As expected…
We define the notion of A-model Lagrangian D-branes as introducing defects in the Calabi-Yau crystal. The crystal melting in the presence of these defects reproduces all genus string amplitudes as well as leads to additional non-perturbative terms.
We consider the compactification of the dual form of $N=1$ $D=10$ supergravity on a six-dimensional Calabi-Yau manifold. An $N=1$ off-shell supergravity effective Lagrangian in four dimensions can be constructed in a dual version of the gravitational sector (new-minimal supergravity form). Superspace duality has a simple interpretation in terms of Poincaré duality of two-form cohomology. The resulting $4D$ Lagrangian may describe the low-energy point-field limit of a five-brane theory, dual to string theory, provided Calabi-Yau spaces are consistent vacua of such dual theory.
