We construct supersymmetric conformal sigma models in three dimensions. Nonlinear sigma models in three dimensions are nonrenormalizable in perturbation theory. We use the Wilsonian renormalization group equation method, which is one of the nonperturbative methods, to find the fixed points. Existence of fixed points is extremely important in this approach to show the renormalizability. Conformal sigma models are defined as the fixed point theories of the Wilsonian renormalization group equation. The Wilsonian renormalization group equation with anomalous dimension coincides with the modified Ricci flow equation. The conformal sigma models are characterized by one parameter which corresponds to the anomalous dimension of the scalar fields. Any Einstein-Kähler manifold corresponds to a conformal field theory when the anomalous dimension is $γ=-1/2$. Furthermore, we investigate the properties of target spaces in detail for two dimensional case, and find the target space of the fixed point theory becomes compact or noncompact depending on the value of the anomalous dimension. This is a proceeding to the conference based on the talk given by E.I.
We review the construction of 4D, N =2 globally supersymmetric off-shell nonlinear sigma models whose target spaces are the cotangent bundles of Kähler manifolds.
The Supersymmetric Dual Sigma Model (SDSM) is a local field theory introduced to be nonlocally equivalent to the Supersymmetric Chiral nonlinear sigma-Model (SCM), this dual equivalence being proven by explicit canonical transformation in tangent space. This model is here reconstructed in superspace and identified as a chiral-entwined supersymmetrization of the Dual Sigma Model (DSM). This analysis sheds light on the Boson-Fermion Symphysis of the dual transition, and on the new geometry of the DSM.
We study D-branes of N=2 supersymmetric sigma models. Supersymmetric nonlinear sigma models with 2-dimensional target space have D0,D1,D2-branes, which are realized as A-,B-type supersymmetric boundary conditions on the worldsheet. When we embed the models in the string theory, the Kahler potential is restricted and leads to a 2-dim black hole metric with a dilaton background. The D-branes in this model are susy cycles and consistent with the analysis of conjugacy classes. The generalized metrics with U(n) isometry is proposed and dynamics on them are realized by linear sigma models. We invest…
We construct a closed form of the action of the supersymmetric $CP^N$ sigma model on noncommutative superspace in four dimensions. We show that this model has $\mathcal{N}={1/2}$ supersymmetry and that the transformation law is not modified. The supersymmetric $CP^N$ sigma model on noncommutative superspace in two dimensions is obtained by dimensional reducing the model in four dimensions.
We discuss the N=2 supersymmetric extension of the gauged O(3) sigma model in (2+1) dimensions with an abelian Chern-Simons term. It is shown that the self-dual potential and the Bogomolny relations naturally appear as consequences of extended supersymmetry.
We construct supersymmetric deformations of general, locally supersymmetric, nonlinear sigma models in three spacetime dimensions, by extending the pure supergravity theory with a Chern-Simons term and gauging a subgroup of the sigma model isometries, possibly augmented with R-symmetry transformations. This class of models is shown to include theories with standard Yang-Mills Lagrangians, with optional moment interactions and topological mass terms. The results constitute a general classification of three-dimensional gauged supergravities.
We find a new class of (2,0)-supersymmetric two-dimensional sigma models with torsion and target spaces almost complex manifolds extending similar results for models with (2,2) supersymmetry. These models are invariant under a new symmetry which is generated by a Noether charge of Lorentz weight one and it is associated to the Nijenhuis tensor of the almost complex structure of the sigma model target manifold. We compute the Poisson bracket algebra of charges of the above (2,0)-and (2,2)-supersymmetric sigma models and show that it closes but it is not isomorphic to the standard (2,0) and (2,2…
This is a review of the relation between supersymmetric non-linear sigma models and target space geometry. In particular, we report on the derivation of generalized Kähler geometry from sigma models with additional spinorial superfields. Some of the results reviewed are: Generalized complex geometry from sigma models in the Lagrangian formulation; Coordinatization of generalized Kähler geometry in terms of chiral, twisted chiral and semi-chiral superfields; Generalized Kähler geometry from sigma models in the Hamiltonian formulation.
In this letter we study supersymmetric sigma models on toric varieties. These manifolds are generalizations of CP^n manifolds. We examine here sigma models, viewed as gauged linear sigma models, on one of the simplest such manifold, the blow-up of P^2_(2,1,1), and determine their properties using the techniques of topological- antitopological fusion. We find that the model contains solitons which become massless at the singular point of the theory where a gauge symmetry remains unbroken.
We clarify the discussion of N=2 supersymmetric boundary conditions for the classical d=2, N=(2,2) Non-Linear Sigma Model on an infinite strip. Our conclusions about the supersymmetric cycles match the results found in the literature. However, we find a constraint on the boundary action that is not satisfied by many boundary actions used in the literature.
We find the solitons of massive (1,1)-supersymmetric sigma models with target space the groups $SO(2)$ and $SU(2)$ for a class of scalar potentials and compute their charge, mass and moduli space metric. We also investigate the massive sigma models with target space any semisimple Lie group and show that some of their solitons can be obtained from embedding the $SO(2)$ and $SU(2)$ solitons.
We derive the Wilsonian renormalization group equation in two dimensional ${\cal N}=2$ supersymmetric nonlinear sigma models. This equation shows that the sigma models on compact Einstein Kähler manifolds are aymptotically free. This result is gerenal and does not depend on the specific forms of the Kähler potentials. We also examine the renormalization group flow in a new model which connects two manifolds with different global symmetries.
We perturbed N=(2,2) supersymmetric WZW and sigma models on Lie groups by adding a term to their actions. Then by using non-coordinate basis we obtain conditions, from the algebraic point of view, under which the N=(2,2) supersymmetry is preserved. By applying this method, we have obtained conditions on the existence of N=(2,2) supersymmetry on the Drinfeld action (master action for the Poisson-Lie T-dual sigma models).
We study the instanton equation of the supersymmetric CP^{N-1} sigma model on non(anti)commutative superspace in two dimensions. We show that the undeformed instanton equation is consistent with the deformed equations of motion. Then we conclude that the instanton equation is not deformed by superspace non(anti)commutativity.
In this paper we investigate the non-linear sigma model arising in the reduction of D = 5 supergravity to D = 3, and present the application of this sigma model to supersymmetric black ring solutions in five-dimensional minimal supergravity. With the ansatz of stationary solutions with $R \times U(1)\times U(1)$ isometry, we obtain a two-dimensional Lagrangian corresponding to geodesic motion of a string-like object on the coset $G_{2(+2)}/SO(4)$, and study the algebra of conserved charges and supersymmetry constraints of supersymmetric black rings. We also obtain the semi-classical wave funct…
Two-dimensional sigma-models describing superstrings propagating on manifolds of special holonomy are characterized by symmetries related to covariantly constant forms that these manifolds hold, which are generally non-linear and close in a field dependent sense. The thesis explores various aspects of the special holonomy symmetries.
After an elementary presentation of the relation between supersymmetric nonlinear sigma models and geometry, I focus on 2D and the target space geometry allowed when there is an extra supersymmetry. This leads to a brief introduction to generalized complex geometry, a notion introduced recently by Hitchin which interpolates between complex and symplectic manifolds. Finally I present worldsheet realizations of this geometry,
A method for quantizing the bidimensional N=2 supersymmetric non-linear sigma model is developed. This method is both covariant under coordinate transformations (concerning the order relevant for calculations) and explicitly N=2 supersymmetric. The OPE of the supercurrent is computed accordingly, including also the dilaton. By imposing the N=2 superconformal algebra the equations for the metric and dilaton are obtained. In particular, they imply that the dilaton is a constant.
Following a suggestion made by Tseytlin, we investigate the case when one replaces the transverse part of the bosonic action by an $n=2$ supersymmetric sigma-model with a symmetric homogeneous Kählerian target space. As conjectured by Tseytlin, the metric is shown to be exactly known since the beta function is known to reduce to its one-loop value.
We study the supersymmetric extension of the gauged $ O(3) $ sigma model in $ 2+1 $ dimensions and find the supersymmetry algebra. We also discuss soliton solutions in case the Maxwell term is replaced by the Born-Infeld term. We show that by appropriate choice of the potential, the self-dual equations in the Born-Infeld case coincide with those of the Maxwell's case.
We discuss the quantization of the ADHM sigma model. We show that the only quantum contributions to the effective theory come from the chiral anomalies and compute the first and second order terms. Finally the limit of vanishing instanton size is discussed.
The general scalar potential of D-dimensional massive sigma-models with eight supersymmetries is found for $D=3,4$. These sigma models typically admit 1/2 supersymmetric domain wall solutions and we find, for a particular hyper-Kähler target, exact 1/4 supersymmetric static solutions representing a non-trivial intersection of two domain walls. We also show that the intersecting domain walls can carry Noether charge while preserving 1/4 supersymmetry. We briefly discuss an application to the D1-D5 brane system.
Renormalization group methods are used to determine the evolution of the low energy Wilson effective action for supersymmetric nonlinear sigma models in four dimensions. For the case of supersymmetric $CP^{(N-1)}$ models, the Kähler potential is determined exactly and is shown to exhibit a nontrivial ultraviolet fixed point in addition to a trivial infrared fixed point. The strong coupling behavior of the theory suggests the possible existence of additional relevant operators or nonperturbative degrees of freedom.
Supersymmetric non-linear sigma-models are described by a field dependent Kaehler metric determining the kinetic terms. In general it is not guaranteed that this metric is always invertible. Our aim is to investigate the symmetry structure of supersymmetric models in four dimensional space-time in which metric singularities occur. For this purpose we study a simple anomaly-free extension of the supersymmetric CP^1 model from a classical point of view. We show that the metric singularities can be regularized by the addition of a soft supersymmetry-breaking mass parameter.
We analyse with the algebraic, regularisation independent, cohomological B.R.S. methods, the renormalisability of torsionless N=2 supersymmetric non-linear sigma models built on Kahler spaces. Surprisingly enough with respect to the common wisdom, we obtain an anomaly candidate, at least in the compact Ricci-flat case. In the compact homogeneous Kahler case, as expected, the anomaly candidate disappears.
We construct the N=2 supersymmetric Grassmannian nonlinear sigma model for the massless case and extend it to massive N=2 model by adding an appropriate superpotential. We then study their BPS equations leading to supersymmetric Q-lumps carrying both topological and Noether charges. These solutions are shown to be always time dependent even sometimes involving multiple frequencies. Thus we illustrate explicitly that the time dependence is consistent with remaining supersymmetries of solitons.
Long ago, Nemeschansky and Sen demonstrated that the Ricci-flat metric on a Calabi-Yau manifold could be corrected, order by order in perturbation theory, to produce a conformally invariant (2,2) nonlinear sigma model. Here we extend this result to (0,2) sigma models for stable holomorphic vector bundles over Calabi-Yaus.
For the most general off-shell N = 2 supersymmetric sigma model in projective superspace, we elaborate on its formulation in terms of N = 1 chiral superfields. A universal (model-independent) expression is obtained for the holomorphic symplectic two-form, which determines the second supersymmetry transformation. This two-form is associated with the two complex structures of the hyperkahler target space, which are complimentary to the one used to realize the target space as a Kahler manifold.
Riemannian coordinates for flat metrics corresponding to three--dimensional conformal Poisson--Lie T--dualizable sigma models are found by solving partial differential equations that follow from the transformations of the connection components. They are then used for finding general forms of the dilaton fields satisfying the vanishing beta equations of the sigma models.
We construct the most general N=2 supersymmetric nonlinear sigma-model in four-dimensional anti-de Sitter (AdS) space in terms of N=1 chiral superfields. The target space is shown to be a non-compact hyperkahler manifold restricted to possess a special Killing vector field. A remarkable property of the sigma-model constructed is that the algebra of OSp(2|4) transformations is closed off the mass shell.
Massive D=4 N=2 supersymmetric sigma models typically admit domain wall (Q-kink) solutions and string (Q-lump) solutions, both preserving 1/2 supersymmetry. We exhibit a new static 1/4 supersymmetric `kink-lump' solution in which a string ends on a wall, and show that it has an effective realization as a BIon of the D=4 super DBI-action. It is also shown to have a time-dependent Q-kink-lump generalization which reduces to the Q-lump in a limit corresponding to infinite BI magnetic field. All these 1/4 supersymmetric sigma-model solitons are shown to be realized in M-theory as calibrated, or `Q…
We study the pseudoduality transformation in supersymmetric sigma models. We generalize the classical construction of pseudoduality transformation to supersymmetric case. We perform this both by component expansion method on manifold M and by orthonormal coframe method on manifold SO(M). The component expansion method yields the result that pseudoduality tranformation is not invertible at all points and occurs from all points on one manifold to only one point where riemann normal coordinates are valid on the second manifold. Torsion of the sigma model on M must vanish while it is nonvanishing …
Using (1,0) superfield methods, we determine the general scalar potential consistent with off-shell (p,0) supersymmetry and (1,1) supersymmetry in two-dimensional non-linear sigma models with torsion. We also present an extended superfield formulation of the (p,0) models and show how the (1,1) models can be obtained from the (1,1)-superspace formulation of the gauged, but massless, (1,1) sigma model.
Power-counting arguments based on extended superfields have been used to argue that two-dimensional supersymmetric sigma models with (4,0) supersymmetry are finite. This result is confirmed up to three loop order in pertubation theory by an explicit calculation using (1,0) superfields. In particular, it is shown that the finite counterterms which must be introduced into the theory in order to maintain (4,0) supersymmetry are precisely the terms that are required to establish ultra-violet finiteness.
We study the conditions under which N=(1,1) generalized sigma models support an extension to N=(2,2). The enhanced supersymmetry is related to the target space complex geometry. Concentrating on a simple situation, related to Poisson sigma models, we develop a language that may help us analyze more complicated models in the future. In particular, we uncover a geometrical framework which contains generalized complex geometry as a special case.
We discuss the structure of the non-anticommutative N=2 non-linear sigma-model in two dimensions, constructing differential operators which implement the deformed supersymmetry generators and using them to reproduce the classical action. We then compute the one-loop quantum corrections and express them in a more compact form using the differential operators.
We analyse the structure of N=1 and N=2 supersymmetric non-linear sigma-models built up with a pair of real superfields defined in the superspace of Atiyah-Ward space-time. The geometry arising has new features such as the existence of a locally product structure (N=1 case) and a set of automorphisms of the tangent space that is isomorphic to the split-quaternionic algebra (N=2 case).
In the IR limit the Matrix string theory is expected to be described by the $S^N\R^{8}$ supersymmetric orbifold sigma model. Recently Dijkgraaf, Verlinde and Verlinde proposed a vertex that may describe the type IIA string interaction. In this paper using this interaction vertex we derive the four graviton scattering amplitude from the orbifold model in the large $N$ limit.
Field theoretic and geometric ideas are used to construct a chiral supersymmetric field theory whose ground state is a specified irreducible representation of a centrally extended loop group. The character index of the associated supercharge (an appropriate Dirac operator on $LG/T$) is the Weyl-Kǎc character formula which we compute explicitly by the steepest descent approximation.
Auxiliary field methods in D=2 (or 3), N=2 supersymmetric (SUSY) nonlinear sigma models (NLSMs) are studied. For these models auxiliary fields as Lagrange multipliers belong to a vector or a chiral superfield, which gives a Kahler quotient of complexified gauge group or a holomorphic constraint on it, respectively. Using these, NLSMs on all Hermitian symmetric spaces were formulated previously. In this paper, we formulate new SUSY NLSMs on some rank-two Kahler coset spaces as SUSY gauge theories with two Fayet-Iliopoulos parameters.
Recently, we have studied the general Virasoro construction at one loop in the background of the general non-linear sigma model. Here, we find the action formulation of these new conformal field theories when the background sigma model is itself conformal. In this case, the new conformal field theories are described by a large class of new spin-two gauged sigma models. As examples of the new actions, we discuss the spin-two gauged WZW actions, which describe the conformal field theories of the generic affine-Virasoro construction, and the spin-two gauged g/h coset constructions. We are able to…
We rewrite the N=(2,2) non-linear sigma model using auxiliary spinorial superfields defining the model on ${\cal T}\oplus^ *{\cal T}$, where ${\cal T}$ is the tangent bundle of the target space. This is motivated by possible connections to Hitchin's generalized complex structures. We find the general form of the second supersymmetry compatible with that of the original model.
A three-dimensional simple N=1 supergravity theory with a supersymmetric sigma-model on the coset E_{8(+8)} / SO(16) is constructed. Both bosons and fermions in the matter multiplets are in the spinorial 128-representation of SO(16) with the same chirality. Due to their common chirality, this model can not be obtained from the maximal N=16 supergravity. By introducing an independent vector multiplet, we can also gauge an arbitrary subgroup of SO(16) together with a Chern-Simons term. Similar N=1 supersymmetric sigma-models coupled to supergravity are also constructed for the cosets F_{4(-20)} …
Two dimensional N=2 supersymmetric nonlinear sigma models on hermitian symmetric spaces are formulated in terms of the auxiliary superfields. If we eliminate auxiliary vector and chiral superfields, they give D- and F-term constraints to define the target manifolds. The integration over auxiliary vector superfields, which can be performed exactly, is equivalent to the elimination of the auxiliary fields by the use of the classical equations of motion.
We construct novel conformal sigma models in three dimensions. Nonlinear sigma models in three dimensions are nonrenormalizable in perturbation theory. We use Wilsonian renormalization group equation method to find the fixed points. Existence of fixed points is extremely important in this approach to show the renormalizability. Conformal sigma models are defined as the fixed point theories of the Wilsonian renormalization group equation. The Wilsonian renormalization group equation with anomalous dimension coincides with the modified Ricci flow equation. The conformal sigma models are characte…
We study dualities in off-shell 4D N = 2 supersymmetric sigma-models, using the projective superspace approach. These include (i) duality between the real O(2n) and polar multiplets; and (ii) polar-polar duality. We demonstrate that the dual of any superconformal sigma-model is superconformal. Since N = 2 superconformal sigma-models (for which target spaces are hyperkahler cones) formulated in terms of polar multiplets are naturally associated with Kahler cones (which are target spaces for N = 1 superconformal sigma-models), polar-polar duality generates a transformation between different Kahl…
We discuss the target space pseudoduality in supersymmetric sigma models on symmetric spaces using two different methods, orthonormal coframe and component expansion. These two methods yield similar results to the classical cases with the exception that commuting bracket relations in classical case turns out to be anticommuting ones because of the appearance of grassmann numbers. In component expansion method it is understood that constraint relations in case of non-mixing pseudoduality are the remnants of mixing pseudoduality. Once mixing terms are included in the pseudoduality relations the …
We calculate the beta-functions of the general massive (p,q) supersymmetric sigma model to two loop order using (1,0) superfields. The conditions for finiteness are discussed in relation to (p,q) supersymmetry. We also calculate the effective potential using component fields to one loop order and consider the possibility of perturbative breaking of supersymmetry. The effect of one loop finite local counter terms and the ultra-violet behaviour of the off-shell (p,q) models to all orders in perturbation theory are also addressed.
There exists a certain argument that in even dimensions, scale invariant quantum field theories are conformal invariant. We may try to extend the argument in $2n + ε$ dimensions, but the naive extension has a small loophole, which indeed shows an obstruction in non-linear sigma models in $2+ε$ dimensions. Even though it could have failed due to the loophole, we show that scale invariance does imply conformal invariance of non-linear sigma models in $2+ε$ dimension from the seminal work by Perelman on the Ricci flow.
We present a superfield Lax formalism of superspace sigma model based on the target space ${\cal G}/{\cal H}$ and show that a one-parameter family of flat superfield connections exists if the target space ${\cal G}/{\cal H}$ is a symmetric space. The formalism has been related to the existences of an infinite family of local and non-local superfield conserved quantities. Few examples have been given to illustrate the results.
We consider classical and quantum integrable sigma models and their relations with the solutions of renormalization group equations. We say that an integrable sigma model possesses the "nice" duality property if the dual quantum field theory has the weak coupling region. As an example, we consider the deformed $CP(n-1)$ sigma model with additional quantum degrees of freedom. We formulate the dual integrable field theory and use perturbed conformal field theory, perturbation theory, $S$-matrix, Bethe Ansatz and renormalization group methods to show that this field theory has the "nice" duality …
We provide a Lagrangian formulation of \Nf supersymmetric mechanics describing the motion of an isospin carrying particle on conformal to hyper-Kähler spaces in a non-Abelian background gauge field. In two examples we discuss in details, this background field is identified with the field of BPST instantons in the flat and Taub-NUT spaces.
There is a remarkable connection between the number of quantum states of conformal theories and the sequence of dimensions of Lie algebras. In this paper, we explore this connection by computing the asymptotic expansion of the elliptic genus and the microscopic entropy of black holes associated with (supersymmetric) sigma models. The new features of these results are the appearance of correct prefactors in the state density expansion and in the coefficient of the logarithmic correction to the entropy.
We use the large $N$ self consistency method to compute the critical exponents of the fields and coupling of the supersymmetric CP(N) sigma model at leading order in $1/N$ in various dimensions. We verify that the correction to the critical beta-function slope vanishes at O(1/N) which is consistent with supersymmetry. The three dimensional model is investigated explicitly when a Chern Simons term is included supersymmetrically. We determine the modification that this has on the gauge independent quantity β^\prime(g_c) as a function of the Chern Simons coupling, \vartheta. For an N = 2 supersym…
We study the new case of the application of the JKLMR conjecture on the connection between the exact partition functions of $\mathcal{N}=(2,2)$ supersymmetric gauged linear sigma models (GLSM) on $S^2$ and special Kähler geometry on the moduli spaces of Calabi-Yau manifold $Y$. The last ones arise as manifolds of the supersymmetric vacua of the GLSM. We establish this correspondence using the Mirror symmetry in Batyrev's approach. Namely, starting from the two-moduli non-Fermat Calabi-Yau manifold $X$ we construct the dual GLSM with the supersymmetric vacua $Y$, which is the mirror for $X$. Kn…
We show that it is possible to construct a supersymmetric mechanics with four supercharges possessing not conformally flat target space. A general idea of constructing such models is presented. A particular case with Eguchi--Hanson target space is investigated in details: we present the standard and quotient approaches to get the Eguchi--Hanson model, demonstrate their equivalence, give a full set of nonlinear constraints, study their properties and give an explicit expression for the target space metric.
Massive maximally-supersymmetric sigma models are shown to exhibit multiple static kink-domain wall solutions that preserve 1/2 of the supersymmetry. The kink moduli space admits a natural Kahler metric. We examine in some detail the case when the target of the sigma model is given by the co-tangent bundle of CP^n equipped with the Calabi metric, and we show that there exist BPS solutions corresponding to n kinks at arbitrary separation. We also describe how 1/4-BPS charged and intersecting domain walls are described in the low-energy dynamics on the kink moduli space. We comment on the simila…
For the minimal O(N) sigma model, which is defined to be generated by the O(N) scalar auxiliary field alone, all n-point functions, till order 1/N included, can be expressed by elementary functions without logarithms. Consequently, the conformal composite fields of m auxiliary fields possess at the same order such dimensions, which are m times the dimension of the auxiliary field plus the order of differentiation.
