Chiral formulation for hyperkaehler sigma-models on cotangent bundles of symmetric spaces

Supersymmetric Yang-Mills, octonionic instantons and triholomorphic curves

In four-dimensional gauge theory there exists a well-known correspondence between instantons and holomorphic curves, and a similar correspondence exists between certain octonionic instantons and triholomorphic curves. We prove that this latter correspondence stems from the dynamics of various dimensional reductions of ten-dimensional supersymmetric Yang-Mills theory. More precisely we show that the dimensional reduction of the (5+1)-dimensional supersymmetric sigma model with hyperkaehler (but otherwise arbitrary) target X to a four-dimensional hyperkaehler manifold M is a topological sigma mo…

Cotangent bundle over Hermitian symmetric space $E_7/E_6 \times U(1)$ from projective superspace

We construct an $\mathcal{N}$ supersymmetric sigma model on the cotangent bundle over the Hermitian symmetric space $E_7/(E_6\times U(1))$ in the projective superspace formalism, which is a manifest $\mathcal{N}=2$ off-shell superfield formulation in four-dimensional spacetime. To obtain this model we elaborate on results developed in arXiv:0811.0218 and present a new closed formula for the cotangent bundle action, which is valid for all Hermitian symmetric spaces. We show that the structure of cotangent bundle action is intimately related to the analytic structure of the Kähler potential with…

On superpotentials for nonlinear sigma-models with eight supercharges

Using projective superspace techniques, we consider 4D N = 2 and 5D N = 1 gauged supersymmetric nonlinear sigma-models for which the hyper-Kahler target space is (an open domain of the zero section of) the cotangent bundle of a real-analytic Kahler manifold. As in the 4D N = 1 case, one may gauge those holomorphic isometries of the base Kahler manifold (more precisely, their lifting to the cotangent bundle) which are generated by globally defined Killing potentials. In the U(1) case, by freezing the background vector (tropical) multiplet to a constant value of its gauge-invariant superfield st…

Junctions of mass-deformed nonlinear sigma models on $SO(2N)/U(N)$ and $Sp(N)/U(N)$ I

We construct on-shell ${\mathcal{N}}=2$ nonlinear sigma models on $SO(2N)/U(N)$ and $Sp(N)/U(N)$ by holomorphically embedding the models in the hyper-Kähler nonlinear sigma model on the cotangent bundle of the Grassmann manifold $T^\ast G_{2N,N}$ in the ${\mathcal{N}}=1$ superspace formalism. We apply the moduli matrix formalism to the mass-deformed nonlinear sigma models on the quadrics to study three-pronged junctions by using a recently proposed diagram method.

N = 2 supersymmetric sigma-models and duality

For two families of four-dimensional off-shell N = 2 supersymmetric nonlinear sigma-models constructed originally in projective superspace, we develop their formulation in terms of N = 1 chiral superfields. Specifically, these theories are: (i) sigma-models on cotangent bundles T*M of arbitrary real analytic Kaehler manifolds M; (ii) general superconformal sigma-models described by weight-one polar supermultiplets. Using superspace techniques, we obtain a universal expression for the holomorphic symplectic two-form ω^{(2,0)} which determines the second supersymmetry transformation and is assoc…

Hyper-Kahler Sigma Models on (Co)tangent Bundles with SO(n) isometry

We construct N=2 supersymmetric nonlinear sigma models whose target spaces are tangent as well as cotangent bundles over the quadric surface Q^{n-2} = SO(n)/[SO(n-2)\times U(1)]. We use the projective superspace framework, which is an off-shell formalism of N=2 supersymmetry.

Wick Quantization of Cotangent Bundles over Riemannian Manifolds

A simple geometric procedure is proposed for constructing Wick symbols on cotangent bundles to Riemannian manifolds. The main ingredient of the construction is a method of endowing the cotangent bundle with a formal Kähler structure. The formality means that the metric is lifted from the Riemannian manifold $Q$ to its phase space $T^\ast Q$ in the form of formal power series in momenta with the coefficients being tensor fields on the base. The corresponding Kähler two-form on the total space of $T^\ast Q$ coincides with the canonical symplectic form, while the canonical projection of the Kähle…

Exact solutions for some sigma models in quantum field theory

We invistigate exact solutions for the two-dimensional quantum field theories called Wess-Zumino-Novikov-Witten (WZNW) models. A WZNW model is a sigma model whose classical fields are applications from a bidimensional space-time (a Riemann surface in the euclidian case) to a Lie group, the target space. We construct (and we compute in genus zero and one) the metric connection, called the Knizhnik-Zamolodchikov-Bernard (KZB) connection, on the bundle of conformal blocks of the WZNW model. The KZB connection may be viewed as a quantization of Hitchin integrable systems whose configuration space …

Polar supermultiplets, Hermitian symmetric spaces and hyperkahler metrics

We address the construction of four-dimensional N=2 supersymmetric nonlinear sigma models on tangent bundles of arbitrary Hermitian symmetric spaces starting from projective superspace. Using a systematic way of solving the (infinite number of) auxiliary field equations along with the requirement of supersymmetry, we are able to derive a closed form for the Lagrangian on the tangent bundle and to dualize it to give the hyperkahler potential on the cotangent bundle. As an application, the case of the exceptional symmetric space E_6/SO(10) \times U(1) is explicitly worked out for the first time.

Hamiltonian Algebroid Symmetries in W-gravity and Poisson sigma-model

Starting from a Lie algebroid ${\cal A}$ over a space V we lift its action to the canonical transformations on the principle affine bundle ${\cal R}$ over the cotangent bundle $T^*V$. Such lifts are classified by the first cohomology $H^1({\cal A})$. The resulting object is the Hamiltonian algebroid ${\cal A}^H$ over ${\cal R}$ with the anchor map from $\G({\cal A}^H)$ to Hamiltonians of canonical transformations. Hamiltonian algebroids generalize the Lie algebras of canonical transformations. We prove that the BRST operator for ${\cal A}^H$ is cubic in the ghost fields as in the Lie algebra c…

Membrane Sigma-Models and Quantization of Non-Geometric Flux Backgrounds

We develop quantization techniques for describing the nonassociative geometry probed by closed strings in flat non-geometric R-flux backgrounds M. Starting from a suitable Courant sigma-model on an open membrane with target space M, regarded as a topological sector of closed string dynamics in R-space, we derive a twisted Poisson sigma-model on the boundary of the membrane whose target space is the cotangent bundle T^*M and whose quasi-Poisson structure coincides with those previously proposed. We argue that from the membrane perspective the path integral over multivalued closed string fields …

On Five-dimensional Superspaces

Recent one-loop calculations of certain supergravity-mediated quantum corrections in supersymmetric brane-world models employ either the component formulation (hep-th/0305184) or the superfield formalism with only half of the bulk supersymmetry manifestly realized (hep-th/0305169 and hep-th/0411216). There are reasons to expect, however, that 5D supergraphs provide a more efficient setup to deal with these and more involved (in particular, higher-loop) calculations. As a first step toward elaborating such supergraph techniques, we develop in this letter a manifestly supersymmetric formulation …

$N=(2,2)$ 超场与几何再探

$N=(2,2)$ superfields and geometry revisited

本文通过结合不同超场的对偶表述，将二维$N=(2,2)$超对称sigma模型与广义Kähler几何的关系重新表述为加倍靶空间上的几何结构，并用Lagrangian子流形的交解释运动方程，进一步推广到四倍化几何。