6d Tensor Research Articles

Non-flat elliptic four-folds, three-form cohomology and strongly coupled theories in four dimensions

In this note we consider smooth elliptic Calabi-Yau four-folds whose fiber ceases to be flat over compact Riemann surfaces of genus g in the base. These non-flat fibers contribute Kähler moduli to the four-fold but also add to the three-form cohomology for g > 0. In F-/M-theory these sectors are to be interpreted as compactifications of six/five dimensional mathcal{N} = (1, 0) superconformal matter theories. The three-form cohomology leads to additional chiral singlets proportional to the dimension of five dimensional Coulomb branch of those sectors. We construct explicit examples for E-string theories as well as higher rank cases. For the E-string theories we further investigate conifold transitions that remove those non-flat fibers. First we show how non-flat fibers can be deformed from curves down to isolated points in the base. This removes the chiral singlet of the three-forms and leads to non-perturbative four-point couplings among matter fields which can be understood as remnants of the former E-string. Alternatively the non-flat fibers can be avoided by performing birational base changes analogous to 6D tensor branches. For compact bases these transitions alternate all Hodge numbers but leave the Euler number invariant.

Witten indices of abelian M5 brane on ℝ × S 5 $$ \mathbb{R}\times {S}^5 $$

Witten indices and partition functions are computed for abelian 6d tensor and hypermultiplets on $\mathbb{R}\times S^5$ in Lorentzian signature in an R gauge field background which preserves some supersymmetry. We consider a generic supersymmetric squashing that also admits squashing of the Hopf fiber. Wick rotation to Euclidean M5 brane amounts to Wick rotation of squashing parameters and the hypermultiplet mass parameter. We compute Casimir energies for tensor and hypermultiplets separately for general squashing, and match these with the corresponding gravitational anomaly polynomials. We extract Witten indices on $\mathbb{R}\times \mathbb{CP}^2$ and find that this is zero, again matching with the vanishing anomaly polynomial on an odd dimensional space.

Electric-magnetic duality of Abelian gauge theory on the four-torus, from the fivebrane on T 2 × T 4, via their partition functions

We compute the partition function of four-dimensional abelian gauge theory on a general four-torus T 4 with flat metric using Dirac quantization. In addition to an $$ \mathrm{S}\mathrm{L}\left(4,\;\mathcal{Z}\right) $$ symmetry, it possesses $$ \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right) $$ symmetry that is electromagnetic S-duality. We show explicitly how this $$ \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right) $$ S-duality of the 4d abelian gauge theory has its origin in symmetries of the 6d (2, 0) tensor theory, by computing the partition function of a single fivebrane compactified on T 2 times T 4, which has $$ \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right)\times \mathrm{S}\mathrm{L}\left(4,\;\mathcal{Z}\right) $$ symmetry. If we identify the couplings of the abelian gauge theory $$ \tau =\frac{\theta }{2\pi }+i\frac{4\pi }{e^2} $$ with the complex modulus of the T 2 torus $$ \tau ={\beta}^2+i\frac{R_1}{R_2} $$ , then in the small T 2 limit, the partition function of the fivebrane tensor field can be factorized, and contains the partition function of the 4d gauge theory. In this way the $$ \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right) $$ symmetry of the 6d tensor partition function is identified with the S-duality symmetry of the 4d gauge partition function. Each partition function is the product of zero mode and oscillator contributions, where the $$ \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right) $$ acts suitably. For the 4d gauge theory, which has a Lagrangian, this product redistributes when using path integral quantization.

Six-dimensional (1,0) effective action of F-theory via M-theory on Calabi-Yau threefolds

The 6d effective action of F-theory compactified on an elliptically fibred CalabiYau threefold is determined using an M-theory lift. The non-Abelian gauge theory arises on a stack of seven-branes located on the singularities of the threefold. The gauge theory is analysed in the 5d Coulomb branch by resolving the Calabi-Yau threefold. The reduction includes an arbitrary number of 6d tensor multiplets for which the self- and anti-self duality constraints can be imposed in the 5d M-theory set-up. Higher curvature terms are crucial for anomaly cancellation in six dimensions, and are obtained from corrections to the 11d supergravity action. The chiral spectrum in a 6d supergravity theory is linked to the form of the Green-Schwarz couplings via anomalies. We find that these data are encoded in the 5d effective action via one loop corrections.

Brain Differences Visualized in the Blind using Tensor Manifold Statistics and Diffusion Tensor Imaging.

Diffusion tensor magnetic resonance imaging (DTI) reveals the local orientation and integrity of white matter fiber structure based on imaging multidirectional water diffusion. Group differences in DTI images are often computed from single scalar measures, e.g., the Fractional Anisotropy (FA), discarding much of the information in the 6-parameter symmetric diffusion tensor. Here, we compute multivariate 6D tensor statistics to detect brain morphological changes in 12 blind subjects versus 14 sighted controls. After Log-Euclidean tensor denoising, images were fluidly registered to a common template. Fluidly-convected tensor signals were re-oriented by applying the local rotational and translational component of the deformation. Since symmetric, positive-definite matrices form a non-Euclidean manifold, we applied a Riemannian manifold version of the Hotelling's T2 test to the logarithms of the tensors, using a log-Euclidean metric. Statistics on the full 6D tensor-valued images outperformed univariate analysis of scalar images, such as the FA and the geodesic anisotropy (GA).