On AGT-W Conjecture and q-Deformed W-Algebra

We propose an extension of the Alday-Gaiotto-Tachikawa-Wyllard conjecture to 5d SU(N) gauge theories. A Nekrasov partition function then coincides with the scalar product of the corresponding Gaiotto-Whittaker vectors of the q-deformed W_N algebra.Comment: 18 page

Seiberg Duality, 5d SCFTs and Nekrasov Partition Functions

It is known that a 4d N = 1 SCFT lives on D3-branes probing a local del Pezzo Calabi-Yau singularity. The Seiberg (or toric) duality of this SCFT arises from the Picard-Lefshetz transformation of the affine E_N 7-brane background that is associated with the Calabi-Yau threefold. In this paper we study the duality of the affine E_N background itself and a 5-brane probing it. We then find that many different Type IIB 5-brane webs describe the same SCFT in 5d. We check this duality by comparing the Nekrasov partition functions of these 5-brane web configurations.Comment: 66 pages, 37 figure

Notes on Enhancement of Flavor Symmetry and 5d Superconformal Index

The UV fixed point theory of SU(2) gauge theory with N_f = 0,1,...,7 flavors is believed to have the enlarged E_{N_f +1} flavor symmetry. Actually it is not easy to check this conjecture because the UV theory is strongly-coupled, however, computation of certain SUSY protected quantities provides strong evidence for the enhancement of flavor symmetry. We study the superconformal index for SU(2) gauge theory with N_f = 0, 1 flavors in details, and we give a support for the enhancement by studying combinatorial structure of the superconformal indexes of these theories. We also give a nontrivial evidence that the local F_2 geometry leads to the E_1 superconformal field theory.Comment: 32 pages, 4 figure

On AGT Conjecture for Pure Super Yang-Mills and W-algebra

Recently Alday, Gaiotto and Tachikawa have proposed relation between 2- and 4-dimensional conformal field theories. The relation implies that the Nekrasov partition functions of N=2 superconformal gauge theories are equal to conformal blocks associated with the conformal algebra. Likewise, a counterpart in pure super Yang-Mills theory exists in conformal field theory. We propose a simple relation between the Shapovalov matrix of the W_3-algebra and the Nekrasov partition function of N=2 SU(3) Yang-Mills theory.Comment: 22 page

Deep Residual Networks and Weight Initialization

Residual Network (ResNet) is the state-of-the-art architecture that realizes successful training of really deep neural network. It is also known that good weight initialization of neural network avoids problem of vanishing/exploding gradients. In this paper, simplified models of ResNets are analyzed. We argue that goodness of ResNet is correlated with the fact that ResNets are relatively insensitive to choice of initial weights. We also demonstrate how batch normalization improves backpropagation of deep ResNets without tuning initial values of weights.Comment: 10 pages, 4 figure

Generalized Whittaker states for instanton counting with fundamental hypermultiplets

M-theoretic construction of N=2 gauge theories implies that the instanton partition function is expressed as the scalar product of coherent states (Whittaker states) in the Verma module of an appropriate two dimensional conformal field theory. We present the characterizing conditions for such states that give the partition function with fundamental hypermultiplets for SU(3) theory and SU(2) theory with a surface operator. We find the states are no longer the coherent states in the strict sense but we can characterize them in terms of a few annihilation operators of lower levels combined with the zero mode (Cartan part) of the Virasoro algebra L_0 or the sl(2) current algebra J_0^0.Comment: 46 pages; minor change

Flop Invariance of Refined Topological Vertex and Link Homologies

It has been proposed recently that the topological A-model string theory on local toric Calabi-Yau manifolds has a two parameter extension. Amplitudes of the two parameter topological strings can be computed using a diagrammatic method called the refined topological vertex. In this paper we study properties of the refined amplitudes under the flop transition of toric Calabi-Yau three-folds. We also discuss that the slicing invariance and the flop transition imply a simple formula for the homological sl(N) invariants of the Hopf link. The new expression for the invariants gives a simple refinement of the Hopf link invariant of Chern-Simons theory.Comment: 15 pages, 3 figure

M2-branes Theories without 3+1 Dimensional Parents via Un-Higgsing

N=2 quiver Chern-Simons theory has lately attracted attention as the world volume theory of multiple M2 branes on a Calabi-Yau 4-fold. We study the connection between the stringy derivation of M2 brane theories and the forward algorithm which gives the toric Calabi-Yau 4-fold as the moduli space of the quiver theory. Then the existence of the 3+1 dimensional parent, which is the consistent 3+1 dimensional superconformal theory with the same quiver diagram, is crucial for stringy derivation of M2 brane theories. We also investigate the construction of M2 brane theories that do not have 3+1 dimensional parents. The un-Higgsing procedure plays a key role to construct these M2 brane theories. We find some N=2 quiver Chern-Simons theories which correspond to interesting Calabi-Yau singularities.Comment: 60 pages, 52 figure

Holomorphic Blocks for 3d Non-abelian Partition Functions

The most recent studies on the supersymmetric localization reveal many non-trivial features of supersymmetric field theories in diverse dimensions, and 3d gauge theory provides a typical example. It was conjectured that the index and the partition function of a 3d N=2 theory are constructed from a single component: the holomorphic block. We prove this conjecture for non-abelian gauge theories by computing exactly the 3d partition functions and holomorphic blocks.Comment: 24 pages, 4 figure

Asymmetric Cloaking Theory Based on Finsler Geometry ~ How to design Harry Potter's invisibility cloak with a scientific method ~

Is it possible to actually make Harry's invisibility cloaks? The most promising approach for realizing such magical cloaking in our real world would be to use transformation optics, where an empty space with a distorted geometry is imitated with a non-distorted space but filled with transformation medium having appropriate permittivity and permeability. An important requirement for true invisibility cloaks is nonreciprocity; that is, a person in the cloak should not be seen from the outside but should be able to see the outside. This invisibility cloak, or a nonreciprocal shield, cannot be created as far as we stay in conventional transformation optics. Conventional transformation optics is based on Riemann geometry with a metric tensor independent of direction, and therefore cannot be used to design the nonreciprocal shield. To overcome this problem, we propose an improved theory of transformation optics that is based on Finsler geometry, an extended version of Riemann geometry. Our theory shows that nonreciprocal shielding can be realized by covering cloaking space with transformation medium having anisotropic, nonreciprocal permittivity and permeability. This theory includes conventional transformation optics as special cases. We show the method for designing the spatial distribution of the permittivity and permeability required to make the nonreciprocal shield.Comment: 10 pages, 2 figure