Integrable Hierarchies and Contact Terms in u-plane Integrals of Topologically Twisted Supersymmetric Gauge Theories

The $u$-plane integrals of topologically twisted $N = 2$ supersymmetric gauge theories generally contain contact terms of nonlocal topological observables. This paper proposes an interpretation of these contact terms from the point of view of integrable hierarchies and their Whitham deformations. This is inspired by Mari\~no and Moore's remark that the blowup formula of the $u$-plane integral contains a piece that can be interpreted as a single-time tau function of an integrable hierarchy. This single-time tau function can be extended to a multi-time version without spoiling the modular invariance of the blowup formula. The multi-time tau function is comprised of a Gaussian factor $e^{Q(t_1,t_2,...)}$ and a theta function. The time variables $t_n$ play the role of physical coupling constants of 2-observables $I_n(B)$ carried by the exceptional divisor $B$. The coefficients $q_{mn}$ of the Gaussian part are identified to be the contact terms of these 2-observables. This identification is further examined in the language of Whitham equations. All relevant quantities are written in the form of derivatives of the prepotential.Comment: latex, 17 pages, no figures; final version for publicatio

Non-Abelian Stokes Theorem for Wilson Loops Associated with General Gauge Groups

A formula constituting the non-Abelian Stokes theorem for general semi-simple compact gauge groups is presented. The formula involves a path integral over a group space and is applicable to Wilson loop variables irrespective of the topology of loops. Some simple expressions analogous to the 't Hooft tensor of a magnetic monopole are given for the 2-form of interest. A special property in the case of the fundamental representation of the group SU(N) is pointed out.Comment: 11 pages, PTPTEX, corrected some typo

Consistency Conditions of the Faddeev-Niemi-Periwal Ansatz for the SU(N) Gauge Field

The consistency condition of the Faddeev-Niemi ansatz for the gauge-fixed massless SU(2) gauge field is discussed. The generality of the ansatz is demonstrated by obtaining a sufficient condition for the existence of the three-component field introduced by Faddeev and Niemi. It is also shown that the consistency conditions determine this three-component field as a functional of two arbitrary functions. The consistency conditions corresponding to the Periwal ansatz for the SU(N) gauge field with N larger than 2 are also obtained. It is shown that the gauge field obeying the Periwal ansatz must satisfy extra (N-1)(N-2)/2 conditions.Comment: PTP Tex, 15 pages, Eq.(3.18) inserte

Integrable Top Equations associated with Projective Geometry over Z_2

We give a series of integrable top equations associated with the projective geometry over Z_2 as a (2^n-1)-dimensional generalisation of the 3D Euler top equations. The general solution of the (2^n-1)D top is shown to be given by an integration over a Riemann surface with genus (2^{n-1}-1)^2.Comment: 8 pages, Late

Non-Abelian Stokes Theorem for Loop Variables Associated with Nontrivial Loops

The non-Abelian Stokes theorem for loop variables associated with nontrivial loops (knots and links) is derived. It is shown that a loop variable is in general different from unity even if the field strength vanishes everywhere on the surface surrounded by the loop.Comment: 18 pages,10 Postscript figures, PTP Tex, Journal-ref adde

The correspondence between Tracy-Widom (TW) and Adler-Shiota-van Moerbeke (ASvM) approaches in random matrix theory: the Gaussian case

Two approaches (TW and ASvM) to derivation of integrable differential equations for random matrix probabilities are compared. Both methods are rewritten in such a form that simple and explicit relations between all TW dependent variables and $\tau$-functions of ASvM are found, for the example of finite size Gaussian matrices. Orthogonal function systems and Toda lattice are seen as the core structure of both approaches and their relationship.Comment: 20 pages, submitted to Journal of Mathematical Physic

Using data from connected thermostats to track large power outages in the United States

The detection of power outages is an essential activity for electric utilities. A large, national dataset of Internet-connected thermostats was used to explore and illustrate the ability of Internet-connected devices to geospatially track outages caused by hurricanes and other major weather events. The method was applied to nine major outage events, including hurricanes and windstorms. In one event, Hurricane Irma, a network of about 1000 thermostats provided quantitatively similar results to detailed utility data with respect to the number of homes without power and identification of the most severely affected regions. The method generated regionally uniform outage data that would give emergency authorities additional visibility into the scope and magnitude of outages. The network of thermostat-sensors also made it possible to calculate a higher resolution version of outage duration (or SAIDI) at a level of customer-level visibility that was not previously available

Toda Lattice Hierarchy and Zamolodchikov's Conjecture

In this letter, we show that certain Fredholm determinant $D(\lambda;t)$, introduced by Zamolodchikov in his study of 2D polymers, is a continuum limit of soliton solution for the Toda lattice hierarchy with 2-periodic reduction condition.Comment: 6 pages, LaTeX file, no figure

Higher-dimensional WZW Model on K\"ahler Manifold and Toroidal Lie Algebra

We construct a generalization of the two-dimensional Wess-Zumino-Witten model on a $2n$-dimensional K\"ahler manifold as a group-valued non-linear sigma model with an anomaly term containing the K\"ahler form. The model is shown to have an infinite-dimensional symmetry which generates an $n$-toroidal Lie algebra. The classical equation of motion turns out to be the Donaldson-Uhlenbeck-Yau equation, which is a $2n$-dimensional generalization of the self-dual Yang-Mills equation.Comment: 12 pages, Late