 Award ID(s):
 2209700
 NSFPAR ID:
 10411950
 Date Published:
 Journal Name:
 Journal of High Energy Physics
 Volume:
 2022
 Issue:
 11
 ISSN:
 10298479
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

A bstract Motivated by applications to soft supersymmetry breaking, we revisit the expansion of the SeibergWitten solution around the multimonopole point on the Coulomb branch of pure SU( N ) $$ \mathcal{N} $$ N = 2 gauge theory in four dimensions. At this point N − 1 mutually local magnetic monopoles become massless simultaneously, and in a suitable duality frame the gauge couplings logarithmically run to zero. We explicitly calculate the leading threshold corrections to this logarithmic running from the SeibergWitten solution by adapting a method previously introduced by D’Hoker and Phong. We compare our computation to existing results in the literature; this includes results specific to SU(2) and SU(3) gauge theories, the large N results of Douglas and Shenker, as well as results obtained by appealing to integrable systems or topological strings. We find broad agreement, while also clarifying some lingering inconsistencies. Finally, we explicitly extend the results of Douglas and Shenker to finite N , finding exact agreement with our first calculation.more » « less

Embedding properties of network realizations of dissipative reduced order models Jörn Zimmerling, Mikhail Zaslavsky,Rob Remis, Shasri Moskow, Alexander Mamonov, Murthy Guddati, Vladimir Druskin, and Liliana Borcea Mathematical Sciences Department, Worcester Polytechnic Institute https://www.wpi.edu/people/vdruskin Abstract Realizations of reduced order models of passive SISO or MIMO LTI problems can be transformed to tridiagonal and blocktridiagonal forms, respectively, via dierent modications of the Lanczos algorithm. Generally, such realizations can be interpreted as ladder resistorcapacitorinductor (RCL) networks. They gave rise to network syntheses in the rst half of the 20th century that was at the base of modern electronics design and consecutively to MOR that tremendously impacted many areas of engineering (electrical, mechanical, aerospace, etc.) by enabling ecient compression of the underlining dynamical systems. In his seminal 1950s works Krein realized that in addition to their compressing properties, network realizations can be used to embed the data back into the state space of the underlying continuum problems. In more recent works of the authors Krein's ideas gave rise to socalled nitedierence Gaussian quadrature rules (FDGQR), allowing to approximately map the ROM statespace representation to its full order continuum counterpart on a judicially chosen grid. Thus, the state variables can be accessed directly from the transfer function without solving the full problem and even explicit knowledge of the PDE coecients in the interior, i.e., the FDGQR directly learns" the problem from its transfer function. This embedding property found applications in PDE solvers, inverse problems and unsupervised machine learning. Here we show a generalization of this approach to dissipative PDE problems, e.g., electromagnetic and acoustic wave propagation in lossy dispersive media. Potential applications include solution of inverse scattering problems in dispersive media, such as seismic exploration, radars and sonars. To x the idea, we consider a passive irreducible SISO ROM fn(s) = Xn j=1 yi s + σj , (62) assuming that all complex terms in (62) come in conjugate pairs. We will seek ladder realization of (62) as rjuj + vj − vj−1 = −shˆjuj , uj+1 − uj + ˆrj vj = −shj vj , (63) for j = 0, . . . , n with boundary conditions un+1 = 0, v1 = −1, and 4n real parameters hi, hˆi, ri and rˆi, i = 1, . . . , n, that can be considered, respectively, as the equivalent discrete inductances, capacitors and also primary and dual conductors. Alternatively, they can be viewed as respectively masses, spring stiness, primary and dual dampers of a mechanical string. Reordering variables would bring (63) into tridiagonal form, so from the spectral measure given by (62 ) the coecients of (63) can be obtained via a nonsymmetric Lanczos algorithm written in Jsymmetric form and fn(s) can be equivalently computed as fn(s) = u1. The cases considered in the original FDGQR correspond to either (i) real y, θ or (ii) real y and imaginary θ. Both cases are covered by the Stieltjes theorem, that yields in case (i) real positive h, hˆ and trivial r, rˆ, and in case (ii) real positive h,r and trivial hˆ,rˆ. This result allowed us a simple interpretation of (62) as the staggered nitedierence approximation of the underlying PDE problem [2]. For PDEs in more than one variables (including topologically rich datamanifolds), a nitedierence interpretation is obtained via a MIMO extensions in block form, e.g., [4, 3]. The main diculty of extending this approach to general passive problems is that the Stieltjes theory is no longer applicable. Moreover, the tridiagonal realization of a passive ROM transfer function (62) via the ladder network (63) cannot always be obtained in portHamiltonian form, i.e., the equivalent primary and dual conductors may change sign [1]. 100 Embedding of the Stieltjes problems, e.g., the case (i) was done by mapping h and hˆ into values of acoustic (or electromagnetic) impedance at grid cells, that required a special coordinate stretching (known as travel time coordinate transform) for continuous problems. Likewise, to circumvent possible nonpositivity of conductors for the nonStieltjes case, we introduce an additional complex sdependent coordinate stretching, vanishing as s → ∞ [1]. This stretching applied in the discrete setting induces a diagonal factorization, removes oscillating coecients, and leads to an accurate embedding for moderate variations of the coecients of the continuum problems, i.e., it maps discrete coecients onto the values of their continuum counterparts. Not only does this embedding yields an approximate linear algebraic algorithm for the solution of the inverse problems for dissipative PDEs, it also leads to new insight into the properties of their ROM realizations. We will also discuss another approach to embedding, based on KreinNudelman theory [5], that results in special datadriven adaptive grids. References [1] Borcea, Liliana and Druskin, Vladimir and Zimmerling, Jörn, A reduced order model approach to inverse scattering in lossy layered media, Journal of Scientic Computing, V. 89, N1, pp. 136,2021 [2] Druskin, Vladimir and Knizhnerman, Leonid, Gaussian spectral rules for the threepoint second dierences: I. A twopoint positive denite problem in a semiinnite domain, SIAM Journal on Numerical Analysis, V. 37, N 2, pp.403422, 1999 [3] Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Distance preserving model order reduction of graphLaplacians and cluster analysis, Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Journal of Scientic Computing, V. 90, N 1, pp 130, 2022 [4] Druskin, Vladimir and Moskow, Shari and Zaslavsky, Mikhail LippmannSchwingerLanczos algorithm for inverse scattering problems, Inverse Problems, V. 37, N. 7, 2021, [5] Mark Adolfovich Nudelman The Krein String and Characteristic Functions of Maximal Dissipative Operators, Journal of Mathematical Sciences, 2004, V 124, pp 49184934 Go back to Plenary Speakers Go back to Speakers Go backmore » « less

We describe the confining instabilities of a proposed quantum spin liquid underlying the pseudogap metal state of the holedoped cuprates. The spin liquid can be described by a SU(2) gauge theory of
N _{f}= 2 massless Dirac fermions carrying fundamental gauge charges—this is the lowenergy theory of a meanfield state of fermionic spinons moving on the square lattice withπ flux per plaquette in the ℤ_{2}center of SU(2). This theory has an emergent SO(5)_{f}global symmetry and is presumed to confine at low energies to the Néel state. At nonzero doping (or smaller Hubbard repulsionU at halffilling), we argue that confinement occurs via the Higgs condensation of bosonic chargons carrying fundamental SU(2) gauge charges also moving inπ ℤ_{2}flux. At halffilling, the lowenergy theory of the Higgs sector hasN _{b}= 2 relativistic bosons with a possible emergent SO(5)_{b}global symmetry describing rotations between ad wave superconductor, period2 charge stripes, and the timereversal breaking “d density wave” state. We propose a conformal SU(2) gauge theory withN _{f}= 2 fundamental fermions,N _{b}= 2 fundamental bosons, and a SO(5)_{f}×SO(5)_{b}global symmetry, which describes a deconfined quantum critical point between a confining state which breaks SO(5)_{f}and a confining state which breaks SO(5)_{b}. The pattern of symmetry breaking within both SO(5)s is determined by terms likely irrelevant at the critical point, which can be chosen to obtain a transition between Néel order andd wave superconductivity. A similar theory applies at nonzero doping and largeU , with longerrange couplings of the chargons leading to charge order with longer periods. 
We revisit the correspondence between CalabiYau (CY) threefoldisolated singularities \mathbf{X} 𝐗 and fivedimensional superconformal field theories (SCFTs), which ariseat low energy in Mtheory on the spacetime transverse to \mathbf{X} 𝐗 .Focussing on the case of toric CY singularities, we analyze the“gaugetheory phases” of the SCFT by exploiting fiberwise Mtheory/typeIIA duality. In this setup, the lowenergy gauge group simply arises onstacks of coincident D6branes wrapping 2cycles in some ALE space oftype A_{M1} A M − 1 fibered over a real line, and the map between the Kähler parameters of \mathbf{X} 𝐗 and the Coulomb branch parameters of the field theory (masses and VEVs)can be read off systematically. Different type IIA “reductions” giverise to different gauge theory phases, whose existence depends on theparticular (partial) resolutions of the isolated singularity \mathbf{X} 𝐗 .We also comment on the case of nonisolated toric singularities.Incidentally, we propose a slightly modified expression for theCoulombbranch prepotential of 5d \mathcal{N}=1 𝒩 = 1 gauge theories.more » « less

A bstract We develop an approach to the study of Coulomb branch operators in 3D $$ \mathcal{N} $$ N = 4 gauge theories and the associated quantization structure of their Coulomb branches. This structure is encoded in a onedimensional TQFT subsector of the full 3D theory, which we describe by combining several techniques and ideas. The answer takes the form of an associative and noncommutative star product algebra on the Coulomb branch. For “good” and “ugly” theories (according to the GaiottoWitten classification), we also exhibit a trace map on this algebra, which allows for the computation of correlation functions and, in particular, guarantees that the star product satisfies a truncation condition. This work extends previous work on abelian theories to the nonabelian case by quantifying the monopole bubbling that describes screening of GNO boundary conditions. In our approach, monopole bubbling is determined from the algebraic consistency of the OPE. This also yields a physical proof of the BullimoreDimofteGaiotto abelianization description of the Coulomb branch.more » « less