Two-dimensional gauge theory and matrix model

We study a matrix model obtained by dimensionally reducing Chern–Simons theory on S³$S^3$. We find that the matrix integration is decomposed into sectors classified by the representation of SU(2)$\mathit{SU}(2)$. We show that the N  -block sectors reproduce SU(N)$\mathit{SU}(N)$ Yang–Mills theory on S²$S^2$ as the matrix size goes to infinity.     

The exact decomposition of gauge variables in lattice Yang–Mills theory

In this Letter, we consider lattice versions of the decomposition of the Yang–Mills field a la Cho–Faddeev–Niemi, which was extended by Kondo, Shinohara and Murakami in the continuum formulation. For the SU(N)$\mathit{SU}(N)$ gauge group, we propose a set of defining equations for specifying the decomposition of the gauge link variable and solve them exactly without using the ansatz adopted in the previous studies for SU(2)$\mathit{SU}(2)$ and SU(3)$\mathit{SU}(3)$. As a result, we obtain the general form of the decomposition for SU(N)$\mathit{SU}(N)$ gauge link variables and confirm the previous results obtained for SU(2)$\mathit{SU}(2)$ and SU(3)$\mathit{SU}(3)$.     

The 't Hooft–Polyakov monopole in the presence of a 't Hooft operator

We present explicit BPS field configurations representing one non-Abelian monopole with one minimal weight 't Hooft operator insertion. We explore the SO(3)$\mathit{SO}(3)$ and SU(2)$\mathit{SU}(2)$ gauge groups. In the case of SU(2)$\mathit{SU}(2)$ gauge group the minimal 't Hooft operator can be completely screened by the monopole. If the gauge group is SO(3)$\mathit{SO}(3)$, however, such screening is impossible. In the latter case we observe a different effect of the gauge symmetry enhancement in the vicinity of the 't Hooft operator.     

Matrix regularization of embedded 4-manifolds

We consider products of two 2-manifolds such as S²×S²$S^2\times S^2$, embedded in Euclidean space and show that the corresponding 4-volume preserving diffeomorphism algebra can be approximated by a tensor product SU(N)⊗SU(N)$\mathit{SU}(N)\otimes \mathit{SU}(N)$ i.e. functions on a manifold are approximated by the Kronecker product of two SU(N)$\mathit{SU}(N)$ matrices.     

The puzzle of apparent linear lattice artifacts in the 2d non-linear σ-model and Symanzik's solution

Lattice artifacts in the 2d O(n) non-linear σ  -model are expected to be of the form O(a²)$\mathrm{O}(a^2)$, and hence it was (when first observed) disturbing that some quantities in the O(3)$\mathrm{O}(3)$ model with various actions show parametrically stronger cutoff dependence, apparently O(a)$\mathrm{O}(a)$, up to very large correlation lengths. In a previous letter Balog et al. (2009) [1] we described the solution to this puzzle. Based on the conventional framework of Symanzik's effective action, we showed that there are logarithmic corrections to the O(a²)$\mathrm{O}(a^2)$ artifacts which are especially large (ln³a${\ln }^{3}a$) for n=3$n=3$ and that such artifacts are consistent with the data. In this paper we supply the technical details of this computation. Results of Monte Carlo simulations using various lattice actions for O(3)$\mathrm{O}(3)$ and O(4)$\mathrm{O}(4)$ are also presented.     

Yang–Mills instantons and dyons on group manifolds

We consider Euclidean SU(N)$\mathrm{SU}(N)$ Yang–Mills theory on the space G×R$G\times \mathbb{R}$, where G is a compact semisimple Lie group, and introduce first-order BPS-type equations which imply the full Yang–Mills equations. For gauge fields invariant under the adjoint G  -action these BPS equations reduce to first-order matrix equations, to which we give instanton solutions. In the case of G=SU(2)≅S³$G=\mathrm{SU}(2)\cong S^3$, our matrix equations are recast as Nahm equations, and a further algebraic reduction to the Toda chain equations is presented and solved for the SU(3) example. Finally, we change the metric on G×R$G\times \mathbb{R}$ to Minkowski and construct finite-energy dyon-type Yang–Mills solutions. The special case of G=SU(2)×SU(2)$G=\mathrm{SU}(2)\times \mathrm{SU}(2)$ may be used in heterotic flux compactifications.     

Fixed point resolution in extensions of permutation orbifolds

We determine the simple currents and fixed points of the orbifold theory CFT⊗CFT/Z₂$\mathit{CFT}\otimes \mathit{CFT}/{\mathbb{Z}}_2$, given the simple currents and fixed point of the original CFT  . We see in detail how this works for the SUk₍₂₎$\mathit{SU}{(2)}_k$ WZW model, focusing on the field content (i.e. h  -spectrum of the primary fields) of the theory. We also look at the fixed point resolution of the simple current extended orbifold theory and determine the SJ$S^J$ matrices associated to each simple current for SU₂₍₂₎$\mathit{SU}{(2)}_2$ and for the B_1(n)$B{(n)}_1$ and D_1(n)$D{(n)}_1$ series.     

Zero modes for the boundary giant magnons

We study the fermion zero-mode dynamics for open strings ending on the giant graviton branes. For the open string ending on the Z=0$Z=0$ brane, the quantization of the fermion zero-modes of boundary giant magnons reproduces the 256 states of the boundary degrees with the precise realization of the SU(2|2)×SU(2|2)$\mathit{SU}(2|2)\times \mathit{SU}(2|2)$ symmetry algebra. Also for the open string ending on the Y=0$Y=0$ brane, we reproduce the unique vacuum state from the fermion zero-modes.     

Multiple layer structure of non-Abelian vortex

Bogomolny–Prasad–Sommerfield (BPS) vortices in U(N)$U(N)$ gauge theories have two layers corresponding to non-Abelian and Abelian fluxes, whose widths depend nontrivially on the ratio of U(1)$U(1)$ and SU(N)$\mathit{SU}(N)$ gauge couplings. We find numerically and analytically that the widths differ significantly from the Compton lengths of lightest massive particles with the appropriate quantum number.     

Dynamical SUSY breaking in heterotic M-theory

It is shown that four-dimensional N=1$N=1$ supersymmetric QCD with massive flavors in the fundamental representation of the gauge group can be realized in the hidden sector of E₈×E₈$E_8\times E_8$ heterotic string vacua. The number of flavors can be chosen to lie in the range of validity of the free-magnetic dual, using which one can demonstrate the existence of long-lived meta-stable non-supersymmetric vacua. This is shown explicitly for the gauge group Spin(10)$\mathit{Spin}(10)$, but the methods are applicable to Spin(Nc)$\mathit{Spin}(N_c)$, SU(Nc)$\mathit{SU}(N_c)$ and Sp(Nc)$\mathit{Sp}(N_c)$ for a wide range of color index Nc$N_c$. Hidden sectors of this type can potentially be used as a mechanism to break supersymmetry within the context of heterotic M-theory.     

A dual non-Abelian Yang–Mills amplitude in four dimensions

We derive an explicit formula for the vertex amplitude of dual SU(2)$\mathit{SU}(2)$ Yang–Mills theory in four dimensions on the lattice, and provide an efficient algorithm (of order j⁴$j^4$) for its computation. This opens the way for both numerical and analytical development of dual methods, previously limited to the case of three dimensions.     

Rational r-matrices,higher rank Lie algebras and integrable proton–neutron BCS models

We study integrable cases of pairing BCS hamiltonians containing several types of fermions. We prove that there exist three classes of such integrable models associated with classical rational r  -matrices and Lie algebras gl(2m)$\mathit{gl}(2m)$, sp(2m)$\mathit{sp}(2m)$ and so(2m)$\mathit{so}(2m)$ correspondingly. We diagonalize the constructed hamiltonians by means of the algebraic Bethe ansatz. In the partial case of two types of fermions (m=2$m=2$) the obtained models may be interpreted as N=Z$N=Z$ proton–neutron integrable models. In particular, in the case of sp(4)$\mathit{sp}(4)$ we recover the famous integrable proton–neutron model of Richardson.     

Self-duality of the compactified Ruijsenaars–Schneider system from quasi-Hamiltonian reduction

The Delzant theorem of symplectic topology is used to derive the completely integrable compactified Ruijsenaars–Schneider _IIIb${\mathrm{III}}_{\mathrm{b}}$ system from a quasi-Hamiltonian reduction of the internally fused double SU(n)×SU(n)$\mathit{SU}(n)\times \mathit{SU}(n)$. In particular, the reduced spectral functions depending respectively on the first and second SU(n)$\mathit{SU}(n)$ factor of the double engender two toric moment maps on the _IIIb${\mathrm{III}}_{\mathrm{b}}$ phase space CP(n−1)$\mathbb{C}P(n-1)$ that play the roles of action-variables and particle-positions. A suitable central extension of the SL(2,Z)$\mathit{SL}(2,\mathbb{Z})$ mapping class group of the torus with one boundary component is shown to act on the quasi-Hamiltonian double by automorphisms and, upon reduction, the standard generator S   of the mapping class group is proved to descend to the Ruijsenaars self-duality symplectomorphism that exchanges the toric moment maps. We give also two new presentations of this duality map: one as the composition of two Delzant symplectomorphisms and the other as the composition of three Dehn twist symplectomorphisms realized by Goldman twist flows. Through the well-known relation between quasi-Hamiltonian manifolds and moduli spaces, our results rigorously establish the validity of the interpretation [going back to Gorsky and Nekrasov] of the _IIIb${\mathrm{III}}_{\mathrm{b}}$ system in terms of flat SU(n)$\mathit{SU}(n)$ connections on the one-holed torus.     

Supersymmetric grand unification with light color-triplet

We construct a natural model of the supersymmetric SU(6)$\mathit{SU}(6)$ unification, in which the symmetry breaking, down to the standard model gauge group, results in the number of pseudo-Nambu–Goldstone superfields with interesting properties. Namely, besides the Higgs doublet–antidoublet pair which is responsible for the electroweak phase transition, the Nambu–Goldstone sector consists of multiplets in the anti- and fundamental representations of SU(5)$\mathit{SU}(5)$. While being strictly massless in the supersymmetric limit, they acquire the weak scale masses as a result of its breaking. The color-triplet components of this light sector could, in principle, mediate an unacceptably fast proton decay; however, because of the natural TeV/_MGUT$\text{TeV}/M_{\mathrm{GUT}}$ suppression of the Yukawa couplings to the light quarks and leptons, their existence is compatible with the experimental bound on proton lifetime. This suppression is made further interesting, since it results in the lifetime, of the lightest of the above-mentioned colored particles from 1 s to 1 day$1\text{day}$, long enough for it to appear stable in the detector. Furthermore, we argue that the accommodation of the color-triplet pseudo-Nambu–Goldstones, without fine-tuning or contradicting observations, implies SU(6)$\mathit{SU}(6)$ unification.     

A note on embedding non-Abelian finite flavor groups in continuous groups

A non-Abelian finite flavor group G⊂SO(3)$G\subset \mathit{SO}(3)$ can have double covering G^′⊂SU(2)$G^{\prime }\subset \mathit{SU}(2)$ such that G⊄G^′$G\not\subset G^{\prime }$. This situation is not contradictory, but quite natural, and we give explicit examples such as G=Dn$G=D_n$, G^′=Q_2n$G^{\prime }=Q_{2n}$ and G=T$G=T$, G^′=T^′$G^{\prime }=T^{\prime }$. This observation can be crucial in particle theory model building.