Spinning the fuzzy sphere

Published for SISSA by Springer Received: June 20, 2015 Accepted: August 5, 2015 Published: August 27, 2015 David Berenstein,a,b Eric Dzienkowskib and Robin Lashof-Regasb a Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, U.K. b Department of Physics, University of California Santa Barbara, Santa Barbara, California 93106, U.S.A. E-mail: , , Abstract: We construct various exact analytical solutions of the SO(3) BMN matrix model that correspond to rotating fuzzy spheres and rotating fuzzy tori. These are also solutions of Yang Mills theory compactified on a sphere times time and they are also translationally invariant solutions of the N = 1∗ field theory with a non-trivial charge density. The solutions we construct have a ZN symmetry, where N is the rank of the matrices. After an appropriate ansatz, we reduce the problem to solving a set of polynomial equations in 2N real variables. These equations have a discrete set of solutions for each value of the angular momentum. We study the phase structure of the solutions for various values of N . Also the continuum limit where N → ∞, where the problem reduces to finding periodic solutions of a set of coupled differential equations. We also study the topology change transition from the sphere to the torus. Keywords: Matrix Models, Non-Commutative Geometry, Supersymmetric gauge theory, AdS-CFT Correspondence ArXiv ePrint: 1506.01722v2 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP08(2015)134 JHEP08(2015)134 Spinning the fuzzy sphere Contents 1 Introduction 1 2 The Hamiltonian and the Ansatz 2.1 Relation to SYM and N = 1∗ 2 4 3 Symmetry considerations 6 5 The case of 2 × 2 matrices 11 6 The case of 3 × 3 matrices 13 7 Other examples 19 8 Large N 21 9 Topology change 25 10 Conclusion 28 1 Introduction The BMN matrix model [1] is a massive deformation of the BFSS matrix model [2]. The BFSS matrix model describes the discrete lighten quantization of M-theory on flat space. The BMN matrix model analogously describes the discrete light cone quantization of Mtheory on a maximally supersymmetric plane wave. These maximally supersymmetric plane wave geometries were constructed by taking a Penrose limit of supersymmetric AdS × S spaces in [3]. The BFSS matrix model results from the dimensional reduction of N = 1 SYM from ten dimensions downs to 0+1 dimensions and corresponds to the dynamics of D0-branes [4]. It has an SO(9) symmetry of the transverse directions and a gauged U(N ) symmetry, where N is the rank of the matrices. The BMN matrix model splits these transverse directions into two sets with different masses, so that the bosonic symmetry reduces to an SO(3) × SO(6) subgroup of SO(9). The subset of the theory where only the SO(3) charged scalars are excited is the SO(3) BMN matrix model. The full model also results from considering an SU(2)L invariant set of configurations in Yang-Mills theory on an S 3 × R geometry [5]. Thus, any classical solution of the SO(3) BMN matrix model is also a classical solution of Yang Mills theory on a sphere times time. One can also show that any solution of the –1– JHEP08(2015)134 4 The solutions of the ansatz are a set of critical points of an energy function 8 2 The Hamiltonian and the Ansatz The SO(3) BMN matrix model is a dynamical system with three N ×N Hermitian matrices X 1,2,3 , or alternatively X, Y, Z. The conjugate momentum matrices are P1,2,3 , and PX,Y,Z respectively. The BMN Hamiltonian is given by   3 X 1 1 H = Tr(P12 + P22 + P32 ) + Tr  (X j + ijmn X m X n )2  (2.1) 2 2 j=1 The system possesses a U(N ) gauge symmetry where X i and Pi both transform in the adjoint, X i → U X i U −1 and Pi → U Pi U −1 . The presentation of the Hamiltonian (2.1) is in the gauge A0 = 0. The generators of gauge transformations are the matrix of functions on phase space given by 3 X G= [Pj , X j ] (2.2) j=1 The dynamics need to be supplemented by the Gauss’ law constraint G = 0. The system also enjoys an SO(3) symmetry of rotations of the matrices X, Y, Z into each other. The generator of angular momentum along the Z direction is J = LZ = Tr(XPY − Y PX ) –2– (2.3) JHEP08(2015)134 BMN matrix model corresponds to a classical solution of the N = 1∗ field theory ( see [6] and references therein). It is known that for generic initial conditions, the BMN matrix model is chaotic [7– 9]. This can be understood from the chaotic dynamics of dimensionally reduced YangMills [10–13]. However, special initial conditions can in principle be soluble analytically. It is expected quite generally that solutions that minimize the energy with an additional conserved quantity turned on and constrained can be stationary. This is usually handled with a Routhian if the conjugate variable to the conserved quantity can be separated. Many of these states that minimize the energy given some conserved central charge have interpretations in terms of BPS states in supersymmetric field theories. As it turns out, the BMN matrix model has exact, supersymmetric solutions with zero energy [1]. These matrix configurations are characterized by all adjoint representations of su(2). They have an interpretation as giant gravitons [14]. The spectrum of fluctuations around these solutions is known [15] (see also [16] for an alternative derivation of the spectrum) and one can argue that there is a large tower of protected states that are available to study [17]. Unfortunately the nonlinear structure of the classical solutions that make this tower of BPS states is not known. It is expected that adding angular momentum to the fuzzy sphere states can induce topology changes from a sphere to a torus [18]. Our purpose in this paper is to investigate this topology transition with a special family of matrix solutions at finite angular momentum. The paper is mostly devoted to constructing these solutions. Once the solutions are found, the geometry of the corresponding fuzzy membrane is analyzed using the techniques in [19]. with similar expressions for the other two SO(3) generators. Lastly, the equations of motion are ∂H Ẋ j = = Pj , (2.4) ∂P j ∂H Ṗj = − = −X j − 3ijmn X m X n − [[X j , X m ], X m ] (2.5) ∂X j The solutions with H = 0 are given by fuzzy spheres. These are solutions of the equations [X i , X j ] = iijk X k (2.6) The cross terms between X2 , P1 and X1 , P2 in the squares generate a copy of J that needs to be subtracted. The cross terms with P1 and [X 3 , X 1 ] lead to something that does not automatically cancel for generic matrices, but after a bit of reshuffling can be shown to be proportional to X 3 ([X 1 , P 1 ] + [X 2 , P 2 ]) (2.8) and we recognize the Gauss’ law constraint starting to arise. After imposing the full Gauss’ law constraint, we get Tr(X 3 [X 3 , P 3 ]) that does vanish identically. The BPS bound is not directly related to supersymmetry. Instead it is derived from the co (...truncated)