Doubletons and 5D Higher Spin Gauge Theory

We use Grassmann even spinor oscillators to construct a bosonic higher spin extension hs(2,2) of the five-dimensional anti-de Sitter algebra SU(2,2), and show that the gauging of hs(2,2) gives rise to a spectrum S of physical massless fields with spin s=0,2,4,... that is a UIR of hs(2,2). In addition to a master gauge field which contains the massless s=2,4,.. fields, we construct a scalar master field containing the massless s=0 field, the generalized Weyl tensors and their derivatives. We give the appropriate linearized constraint on this master scalar field, which together with a linearized curvature constraint produces the correct linearized field equations. A crucial step in the construction of the theory is the identification of a central generator K which is eliminated by means of a coset construction. Its charge vanishes in the spectrum S, which is the symmetric product of two spin zero doubletons. We expect our results to pave the way for constructing an interacting theory whose curvature expansion is dual to a CFT based on higher spin currents formed out of free doubletons in the large N limit. Thus, extending a recent proposal of Sundborg (hep-th/0103247), we conjecture that the hs(2,2) gauge theory describes a truncation of the bosonic massless sector of tensionless Type IIB string theory on AdS_5 x S^5 for large N. This implies AdS/CFT correspondence in a parameter regime where both boundary and bulk theories are perturbative.Comment: 31 pages, late

Higher Spin Gauge Theories in Various Dimensions

Properties of nonlinear higher spin gauge theories of totally symmetric massless higher spin fields in anti-de Sitter space of any dimension are discussed with the emphasize on the general aspects of the approach.Comment: LaTeX, 20 pages, Unified version of the talks delivered at International Workshop "Supersymmetries and Quantum Symmetries" (SQS'03, Dubna, Russia, July 24-29, 2003), 27th Johns Hopkins Workshop (Goteborg, Sweden, August 24-26) and 36th International Symposium Ahrenshoop on the Theory of Elementary Particles (Berlin, Germany, 26-30 Aug 2003); typos corrected, reference adde

Higher Spins and Stringy AdS5xS5

In this lecture I review recent work on higher spin holography. After a notational flash on the AdS/CFT correspondence, I will discuss HS symmetry enhancement and derive the spectrum of perturbative type IIB superstring excitations on AdS5 in this limit. I will then successfully compare it with the free N=4 SYM spectrum obtained by means of Polya theory. Decomposing the spectrum in HS multiplets, I will eventually sketch how ``La Grande Bouffe'' can be formulated a` la Stueckelberg.Comment: 40 pages, Latex, uses youngtab.sty. Lecture delivered at the RTN Workshop "The quantum structure of spacetime and the geometric nature of fundamental interactions", EXT Workshop "Fundamental Interactions and the Structure of Spacetime" in Kolymbari, Crete, 5-10 September 200

A generalized boundary condition applied to Lieb-Schultz-Mattis type ingappabilities and many-body Chern numbers

We introduce a new boundary condition which renders the flux-insertion argument for the Lieb-Schultz-Mattis type theorems in two or higher dimensions free from the specific choice of system sizes. It also enables a formulation of the Lieb-Schultz-Mattis type theorems in arbitrary dimensions in terms of the anomaly in field theories of $1+1$ dimensions with a bulk correspondence as a BF-theory in 2+1 dimensions. Furthermore, we apply the anomaly-based formulation to the constraints on a half-filled spinless fermion on a square lattice with $\pi$ flux, utilizing time-reversal, the magnetic translation and on-site internal $U(N)$ symmetries. This demonstrates the role of time-reversal anomaly on the ingappabilities of a lattice model.Comment: 4 figure

A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank

For even $k$, the matchings connectivity matrix $\mathbf{M}_k$ encodes which pairs of perfect matchings on $k$ vertices form a single cycle. Cygan et al. (STOC 2013) showed that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_2$ is $\Theta(\sqrt 2^k)$ and used this to give an $O^*((2+\sqrt{2})^{\mathsf{pw}})$ time algorithm for counting Hamiltonian cycles modulo $2$ on graphs of pathwidth $\mathsf{pw}$. The same authors complemented their algorithm by an essentially tight lower bound under the Strong Exponential Time Hypothesis (SETH). This bound crucially relied on a large permutation submatrix within $\mathbf{M}_k$, which enabled a "pattern propagation" commonly used in previous related lower bounds, as initiated by Lokshtanov et al. (SODA 2011). We present a new technique for a similar pattern propagation when only a black-box lower bound on the asymptotic rank of $\mathbf{M}_k$ is given; no stronger structural insights such as the existence of large permutation submatrices in $\mathbf{M}_k$ are needed. Given appropriate rank bounds, our technique yields lower bounds for counting Hamiltonian cycles (also modulo fixed primes $p$) parameterized by pathwidth. To apply this technique, we prove that the rank of $\mathbf{M}_k$ over the rationals is $4^k / \mathrm{poly}(k)$. We also show that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_p$ is $\Omega(1.97^k)$ for any prime $p\neq 2$ and even $\Omega(2.15^k)$ for some primes. As a consequence, we obtain that Hamiltonian cycles cannot be counted in time $O^*((6-\epsilon)^{\mathsf{pw}})$ for any $\epsilon>0$ unless SETH fails. This bound is tight due to a $O^*(6^{\mathsf{pw}})$ time algorithm by Bodlaender et al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be counted modulo primes $p\neq 2$ in time $O^*(3.97^\mathsf{pw})$, indicating that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in SODA 201