Distinct quantum vacua of topologically ordered states can be tunneled into each other via extended operators. The possible applications include condensed matter and quantum cosmology. We present a straightforward approach to calculate the partition function on various manifolds and ground state degeneracy (GSD), mainly based on continuum/cochain Topological Quantum Field Theories (TQFT), in any dimension. This information can be related to the counting of extended operators of bosonic/fermionic TQFT. On the lattice scale, anyonic particles/strings live at the ends of line/surface operators. Certain systems in different dimensions are related to each other through dimensional reduction schemes, analogous to (de)categorification. Examples include spin TQFTs derived from gauging the interacting fermionic symmetry protected topological states (with fermion parity Z f 2 ) of symmetry group Z 4 × Z 2 and (Z 4 ) 2 in 3+1D, also Z 2 and (Z 2 ) 2 in 2+1D. Gauging the last three cases begets non-Abelian spin TQFT (fermionic topological order). We consider situations where a TQFT lives on (1) a closed spacetime or (2) a spacetime with boundary, such that the bulk and boundary are fully-gapped and short or long-range entangled (SRE/LRE). Anyonic excitations can be deconfined on the boundary. We introduce new exotic topological interfaces on which neither particle nor string excitations alone condensed, but only fuzzy-composite objects of extended operators can end (e.g. a stringlike composite object formed by a set of particles can end on a special 2+1D boundary of 3+1D bulk). We explore the relations between group extension constructions and partially breaking constructions (e.g. 0-form/higher-form/"composite" breaking) of topological boundaries, after gauging. We comment on the implications of entanglement entropy for some of such LRE systems. 1 We denote the spacetime dimensions as d + 1D 2 One can also consider a generalization of this relation by turning on a background flat connection A (G) for a global symmetry G. First, non-trivial holonomies along 1-cycles of M d will result in replacement H M d by the corresponding twisted Hilbert space H tw M d . Second, a non-trivial holonomy g ∈ G along the time S 1 will result in insertion of ρ(g) into the trace, where ρ is the representation of G on the Hilbert space:( 1.2) In condensed matter, this is related to the symmetry twist inserted on M d to probe the Symmetry Protected/Enriched Topological states (SPTs/SETs) [2,12]. In this work, instead we mainly focus on eqn. (1.1).
A supersymmetric vacuum has to obey a set of constraints on fluxes as well as first order differential equations defined by the G-structures of the internal manifold. We solve these equations for type IIB supergravity with SU(3) structures. The 6-dimensional internal manifold has to be complex, the axion/dilaton is in general non-holomorphic and a cosmological constant is only possible if the SU(3) structures are broken to SU(2) structures. The general solution is expressed in terms of one function which is holomorphic in the three complex coordinates and if this holomorphic function is constant, we obtain a flow-type solution and near poles and zeros we find the so-called type-A and type-B vacuum.
SL(2, Z) duality transformations in asymptotically AdS 4 × S 7 act non-trivially on the three-dimensional SCFT of coincident M2-branes on the boundary. We show how S-duality acts away from the IR fixed point. We develop a systematic method to holographically obtain the deformations of the boundary CFT and show how electric-magnetic duality relates different deformations. We analyze in detail marginal deformations and deformations by dimension 4 operators. In the case of massive deformations, the RG flow relates S-dual CFT's. Correlation functions in the CFT are computed by varying magnetic bulk sources, whereas correlation functions in the dual CFT are computed by electric bulk sources. Under massive deformations, the boundary effective action is generically minimized by massive self-dual configurations of the U(1) gauge field. We show that a self-dual choice of boundary conditions exists, and it corresponds to the self-dual topologically massive gauge theory in 2+1 dimensions. Thus, self-duality in three dimensions can be understood as a consequence of electric-magnetic invariance in the bulk of AdS 4 . B.3 Regularity without gauge fixing .
We prove some Hardy-type inequalities via an approach that involves constructing auxiliary sequences.
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