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Register a new userA semi-classical limit of the gauge/string correspondence
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A world-sheet sigma model approach is applied to string theories dual to four-dimensional gauge theories, and semi-classical soliton solutions representing highly excited string states are identified which correspond to gauge theory operators with relatively small anomalous dimensions. The simplest class of such states are strings on the leading Regge trajectory, with large spin in AdS_5. These correspond to operators with many covariant derivatives, whose anomalous dimension grows logarithmically with the space-time spin. In the gauge theory, the logarithmic scaling violations are similar to those found in perturbation theory. Other examples of highly excited string states are also considered.
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Following earlier work, we view two dimensional non-linear sigma model with target space $cM$ as a single particle relativistic quantum mechanics in the corresponding free loop space $cLM$. In a natural semi-classical limit ($hbar=alpha to 0$) of this model the wavefunction localizes on the submanifold of vanishing loops which is isomorphic to $cM$. One would expect that the relevant semi-classical expansion should be related to the tubular expansion of the theory around the submanifold and an effective dynamics on the submanifold is obtainable using Born-Oppenheimer approximation. In this work we develop a framework to carry out such an analysis at the leading order in $alpha$-expansion. In particular, we show that the linearized tachyon effective equation is correctly reproduced up to divergent terms all proportional to the Ricci scalar of $cM$. The steps leading to this result are as follows: first we define a finite dimensional analogue of the loop space quantum mechanics (LSQM) where we discuss its tubular expansion and how that is related to a semi-classical expansion of the Hamiltonian. Then we study an explicit construction of the relevant tubular neighborhood in $cLM$ using exponential maps. Such a tubular geometry is obtained from a Riemannian structure on the tangent bundle of $cM$ which views the zero-section as a submanifold admitting a tubular neighborhood. Using this result and exploiting an analogy with the toy model we arrive at the final result for LSQM.
The 2d gauged linear sigma model (GLSM) gives a UV model for quantum cohomology on a Kahler manifold X, which is reproduced in the IR limit. We propose and explore a 3d lift of this correspondence, where the UV model is the N=2 supersymmetric 3d gauge theory and the IR limit is given by Giventals permutation equivariant quantum K-theory on X. This gives a one-parameter deformation of the 2d GLSM/quantum cohomology correspondence and recovers it in a small radius limit. We study some novelties of the 3d case regarding integral BPS invariants, chiral rings, deformation spaces and mirror symmetry.
We consider a general gauge theory with independent generators and study the problem of gauge-invariant deformation of initial gauge-invariant classical action. The problem is formulated in terms of BV-formalism and is reduced to describing the general solution to the classical master equation. We show that such general solution is determined by two arbitrary generating functions of the initial fields. As a result, we construct in explicit form the deformed action and the deformed gauge generators in terms of above functions. We argue that the deformed theory must in general be non-local. The developed deformation procedure is applied to Abelian vector field theory and we show that it allows to derive non-Abelain Yang-Mills theory. This procedure is also applied to free massless integer higher spin field theory and leads to local cubic interaction vertex for such fields.
We study the time evolution of the expectation value of a rectangular Wilson loop in strongly anisotropic time-dependent plasma using gauge-gravity duality. The corresponding gravity theory is given by describing time evolution of a classical string in the Lifshitz-Vaidya background. We show that the expectation value of the Wilson loop oscillates about the value of the static potential with the same parameters after the energy injection is over. We discuss how the amplitude and frequency of the oscillation depend on the parameters of the theory. In particular, by raising the anisotropy parameter, we observe that the amplitude and frequency of the oscillation increase.
We consider the quantum evolution $e^{-ifrac{t}{hbar}H_{beta}} psi_{xi}^{hbar}$ of a Gaussian coherent state $psi_{xi}^{hbar}in L^{2}(mathbb{R})$ localized close to the classical state $xi equiv (q,p) in mathbb{R}^{2}$, where $H_{beta}$ denotes a self-adjoint realization of the formal Hamiltonian $-frac{hbar^{2}}{2m},frac{d^{2},}{dx^{2}} + beta,delta_{0}$, with $delta_{0}$ the derivative of Diracs delta distribution at $x = 0$ and $beta$ a real parameter. We show that in the semi-classical limit such a quantum evolution can be approximated (w.r.t. the $L^{2}(mathbb{R})$-norm, uniformly for any $t in mathbb{R}$ away from the collision time) by $e^{frac{i}{hbar} A_{t}} e^{it L_{B}} phi^{hbar}_{x}$, where $A_{t} = frac{p^{2}t}{2m}$, $phi_{x}^{hbar}(xi) := psi^{hbar}_{xi}(x)$ and $L_{B}$ is a suitable self-adjoint extension of the restriction to $mathcal{C}^{infty}_{c}({mathscr M}_{0})$, ${mathscr M}_{0} := {(q,p) in mathbb{R}^{2},|,q eq 0}$, of ($-i$ times) the generator of the free classical dynamics. While the operator $L_{B}$ here utilized is similar to the one appearing in our previous work [C. Cacciapuoti, D. Fermi, A. Posilicano, The semi-classical limit with a delta potential, Annali di Matematica Pura e Applicata (2020)] regarding the semi-classical limit with a delta potential, in the present case the approximation gives a smaller error: it is of order $hbar^{7/2-lambda}$, $0 < lambda < 1/2$, whereas it turns out to be of order $hbar^{3/2-lambda}$, $0 < lambda < 3/2$, for the delta potential. We also provide similar approximation results for both the wave and scattering operators.
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