1,721,064 research outputs found
The cocycle of the quantum HJ equation and the stress tensor of CFT
Full text linkWe consider two theorems formulated in the derivation of the Quantum Hamilton-Jacobi Equation from the EP. The first one concerns the proof that the cocycle condition uniquely defines the Schwarzian derivative. This is equivalent to show that the infinitesimal variation of the stress tensor "exponentiates" to the Schwarzian derivative. The cocycle condition naturally defines the higher dimensional version of the Schwarzian derivative suggesting a role in the transformation properties of the stress tensor in higher dimensional CFT The other theorem shows that energy quantization is a direct consequence of the existence of the quantum Hamilton-Jacobi equation under duality trans formations as implied by the EP
SEIBERG-WITTEN DUALITY IN DIJKGRAAF-VAFA THEORY
Full text linkWe show that a suitable rescaling of the matrix model coupling constant makes manifest the duality group of the N=2 SYM theory with gauge group SU(2). This is done by first identifying the possible modifications of the SYM moduli preserving the monodromy group. Then we show that in matrix models there is a simple rescaling of the pair which makes them dual variables with monodromy. We then show that, thanks to a crucial scaling property of the free energy derived perturbatively by Dijkgraaf, Gukov, Kazakov and Vafa, this redefinition corresponds to a rescaling of the free energy which in turn fixes the rescaling of the coupling constant. Next, we show that in terms of the rescaled free energy one obtains a nonperturbative relation which is the matrix model counterpart of the relation between the --modulus and the prepotential of N=2 SYM. This suggests considering a dual formulation of the matrix model in which the expansion of the prepotential in the strong coupling region, whose QFT derivation is still unknown, should follow from perturbation theory. The investigation concerns the SU(2) gauge group and can be generalized to higher rank groups
THE HIGGS MODEL FOR ANYONS AND LIOUVILLE ACTION: CHAOTIC SPECTRUM, ENERGY GAP AND EXCLUSION PRINCIPLE
No full textThe requirements of geodesic completeness and self-adjointness imply that the Hamiltonian for anyons is the Laplacian with respect to the Weil-Petersson metric. This metric is complete on the Deligne-Mumford compactification of moduli (configuration) space. The structure of this compactification fixes the possible anyon configurations. This allows us to identify anyons with singularities (elliptic points with ramification q-1) in the Poincare metric implying that anyon spectrum is chaotic for n≥3. Furthermore, the bound on the holomorphic sectional curvature of moduli spaces implies a gap in the energy spectrum. For q=0 (punctures) anyons are infinitely separated in the Poincare metric (hard core). This indicates that the exclusion principle has a geometrical interpretation. Finally we give the differential equation satisfied by the generating function for volumes of the configuration space of anyons
THE AFFINE CONNECTION OF SUPERSYMMETRIC SO(N)/SP(N) THEORIES
Full text linkWe study the covariance properties of the equations satisfied by the generating functions of the chiral operators R and T of supersymmetric SO(N)/Sp(N) theories with symmetric/antisymmetric tensors. It turns out that T is an affine connection. As such it cannot be integrated along cycles on Riemann surfaces. This explains the discrepancies observed by Kraus and Shigemori. Furthermore, by means of the polynomial defining the Riemann surface, seen as quadratic-differential, one can construct an affine connection that added to T leads to a new generating function which can be consistently integrated. Remarkably, thanks to an identity, the original equations are equivalent to equations involving only one-differentials. This provides a geometrical explanation of the map recently derived by Cachazo in the case of Sp(N) with antisymmetric tensor. Finally, we suggest a relation between the Riemann surfaces with rational periods which arise in studying the laplacian on special Riemann surfaces and the integrality condition for the periods of T
INSTANTONS AND RECURSION RELATIONS IN N=2 SUSY GAUGE THEORY
Full text linkWe find the transformation properties of the prepotential F of N = 2 SUSY gauge theory with gauge group SU(2). Next we show that G(a) = πi(F(a) -1/2a ∂aF(a)) is modular invariant. We also show that u = G(a), so that F() =1/πi + 1/2. This implies thatG (a) satisfies the non-linear differential equation (1 - G2) G'' +1/4aG '3 = 0. We use this equation to derive recursion relations for the instanton contributions. These results can be extended to more general cases
NONPERTURBATIVE MODEL OF LIOUVILLE GRAVITY
Full text linkWe formulate nonperturbative 2D gravity in the framework of Liouville theory. In particular, we express the specific heat Z of pure gravity in terms of an expansion of integrals on moduli spaces of punctured Riemann spheres. We recognize the relevant divisors on moduli spaces and write the integrands in terms of the Liouville action. We evaluate the integrals (rational intersections) and show that Z satisfies the Painlevé I
CONTINUUM LIMIT OF SU(2) LATTICE GAUGE THEORY IN FIVE-DIMENSIONS
No full textUsing Monte Carlo simulations, we study the phase diagram of a two-parameter action for a SU (2) lattice gauge theory in five dimensions. The system has only first-order critical points, so that the continuum limit does not exist
UNIFORMIZATION THEORY AND 2-D GRAVITY. 1. LIOUVILLE ACTION AND INTERSECTION NUMBERS
No full textThis is the first part of an investigation concerning the formulation of 2D gravity in the framework of the uniformization theory of Riemann surfaces. As a first step in this direction we show that the classical Liouville action appears in the expression for the correlators of topological gravity. Next we derive an inequality involving the cutoff of 2D gravity and the background geometry. Another result, still related to uniformization theory, concerns a relation between the higher genus normal ordering and the Liouville action. We introduce operators covariantized by means of the inverse map of uniformization. These operators have interesting properties, including holomorphicity. In particular, they are crucial for showing that the chirally split anomaly of CFT is equivalent to the Krichever-Novikov cocycle and vanishes for deformation of the complex structure induced by the harmonic Beltrami differentials. By means of the inverse map we propose a realization of the Virasoro algebra on arbitrary Riemann surfaces and find the eigenfunctions for the holomorphic covariant operators defining higher order cocycles and anomalies which are related to W algebras. Finally we face the problem of considering the positivity of eσ, with σ the Liouville field, by proposing an explicit construction for the Fourier modes on compact Riemann surfaces. These functions, whose underlying number-theoretic structure seems related to Fuchsian groups and to the eigenvalues of the Laplacian, are quite basic and may provide the building blocks for properly investigating the long-standing uniformization problem posed by Klein, Koebe and Poincaré
Classification of commutator algebras leading to the new type of closed Baker-Campbell-Hausdorff formulas
No full textWe show that there are 13 types of commutator algebras leading to the new closed forms of the Baker–Campbell–Hausdorff (BCH) formula
exp(X)exp(Y)exp(Z)=exp(AX+BZ+CY+DI),
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derived in Matone (2015). This includes, as a particular case, exp(X)exp(Z), with [X,Z] containing other elements in addition to X and Z. The algorithm exploits the associativity of the BCH formula and is based on the decomposition exp(X)exp(Y)exp(Z)=exp(X)exp(αY)exp((1−α)Y)exp(Z), with α fixed in such a way that it reduces to View the MathML source, with View the MathML source and View the MathML source satisfying the Van-Brunt and Visser condition View the MathML source. It turns out that eα satisfies, in the generic case, an algebraic equation whose exponents depend on the parameters defining the commutator algebra. In nine types of commutator algebras, such an equation leads to rational solutions for α. We find all the equations that characterize the solution of the above decomposition problem by combining it with the Jacobi identity
Superluminality and a curious phenomenon in the relativistic quantum Hamilton–Jacobi equation
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