p-adic superstrings

Different possibilities to construct p-adic superstring amplitudes are discussed. To describe one of these possibilities we formulate a p-adic generalization of the conformal technique. p-adic conformal rules allow us to explicitly write down the amplitude for massless particles. They do not have the form of the usual kinematical factors multiplied by the p-adic modification of the usual amplitudes. The relation between open and closed p-adic superstring amplitudes is discussed.     

p-adic analyticity and virasoro algebras for conformal theories in more than two dimensions

The field of p-adic complex numbers has a much richer structure than the field of ordinary complex numbers. This is used in order to extend the powerful tools of two-dimensional conformal field theories to higher dimensions. It is thus proposed that critical systems in more than two dimensions be first studied over the p-adics and then, if possible, recovered by the adelic construction. It is further argued that this higher-dimensional p-adic analyticity may be the key to membrane theories. A natural ansatz for three-brane tree-scattering amplitudes, where p-adic analyticity is instrumental, is given as an explicit example.     

p-adic quantum mechanics

An extension of the formalism of quantum mechanics to the case where the canonical variables are valued in a field ofp-adic numbers is considered. In particular the free particle and the harmonic oscillator are considered. In classicalp-adic mechanics we consider time as ap-adic variable and coordinates and momentum orp-adic or real. For the case ofp-adic coordinates and momentum quantum mechanics with complex amplitudes is constructed. It is shown that the Weyl representation is an adequate formulation in this case. For harmonic oscillator the evolution operator is constructed in an explicit form. For primesp of the form 4l+1 generalized vacuum states are constructed. The spectra of the evolution operator have been investigated. Thep-adic quantum mechanics is also formulated by means of probability measures over the space of generalized functions. This theory obeys an unusual property: the propagator of a massive particle has power decay at infinity, but no exponential one.     

On the adelic string amplitudes

Some remarkable properties of the adelic string amplitudes for the physical domain of the Mandelstam variables are considered. It is shown that the p-adic four-point functions are always negative. Also, a formula is obtained which expresses the product of moduli of the p-adic amplitudes and the Veneziano amplitude in terms of the zeta functions. This product is absolutely convergent unlike the divergent product of these amplitudes without moduli, recently considered by Freund and Witten. Using the zeta function representation, p-adic interpolation of the Veneziano amplitude is also considered.     

Some <Emphasis Type="Italic">p</Emphasis>-Adic Integrals on ℤ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> Associated with Trigonometric Functions

In this paper, we study some p-adic invariant and fermionic p-adic integrals on ?_p associated with trigonometric functions. By using these p-adic integrals we represent several trigonometric functions as a formal power series involving either Bernoulli or Euler numbers. In addition, we obtain some identities relating various special numbers like zigzag, some ‘trigonometric’, Bernoulli, Euler numbers, and Euler numbers of the second kind.     

Some identities on the q-Euler polynomials of higher order and q-stirling numbers by the fermionic p-adic integral on ℤ p

A systemic study of some families of q-Euler numbers and families of polynomials of Nörlund type is presented by using the multivariate fermionic p-adic integral on ? _p . The study of these higher-order q-Euler numbers and polynomials yields an interesting q-analog of identities for Stirling numbers.     

Airy Functions Over Local Fields

Airy integrals are very classical but in recent years they have been generalized to higher dimensions and these generalizations have proved to be very useful in studying the topology of the moduli spaces of curves. We study a natural generalization of these integrals when the ground field is a non-archimedean local field such as the field of p-adic numbers. We prove that the p-adic Airy integrals are locally constant functions of moderate growth and present evidence that the Airy integrals associated with compact p-adic Lie groups also have these properties.     

A note on p-adic Euler measure on ℤp

In this paper, we give some p-adic approximation of E _n,x for certain n. Finally we will treat p-adic l-function of Kubota-Leopoldt’s type Euler numbers and p-adic measure for Euler numbers.     

Frobenius transformation,mirror map and instanton numbers

We show that one can express Frobenius transformation on middle-dimensional p-adic cohomology of Calabi–Yau threefold in terms of mirror map and instanton numbers. We express the mirror map in terms of Frobenius transformation on p-adic cohomology. We discuss a p-adic interpretation of the conjecture about integrality of Gopakumar–Vafa invariants.     

Non-archimedean string dynamics

Explicit formulas for the N-point tree amplitudes of the non-archimedean open string are derived. These amplitudes can be generated from a simple non-local lagrangian involving a single scalar field (the tachyon) in ambient space-time. This lagrangian is studied and is found to possess a tachyon free vacuum with no “particles” but with soliton solutions. The question of generalizing the adelic product formular to N-point amplitudes is taken up. The infinite product of 5-point amplitudes is shown to converge in a suitably chosen kinematic region whence it can be analytically continued. Though the precise form of the product formula for the 5-point (and N-point)amplitudes is not found, it is shown that the product is not equal to one as it is for the 4-point amplitudes but rather involves the famous zeros of the Riemann zeta function. Chan-Paton rules for non-archimedean open strings are given. A string over the (global) field of rational numbers is constructed. Other problems that are addressed are the introduction of supersymmetry, the nature of a p-adic string lagrangian, and the possibility of strings over other locally compact fields.     

Covariant superstring fermion amplitudes from the sum over fermionic surfaces

Covariant fermion amplitudes in the NSR string are obtained from the sum over fermionic surfaces with marked points, world sheet fermionic fields being double valued at these points. It is shown that under such boundary conditions one cannot choose a superconformal gauge when the number N = 2p + 4 of marked points is more than 4. The gauge unequivalent supermetrics on a string world sheet are parametrized by grassmannian parameters (supermoduli) and the integration over them produces the transformation of p vertices into the form with the opposite ghost charge. The ghost contribution to the amplitudes is also computed.     

Extended q -euler numbers and polynomials associated with fermionic p -adic q -integral on Z p

The purpose of this paper is to construct extended q-Euler numbers and polynomials related to fermionic p-adic q-integral on ℤ _p . By evaluating a multivariate p-adic q-integral on ℤ _p , we give new explicit formulas related to these numbers and polynomials.     

Adelic conformal field theory

We stress the use of modular forms in obtaining adelic formulations of field theoretical problems. Supersymmetry then appears in the real section with thep-adic parts as arithmetic completions. We first show how the Casimir effect is naturally interpreted adelically and the coefficient arises from dimensional analysis. We then suggest looking at the zero slope limit of adelic string amplitudes. Finally, we interpret the rationality of the critical exponents for conformal field theories in terms of p-adic analyticity of correlation functions.     

Calculation of generalp-adic Feynman amplitude

The generaln-point masslessp-adic Feynman amplitude with arbitrary parameters of analytic regularization for each line is calculated. This result is presented in the form of a sum over hierarchies of a given graph. The structure of ultraviolet and infrared divergences ofp-adic Feynman amplitudes is characterized and the startriangle uniqueness identity in thep-adic case is derived.Supported by Alexander von Humboldt-Stiftung     

Invariance in adelic quantum mechanics

Adelic quantum mechanics is form-invariant under an interchange of real andp-adic number fields as well as rings ofp-adic integers. We also show that in adelic quantum mechanics Feynman’s path integrals for quadratic actions with rational coefficients are invariant under changes of their entries within nonzero rational numbers.     

Multiloop calculations inp-adic string theory and Bruhat-Tits trees

We treat the openp-adic string world sheet as a coset spaceF=T/, whereT is the Bruhat-Tits tree for thep-adic linear groupGL(2, _p ) and PGL(2, _p ) is some Schottky group. The boundary of this world sheet corresponds to ap-adic Mumford curve of finite genus. The string dynamics is governed by the local gaussian action on the coset spaceF. The tachyon amplitudes expressed in terms ofp-adic -functions are proposed for the Mumford curve of arbitrary genus. We compare them with the corresponding usual archimedean amplitudes. The sum over moduli space of the algebraic curves is conjectured to be expressed in the arithmetic surface terms. We also give the necessary mathematical background including the Mumford approach top-adic algebraic curves. The connection of the problem of closedp-adic strings with the considered topics is discussed.     

p-Adic string compactified on a torus

U(1) ^xD model with the Villain action on ag-loop generalizationF _g of the Bruhat-Tits tree for thep-adic linear groupGL(2, _p ) is considered. All correlation functions and the statistical sum are calculated. We compute also the averages of these correlation functions forN vertices attached to the boundary ofF _g. When the compactification radius tends to infinity the averages provide theg-loopN-point amplitudes of the uncompactifiedp-adic string theory, in particular forg=0 the Freund-Olson amplitudes.     

Lebesgue-Radon-Nikodym theorem with respect to fermionic p-adic invariant measure on ? p

In the paper, we derive an analog of the Lebesgue-Radon-Nikodym theorem with respect to fermionic p-adic invariant measure on ? _p .     

Non-archimedean strings and Bruhat-Tits trees

It is shown that the Bruhat-Tits tree for thep-adic linear groupGL(2) is a natural non-archimedean analog of the open string world sheet. The boundary of the tree can be identified with the field ofp-adic numbers. We construct a lattice quantum field theory on the Bruhat-Tits tree with a simple local lagrangian and show that it leads to the Freund-Olson amplitudes for emission processes of the particle states from the boundary.