Higher derivative modifications ofD=10 superspace Yang-Mills theories

We extend the superspace treatment of higher derivative supersymmetric Yang-Mills theories, emphasizing the role played by manifestD=10 Lorentz covariance and supersymmetry invariance. As an example, the superspace formulation of the effective action is considered for the massless fields in the open superstring, as well as in theSO(32) andE ₈×E₈ superstring theories. We show that there exists a unique modification of thef-tensor supercurrent which can provide the embedding of theO(α'³) string corrections in the slope parameter expansion for the Yang-Mills sector inD=10 superspace. The new correction term is relevant for the understanding of nonperturbative effects.     

Superstring theory

Dual string theories, initially developed as phenomenological models of hadrons, now appear more promising as candidates for a unified theory of fundamental interactions. Type I superstring theory (SST I), is a ten-dimensional theory of interacting open and closed strings, with one supersymmetry, that is free from ghosts and tachyons. It requires that an SO(n) or Sp(2n) gauge group be used. A light-cone-gauge string action with space-time supersymmetry automatically incorporates the superstring restrictions and leads to the discovery of type II superstring theory (SST II). SST II is an interacting theory of closed strings only, with two D = 10 supersymmetries, that is also free from ghosts and tachyons. By taking six of the spatial dimensions to form a compact space, it becomes possible to reconcile the models with our four-dimensional perception of spacetime and to define low-energy limits in which SST I reduces to N = 4, D = 4 super Yang-Mills theory and SST II reduces to N = 8, D = 4 supergravity theory. The superstring theories can be described by a light-cone-gauge action principle based on fields that are functionals of string coordinates. With this formalism any physical quantity should be calculable. There is some evidence that, unlike any conventional field theory, the superstring theories provide perturbatively renormalizable (SST I) or finite (SST II) unifications of gravity with other interactions.     

Double field formulation of Yang-Mills theory

Based on our previous work on the differential geometry for the closed string double field theory, we construct a Yang-Mills action which is covariant under O(D,D) T-duality rotation and invariant under three-types of gauge transformations: non-Abelian Yang-Mills, diffeomorphism and one-form gauge symmetries. In double field formulation, in a manifestly covariant manner our action couples a single O(D,D) vector potential to the closed string double field theory. In terms of undoubled component fields, it couples a usual Yang-Mills gauge field to an additional one-form field and also to the closed string background fields which consist of a dilaton, graviton and a two-form gauge field. Our resulting action resembles a twisted Yang-Mills action.     

Partially massless higher-spin theory II: one-loop effective actions

We continue our study of a generalization of the D-dimensional linearized Vasiliev higher-spin equations to include a tower of partially massless (PM) fields. We compute one-loop effective actions by evaluating zeta functions for both the “minimal” and “non-minimal” parity-even versions of the theory. Specifically, we compute the log-divergent part of the effective action in odd-dimensional Euclidean AdS spaces for D = 7 through 19 (dual to the a-type conformal anomaly of the dual boundary theory), and the finite part of the effective action in even-dimensional Euclidean AdS spaces for D = 4 through 8 (dual to the free energy on a sphere of the dual boundary theory). We pay special attention to the case D = 4, where module mixings occur in the dual field theory and subtlety arises in the one-loop computation. The results provide evidence that the theory is UV complete and one-loop exact, and we conjecture and provide evidence for a map between the inverse Newton’s constant of the partially massless higher-spin theory and the number of colors in the dual CFT.     

New heterotic string theories in uncompactified dimensions < 10

It is shown that infinitely many heterotic string theories exist in uncompactified dimensions less than 10, that are one-loop finite (for massless external legs). Tachyons are removed by compactifying into tori (10 − d) and (26 −d) dimensions of the right-moving superstring and left-moving bosonic string sectors, respectively. The condition for modular invariance is shown to be equivalent to self-duality condition on even lorentzian lattices with (10 −d) and (26 −d) timelike and spacelike directions, respectively. The construction results in a (10 −d)(26 −d) parameter family of one-loop finite string theories. The zero mass sector of these theories for d = 4 and 6 correspond to N = 4 and 2 supergravity coupled to super Yang-Mills with many possible groups, some of which cannot be obtained by compactifying d = 10 heterotic string theory.     

Holonomies of gauge fields in twistor space 4: Functional MHV rules and one-loop amplitudes

We consider generalization of the Cachazo-Svrcek-Witten (CSW) rules to one-loop amplitudes of N=4 super Yang-Mills theory in a recently developed holonomy formalism in twistor space. We first reconsider off-shell continuation of the Lorentz-invariant Nair measure for the incorporation of loop integrals. We then formulate an S-matrix functional for general amplitudes such that it implements the CSW rules at quantum level. For one-loop MHV amplitudes, the S-matrix functional correctly reproduces the analytic expressions obtained in the Brandhuber-Spence-Travaglini (BST) method. Motivated by this result, we propose a novel regularization scheme by use of an iterated-integral representation of polylogarithms and obtain a set of new analytic expressions for one-loop NMHV and N²MHV amplitudes in a conjectural form. We also briefly sketch how the extension to one-loop non-MHV amplitudes in general can be carried out.     

Quantum properties of higher dimensional and dimensionally reduced supersymmetric theories

We discuss the relation of quantum properties of d-dimensional and dimensionally reduced theories with a special emphasis on the use of a regularization accounting for power divergences. The main examples are N=1, d=10 super Yang-Mills (SYM₁₀¹) and N=1, d=11 supergravity (SG₁₁¹) and their reductions. In particular, we understand the vanishing of the one-loop β-function in SYM₄⁴ (zero conformal anomalies in SG₄⁸) as a consequence of the absence of L⁶ (L⁷) power divergences in SYM₁₀¹(SG₁₁¹). The heat kernel expansion coefficients b_2k, k=0,…, 3 are found to be zero for SG₁₁¹ (as well as for SYM₁₀¹ and SG₁₀¹ and thus the one-loop finiteness of maximal SGs in d?7 is explicity demonstrated. We also present the expressions for the one-loop constant gauge field effective lagrangian and scalar effective potential in the SYM₄⁴ theory and analyse the problem of N=4 supersymmetry breaking.     

Pseudo-supergravity extension of the bosonic string

We construct a “pseudo-supersymmetric” fermionic extension of the effective action of the bosonic string in arbitrary spacetime dimension D. The theory is invariant under pseudo-supersymmetry transformations up to the quadratic fermion order, which is sufficient in order to be able to derive Killing spinor equations in bosonic backgrounds, and hence to define BPS type solutions determined by a system of first-order equations. The pseudo-supersymmetric theory can be extended by coupling it to a Yang-Mills pseudo-supermultiplet. This also allows us to construct “α^′ corrections” involving quadratic curvature terms. An exponential dilaton potential term, associated with the conformal anomaly for a bosonic string outside its critical dimension, can also be pseudo-supersymmetrised.     

On the low-energy effective action of N = 2 supersymmetric Yang-Mills theory

We investigate the perturbative part of Seiberg's low-energy effective action of N = 2 supersymmetric Yang-Mills theory in Wess-Zumino gauge in the conventional effective field theory technique. Using the method of constant field approximation and restricting the effective action with at most two derivatives and not more than four-fermion couplings, we show some features of the low-energy effective action given by Seiberg based on U(1)_R anomaly and non-perturbative β-function arguments.     

Elliptic index and superstring effective actions

Certain N = 1 supersymmetric string one-loop effective actions can be obtained directly from the path integral. As the computation is essentially the same as the one leading to the index of the Dirac-Ramond operator, they are determined by the gauge and gravitational anomaly structure of the theory. Specifically, we calculate the four-point effective action in ten dimensions, the corrections to the kinetic terms in d = 6 (including auxiliary fields) and the Fayet-Iliopoulos D-term in d = 4. We also compute the β-function of four-dimensional N = 2 theories from the elliptic genus in d = 6. Furthermore, we derive supersymmetry Ward type identities in terms of Kac-Moody characters, relating parity conserving with parity violating amplitudes.     

Instanton zero modes and β-functions in conformal supergravity

We show that one-loop infinities in N-extended conformal supergravities are produced only by zero modes when calculated on gravitational self-dual (N?1), de Sitter (N?3) and SU_N gauge instanton (N?3) backgrounds. Guided by analogy with the super Yang-Mills case we then discuss how to extract information about exact β-functions. We also establish a part of the N?4 lagrangian sufficient for one loop calculations in the U_N sector and carry out independent calculations of one-loop N?4 gauge β-functions separately in U₁ and SU_N background sectors.     

One-loop string corrections to the effective field theory

We calculate the parity-conserving one-loop string corrections to the bosonic part of the 10-dimensional effective field theory for the heterotic string. There is no renormalization of the Chamseddine-Chapline-Manton supergravity action, but terms of fourth order in the curvature tensors are generated. Most of these are of the same forms as those found at the tree level of the string topological expansion, such as (TrR²)², (TrR²)(TrF²) and (TrF²)². However, we also find a non-zero Tr(F⁴) coupling which was not present at the tree level. We comment on the phenomenological implications of these results, which include absences of renormalization of the Kähler metric and of the gauge kinetic function of the effective field theory in four dimensions after compactification, and a modification of the classical equations to be obeyed by the manifold of compactification.     

Infrared regularization of superstring theory and the one-loop calculation of coupling constants

Infrared regularized versions of 4-D N = 1 superstring ground states are constructed by curving the spacetime. A similar regularization can be performed in field theory. For the IR regularized string ground states we derive the exact one-loop effective action for non-zero U(1) or chromomagnetic fields as well as gravitational and axionic-dilatonic fields. This effective action is IR and UV finite. Thus, the one-loop corrections to all couplings (gravitational, gauge and Yukawas) are unambiguously computed. These corrections are necessary for quantitative string superunification predictions at low energies. The one-loop corrections to the couplings are also found to satisfy Infrared Flow Equations.     

The Born-Infeld action from conformal invariance of the open superstring

We show that the one-loop approximation to sigma-model perturbation theory for the open-superstring leads to the low-energy effective action of Born-Infeld for the gauge field coupled to the ends of the string. Thus the bosonic part of the low-energy open superstring effective action is just that of the open bosinic string. We also that these results are exact to all loop orders of sigma-model perturbation theory.     

Effective action in higher derivative (Super)gravities

'The one-loop effective action (EA) with an accuracy up to linear curvature terms ind=4R ²-gravity, conformal gravity, andN=1,d=4 conformal supergravity on the backgroundR ₄×T_4–k,k=1, 2, 3 is calculated. (Here,R _k is thek-dimensional curved space, T_n is then-dimensional torus). The one-loop EA in multidimensionalR ²-gravity and ind=10 conformal supergravity on the backgroundR ₄ ×T _d–4 is also obtained. The mechanism of inducing the Einstein gravity from the EA of considered theories of higher derivative (super)gravity is presented.We are grateful to I. L. Bukhbinder for the numerous discussions of considered questions.     

Vector field effective action in the open superstring theory

Using a path integral technique we find a closed expression (to all orders in α′) for the abelian, constant field strenght limit of the (tree) effective action for the massless vector field in the open superstring theory. The result is a modification of the Born-Infeld action found in the Bose string theory case. One-loop correction to the effective action is computed and shown to be finite if the gauge group is SO(32). It is demonstrated how the on-shell superstring scattering amplitudes can be calculated in the path-integral approach. We determine the leading (O(α') and O(α′²)) terms in the full non-abelian effective action starting from the known results for the 3-point and 4-point amplitudes. We find that because of the equivalence theorem the coefficients of some of the invariant structures in the effective lagrangian cannot be fixed from the S-matrix. In the path integral approach this ambiguity manifests itself as a 2d renormalization scheme (and Weyl gauge choice) ambiguity. We also discuss the leading terms in the gravitational effective actions in the closed (super) string theories and point out that whether or not the R² terms form the “Gauss-Bonnet” combination depends on choice of a renormalization (massless exchange subtraction) scheme.     

The one-loop Green’s functions of dimensionally reduced gauge theories

We apply the dimensional regularization technique as well as that by dimensional reduction to the calculation of the regularized one-loop Green’s functions ind ₀-dimensional Yang-Mills theory with real massless scalars and spinors in arbitrary (real) representations of a gauge groupG. As a particular example, the super-symmetrically regularized one-loop Green’s functions of theN=4 supersymmetric Yang-Mills model are derived.     

String loop effect on the BRST charge

An effective BRST charge Q_BRST which incorporates the string one-loop corrections is presented for the closed basonic string in an arbitrary background. The effective σ-model action which leads to such a Q_BRST is obtained and some consequences are discussed.