The tadpole conjecture proposes that complex structure moduli stabilisation by fluxes that have low tadpole charge can be realised only at special points in moduli space, leading generically to (large) gauge symmetries. Here we provide an exhaustive survey of the gauge symmetries arising in F-theory flux compactifications on products of attractive $\mbox{K3}$ surfaces, with complex structure moduli fully stabilised. We compute the minimal rank of the left-over non-abelian gauge group for all flux configurations within the tadpole bound, finding that it is always non-zero. It decreases in a roughly linear fashion with the tadpole charge, reaching zero at charge 30. By working out possible gauge algebras for different values of the tadpole, we find that all simple ADE Lie algebras of rank $\le 18$ appear.
We use the Witten index in the open string sector to determine tadpole charges of orientifold planes and D-branes. As specific examples we consider type I compactifications on Calabi Yau manifolds and noncompact orbifolds. The tadpole constraints suggest that the standard embedding is not a natural choice for the gauge bundle. Rather there should be a close connection of the gauge bundle and the spin bundle. In the case of a four fold, the standard embedding does not in general fulfill the tadpole conditions. We show that this agrees with the Green-Schwarz mechanism. In the case of noncompact …
We investigate the properties of localized anomalous U(1)'s in heterotic string theory on the orbifold T^6/Z_3. We argue that the local four dimensional and original ten dimensional Green-Schwarz mechanisms can be implemented simultaneously, making the theory manifestly gauge invariant everywhere, in the bulk and at the fixed points. We compute the shape of the Fayet-Iliopoulos tadpoles, and cross check this derivation for the four dimensional auxiliary fields by a direct calculation of the tadpoles of the internal gauge fields. Finally we study some resulting consequences for spontaneous symm…
Orbifolds in field theory are potentially singular objects for at their fixed points the curvature becomes infinite, therefore one may wonder whether field theory calculations near orbifold singularities can be trusted. String theory is perfectly well defined on orbifolds and can therefore be taken as a UV completion of field theory on orbifolds. We investigate the properties of field and string theory near orbifold singularities by reviewing the computation of a one loop gauge field tadpole. We find that in string theory the twisted states give contributions that have a spread of a couple of …
Closed string theories on orbifolds contain both untwisted and twisted states. The latter are normally assumed to live exactly at the orbifold fixed points. We perform a calculation of a gauge field tadpole amplitude and show that off-shell both the twisted and untwisted states give rise to non-trivial momentum profiles over the orbifold C^3/Z_3. These profiles take the form of Gaussian distributions integrated over the fundamental domain of the modular parameter of the torus. The propagators of the internal coordinate fields on the torus world sheet determine the width of the Gaussian profile…
In orbifold gauge theories localized tadpoles can be radiatively generated at the fixed points where U(1) subgroups are conserved. If the Standard Model Higgs fields are identified with internal components of the bulk gauge fields (Higgs-gauge unification) in the presence of these tadpoles the Higgs mass becomes sensitive to the UV cutoff and electroweak symmetry breaking is spoiled. We find the general conditions, based on symmetry arguments, for the absence/presence of localized tadpoles in models with an arbitrary number of dimensions D. We show that in the class of orbifold compactificatio…
Certain classes of chiral four-dimensional gauge theories may be obtained as the worldvolume theories of D5-branes suspended between networks of NS5-branes, the so-called brane box models. In this paper, we derive the stringy consistency conditions placed on these models, and show that they are equivalent to anomaly cancellation of the gauge theories. We derive these conditions in the orbifold theories which are T-dual to the elliptic brane box models. Specifically, we show that the expression for tadpoles for unphysical twisted Ramond-Ramond 4-form fields in the orbifold theory are proportion…
We revisit models of gauge mediated supersymmetry breaking where messenger parity is violated. Such a symmetry is usually invoked in order to set to zero potentially dangerous hypercharge D-term tadpoles. A milder hypothesis is that the D-tadpole vanishes only at the first order in the gauge coupling constant. Then the next order leads to a contribution to the sfermion masses which is of the same magnitude as the usual radiative one. This enlarges the parameter space of gauge mediated models. We first give a completely general characterization of this contribution, in terms of particular three…
It is often stated in the literature concerning D=4,6 compact Type IIB orientifolds that tadpole cancellation conditions i) uniquely fix the gauge group (up to Wilson lines and/or moving of branes) and ii) are equivalent to gauge anomaly cancellation. We study the relationship between tadpole and anomaly cancellation conditions and qualify both statements. In general the tadpole cancellation conditions imply gauge anomaly cancellation but are stronger than the latter conditions in D=4, N=1 orientifolds. We also find that tadpole cancellation conditions in Z_N D=4,6 compact orientifolds do not …
We study RR charge cancellation consistency conditions in string compactifications with open string sectors, by introducing D-brane probes in the configuration. We show that uncancelled charges manifest as chiral gauge anomalies in the world-volume of suitable probes. RR tadpole cancellation can therefore be described as the consistency of the effective compactified theory not just in the vacuum, but also in all topological sectors (presence of D-brane probes). The result explains why tadpole cancellation is usually much stronger than anomaly cancellation of the compactified theory (in the vac…
In the 5-dimensional twisted $U(1)$-like gauge theory on $κ$-Minkowski, the one-loop one-point (tadpole) function was computed in arXiv:2107.14462. This article summarizes the construction of such a gauge theory and discusses the non-vanishing of the tadpole.
Recent work has provided a direct string calculation of the internal coordinate dependence of gauge field tadpoles on the orbifold C^3/Z_3. We investigate the structure of these profiles in momentum and coordinate space representations using both analytical and numerical methods. The twisted states are determined to be localized within a few string lengths of the singularity. This provides an example of how delta-like singularities, which are typical for field theories on orbifolds, are avoided in string theory. A systematic expansion of the full string result allows us to speculate on how the…
We discuss the role played by the divergences appearing in the interaction between a fractional D3 brane dressed with an SU(N) gauge field and a stack of N fractional D3 branes on the orbifolds C^2/Z_2 and C^3/(Z_2 x Z_2). In particular we show that the logarithmic divergences in the closed string channel, interpreted as due to twisted massless tadpoles, are mapped, under open/closed string duality, in the logarithmic ones in the open string channel, due to the massless states circulating in the annulus diagram and corresponding to the one-loop divergences that one finds in the gauge theory li…
We compute a class of gauge invariant observables for marginal solutions and the tachyon vacuum. In each case we find that the observables are related in a simple way to the closed-string tadpole on a disk with appropriate boundary conditions. We give a sketch of an argument that this result should hold in general using the BRST invariance of the closed string two-point function. Finally, we discuss the analogous set of invariants in the Berkovits superstring field theory.
Light-front gauge is the most popular one to work with fundamental interactions, due to its characteristic maximum kinematical Poincare operators that it allows. However, it is also known to be one of the trickiest gauges one can work with for gauge theories, due to its singular nature. So, in terms of perturbative calculations in the light-front, there are only a few published and tabulated results for the pertinent Feynman integrals, mostly involving massless integrals. And the majority of the results given are only for the divergent parts of them and the complete closed form (or with the fi…
Lepage and Mackenzie have shown that tadpole renormalization and systematic improvement of lattice perturbation theory can lead to much improved numerical results in lattice gauge theory. It is shown that lattice perturbation theory using the Cayley parametrization of unitary matrices gives a simple analytical approach to tadpole renormalization, and that the Cayley parametrization gives lattice gauge potentials gauge transformations close to the continuum form. For example, at the lowest order in perturbation theory, for SU(3) lattice gauge theory, at $β=6,$ the `tadpole renormalized' couplin…
In this contribution to the Proceedings of the Corfu Summer Institute 2015 we present our study on photon self-interaction in the deformed U(1) gauge field theory defined via Seiberg-Witten map. We derive the theta-exact expressions for the four photon self-coupling in this theory with a large number of gauge symmetry inspired freedom parameters included. The physical effect of four photon self-coupling is accessed by its contribution to photon polarization tensor via 1-loop four-photon-tadpole-diagram. The four-photon tadpole diagram consists the same tensorial structures as the photon self-i…
We discuss non-conformal gauge theories from type IIB D3-branes embedded in orbifolded space-times. Such theories can be obtained by allowing some non-vanishing logarithmic twisted tadpoles. In certain cases with N=0,1 supersymmetry correlation functions in the planar limit are the same as in the parent N=2 supersymmetric theories. In particular, the effective action in such theories perturbatively is not renormalized beyond one loop in the planar limit. In the N=2 as well as such N=0,1 theories quantum corrections in the D3-brane gauge theories are encoded in the corresponding classical highe…
A gauge theory with gauge group G defined in D>4 space-time dimensions can be broken to a subgroup H on four dimensional fixed point branes, when compactified on an orbifold. Mass terms for extra dimensional components of gauge fields A_i (brane scalars) might acquire (when allowed by the brane symmetries) quadratically divergent radiative masses and thus jeopardize the stability of the four-dimensional theory. We have analyzed Z_2 compactifications and identified the brane symmetries remnants of the higher dimensional gauge invariance. No mass term is allowed for D=5 while for D>5 a tadpole ε…
We consider a gauge theory on the 5-d $κ$-Minkowski which can be viewed as the noncommutative analog of a $U(1)$ gauge theory. We show that the Hermiticity condition obeyed by the gauge potential $A_μ$ is necessarily twisted. Performing a BRST gauge-fixing with a Lorentz-type gauge, we carry out a first exploration of the one loop quantum properties of this gauge theory. We find that the gauge-fixed theory gives rise to a non-vanishing tadpole for the time component of the gauge potential, while there is no non-vanishing tadpole 1-point function for the spatial components of $A_μ$. This signal…
In this work we classify the subalgebras satisfied by non-geometric Q-fluxes in type IIB orientifolds on T^6/(Z_2 x Z_2) with three moduli (S,T,U). We find that there are five subalgebras compatible with the symmetries, each one leading to a characteristic flux-induced superpotential. Working in the 4-dimensional effective supergravity we obtain families of supersymmetric AdS_4 vacua with all moduli stabilized at small string coupling g_s. Our results are mostly analytic thanks to a judicious parametrization of the non-geometric, RR and NSNS fluxes. We are also able to leave the flux-induced C…
A new version of differential renormalization is presented. It is based on pulling out certain differential operators and introducing a logarithmic dependence into diagrams. It can be defined either in coordinate or momentum space, the latter being more flexible for treating tadpoles and diagrams where insertion of counterterms generates tadpoles. Within this version, gauge invariance is automatically preserved to all orders in Abelian case. Since differential renormalization is a strictly four-dimensional renormalization scheme it looks preferable for application in each situation when dimens…
We discuss the structure of 4D gauged supergravity algebras corresponding to globally non-geometric compactifications of F-theory, admitting a local geometric description in terms of 10D supergravity. By starting with the well known algebra of gauge generators associated to non-geometric type IIB fluxes, we derive a full algebra containing all, closed RR and NSNS, geometric and non-geometric dual fluxes. We achieve this generalization by a systematic application of SL(2,Z) duality transformations and by taking care of the spinorial structure of the fluxes. The resulting algebra encodes much in…
We discuss the first string theory examples of three generation non-supersymmetric SU(5) and {\em flipped} SU(5) GUTS, which break to the Standard model at low energy, without extra matter and/or gauge group factors. Our GUT examples are based on IIA $Z_3$ orientifolds with D6-branes intersecting at non-trivial angles. These theories necessarily satisfy RR tadpoles and are free of NSNS tadpoles as the complex structure moduli are frozen (even though a dilaton tadpole remains) to discrete values. We identify appropriately the bifundamental Higgses responsible for electroweak symmetry breaking. …
If the fundamental type-I string scale is of the order of few TeV, the problem of the gauge hierarchy is that of understanding why some dimensions transverse to our brane-world are so large. The technical aspect of this problem, as usually formulated, is `why quantum corrections do not modify drastically the masses and other parameters of the Standard Model'. We argue that within type-I perturbation theory, the technical hierarchy problem is solved (a) if all massless tadpoles cancel locally over distances of order the string length in the transverse space, or (b) if the massless fields with u…
We study F-Theory compactifications to four dimensions that exhibit discrete gauge symmetries. Geometrically these arise by deforming elliptic fibrations with two sections to a genus-one fibration with a bi-section. From a four-dimensional field-theory perspective they are remnant symmetries from a Higgsed U(1) gauge symmetry. We implement such symmetries in the presence of an additional SU(5) symmetry and associated matter fields, giving a geometric prescription for calculating the induced discrete charge for the matter curves and showing the absence of Yukawa couplings that are forbidden by …
We classify discrete modular symmetries in the effective action of Type IIB string on toroidal orientifolds with three-form fluxes, emphasizing on $T^6/\mathbb{Z}_2$ and $T^6/(\mathbb{Z}_2\times \mathbb{Z}_2^\prime)$ orientifold backgrounds. On the three-form flux background, the modular group is spontaneously broken down to its congruence subgroup whose pattern is severely constrained by a quantization of fluxes and tadpole cancellation conditions. We explicitly demonstrate that the congruence subgroups appearing in the effective action arise on magnetized D-branes wrapping certain cycles of …
Given the important role of discrete gauge symmetries in viable models, we discuss these symmetries in intersecting D6-brane trinification model where the ZN symmetry is investigated and its identification is shown.
We study compactified pure gauge/gravitational theories with gauge-fixing terms and show that these theories possess quantum mechanical SUSY-like symmetries between unphysical degrees of freedom. These residual symmetries are global symmetries and generated by quantum mechanical N=2 supercharges. Also, we establish new one-parameter family of gauge choices for higher-dimensional gravity, and calculate as a check of its validity one graviton exchange amplitude in the lowest tree-level approximation. We confirm that the result is indeed $ξ$-independent and the cancellation of the $ξ$-dependence …
We study the geometry of the Lagrangian Batalin--Vilkovisky theory on an antisymplectic manifold. We show that gauge symmetries of the BV-theory are essentially the symmetries of an even symplectic structure on the stationary surface of the master action.
A Hamiltonian formulation of gauge symmetries on noncommutative ($θ$ deformed) spaces is discussed. Both cases- star deformed gauge transformation with normal coproduct and undeformed gauge transformation with twisted coproduct- are considered. While the structure of the gauge generator is identical in either case, there is a difference in the computation of the graded Poisson brackets that yield the gauge transformations. Our analysis provides a novel interpretation of the twisted coproduct for gauge transformations.
Field theories with p-form gauge potentials can possess ``hidden'' symmetries leaving the field strengths invariant on-shell without being gauge symmetries on-shell. The relevance of such symmetries to supersymmetric models is discussed. They provide central charges of supersymmetry algebras, play a particular role in duality relations, and lead to peculiar interactions. A multiplet of N=2 supersymmetry in four dimensions with two hidden central charges is presented.
In this paper we study U(1) gauge transformations on the space-time coordinates and on the background fields $g_{μν}$ and $ϕ$. For some special gauge functions, gauged coordinates and gauged U(1) field are equivalent to the rotated coordinates and rotated gauge field. We find gauge transformations that are symmetries of the string action. Also we obtain general $T$-duality transformations for the background fields. For special background fields this duality is equivalent to a gauge transformation.
Anomalous U(1) gauge symmetries can appear both in heterotic and type I string theories. In the heterotic case we find a single anomalous U(1), while in open string theories several such symmetries can appear. Nonetheless, there is a conjectured duality symmetry that might connect these two theories. We review the properties of anomalous gauge symmetries in various string theories as well as the status of this heterotic-type I/II duality. We also comment on the possible phenomenological applications of anomalous gauge symmetries in string theory.
We study gauge threshold corrections for local GUT models in IIB/F-theory. Consistency with holomorphy requirements of supergravity and the Kaplunovsky-Louis formula implies that the unification scale is enhanced by the bulk radius R from the string scale to M_X =RM_S. We argue that the stringy interpretation of this is via a locally uncancelled tadpole sourced by the hypercharge flux. This sources closed string modes propagating into the bulk; equivalently open string gauge coupling running up to the winding scale M_X. The enhancement to R M_s is tied to GUT breaking by a globally trivial hyp…
We reconsider the problem of finding all local symmetries of a Lagrangian. Our approach is completely Hamiltonian without any reference to the associated action. We present a simple algorithm for obtaining the restrictions on the gauge parameters entering in the definition of the generator of gauge transformations.
We study the enhanced gauge symmetries at constant couplings in the moduli space of the Gimon-Polchinski Orientifold. We show that whenever there is a singularity of E_6, E_7 and E_8 type we get enhanced gauge symmetry U(5), U(6) and U(8) respectively. This is due to the additional twisting involved in the definition of the Gimon-Polchiniski orientifold.
This review is dedicated to some modern applications of the remarkable paper written in 1918 by E. Noether. On a single paper, Noether discovered the crucial relation between symmetries and conserved charges as well as the impact of gauge symmetries on the equations of motion. Almost a century has gone since the publication of this work and its applications have permeated modern physics. Our focus will be on some examples that have appeared recently in the literature. This review is aim at students, not researchers. The main three topics discussed are (i) global symmetries and conserved charge…
Using well known Lagrangean techniques for uncovering the gauge symmetries of a Lagrangean, we derive the transformation laws for the phase space variables corresponding to local symmetries of the Hamilton equations of motion. These transformation laws are shown to coincide with those derived by Hamiltonian methods based on the Dirac conjecture. The connection between the Lagrangean and Hamiltonian approach is illustrated for first class systems involving one primary constraint.
We study the underlying gauge symmetry algebra of the $N=2$ string, which is broken down to a subalgebra in any spacetime background. For given toroidal backgrounds, the unbroken gauge symmetries (corresponding to holomorphic and antiholomorphic worldsheet currents) generate area-preserving diffeomorphism algebras of null 2-tori. A minimal Lie algebraic closure containing all the gauge symmetries that arise in this way, is the background--independent volume--preserving diffeomorphism algebra of the target Narain torus $T^{4,4}$. The underlying symmetries act on the ground ring of functions on …
Duality symmetries are discussed for non-linear gauge theories of (n-1)-th rank antisymmetric tensor fields in general even dimensions d=2n. When there are M field strengths and no scalar fields, the duality symmetry groups should be compact. We find conditions on the Lagrangian required by compact duality symmetries and show an example of duality invariant non-linear theories. We also discuss how to enlarge the duality symmetries to non-compact groups by coupling scalar fields described by non-linear sigma models.
Hidden symmetries in a covariant Hamiltonian formulation are investigated involving gauge covariant equations of motion. The special role of the Stackel-Killing tensors is pointed out. A reduction procedure is used to reduce the original phase space to another one in which the symmetries are divided out. The reverse of the reduction procedure is done by stages performing the unfolding of the gauge transformation followed by the Eisenhart lift in connection with scalar potentials.
Recent investigations into asymptotic symmetries of gauge theory and gravity have illuminated connections between gauge field zero-mode sectors, the corresponding soft factors, and their classically observable counterparts -- so called "memories." Here we complete this triad for the case of large U(1) gauge symmetries at null infinity.
In string theory, there are no continuous global symmetries. Discrete symmetries frequently appear, and these can often be understood as unbroken subgroups of larger, spontaneously broken gauge symmetries (discrete gauge symmetries). In cases which have been studied previously, anomalies in these symmetries could always be cancelled by a Green-Schwarz mechanism. In the present work, we describe results of an investigation of a large number of $\ZZ_3$ orbifold models with two and three Wilson lines. We find that the discrete gauge anomalies can always be cancelled.}
In particle physics model building discrete symmetries are often invoked to forbid unwanted or dangerous couplings. A classical example is the R-parity of the MSSM, which guarantees the absence of dimension four baryon- and lepton-number violating operators. Although phenomenologically useful, these discrete symmetries are, in the context of field theory, poorly motivated at a more fundamental level. Moreover, discrete {\em global} symmetries are expected to be violated in consistent couplings to quantum gravity, while their {\em gauged} versions are expected to actually exist. In this paper w…
We consider the general case of a type IIA string compactified on a Calabi-Yau manifold which has a heterotic dual description. It is shown that the nonabelian gauge symmetries which can appear nonperturbatively in the type II string but which are understood perturbatively in the heterotic string are purely a result of string-string duality in six dimensions. We illustrate this with some examples.
Krauss and Wilczek have shown that an unbroken discrete gauge symmetry is respected by gravitationally mediated processes. This has led to a search for such a symmetry compatible with the standard model or MSSM that would protect protons from gravitationally mediated decay in a universe with a low scale for quantum gravity (large extra dimensions). The fact that the discrete symmetry must remain unbroken and have a gauge origin puts important restrictions on the space of possible discrete symmetries.
We construct a modification of the Poisson bracket which is suitable for a canonical analysis of space-time noncommutative field theories. We show that this bracket satisfies the Jacobi identities and generates equations of motion. In this modified canonical formalism one can define the notion of the first-class constraints, demonstrate that they generate gauge symmetries, and derive an explicit form of these symmetry transformations.
We study the space-time symmetries and transformation properties of the non-commutative U(1) gauge theory, by using Noether charges. We carry out our analysis by keeping an open view on the possible ways $θ^{μν}$ could transform. We conclude that $θ^{μν}$ cannot transform under any space-time transformation since the theory is not invariant under the conformal transformations, with the only exception of space-time translations. The same analysis applies to other gauge groups.
We show how the Newton-Hooke (NH) symmetries, representing a nonrelativistic version of de-Sitter symmetries, can be enlarged by a pair of translation vectors describing in Galilean limit the class of accelerations linear in time. We study the Cartan-Maurer one-forms corresponding to such enlarged NH symmetry group and by using cohomological methods we determine the general 2-parameter (in D=2+1 4-parameter)central extension of the corresponding Lie algebra. We derive by using nonlinear realizations method the most general group - invariant particle dynamics depending on two (in D=2+1 on four)…
A superspace formulation is proposed for the osp(1,2)-covariant Lagrangian quantization of general massive gauge theories. Thereby, osp(1,2) is considered as subalgebra of the superalgebra sl(1,2) which is interpreted as conformal algebra acting on two anticommuting coordinates. The mass-dependent (anti)BRST symmetries of the quantum action in the osp(1,2) superfield formalism are realized as translations associated by mass-dependent special conformal transformations.
The gauge symmetries of a general dynamical system can be systematically obtained following either a Hamiltonean or a Lagrangean approach. In the former case, these symmetries are generated, according to Dirac's conjecture, by the first class constraints. In the latter approach such local symmetries are reflected in the existence of so called gauge identities. The connection between the two becomes apparent, if one works with a first order Lagrangean formulation. We thereby confirm Dirac's conjecture. Our analysis applies to arbitrary constrained systems with first and second class constraints…
We investigate the relation between gauge theories and brane configurations described by brane tilings. We identify U(1)_B (baryonic), U(1)_M (mesonic), and U(1)_R global symmetries in gauge theories with gauge symmetries in the brane configurations. We also show that U(1)_MU(1)_B^2 and U(1)_RU(1)_B^2 't Hooft anomalies are reproduced as gauge transformations of the classical brane action.
We study new symmetries between A and D type quiver gauge theories with different numbers of colors. We realize these gauge theories with twisted masses via a brane construction that reproduces all the parameters of the Gauge/Bethe correspondence.
String-string duality dictates that type IIA strings compactified on a K3 surface acquire non-abelian gauge groups for certain values of the K3 moduli. We argue that, contrary to expectation, the theories for which such enhanced gauge symmetries appear are not orbifolds in the string sense. For a specific example we show that a theory with enhanced gauge symmetry and an orbifold theory have the same classical K3 surface as a target space but the value of the ``B-field'' differs. This raises the possibility that the conformal field theory associated to a string theory with an enhanced gauge gro…
Picture changed operators are discussed in $N=2$ strings with space-time signature $(2,2)$. A gauge symmetry algebra is derived in a background of torus space-time $T^{2,2}$ and its simple representation on the picture changed operators is given. Simple Ward identities associated with the gauge algebra and their consequences for three and four point amplitudes of arbitrary loops are also discussed.
