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Derived Categories E.R. Sharpe, in Encyclopedia of Mathematical Physics, 2006 Bondal–Kapranov Enhancements Mathematically derived categories are not quite as ideal as one would like. For example, the cone construction used in triangulated categories does not behave functorially – the cone depends upon the representative of the equivalence class defining an object in a derived category, and not just the object itself. Physically, our discussion of brane/antibrane systems was not the most general possible. One can give vacuum expectation values to more general vertex operators, not just the tachyons. Curiously, these two issues solve each other. By incorporating a more general class of boundary vertex operators, one realizes a more general mathematical structure, due to Bondal and Kapranov, which repairs many of the technical deficiencies of ordinary derived categories. Ordinary complexes are replaced by generalized complexes in which arrows can map between non-neighboring elements of the complex. Schematically, the BRST operator is deformed by boundary vacuum expectation values to the form and demanding that the BRST operator square to zero implies that which is the same as the condition for a generalized complex. Note that for ordinary complexes, the condition above factors into which yields an ordinary complex of sheaves (Figure 1). Sign in to download full-size image Figure 1. 1. Example of generalized complex. Each arrow is labeled by the degree of the corresponding vertex operator. THE CONCEPTUAL BASIS OF QUANTUM FIELD THEORY Gerard't Hooft, in Philosophy of Physics, 2007 2.2 Spontaneous symmetry breaking: Goldstone modes In the classical theory, the Hamilton density is (12) The theory is invariant under the group of transformations (13) if A is orthogonal and the potential function V (ϕ) is invariant under that group. The simplest example is the transformation ϕ ⟷ -ϕ: (14) There are two cases to consider: i) a > 0. In this case, ϕ = 0 is the absolute minimum of V. We write (15) and find that m indeed describes the mass of the particle. All Feynman diagrams have an even number of external lines. Since, in the quantum theory, these lines will be associated with particles, we find that states with an odd number of particles can never evolve into states with an even number of particles, and vice versa. If we define the quantum number PC = (−1)N, where N is the number of ϕ particles, then we find that PC is conserved during interactions. ii) a < 0. In this case, we see that: — trying to identify the mass of the particle using Eq. (15) yields the strange result that the mass would be purely imaginary. Such objects (“tachyons”) are not known to exist and probably difficult to reconcile with causality, and furthermore: — the configuration ϕ = 0 does not correspond to the lowest energy configuration of the system. The lowest energy is achieved when (16) It is now convenient to rewrite the potential V as (17) where we did not bother to write down the value of the constant C, since it does not occur in the evolution equations (2.1). There are now two equivalent vacuum states, the minima of V. Choosing one of them, we introduce a new field variable to write (18) and we see that a) for the new field , the mass-squared is positive, and b) a three-prong vertex appeared, with associated factor λF. The quantum number PC is no longer apparently conserved. This phenomenon is called ‘spontaneous symmetry breaking’, and it plays an important role in Quantum Field Theory. Next, let us consider the case of a continuous symmetry. The prototype example is the U (1) symmetry of a complex field. The symmetry group consists of the transformations A (θ), where θ is an angle: (19) Again, the most general potential3 invariant under these transformations is (20) In the case where the U (1) symmetry is apparent, one can rewrite the Feynman rules to apply directly to the complex field ϕ, noticing that one can write the potential V as a real function of the two independent variables ϕ and ϕ*. With (21) one notices that the Feynman propagators can be written with an arrow in them: an arrow points towards a point x where the function ϕ(x) is called for, and away from a point x′ where a factor ϕ*(x′) is extracted from the potential V. At every vertex, as many arrows enter as they leave, and so, during an interaction, the total number of lines pointing forward in time minus the number of lines pointing backward is conserved. This is an additively conserved quantum number, to be interpreted as a ‘charge’ Q. According to Noether's theorem, every symmetry is associated to such a conservation law. However, if a < 0, this U (1) symmetry is spontaneously broken. Then we write (22) This time, the stable vacuum states form a closed circle in the complex plane of ϕ values. Let us write (23) The striking thing about this potential is that the mass term for the field is missing. The mass squared for the field is . The fact that one of the effective fields is massless is a fundamental consequence of the fact that we have spontaneous breakdown of a continuous symmetry. Quite generally, there is a theorem, called the Goldstone theorem: If a continuous symmetry whose symmetry group has N independent generators, is broken down spontaneously into a (residual) symmetry whose group has N1 independent generators, then N – N1 massless effective fields emerge. The propagators for massless fields obey Eq. (6) without the m2 term, which gives these expressions an ‘infinite range’: such a Green's function drops off only slowly for large spatial or timelike separations. These massless oscillating modes are called ‘Goldstone modes’. String Theory and the Real World: From Particle Physics to Astrophysics J.L.F. Barbón, E. Rabinovici, in Les Houches, 2008 2 Introduction In general a system described by a local field theory in d + 1 spacetime dimensions containing a finite number of mass thresholds has an asymptotic density of states Ω (E) that increases according to the entropic law S(E) = log Ω (E) ∝, where the energy E is assumed to be well above all the thresholds. The fact that the power of the energy is positive, but less than unity, for any finite dimension ensures that the temperature of the system can be increased at will and the derived specific heat is always positive. Free string theory does not have a finite number of mass thresholds and indeed its entropy for energies well above the string scale, ms, is larger than that of a field theory at a given energy, it is S(E) ∝ E/ms. This results in a bound on the largest possible temperature, Ts ∼ ms = 1/ℓs, called the Hagedorn temperature as well as in zero specific heat [1]. Even for the free string such a behavior would lead to the emergence of tachyons unless the system is non generic, essentially supersymmetric [9]. The behavior of the system at a finite value of the string coupling gs it at question. Schwarzschild black holes in flat space have an even larger density of states: the entropy of a black hole of mass E grows as S(E) ∝. This large density of states leads not only to a maximal temperature but to a system which has negative specific heat and eventually such an asymptotic behavior does not allow to discuss a thermodynamical limit with non-zero energy density states. It also leads to other types of highly non-standard behavior [10]. One may well need to eventually readjust one's thinking to accommodate these features. After all, if our universe turns out to be metastable many issues need to be reevaluated. However another possibility is that the excessive number of degrees of freedom is a cry for help. Perhaps above a certain energy the nature of the weakly coupled—and thus reliable—degrees of freedom describing the system is very different. Once the system is diagnosed in terms of the new degrees of freedom the peculiarities may disappear. Historically this was the case for strings. At the onset of string theory it was thought that the spectrum of strings could be related to the hadronic one. The hadrons had a Hagedorn spectrum but it was realized that, once they interact, they are no longer the relevant degrees of freedom at high energies. It is the hadronic constituents, the gluons and quarks, which describe better the system. Their entropy is field theoretic, thus removing the upper bound on the temperature. The asymptotic density of states was field theoretic but there is a range of energies over which the Hagedorn spectrum is the appropriate description of the system. Also for finite gs—that is in the presence of gravity—it was suggested [14] that the long excited strings cease to be the relevant degrees of freedom above a certain energy and that the black hole description takes over. This however only made the behavior of the system more puzzling from the point of view discussed above. For black holes the situation was emeliorated once the black holes were embedded in an AdS space [15]. Such black holes have asymptotically a positive specific heat and no limiting temperature. In fact their entropy is of the field-theoretic class, something that was only fully appreciated with the advent of the AdS/CFT correspondence [16]. For strings propagating in an AdS5 × S5 background one can identify in the density of states intermediate energy ranges for which there exist effectively both a Hagedorn type and black-hole type entropy function reflecting unstable states. But above a certain energy scale the system has a field-theoretic type entropy, i.e. the system has chosen field-theory degrees of freedom as its UV exit strategy. This is reviewed in detail in Section 3. The AdS/CFT correspondence was derived in the context of a 1/N expansion. The leading order relates a small curvature, classical, ten-dimensional gravitational background to a strongly coupled four dimensional gauge theory. This relation is an example of the property of holography attributed to some gravitational systems [11]. While this leading order was tested successfully from many points of view, new issues come up when one examines the next to leading order. One component of the 1/N corrections is the emergence of quantum radiation in the given gravitational background. One needs to understand if and how also the radiation becomes holographic, this is done in Section 4 with an emphasis on the physical qualitative arguments. It was suggested [24] that there exists a special type of theory which is in some sense more than a field theory but less than a full fledged string theory called Little String Theory (LST). The system is supposed to have a Hagedorn-type entropy at asymptotic energies. In Section 5 we review the construction of the theory and the emergence of the Hagedorn behavior in the classical gravitational approximation and then we go on to discuss the holographic properties of the radiation correction. We find that they are very different from the AdS case. One has to enforce an infrared volume cutoff on the gravitational theory and test the various limits: large energy, large volume and large N. We find the limits do not commute in the bulk and thus holography is difficult to impose. In Section 7 we discuss the construction of a system which is in a sense a UV completion of LST. At intermediate scales the Hagedorn LST behavior is demonstrated. The system is described by field-theoretical fixed points at both high and low energies. We discuss both the classical and quantum aspects from the point of view of the bulk theory and discover a first-order phase transition similar to the Hawking-Page one, however it occurs only after quantum corrections are identified. In the process we analyse the way holography is imposed or not on the radiation of various systems including those having noncommutative dimensions, which emerge as a byproduct of some types of UV completion. All in all we find that for all systems studied up till now the Hagedorn behavior is allowed at intermediate but not asymptotic energies. We find that a negative specific heat can be an indication of a first-order phase transition that can be actualized once an explicit UV completion is found. Spontaneous symmetry breaking L.B. OKUN, in Leptons and Quarks, 1984 20.1 Spontaneous breaking of discrete symmetry Let us look first at the simplest case: the ordinary, scalar real field with the lagrangian and the hamiltonian where m is the mass of particles described by the field φ, λ being a dimensionless constant characterizing the interaction between these particles. Consider a field which is constant in time and space. For this field The function V(φ) is plotted in fig. 20.1. It has a minimum at φ = 0 which means that there is no field in vacuum (in the minimum-energy state). The expression for the lagrangian and fig. 20.1 show that the lagrangian is symmetric with respect to a discrete transformation φ →  – φ. Sign in to download full-size image Fig. 20.1. Now let us consider the same lagrangian but with a different sign in front of m2. At first glance, we obtain a particle for which E2 = p2 – m2 and which therefore moves at a speed exceeding that of light: In the literature such particles are called tachyons. In fact no tachyon is implied since the state with φ = 0 is not the vacuum any more. In order to demonstrate this, let us turn to V(φ). Fig. 20.2 gives the potential energy Sign in to download full-size image Fig. 20.2. We see that at φ = 0 this function has a maximum, not a minimum. Small perturbations of the field φ in the vicinity of φ = 0 cannot therefore be considered as particles. Here the system is unstable, tending to slip into one of the stable minima, φ = m/λ or φ = −m/λ. Instead of a single vacuum at φ = 0, as in the usual case of positive sign in front of m2, the system has two degenerate vacua at φ = ±m/λ. We call them degenerate because they have equal energies. By adding a constant to V(φ) (this does not change the field equations!) we rewrite it in a form in which the degenerate vacua are found at the points where V(φ) = 0. The new expression for V(φ) takes the form (fig. 20.3) Sign in to download full-size image Fig. 20.3. where η = m/λ. It seems more logical to consider V(φ) as shown in fig. 20.3 and not as in fig. 20.2, since in the latter case the energy density of the vacuum is –m4/4λ2. This is a fantastically high negative energy (1041 GeV/cm3 for λ ≃ 1 and m ≃ 1 GeV) as compared with the mean energy density in the universe, 10−6 GeV/cm3 (one proton per cubic meter, i.e. 10−30 g/cm3). It can be concluded from the observed data on the expansion of the universe that the vacuum energy density (the so-called cosmological term in Einstein's equations) is unlikely to be much larger than the observed density of matter (hardly by a factor more than 10). Therefore in what follows we shall use the form . The choice of a specific vacuum is decided by microscopic perturbations in the first moments of the life of the universe. But after the system has spontaneously “slipped” into one of the vacua, it cannot change to another one. The amplitude of a below-barrier transition from the state φ = +η to the state φ = −η is zero. It has the form eiS where S is the action. In the case in question the action is imaginary (the transition is below-barrier, that is classically forbidden) and infinitely large (since the action must be found as an integral over the whole space of the universe), so that we obtain e−∞. Could it be possible to realize the transition to another vacuum in only a part of the universe? For example, could we create the new vacuum in a volume of about 1 m3, in laboratory conditions? Unfortunately, it is very difficult to generate even a very small bubble of the new vacuum, and even a small bubble is unstable. It must collapse on a nuclear time scale, emitting mesons. This would happen because the interface between the two vacua must be a wall with very high surface density σ of the order of λη3 and with thickness δ of the order of 1/λη. For λ ≃ 1 and η = 1 GeV, we obtain δ ≃ 10−14 cm, and σ ≃ 1 kg/cm2. These estimates are easily derived if we minimize the wall energy density which is equal to the sum of two terms: By substituting into it the estimates we find that the minimum of δ is reached at δ ≃ 1/λη, so that σ ≃ λη3. In the case we therefore encounter a lagrangian with mirror symmetry (with respect to the φ → –φ transformation) and a vacuum (let it be φ = η) which has no such symmetry. This is a typical example of the so-called spontaneous symmetry breaking. Abdus Salam once gave an illustration of spontaneous breaking of a discrete symmetry. Imagine a banquet with guests seated at a large round table. A person could equally well take a serviette from his right or from his left. But this symmetry is spontaneously broken once one of the guests decides to pick up the serviette, say, on his left; other guests are left no choice. Obviously, the symmetric state is unstable, especially with hungry guests. If we write the field φ in the form φ = η + χ, then χ will describe excitations of the field (particles) with respect to the vacuum φ = η. In the new variables the lagrangian does not possess the mirror symmetry: Note that the field χ is not tachyonic: its mass μ is equal to λη, and the mass term has the usual sign (minus in the lagrangian and plus in the hamiltonian). D-branes in standard model building, gravity and cosmology Elias Kiritsis, in Physics Reports, 2005 A final issue is that of stability of the ground-state alias the absence of tachyons. We should consider separately the open and closed tachyons. Closed string tachyons signal an instability in the closed sector and must be canceled. Open string tachyons signal an instability of the brane configuration but that is what the SM wants! The standard Higgs (as well as other possible Higgses) are tachyons in the symmetric vacuum. One may thus allow the appropriate tachyons at tree level, and this is the case in some semi-realistic models [161,162,106–108,112–115]. Then the rolling of the Higgs to the minimum of its potential can be interpreted as a rearrangement of the brane configuration (brane recombination [131]). The flip side of this is that in the final ground state the branes do not seem flat anymore and further progress is needed in order to be able to calculate. Open strings C. Angelantonj, A. Sagnotti, in Physics Reports, 2002 It is instructive to summarize the low-lying spectra of these theories. Type II superstrings have no tachyons, and their massless modes arrange themselves in the multiplets of the type IIA and type IIB 10-dimensional supergravities. Both include, in the NS–NS sector, the graviton gμν, an antisymmetric tensor Bμν and a dilaton ϕ. Moreover, both contain a pair of gravitinos and a corresponding pair of spinors, in the NS–R and R–NS sectors. In the IIA string the two pairs contain fields of opposite chiralities, while in the IIB string both gravitinos are left handed and both spinors are right handed. Finally, in the R–R sector type IIA contains an Abelian vector Aμ and a three-form potential Cμνρ, while type IIB contains an additional scalar, an additional antisymmetric two-tensor and a four-form potential A+μνρσ with a self-dual field strength. Type IIB spectrum, although chiral, is free of gravitational anomalies [110]. On the other hand, the 0A and 0B strings do not contain any space–time fermions, while their NS–NS sectors comprise two sub-sectors, related to the O8 and V8 characters, so that the former adds a tachyon to the low-lying NS–NS states of the previous models. Finally, for the 0A theory the R–R states are two copies of those of type IIA, i.e. a pair of Abelian vectors and a pair of three-forms, while for the 0B theory they are a pair of scalars, a pair of two-forms and a full, unconstrained, four-form. These two additional spectra are clearly not chiral, and are thus free of gravitational anomalies. Non-compact string backgrounds and non-rational CFT Volker Schomerus, in Physics Reports, 2006 Before we conclude our discussion of boundary Liouville theory, we would like to briefly comment on its possible applications to the condensation of tachyons. Let us recall from our introductory remarks in the second lecture that we need to take the central charge to or, equivalently, our parameter to , (4.20) Here we have allowed for an additional boundary term so as to capture the condensation of both open and closed string tachyons. It is important to keep in mind that any application of Liouville theory to time-dependent processes also requires a Wick rotation, i.e. we need to consider correlation functions with imaginary rather than real momenta . We shall argue below that the two steps of this program, the limit and the Wick rotation, meet quite significant technical difficulties. Unified time analysis of photon and particle tunnelling Vladislav S. OlkhovskyErasmo RecamiJacek Jakiel, in Physics Reports, 2004 Due to the similarities between tunnelling (quantum) packets and evanescent (classical) waves, exactly the same phenomena are to be expected in the case of classical barriers: Namely, in the case, e.g., of the Helmholtz equation. It can be moreover noticed that all the wave equations, as well as Helmholtz's, are relativistic-like, even when one is fixing his attention on waves different from the electromagnetic ones. It is therefore in order to see what it has been possible to predict, with respect to such “negative speeds”, from the more general point of view of Special Relativity. For this purpose, let us here recall that: (i) even if all the ordinary causal paradoxes, invented for tachyons, seem to be solvable within Special Relativity, when it is not restricted to subluminal motions only [86]; (ii) nevertheless, whenever it is met an object travelling at superluminal speed, negative contributions ought to be expected to the tunnelling times [87]: and this should not to be regarded as unphysical [86]. In fact, whenever an object overcomes the infinite speed with respect to a certain observer, it will afterwards appear to the same observer as its anti-object travelling in the opposite space direction: See Ref. [86]. For instance, when going on from the lab to a frame moving in the same direction—with whatever (even very small) speed—as the particles or waves entering the barrier region, the objects penetrating through the final part of the barrier (with almost infinite speed [88]) will appear in the frame as anti-objects crossing that portion of the barrier in the opposite space-direction [86]. In the new frame , therefore, such anti-objects would yield a negative contribution to the tunnelling time: which could even result, in total, to be negative. What we want to stress here is that the appearance of such negative times is predicted by Relativity itself, on the basis of the ordinary postulates [86–88]. From the theoretical point of view, besides Refs. [86–88], see Ref. [89]. From the (quite interesting!) experimental point of view, see Ref. [90]. Non-geometric backgrounds in string theory Erik Plauschinn, in Physics Reports, 2019 Remarks Let us close this section with remarks on the tachyon correlation function and on the tri-product (10.18): • In CFT correlation functions operators are understood to be radially ordered and so changing the order of operators should not change the form of the amplitude. This is known as crossing symmetry which is one of the defining properties of a CFT. In the case of the -flux background, this is reconciled with (10.17) by applying momentum conservation as (10.20) Therefore, scattering amplitudes of three tachyons do not receive any corrections at linear order in both for the - and -flux. (This is analogous to the situation in non-commutative open-string theory, where the two-point function (10.10) does not receive any corrections.) The non-associative behaviour for the closed string should therefore be understood as an off-shell property of the theory (see also [354]). • In the above analysis the flux was assumed to be constant. For a discussion with a non-constant flux in the context of double field theory see [355]. • Using Courant algebroids and regarding closed strings as boundary excitations of more fundamental membrane degrees of freedom, a non-associative star-product has been proposed in [356]. This product can be related to the tri-product introduced in (10.18), which has been established at linear order in the flux in [356] and at all orders in [354] (including extensions towards a non-associative differential geometry). This star-product was also obtained via deformation quantisation of twisted Poisson manifolds in [356], but can also be found by integrating higher Lie-algebra structures [357]. We also note that membrane sigma-models have been used to study non-geometric fluxes [358] and properties of double field theory [359]. • In relation to the open string, we note that the result of a two-form flux inducing non-commutativity of brane coordinates is completely general, and has also been studied for codimension one branes in the WZW model [360]. However, due to a background -flux in this case, it turns out that the obtained structure is not only non-commutative but also non-associative [360–362]. • Using a non-associative star-product, one can try to construct a corresponding non-associative theory of differential geometry and a non-associative theory of gravity. This idea has been proposed in [352], and further been developed in [363–366]. • Properties of non-associative star-products have been studied from a more mathematical point of view also in [367,368] and [369]. Elsevier logo About ScienceDirect Remote access Shopping cart Advertise Contact and support