This work is split into three main parts, the first parts discusses **non**-**perturbative** aspects of the dynamics of Yang-Mills theories, the second part contains a discussion of aspects of black hole dynamics in the context of the graviton condensate picture proposed by Dvali **and** Gomez while the last part of the thesis consists of reprints of the peer reviewed publications of the author. The first part starts with a review of the dynamics of Yang-Mills theories **and** their behaviour in the so called ’t Hooft limit of a large number of colors. We also review the relation of the ’t Hooft expansion to the genus expansion known from string **theory**. From there it proceeds with a short review of **non** local operators **and** topological **field** theories. Following this there is a review of how to localize massless gauge fields on topological defects as well as a short description of the mechanism for localizing topologically massive gauge fields on domain walls invented by the author. The corre- sponding more detailed explanation can be found in the third part of this thesis. The next section deals with a topological **field** **theory** description of the low energy dynamics of ordinary Yang-Mills theories as well as discussing the behaviour of domain walls appearing in these theories extending previous results by Seiberg **and** collaborators. Here we will also be able to shed light on the appearance of topological degrees of freedom on the world volume of these walls. The subsequent two sections are devoted to an extension of these results to supersymmetric gauge theories. We first review the prop- erties of supersymmetric Yang-Mills theories **and** the existing computations of the tension of domain walls in supersymmetric Yang-Mills theories. Then we follow up with an extension of our work in the previous chapter to show that there are topological degrees of freedom living on these domain walls as well. We conclude this part with a summary of the results achieved in this thesis with respect to domain walls in Yang-Mills **theory** **and** furthermore show how several puzzling properties of Yang-Mills theories can be seen to have natural analogs in critical string theories. We also point out a striking analogy to fractional quantum hall systems. Parts of the second part will be basis for an upcoming publication together with Markus Dierigl.

Mehr anzeigen
220 Mehr lesen

that time **and** still do: String **theory** is about replacing point particles by extended one- dimensional objects called strings, which can be either open or closed. These fields can vibrate, **and** different vibrational modes correspond to different particles, like the differ- ent vibrational modes of a violin generate different tones. Importantly, in the spectrum of vibrational modes there is always an excitation, which describes the fluctuation of a background spacetime metric. This was considered as a hint that string **theory** could be a candidate for a consistent **theory** of quantum **gravity**. Indeed, it is astonishing how string **theory** deals with the bad **non**-renormalizable infinities in quantum **field** **theory** associated to gravitational interactions. The extended nature of the string delocalizes interaction vertices, **and** the problematic ultraviolet regime is mapped by a so-called duality to the infrared regime which can be described easily. More precisely, this dual- ity states that the physics of long strings at high energies is the same as the physics of short strings at low energies. Via a precise ’dictionary’ these regimes can be mapped to each other. All these nice properties have already appeared in the early version of bosonic string **theory**. However, the latter suffers from a couple of important drawbacks which makes it impossible to consider it as a **theory** of the world around us. First, it cannot account for spacetime fermions, which are the fundamental building blocks of our world. Second, in the spectrum of the **theory** one finds tachyons, i.e. modes of imaginary mass. These signal an instability of the **theory**. While the tachyon in the sector of open strings is quite well understood (we are sitting at the maximum of a potential, **and** rolling down corresponds to so-called D-brane condensation), the impli- cations of the tachyon in the closed string sector are not clear but might most certainly render spacetime itself unstable. Both issues, the presence of tachyons **and** the absence of spacetime fermions, soon got resolved by moving from bosonic string **theory** to su- perstring **theory**. By introducing a fermionic partner string for the bosonic string the **theory** acquires a new symmetry, namely two-dimensional supersymmetry. The latter is powerful enough to allow for stable solutions, **and** at the same time also leads to spacetime fermions, while keeping the nice properties in the ultraviolet regime. Indeed, it was found that there even exist five different superstring theories, which all require for consistency a total number of exactly ten spacetime dimensions. They are called type I, type IIA, type IIB, SO(32) heterotic **and** E 8 × E 8 heterotic string **theory**.

Mehr anzeigen
262 Mehr lesen

The main purpose of this paper is to investigate the existence of a smooth limit of our model to Einstein **gravity**, when the mass of the graviton vanishes. It was noticed long ago by van Dam, Veltman **and** Zakharov [ 4 , 5 ] that in linearized massive **gravity** the extra scalar mode of the graviton did not disappear **and** remained coupled to matter even in the limit of a vanishing graviton mass. In turn, this spoils predictions of General Relativity either for the perihelion precession or deflection of starlight. This effect is known as the van Dam-Veltman-Zakharov (vDVZ) discontinuity **and** was first thought to be a no-go the- orem for massive theories of **gravity** [ 4 , 5 ]. However, it was pointed out by Vainshtein that the discontinuity could be an artifact due to the breakdown of the perturbation **theory** of massive **gravity** in the massless limit [ 6 ]. He has shown that in the case of gravitational **field** produced by a source of mass M 0 the nonlinear corrections become important at scales r < R V ≡ M 0 1/5 m −4/5 g (in Planck units) **and** conjectured that in the strong coupling regime General Relativity is restored. When the mass of the graviton m g vanishes the Vainshtein radius R V grows **and** becomes infinite, thus providing a continuous limit to General Rel- ativity in case the Vainshtein conjecture is correct. At distances r ≪ R V , around a static spherically symmetric massive source of mass M 0 the full **non**-linear strongly coupled mas- sive **gravity** has to be considered in order to recover the Einstein **theory**, which makes the proof of the Vainshtein conjecture **non** trivial. The question of continuous matching of the solutions below **and** above the Vainshtein radius have been extensively addressed in recent literature. The first model where such a transition was demonstrated is Dvali-Gabadadze- Porrati (DGP) model which imitates many features of massive **gravity** [ 13 , 20 ]. There was a claim that in the bigravity version of massive graviton the corresponding solutions do not match [ 7 ], but it was recently shown that this claim is not justified [ 8 – 10 ].

Mehr anzeigen
178 Mehr lesen

In the case of the κ-deformed space we concentrate on the noncommutative SU (N ) theories. Using the enveloping algebra approach **and** the Seiberg-Witten map [32], [41], the noncommutative gauge **theory** is constructed perturbatively order by order in the deforma- tion parameter. In this way we obtain an effective **theory** which provides corrections to the commutative **theory** up to first order in the deformation parameter. These corrections are given in terms of the commutative fields, so the **field** content of the **theory** is not changed. However, new interactions arise **and** the deformation parameter enters as a coupling con- stant. This approach has been used to construct the noncommutative gauge **theory** on the θ-deformed space [41], [42], as well as the generalisation of the Standard Model [43], [44]. Using these results some new **effects** which do not appear in the commutative Standard Model were calculated in [45], [46]. Also, it was shown that the theories obtained in this **perturbative** way are anomaly free [47], [48], [49]. It is interesting to note that cutting the **theory** at some order in the deformation parameter one avoids the UV/IR mixing. It only appears in the ”summed-up” theories, that is theories to all orders in the deformation pa- rameter. Also, the ”summed-up” models allow generalisation of the U (N ) gauge theories only, with some exceptions [50], [51].

Mehr anzeigen
124 Mehr lesen

There has been a deep historical connection between the pursuit of understanding the physics of strongly coupled systems **and** string **theory**. String **theory** was proposed as an explanation for the Regge trajectory pattern on bound states. This is the noted pattern that when the angular momentum of hadronic excitations J are plotted versus the mass or energy squared they form a pattern of lines. This pattern could be easily explained by representing the mesons for example as two quarks bound together by a string. With the emergence of QCD in the early 70’s this stringy explanation was abandoned however the ideas of string **theory** persisted into a **theory** of quantum **gravity**. In 1997 it seems these ideas finally came full circle. When Maldacena first conjectured the correspondence be- tween a geometrical (gravitational) **theory** in anti-de Sitter space (AdS) **and** a conformal

Mehr anzeigen
159 Mehr lesen

As we discussed, the **theory** undergoes a large particle number phase transition [80]. This phase transition interpolates between a ho- mogeneous phase in the weak coupling limit to a phase dominated by a solitonic bound state in the strong coupling limit, known as a bright soliton. The dynamics of the phase transition has been extensively studied, both using the mean-**field** analysis [80] **and** also by a trunca- tion **and** numerical diagonalization of the Hamiltonian [96, 94, 76, 75]. Another interesting feature of this model is that it is exactly inte- grable [84]. As we’ll see, this implies that the Schr ¨odinger equation of the system can be mapped to a set of algebraic equations - the Bethe equations - which fully determine the complete spectrum of the the- ory. Despite the fact that the system can be in principle solved using this technique, in practice the equations are transcendental **and** cannot be analytically solved without any approximations. The only regime where it is possible to obtain exact solutions is in the c → ∞ limit, where we are in the deep solitonic regime. In this regime, it is possible to explicitly construct exact solutions of the Bethe equations, due to the string hypothesis [87], which we’ll revisit later.

Mehr anzeigen
155 Mehr lesen

Quantum low dimensional systems attract huge interest since they provide a test area for the investigation of the general behaviour of complicated nonlinear systems. Indeed, a lot of inter- esting phenomena in condensed matter, high-energy physics **and** quantum **field** **theory** emerge due to the presence of complicated **non** inear interactions. The study of these systems is often an extremely complicated task that is out of the reach of **perturbative** approaches. However, it is known that in many cases interesting nontrivial many body **effects** prevail even if the system is restricted to 1D. Moreover, the simplified but still nontrivial cases of 1D system allows one to concentrate on key properties of a system while avoiding bulky technical problems. Sometimes this allows to make crucial simplifications because the system will exhibit integrability, i.e. can be solved exactly. We also see that a lot of specific, interesting properties emerge in 1D systems, especially in integrable models, thus making 1D integrable models interesting in their own right. Among these interesting systems it is worth noting t-J [16, 73–79] **and** Hubbard [6, 7, 71, 72] models, which describe lattice gases of strongly-correlated electrons **and** are expected to exhibit the high-T superconductivity behaviour. While this phenomenon was intensively studied over the last 50 years, still most of the mechanisms of high-T superconductivity are unknown. One of the main open questions of the **field** is the mechanism of formation of electron pairs. The derivation of exact solutions of the t-J **and** Hubbard models can help to answer such a question **and** thus to understand the nature of the high temperature superconductivity. These are important models from the perspective of studying the entanglement entropy **and** Kondo model [9] which describes an anomalous low-T behaviour of conductivity of doped metals (Kondo problem).

Mehr anzeigen
126 Mehr lesen

Whether the generalisation of general relativity to N **non**-interacting metrics is consistent was investigated in ref. [ 114 ], to find out indeed, that such a **theory** is inconsistent unless N = 1. Hinterbichler **and** Rosen first developed then a set of theories for N interacting vielbeine [ 91 ]. However, it was later realized that only those theories whose vielbeine interact pair-wisely through the bimetric potential are free of ghost instabilities [ 115 ]. Furthermore, loop interactions, i.e., when a vielbein interacts only with a next one **and** this latter in addition with a next one **and** so on until closing the circle, also lead to in- consistent theories (see refs. [ 91 , 116 – 118 ]). Those consistent theories were first formulated by Hinterbichler **and** Rosen using the language of differential forms for N vielbeine e a (p) , with p = 1, . . . , N . Their equivalent theories in tensors is established when a generalised symmetry condition (e a (p) ) [µ| (e b (q) ) |ν] η ab = 0 is

Mehr anzeigen
153 Mehr lesen

larly interesting: The mass dimension of the running Newton
onstant, [G k ] = 2 − d , vanishes in exa
tly d = 2 spa
etime dimensions, **and** a **perturbative** treatment be-
omes feasible. This approa
h involves
omputing the β -fun
tions (i.e. the ve
tor eld whi
h drives the RG ow) in d = 2 + ε > 2 dimensions **and** expanding them in terms of ε . A general
onsideration [4℄ shows that the β -fun
tion of the dimensionless Newton
onstant, g k ≡ k

326 Mehr lesen

Tables A9–A13 report results for the subsample for which data on vote shares for the environmental party is available. We assign each subject to the group of ”green consumers” **and** ”**non**-green consumers” based on whether she comes from a region with above- or below-median support for the Green party. **Non**-green consumers have larger (structural) misperceptions of energy efficiency. They overvalue LEDs by around 1.25 euros per bulb **and** undervalue **Non**-LEDs by 2.62 euros per bulb. Green consumers undervalue LEDs by 0.43 **and** overvalue **Non**-LEDs by 1.03 euros per bulb. The less informative signal increases the overvaluation for LEDs for both groups to around 1.80 per bulb. It also decreases consumers’ undervaluation of **Non**-LEDs but leaves consumers with weaker environmental preferences with a larger bias of 2.32 euros per bulb (as opposed to 0.89 for green consumers). In terms of consumer surplus, households with lower green preferences benefit more from the fully informative signal **and** are hurt less by partial information disclosure than households with strong green preferences. The tax vector is similar to the scenario in which income is assumed to identify homogeneous subgroups but is characterized by both a slightly lower subsidy **and** a lower tax. Much of the tax burden is borne by green consumers who lose 5.94 euros per bulb per consumer, which is around 3.7 times more than **non**-green consumers. The reason for this asymmetry is again primarily attributable to the difference in own-price elasticities for **Non**-LED products.

Mehr anzeigen
78 Mehr lesen

While the presence **and** direction of the price e¤ect are mostly clear, **theory** provides less guidance on its strength. There are a number of arguments, such as lack of substitutes **and** salience of price in deciding on public goods contribu- tions, that support the notion that the price elasticity for contributing to public goods should be low (Green 1992). Also, both in a pure **and** impure altruism model in the spirit of Andreoni (1988) **and** Andreoni (1990), respectively, the subjects’ strategic interdependence in providing the public good reduces the price elasticity of the Nash contributions as long as subjects believe that all subjects face the same change in price. Support for predicting low price elastic- ity comes from experimental studies that examine a limited number of discrete price variations **and** report low estimates at the extensive margin of contribut- ing: Smith, Kehoe **and** Cremer (1995) …nd that the decision whether to make a charitable contribution for a rural health care facility is insensitive to price. Likewise, examining contribution choices for an unmatched baseline **and** three match ratios, Karlan **and** List (2007) …nd that while the probability of donating 1 2 Standard deviation is 6.4. The smallest group consisted of 31, the largest of 66 sub jects. 1 3 The same authors also discuss that a broader class of models makes for more equivocal

Mehr anzeigen
36 Mehr lesen

Tables A9–A13 report results for the subsample for which data on vote shares for the environmental party is available. We assign each subject to the group of ”green consumers” **and** ”**non**-green consumers” based on whether she comes from a region with above- or below-median support for the Green party. **Non**-green consumers have larger (structural) misperceptions of energy efficiency. They overvalue LEDs by around 1.25 euros per bulb **and** undervalue **Non**-LEDs by 2.62 euros per bulb. Green consumers undervalue LEDs by 0.43 **and** overvalue **Non**-LEDs by 1.03 euros per bulb. The less informative signal increases the overvaluation for LEDs for both groups to around 1.80 per bulb. It also decreases consumers’ undervaluation of **Non**-LEDs but leaves consumers with weaker environmental preferences with a larger bias of 2.32 euros per bulb (as opposed to 0.89 for green consumers). In terms of consumer surplus, households with lower green preferences benefit more from the fully informative signal **and** are hurt less by partial information disclosure than households with strong green preferences. The tax vector is similar to the scenario in which income is assumed to identify homogeneous subgroups but is characterized by both a slightly lower subsidy **and** a lower tax. Much of the tax burden is borne by green consumers who lose 5.94 euros per bulb per consumer, which is around 3.7 times more than **non**-green consumers. The reason for this asymmetry is again primarily attributable to the difference in own-price elasticities for **Non**-LED products.

Mehr anzeigen
80 Mehr lesen

This research paper shows how E-**theory** can be applied to a scalar boson which underlies gravitational **effects** in 4-dimensional spacetime. The low-energy limit, conservation laws **and** the **perturbative** calculation of scattering amplitudes are shown. With these considerations the general procedure of an application of E-**theory** to gravitational physics called “E-**gravity**” is made clear. Similarities between this quantum **gravity** **theory** **and** Topological Dipole **Field** **Theory** (Linker 2015) are also shown in this research paper. Topological Dipole **Field** **Theory** (TDFT) is a model that describes a modification of the dynamics of gauge bosons which implies distinct behavior of quantum fluctuations.

Mehr anzeigen
including a mean atmosphere **and** ocean potential is introduced. Further potential fields generating pertur- bations on a satellite’s orbit are tides. Therefore, also direct tides, solid Earth tides, ocean tides, **and** pole tides are included in the force model. In addition - **and** which is the most important part for this work - short-term atmospheric **and** oceanic mass variations cause potential variations **and** act as disturbance forces on a satellite’s orbit. During GRACE **gravity** **field** analysis these variations are removed by the so-called atmosphere **and** ocean de-aliasing (AOD) product. This is done in order to avoid aliasing due to temporal undersampling, because GRACE is not able to adequately sample these short-term atmospheric **and** oceanic mass variations. Thus, these variations are modelled in 6-hourly intervals (see chapter 3) via geophysical models **and** are, after interpolation on the integration interval which is usually 5 seconds, ’removed’ during **gravity** **field** determination. This process is called de-aliasing. Having modelled them correctly **and** realis- tically, the resulting GRACE **gravity** **field** solutions should not contain any atmospheric **and** oceanic signal with periods shorter than the sampling period of the **gravity** **field** solution (which is usually one month). The mentioned empirical forces are introduced in the disturbance force models in order to treat the mismod- elled or umodelled forces acting on a satellite. The intent of these empirical **non**-gravitational parameters, introduced during **gravity** **field** determination, is to absorb the poorly modelled parts of the force fields **and** to treat the measuring **effects**. In principle there are several types of empirical parameters like dynamic **and** kinematic ones. E.g., these empirical models are often used to absorb the biased accelerometer mea- surements. A more detailed description of the role of the empirical forces within orbit **and** **gravity** **field** determination is given, e.g., in Kim (2000), J¨ aggi (2006) **and** Beutler et al. (2010b). The main aspect con- cerning these empirical forces, **and** being relevant for this work is the fact, that each empirical parameter introduced in the orbit or **gravity** **field** determination might disturb the solution for the other **non**-empirical parameters, e.g. **gravity** **field** coefficients, as these empirical forces may absorb in addition geophysical sig- nals, also model errors **and** unmodelled **effects**. This issue is revisited in chapter III.

Mehr anzeigen
127 Mehr lesen

In this thesis, we investigate different aspects of **gravity** as an effective **field** **theory**. Building on the arguments of self-completeness of Einstein **gravity**, we argue that any sensible **theory**, which does not propagate negative-norm states **and** reduces to General Relativity in the low energy limit is self-complete. Due to black hole formation in high energy scattering experiments, distances smaller than the Planck scale are shielded from any accessibility. Degrees of freedom with masses larger than the Planck mass are mapped to large classical black holes which are described by the already existing infrared **theory**. Since high energy (UV) modifications of **gravity** which are ghost-free can only produce stronger gravitational interactions than Einstein **gravity**, the black hole shielding is even more efficient in such theories. In this light, we argue that conventional attempts of a Wilsonian UV completion are severely constrained. Furthermore, we investigate the quantum picture for black holes which emerges in the low energy description put forward by Dvali **and** Gomez in which black holes are described as Bose-Einstein condensates of many weakly coupled gravitons. Specifically, we investigate a **non**-relativistic toy model which mimics certain aspects of the graviton condensate picture. This toy model describes the collapse of a condensate of attractive bosons which emits particles due to incoherent scattering. We show that it is possible that the evolution of the condensate follows the critical point which is accompanied by the appearance of a light mode.

Mehr anzeigen
166 Mehr lesen

Even though no direct experimental evidence of any new heavy particles is in sight 2 (see, e.g., [KS+16]), important conceptual problems still remain in high energy physics. One example is our lack of detailed understanding of the interplay between quantum mechanics **and** black hole physics. Following a recent proposal by Dvali **and** Gomez [DG13b; DG14], we have pursued the idea that black hole physics may, after all, be a manifestation of collec- tive quantum eﬀects in high energy physics. In this work, we present some of the results obtained for simplified model systems **and** the conclusion we draw for black holes. From this point, we were intrigued to further explore the relevance of quantum collective eﬀects. During our eﬀorts to unravel the phenomena using techniques of integrability, we discovered a surprising equivalence between our model system (Lieb-Liniger) **and** an otherwise seem- ingly unrelated **theory** in two dimensions (Yang-Mills). Finally, we turned to particle collisions, in which many particles are produced. These may only be accessible at future experiments, but until then, we need to dramatically improve upon our capabilities to calculate such processes, where quantum col- lective eﬀects are dominant. In this domain, the present work contains some formal developments that aim to further our understanding of the required mathematical tools.

Mehr anzeigen
149 Mehr lesen

plies that the latter should be composite as well. Using this mapping we explicitly showed how semi-classical instanton results are easily obtained in terms of the coherent state de- scription of the corresponding soliton to leading order in 1/N . This was done in detail in several cases such as instantons in quantum mechanics, Yang-Mills **theory** or 3-dimensional electrodynamics. In addition, in order to make the mapping manifest we constructed the- ories which naturally embed instanton physics in d dimensions into theories in one more dimension describing evolving solitons. Using the insight that instantons should have a quantum description, we further argued that the concept of resurgence should follow as a consequence of the basic principles of quantum mechanics such as unitarity. As a next step, we were concerned with higher order corpuscular **effects** in the case of solitons in SUSY theories. In the example of a Wess-Zumino model in 1 + 1 dimensions we worked out in detail that these **effects** lead to a novel mechanism of SUSY breaking which can never be discovered in the semi-classical treatment. We argued that these correction can naturally be understood in terms a corpuscular renormalization of the classical profile induced by corpuscular loops. Alternatively, we explained that these **effects** can also be understood in the many-body language. Indeed, in Bogoliubov approximation, quantum corrections are encoded in the dynamics of small fluctuations (quasi-particle excitations) around the mean **field** data. Finally, we applied the coherent state picture to the physics of AdS space-time. To leading order in 1/N , we explained how geometric properties such as local flatness or stability of AdS with respect to decay into Minkowski space-time are mapped to the occu- pation of corpuscles in AdS. In addition, we saw that the central charge of the dual CFT can be understood as a collective **effects** of corpuscles constituting a portion of AdS with volume set by the curvature radius. Based on these results, we proceeded with a discussion of higher order correction. In particular, we investigated how corpuscular **effects** correct propagators **and** Wightman functions in an AdS space-time. On the one hand, it was shown that the corpuscular **effects** on the propagator can be resummed in a Dyson-type series. On the other hand, using the KMS condition as a tool, we demonstrated that there are corrections to thermality of the spectrum that an accelerated observer measures in AdS which can never be uncoverer in the semi-classical treatment.

Mehr anzeigen
204 Mehr lesen

Theories of phase transitions have their origin in the early models of van der Waals (1893), Korteweg (1901), Ginzburg-Landau (1950), Cahn-Hilliard (1958), Allen-Cahn (1960), Halperin, Hohenberg **and** Ma (1977). While the early models did not include any spatial resolution, especially the Cahn-Hilliard equation which, for the first time, addressed demixing phenomena using a spatially-resolved approach. The order parameter entering into the equations was the—conserved—concentration of alloy elements. The Allen-Cahn equation then included the option of **non**-conserved order parameters for the first time. It seems essential to highlight that phase transitions are best described by **non**-conserved order parameters such as, for example, the fraction liquid of a system will change from 1 to 0 in a solidification process **and** is thus not a conserved quantity.

Mehr anzeigen
14 Mehr lesen

An important feature of classical **field** **theory** lies in its relation to symmetries. On the one hand, implementing a symmetry into a given **theory** is rather simple, as we shall see in the following section. On the other hand, if a continuous symmetry is present, then there a exist corresponding conserved quantity (or set of conserved quantities), as we shall now see.

Using the characterization of trivial curves as curves with trivial ω-energy we can prove that we indeed have a global obstruction bundle over the compactification of every moduli space of trivial curves. While in Gromov-Witten **theory** the count of elements in the moduli space, more general, the cobordism class of the moduli space, is independent of the chosen abstract perturbation of the Cauchy-Riemann operator, this no longer holds for the moduli spaces in symplectic **field** **theory**. This follows from the fact that the moduli spaces in symplectic **field** **theory** typically have codimension one boundary strata, while in Gromov-Witten **theory** the regular moduli spaces form pseudo-cycles in the sense that the boundary strata have codimension at least two, i.e., from the homological point of view have no boundary. So while in Gromov-Witten **theory** the moduli spaces can be studied separately, the interplay between the different moduli spaces is the reason why the algebraic invariants of symplectic **field** **theory** are defined as differential algebras, which can be shown to be independent of extra choices like the cylindrical almost complex structure **and** the compact perturbation. In our case this problem is expressed by the fact that we have to study sections in vector bundles over moduli spaces with codimension one boundary, so that the count of zeroes in general depends on the choice of sections in the boundary, i.e., the chosen perturbations of the Cauchy-Riemann operator used to define the regular moduli spaces in the boundary. However we outline below that in our case we indeed have a well-defined count of zeroes so that, as in the Gromov-Witten case, we can (iteratively) define Euler numbers for our Fredholm problems.

Mehr anzeigen
113 Mehr lesen