K E Atkins

J M J Travis

‡ Travis

Local adaptation and the evolution of species ranges under climate change

Keywords: local density dependence, environmental gradient, environmental niche, extinction, species' range

The potential impact of climate change on biodiversity is well documented. A well developed range of statistical methods currently exists that projects the possible future habitat of a species directly from the current climate and a species distribution.

However, studies incorporating ecological and evolutionary processes remain limited.

Here, we focus on the potential role that local adaptation to climate may play in driving the range dynamics of sessile organisms. Incorporating environmental adaptation into a stochastic simulation yields several new insights. Counter-intuitively, our simulation results suggest that species with broader ranges are not necessarily more robust to climate change. Instead, species with broader ranges can be more susceptible to extinction as locally adapted genotypes are often blocked from range shifting by the presence of cooler adapted genotypes that persist even when their optimum climate has left them behind. Interestingly, our results also suggest that it will not always be the cold-adapted phenotypes that drive polewards range expansion. Instead, range shifts may be driven by phenotypes conferring adaptation to conditions prevalent towards the centre of a species' equilibrium distribution. This may have important consequences for the conservation method termed predictive provenancing. These initial results highlight the potential importance of local adaptation in determining how species will respond to climate change and we argue that this is an area requiring urgent theoretical and empirical attention.

Introduction

The threat that climate change poses to biodiversity is well documented [START_REF] Parmesan | A globally coherent fingerprint of climate change impacts across natural systems[END_REF][START_REF] Pearson | A spatial evaluation of climate impact on the envelope of species[END_REF][START_REF] Peterson | Future projections for Mexican faunas under global climate change scenarios[END_REF][START_REF] Thomas | Extinction risk from climate change[END_REF]Walther et al., 2002). Many species have already exhibited responses to the changing environment; in a meta-analysis [START_REF] Parmesan | A globally coherent fingerprint of climate change impacts across natural systems[END_REF] found that 434 species out of 893 had shifted in distribution and/or abundance. A key question is whether climate change increases the risk that species will become extinct, and initial results from a study by [START_REF] Thomas | Extinction risk from climate change[END_REF] are alarming. Even for a best-case scenario, using the minimum expected climate change and ignoring the potential for dispersal limitation, the authors predict a 9-13% loss in species abundance. Using the worst climate scenario and with extreme dispersal limitation this figure rises to a staggering 38-52%. Whilst this study represents a useful exercise, not least in terms of focussing minds on the problem at hand, it simultaneously highlights some important deficiencies that it shares with the many bioclimate envelope models (BEMs), widely used to predict future ranges [START_REF] Hampe | Bioclimate envelope models: what they detect and what they hide[END_REF]. There is an urgent need for the development of a new generation of models to make this type of prediction. However, this goal is hindered by a basic lack of understanding of the likely consequences of key ecological and evolutionary processes for range dynamics under climate change. Local adaptation is a good example of one of these processes: many species comprise collections of populations each adapted to local climatic conditions. However, there has been little formal consideration of the impact the degree of local adaptation present within a species might have on range dynamics [START_REF] Polechová | Species' range: Adaptation in space and time[END_REF], and no attempt to incorporate these processes into predictive modelling. It is likely that two aspects of a species' adaptation will be important. First, the steepness with which an individual's fitness declines as it is moved from its optimal climate and second, the total range of environmental conditions under which a species, as a whole, can survive.

In this contribution, we take a first step and develop a relatively simple simulation model to explore the issue of how local adaptation influences a species ability to track changing climate.

A plethora of studies have used bioclimate envelope modelling to predict the future biogeographic ranges of species [START_REF] Bakkenes | Assessing effects of forecasted climate change on the diversity and distribution of European higher plants for 2050[END_REF][START_REF] Broennimann | Do geographic distribution, niche property and life form explain plants' vulnerability to global change?[END_REF][START_REF] Peterson | Future projections for Mexican faunas under global climate change scenarios[END_REF][START_REF] Schwarz | Predicting extinctions as a result of climate change[END_REF][START_REF] Thuiller | Climate change threats to plant diversity in Europe[END_REF]. Refined methods are being developed that entail the use of increasingly sophisticated statistics (e.g. [START_REF] Gavin | Bioclimatic modelling using gaussian mixture distributions and multiscale segmentation[END_REF]), genetic algortithms (e.g. [START_REF] Termansen | The use of genetic algorithms and bayesian classification to model species distributions[END_REF]) and neural networks (e.g. [START_REF] Pearson | Predicting the impact of climate change on the distribution of species: are bioclimate envelope models useful?[END_REF]), and it has even been argued that the optimal approach should use a combination of these approaches in order to look for concordance between them [START_REF] Araujo | Reducing uncertainty in projections of extinction risk from climate change[END_REF]. Intense activity in this field has provided a suite of methods that obtain a relationship between the climate and a species range prior to environmental change: thus defining the 'climate space' occupied by the species. Utilising future climate scenarios, these methods predict the future climate space for the species. However, there is a growing realisation that this correlational approach alone is ill-equipped to predict future species ranges [START_REF] Davis | Evolutionary responses to changing climate[END_REF][START_REF] Hampe | Bioclimate envelope models: what they detect and what they hide[END_REF][START_REF] Lawton | Ecological Consequences of Heterogeneity[END_REF]Woodward and Beerling, 1997).

The dynamics of a species' range during a period of climate change will be dependent upon the existence and interplay of various ecological and evolutionary pro-cesses. Dispersal is perhaps the most obvious of these and it is anticipated that species with greater dispersal ability will be better able to track a changing climate (Travis and Dytham, 2002;[START_REF] Mustin | The dynamics of climateinduced range shifting; perspectives from simulation modelling[END_REF]. [START_REF] Midgley | Assessing the vulnerability of species richness to anthropogenic climate change in a biodiversity hotspot[END_REF] incorporate simple dispersal functions into a model predicting future ranges of Cape Proteaceae.

The description of the spatial population dynamics used in this model is relatively crude but, in linking a model of ecological dynamics with a BEM, [START_REF] Midgley | Assessing the vulnerability of species richness to anthropogenic climate change in a biodiversity hotspot[END_REF] take an important step in the right direction.

Theoretical studies, often focussing on invasion biology rather than range dynamics under climate change, have considered a range of evolutionary dynamics during expansions: Garcia-Ramos and Rodriguez ( 2002) demonstrate the interacting roles played by local adaptation and habitat heterogeneity in determining spread rate; several theoretical studies have now demonstrated that range expansion might be accelerated by the evolution of increased dispersal propensity at expanding fronts (e.g. Travis and Dytham, 2002;[START_REF] Burton | Trade-offs and the evolution of life-histories during range expansion[END_REF] and recent empirical evidence confirms that this is an important effect [START_REF] Phillips | Invasion and the evolution of speed in toads[END_REF]; [START_REF] Klopfstein | The fate of mutations surfing on the wave of a range expansion[END_REF] demonstrate that neutral mutations arising on the edge of a range expansion sometimes 'surf' on the wave of advance and can thus reach a wider spatial distribution and higher frequency than would be expected in a stationary population. [START_REF] Klopfstein | The fate of mutations surfing on the wave of a range expansion[END_REF] suggest that this surfing phenomenon may increase the rate of evolution of spatially expanding populations. More recent extensions to this work have demonstrated that this surfing dynamic can be important for non-neutral mutations. Even deleterious mutations can sometimes attain high abundance at an expanding front (Burton and Travis, 2008a;[START_REF] Excoffier | Surfing during population expansions promotes genetic revolutions and structuration[END_REF][START_REF] Brooker | Modelling species' range shifts in a changing climate: The impacts of biotic interactions, dispersal distance and the rate of climate change[END_REF] and this effect can substantially modify evolutionary dynamics where fitness landscapes are rugged (Burton and Travis, 2008b). In this paper we concentrate on local adaptation, an important evolutionary process that, to date, has rather surprisingly received little attention in terms of its potential role in range expansions and determining a species dynamics during an episode of climate change.

Considerable empirical evidence indicates that locally adapted phenotypes are commonplace within many species [START_REF] Lambrechts | Adaptive differences in the timing of egg laying between different populations of birds result from variation in photoresponsiveness[END_REF][START_REF] Mcneilly | Evolution in closely adjacent plant populations 4: barriers to gene flow[END_REF][START_REF] Riihimaki | Studying genetics of adaptive variation in model organisms: flowering time variation in Arabidopsis lyrata[END_REF][START_REF] Santamaria | Plant performance across latitude: The role of plasticity and local adaptation in an aquatic plant[END_REF]. For example, it has long been known that many plant species show genetic differentiation in the timing of flowering [START_REF] Mcneilly | Evolution in closely adjacent plant populations 4: barriers to gene flow[END_REF]; this variation is frequently clinal with northern populations consistently different from southern counterparts [START_REF] Riihimaki | Studying genetics of adaptive variation in model organisms: flowering time variation in Arabidopsis lyrata[END_REF]. Occurrence of a genotype outside of its usual range may result in suboptimal flowering time and a corresponding reduction in fitness. Similar patterns have been described for a bird species: across Europe, blue tits exhibit adaptive differences in photoresponsiveness that results in laying dates coinciding with local caterpillar availability [START_REF] Lambrechts | Adaptive differences in the timing of egg laying between different populations of birds result from variation in photoresponsiveness[END_REF]. These two examples both involve the timing of key events being optimally aligned to seasonal environmental conditions, but there are other ways in which local adaptation may be manifest. It is likely that within many species there is variability in thermal tolerances with individuals towards warmer parts of the range adapted to function better in the heat while directional selection will have resulted in individuals found in cooler regions having a physiology that equips them to function efficiently in the cold (e.g. fish [START_REF] Feminella | Intraspecific differences in thermal tolerance of Etheostoma spectible (agassiz) in constant versus fluctuating environments[END_REF], amphibians [START_REF] Olsson | Thermal environment, survival and local adaptation in the common frog, Rana temporaria[END_REF] and insects [START_REF] Ayres | Local adaptation to regional climates in Papilio canadensis (Lepidoptera: Papilionidae)[END_REF]).

There is now recognition that local adaptation is likely to be an important determinant of species' responses to climate change and an interdisciplinary treatment of this issue, linking evolution and ecology, has already been called for [START_REF] Pertoldi | Evolutionary aspects of climate-induced changes and the need for multidisciplinarity[END_REF]. We establish how local adaptation drives the dynamics of biogeographic ranges during and following a period of climate change. We utilise a spatially explicit individual based simulation to model the impact of climate change on a single species. This model uses both stochastic demographic parameters and allows inclusion of local density dependent rates for survival of individuals. In our discussion we emphasise the potential implications of our results for conservation biology and highlight where we believe future effort is required to develop improved theory related to the evolutionary ecology of range shifting.

The Model

There has been considerable recent interest in modelling the ecological and evolutionary dynamics of spatially structured populations that live along environmental gradients (e.g. Alleaume-Benharira et al., 2006;[START_REF] Travis | The distribution of positive and negative species interactions across environmental gradients on a dual-lattice model[END_REF][START_REF] Dytham | Evolved dispersal strategies at range margins[END_REF] and some studies have extended this approach to consider the range shifting of populations structured in this way (e.g. [START_REF] Brooker | Modelling species' range shifts in a changing climate: The impacts of biotic interactions, dispersal distance and the rate of climate change[END_REF][START_REF] Mustin | The dynamics of climateinduced range shifting; perspectives from simulation modelling[END_REF]. We adopt a similar approach in this contribution and below we describe, in turn, the structure of the landscape, the ecological and evolutionary features of the model, and the simulation experiments that we conduct.

The Environment

Individuals of the single species inhabit a two-dimensional landscape that, unless specified otherwise, has dimensions 200 rows (x value) columns by 200 columns (y value), which are ordered from the bottom left hand corner (i.e. increasing an individual's x location would move the individual up the grid, whilst increasing its y location would move it to the right of the grid). A value of (x,y) determines an individual's location. We assume that environmental conditions vary latitudonally (i.e. different rows have different conditions, for example due to variation in climate) and the parameter, θ(x), is used to denote this condition. We assume a linear gradient, b, in environmental conditions (i.e. for every value of y, the condition at x, θ(x) is bx). An optimal environmental location, J, is defined as the climate under which the highest potential population growth rates are obtained. Potential growth rate declines with distance from this optimum location. J is defined as a row value, x, on the grid. In all our simulations, we allow populations to first establish under stable environmental conditions and in these periods the environmental conditions within each row remains constant through time. To simulate environmental change (e.g. climate warming), we increase θ(x) for each row, x and increase J by ν each year. The period of climate change lasts for T years. An example of an environmental condition changing latitudonally could be a warming climate, where the local temperature increases in all locations over time. Local adaptation to this condition would allow individuals to reside in the climate in which they are adapted. An optimal location, J, corresponds to a position where it is always favourable to be near, e.g. [START_REF] Pease | A model of population growth, dispersal and evolution in a changing environment[END_REF] give the example of an advancing glacier, from which there exists an optimal place to reside. In the case of a warming climate, this optimal location could be any number of things e.g. a moving front of adverse weather conditions, the moving range of a quick-to-adapt parasite vector or food stuff in response to the changing climate.

Global adaptation would allow individuals to reside near to this optimal location.

The Ecology

We assume that each cell on the lattice can support no more than one individual (i.e.

we are using a patch occupancy model). In each year we simulate, in turn, reproduction, dispersal of offspring and mortality. All individuals first have the opportunity to reproduce. To avoid potential biases towards particular locations or individuals, the order in which individuals are given the opportunity for reproduction is randomised. The probability that an individual reproduces depends upon its fitness (see Local Adaptation and Evolution, below). A single offspring is born to a reproducing adult and this offspring immediately disperses. This once-in-a-lifetime dispersal event occurs via a uniformly distributed Moore neighbourhood (nearest eight cells).

Offspring that disperse to an unoccupied cell automatically establish. In most simulations, dispersal to an occupied cell results in automatic death and there is thus a strong priority effect in relation to space occupancy. However, in some simulations we relax this assumption and allow offspring to displace established individuals with probability, p. We always assume reflective boundary conditions as a way of maintaining the current population size if the individuals are able to find space. After all individuals have had an opportunity to reproduce, and offspring have dispersed, we simulate mortality. All individuals are subject to stochastic mortality with probability, μ. Mortality is independent of both the condition of the environment and an individual's phenotype.

Local Adaptation and Evolution

Each individual carries two alleles. These are represented as integer values and their mean determines the environmental condition to which the individual is adapted. For example, an individual possessing two alleles with values 24 and 28, has phenotype, z=26 (and is optimally adapted to the conditions in a row x such that θ(x) = 26).

The probability that an individual reproduces in any given year depends upon both where the individual is located in relation to the environmental location with highest potential productivity, J, and the degree to which the individual is adapted to its local environment, θ(x). Formally, the probability of reproducing is given by the following expression:

exp [((θ(x) -z) 2 /2V s ) + ((J -x) 2 /2W s )]
where, the first term accounts for the degree to which the individual is locally adapted and the second accounts for the proximity of the individual to the location with highest potential productivity. V s determines the steepness of decline in an individual's fitness as it is moved further from the conditions to which it is best adapted, while W s determines the rate at which maximum potential reproduction declines with distance from J. Note that W s is akin to the notion of environmental tolerance as described in [START_REF] Pease | A model of population growth, dispersal and evolution in a changing environment[END_REF].

It is worth emphasising the distinction between W s and V s and the need for both parameters in our model using a simple illustrative example. Consider first two individuals, one of which is perfectly adapted to the conditions at location J while the other is perfectly adapted to conditions found 10 rows higher on the lattice than J. Initially we will assume both are located in the environments to which they are adapted. Despite the fact that both are optimally adapted, their reproductive fitness will not be the same as the first individual (at location J) is in the inherently more productive environment. Our parameter W s determines just how much lower the reproductive fitness is of the individual found 10 rows away from J. Next consider the case that these same two individuals are both located at the position 10 rows away from J. Now the first individual will have the lower reproductive fitness because, while both are in the same environment, the first is not optimally adapted to that environment. The amount that the first individual's fitness is reduced due to it being less well-adapted is determined by V s .

When a birth event occurs the offspring inherits two alleles: a copy of one of its parent's alleles (chosen randomly from the set of 2 with equal probability) and a copy from an individual, selected at random, in the vicinity of its parent. This vicinity can be considered as the pollen dispersal neighbourhood and we term it h. If the vicinity contains no individuals then the offspring inherits both alleles from its parent (this is akin to 'selfing'). During the copying of alleles that ensues, mutation occurs at each locus with a probability of β. When a mutation occurs, the value of the allele is modified by an amount randomly drawn from the discrete uniform distribution, (-10, 10).

Simulation Experiments

We have conducted a number of simulation experiments to explore the dynamics of the model. In the first set, we run the model under a stable climate (i.e. maintaining a constant position of maximum potential reproduction, J). We run the model for a range of parameter values and, in particular, vary both V s and W s . We output population abundance and mean phenotype across the environmental gradient. Subsequently, we run a large number of simulations where we initially allow the population to obtain quasiequilibrium under stable environmental conditions before simulating a period of climate change. In these simulations we are most interested in the probability that a population survives an episode of climate change. We run sensitivity analyses to ask, in turn, how this depends upon V s and W s , the strength of the priority effect (p), the rate (ν) and duration of climate change (T ), and the rate and size of mutations. In the final set of simulations we vary the probability of 238 mortality (β), the landscape width (by altering the number of columns on the grid), and the dispersal neighbourhood for pollen (h). In all the sensitivity analyses, we 240 run 50 replicate simulations for each parameter combination.

Results

Under stable environmental conditions, population distributions establish as one would expect. For smaller W s (corresponding to a steeper decline in maximum potential reproductive rate with distance from J), smaller ranges emerge (Figures 1a,1b and1c). The corresponding mean phenotype is shown in Figures 1d, 1e and1f. For smaller W s , there is a greater mismatch between the local environment and the mean phenotype away from J (in these figures J=140).

Most of our results relate to range dynamics under climate change. Figure 2 2a. At this stage, we define a 'climate envelope' for our simulated species. We establish the range of environmental conditions within which 99% of the individuals are found. We show the location of this climate envelope during the period of environmental change to illustrate how well the population's range responds. In the example given in Figure 2, as the environment changes, the population's range lags behind its climate envelope. This is most marked at the expanding front, with the leading edge of the range advance falling a substantial distance behind the shifting climate window. However, it is also apparent that at the trailing edge some individuals are found well behind what we defined as the species' climate envelope.

Running the simulation model for a wide range of parameter combinations revealed a rich range of behaviours. We illustrate some of the most important features in Figure 3a which shows mean phenotypes across the environmental gradient on one axis, and how this distribution of phenotypes changes during and subsequent to a period of climate change (see Figure 3). For most runs of the model where the population survives the episode of climate change we observe that the range width following climate change is narrower (Figures 3a, 3b and3d). This is a consequence of the loss of phenotypes adapted to the warmest and coolest extremes that were occupied by the population prior to the onset environmental change. Ultimately phenotypic variation is recovered and the range expands to reoccupy its original niche but this can take many thousands of generations. A common feature of the results is the initial expansion of the cold-adapted phenotypes (shown in green in Figure 3a) after the onset of change, but this is followed by a rapid collapse of these phenotypes (Figures 3b,3c and 3d) and, in some cases, a rapid collapse of the whole population (Figure 3c). Figure 3c illustrates an important result: even when there is a considerable overlap between the range of a species prior to climate change and the climate envelope of the species following change (shown by the white lines in Figure 3) extinction can still occur. Finally, we sometimes observe dynamics where the population splits into distinct subpopulations comprising individuals with quite distinct phenotypes. Because they are adapted to quite different climates, these subpopulations can remain spatially disjunct for quite lengthy periods following climate change (see Figure 3d). Figures 4 to 6 show the results of our sensitivity analyses, conducted to establish to roles of the different model parameters in determining the likelihood that a population is able to survive a period of climate change. Unsurprisingly, steeper declines in individuals' reproductive fitness as they are moved away from their own optima (i.e. small V s ), lead to reduced probabilities of populations surviving climate change (Figure 4a). However, for the sensitivity analysis around W s , the results are a little less expected. Higher probabilities of extinction are observed when the fundamental environment niche of the species is broader (i.e. the chance of extinction is greater for higher W s -see Figure 4b).

Perhaps our most important result is illustrated in Figure 5. This figure shows the results of simulations conducted to establish the role of p in determining the likelihood that species survive an episode of climate change. Recall that p determines the probability that an offspring is able to displace an already established individual from the site to which it disperses. The counterintuitive result shown in Figure 4b was obtained for p = 0.0 (i.e. there is an absolute priority effect). When the opposite is assumed, and offspring always displace existing occupants (i.e. p = 1.0), we find that species that have broader fundamental niches (higher W s ) are now more likely to survive than those with narrower niches (lower W s ). Interestingly, when we assume an offspring has a 50% chance of displacing an existing occupant (p = 0.5), we find that the initial result holds and extinction risk is higher for higher W s .

We find that the probability that a species' range survives a period of climate change is far more sensitive to some of the model's parameters than others. In general we find that the results are largely insensitive to the mutation parameters, neither the mutation rate nor the effect size of a single mutation have noticeable impacts (results not shown). However, the scale of pollen dispersal, the length of the period of climate change, the extrinsic probability of mortality and the width of the

Discussion

In this contribution, we have presented results from a simulation model that integrates key features from similar models exploring species' dynamics in response to environmental change [START_REF] Brooker | Modelling species' range shifts in a changing climate: The impacts of biotic interactions, dispersal distance and the rate of climate change[END_REF][START_REF] Dytham | Evolved dispersal strategies at range margins[END_REF]Mustin et al., 2009, e.g.) and from previous analytical theory exploring local adaptation on environmental gradients [START_REF] Kirkpatrick | Evolution of a species' range[END_REF][START_REF] Pease | A model of population growth, dispersal and evolution in a changing environment[END_REF]. We establish that local adaptation can play a major role in determining the dynamics of range shifting under climate change, and suggest that this is likely to have important implications for predictions of the impact of environmental change on both future species ranges and the rates of extinction.

Two results from our work clearly highlight how local adaptation can alter the nature of range shifting. First, a species may fail to survive a period of climate change, even when there is an overlap between its range prior to climate change and the area where climate is predicted to be suitable following climate change. Second, we find that often a range begins to expand in the direction of climate change before rapidly collapsing to extinction. Both these results have important applied implications.

The first suggests that any predictions based upon treating the species as a single homogenous unit may be flawed and that, for this reason, simple extensions of the bioclimate envelope approach are likely to be inadequate. The second implies that we need to be extremely cautious before inferring that species that are already expanding their ranges in the direction of climate change will continue to do so.

In our initial simulations, we found that species occupying broader niches more at risk of extinction due to climate change. This counterintuitive result required further investigation, and the key assumption driving this difference turned out to be local density dependence in the form of space occupancy. In the initial simulation model, individuals occupy a patch until they die and are never displaced by a newborn individual. When we alter the priority effect within the simulations, such that established individuals are no longer invulnerable to displacement by newborn individuals, we find that species with broader ranges in a stable climate are better able to survive a period of environmental change. [START_REF] Pease | A model of population growth, dispersal and evolution in a changing environment[END_REF] concluded that without local density dependence, a species with a broader range was less likely to become extinct during an episode of climate change. This is agreement with our findings.

Examining the original distributions of the population before climate change in the simulation, it becomes clear why the result has been obtained. The individuals at the periphery of the range (those lying on the lines at the extrema of the current climate) are lower in density and will necessarily have a lower birth rate than those towards the centre of the distribution (given the same local adaptation). The offspring of these individuals will however be able to establish more readily since the density of individuals decreases away from the globally optimal location. Thus, we note that the absolute gradient of the density of individuals away from this global optimum increases with decreasing W s . As a result, we see that decreasing W s increases the availability of patches surrounding the optimum location, into which offpsring can be born. Therefore as the climate starts to shift and the optimal location moves upwards, the high frequency of better adapted individuals with higher birth rates around the centre of the distribution have more space to move forward in the range and the individuals at the front of the distribution are soon optimally located, making them more fecund (despite being less locally adapted) but with fewer density dependent effects limiting their birth rate. This then creates a faster turnover of new individuals, on which selection can act. When W s is large, this movement and selection of phenotypes is much impaired leading to an overall reduction in the adaptability of the population and an increased chance of extinction. The local density-dependence effects have a much more striking effect on extinction probability than mutation rate or mutation size, which do not ameliorate extinction probability if density dependent effects are present.

The important role played by local density dependence, mediated through space occupancy effects, has recently been demonstrated for species assemblages and our results clearly demonstrate that it can also be important within a species. [START_REF] Takenaka | Local coexistence of tree species and the dynamics of global distribution pattern along an environmental gradient: a simulation study[END_REF] uses simulation modelling to explore the distributions of coexisting tree species as they shift their ranges due to climate change. Takenaka's work shows that the replacement of one species by another better adapted to the new climatic conditions can be dramatically slowed if mature individuals continue to block space for lengthy periods even though the climate no longer suits them. [START_REF] Brooker | Modelling species' range shifts in a changing climate: The impacts of biotic interactions, dispersal distance and the rate of climate change[END_REF] also described a similar space blocking effect in a simulation model describing the range dynamics of an assemblage comprising species arranged along an environmental stress gradient. The authors highlighted that the longevity of an organism would determine the strength of the effect. Our work highlights that these effects are not limited to interspecific interactions; prolonged space occupancy beyond the point at which an individual is well adapted to local conditions can similarly limit the range shifting of locally adapted phenotypes within a single species. An important determinant of species' extinction, both within and between species, will be the relative rates at which fecundity and mortality are impacted as the climate shifts. If fecundity declines relatively rapidly, space occupancy effects will be greater, while if mortality increases more rapidly, when environmental conditions are displaced from those to which an individual is adapted, the effect will be weaker. Additionally, higher rates of fitness-independent mortality will reduce the strength of the space blocking effect and allow more successful range shifting (see Figure 6c for an example). In future work, it will be interesting to explore how the partitioning of local adaptation within and between species may influence the spatial dynamics of a species guild under climate change scenarios.

An interesting feature of our results is that a species can become extinct even when the climate envelopes before and after climate change overlap. The realisation of the simulation model illustrated in Figure 3c shows this effect very clearly.

Representation of a species as a single unit, as is the case in the statistical pattern matching methods used in bioclimate envelope modelling [START_REF] Bakkenes | Assessing effects of forecasted climate change on the diversity and distribution of European higher plants for 2050[END_REF][START_REF] Broennimann | Do geographic distribution, niche property and life form explain plants' vulnerability to global change?[END_REF][START_REF] Schwarz | Predicting extinctions as a result of climate change[END_REF][START_REF] Thuiller | Climate change threats to plant diversity in Europe[END_REF] neglects this potentially crucial effect. There is an urgent need for a careful consideration of the taxanomic resolution at which we should be modelling the response of biodiversity to climate change. As further information becomes available on the extent to which populations within species are differentially adapted to local conditions, a critical appraisal of the importance of this evolutionary process will be required. This question of the organisational level at which species are represented also needs addressing within the community using dynamic vegetation models to predict both changes in ecological and biogeochemical processes during climate change [START_REF] Cramer | Global response of terrestrial ecosystem structure and function to CO 2 and climate change: results from six dynamic global vegetation models[END_REF][START_REF] Morales | Comparing and evaluating process-based ecosystem model predictions of carbon and water fluxes in major european forest biomes[END_REF]. Many of these models do not even work at the resolution of the species, instead they group species into functional types and assume that all members of a functional type share an identical environmental niche.

In reality, species (or populations) towards the cooler end of the environmental space that can be occupied by the functional type will not function in warmer conditions as effectively as those species (or populations) that have adapted to the heat, and vice versa. This may have important consequences not only for the projections of future patterns of biodiversity made by this type of model but also for their projections of future biogeochemical cycles and balances.

A new generation of models is required that takes the best parts from bioclimate modelling and incorporates ecological and evolutionary dynamics. The results presented in this paper make a strong case for the inclusion of local adaptation as a matter of urgency. One conservation management option to mitigate some of the effects described in this paper would be the translocation of genotypes within a species range. Our simulations tend to show that the phenotypes adapted to the local conditions found in the core of the range tend to be the most important for range shifting, and from this we would suggest that a potential strategy would be to translocate individuals (for plants, pollen might suffice) from core areas to populations located towards the expanding front. A model paramaterised with species-specific information on dispersal distance and the genetics of local adaptation would be an invaluable aid in developing this type of strategy, and could be used to provide recommendations on the optimal amount and frequency of translocation.

The model considered in this paper has much in common with a model used to explore the dynamics of speciation along environmental gradients (Doebeli and Dieckmann, 2003). Interestingly, we have observed that in some regions of parameter space we find that disruptive selection can results in two (or more) spatially distinct species arising in our simulation model. This is not the main topic of this paper, however we note that both during and following a period of climate change a species' range tends to become fragmented (see Figures 2 and3d). In some cases this spatial separation can last for a considerable period following the stabilisation of climate (Figure 3d) and it is interesting to speculate that in some cases this may act to promote speciation. It would certainly be worthwhile extending recent speciation models [START_REF] Artzy-Randrup | Sympatric speciation under incompatibility selection[END_REF][START_REF] Bolnick | Multi-species outcomes in a common model of sympatric speciation[END_REF]Yukilevich and True, 2006) to consider the potential impacts of environmental change.

Understanding the response of biodiversity at different organisational levels under a period of rapid environmental change represents the greatest challenge faced by ecologists today. Here, we have considered how local adaptation within a sessile species' range can impact on that species dynamics during climate change. By applying ideas and methods that are already well developed in the ecological literature we have obtained several new insights that we believe have important consequences for how we should go about both predicting and also managing changes in biodiversity under climate change. We hope that the results from theoretical studies such as this one make a persuasive case for a closer integration between the statistical modelling that currently dominates the climate impacts literature and dynamic spatial models incorporating biological processes.

  illustrates typical dynamics showing snapshots at various stages in the climate change period. Initially, we allow the population to obtain quasiequilibrium under stable climate and this is what is shown in Figure

Figure 1 :

 1 Figure 1: Frequency distributions for population size and trait value for the equilibrium in a constant environment, where the environment changes with x. The results were averaged over 50 realisations of the model. A set of simulations were run for a different values of 2lnW s given on the graph by different colours (red-2, blue-3, green-4, yellow-5, black-6). An increasing value of W s relates to an increased environmental niche range, seen in Figures (a), (b) and (c). In Figures(d), (e) and (f) the mean phenotype is an average of all allele values (2 per individual) existing on the environmentally invariant line y = const. In the case of zero density, the mean phenotype is given a value 0.

Figure 2 :

 2 Figure 2: Snapshots of one realisation of the model showing population inhabiting the 2d lattice landscape. The grid corresponds to the individuals on the grid with the invariant environment of each row and changing environmental gradient through each column. The colour grid shows the optimal trait value for each location, θ(x), running from hot (red) to cool (blue). Individual patch colour shows trait value to which the individual is adapted, z. The climate envelope (shown by black lines) indicates the area holding approximately 98% of the population at pre-climate change equilibrium, with the median at the optimal. (a) shows the equilibrium in a constant environment before any climate change has occurred; (b), (c), (d) and (e) show the population at a specified time after change onset and (f) shows final distribution after climate change. Illustrative parameters are chosen as W s =exp(5)/2 and V s =25, under a climate change rate of ν=0.3 over T =200 years to emphasise the range shift. During the climate shift this envelope tracks the environmental optimal for the population. The population splits into clusters before combining once the shift ends.

Figure 3 :

 3 Figure 3: Four realisations of the model through a period of climate change. The colours show the population and its local adapation, z (blue through to red). (a) shows population persistence with the majority of individuals tracking the climate; T =300,V s =100, W s =500, ν=25 leading to the subpopulation at the rear and very front of the range being lost but population persisting due to the middle third of the pre shift population moving. (b) shows population persistence through climate change with only a small subpopulation tracking the climate; T =300,V s =10, W s =500, ν=23 allowing population movement due to the middle of the front half of the range shifting, although front margin does show signs of initial movement. (c) shows population extinction during climate change; T =100, V s =50, W s =250, ν=40 produces a climate shift too fast for adaptation. (d) shows population persistence with two small subpopulations tracking the climate; T =300,V s =7.5, W s =500, ν=23 produces similar subpopulation shifting its range as seen in (b) but shift occurs in two geographically distinct subpopulations, which eventually rejoin for good on reaching the new equilibrium.

Figure 4 :

 4 Figure 4: Sensitivity analysis for variance revealing both population variances (V s and W s ) having opposite effects on extinction probability in the model with (a) showing the effect of variance around local optimal adaptation and (b) the effect of variance around the optimal location. The extinction probability for any population can be noted as a non-decreasing function of the rate of climate change, ν. Notice that an increase in the environmental niche (an increase in W s ) leads to a increase, or no change, in the extinction probability.

Figure 5 :

 5 Figure 5: The effect of decreased density dependent effects on proportion of populations extinct under climate change. Red to Blue scale corresponds as before to 100% to 0% of populations extinct after climate shift. Note the qualitative difference in the two trends: in (a) where p=0.5, a large range width corresponds to a higher extinction probability, whereas in (b) a large range width allows better population persistence.

broader (Figure6d).

funded by NERC and ALTERNET.

Acknowledgements

Thanks to Zaq Coelho and Gustav Delius for continued support. This research was

Travis, J. M. J., Munkemuller, T., Burton, O. J., Best, A. S., Dytham, C., Johst, K., 2007. The relative importance of assortative mating and migration modification. American Naturalist 167, 638-654.


A Blasi

N Maggiore
email: nicola.maggiore@ge.infn.it

N Magnoli

S Storace

Maxwell-Chern-Simons Theory With Boundary

Keywords: Quantum Field Theory, Boundary Conditions, Boundary Algebra PACS Nos: 03.70.+k Theory of Quantized Fields, 11.15.Wx Topologically massive gauge theories 11.15.Yc Chern-Simons gauge theory

The Maxwell-Chern-Simons (MCS) theory with planar boundary is considered. The boundary is introduced according to Symanzik's basic principles of locality and separability. A method of investigation is proposed, which, avoiding the straight computation of correlators, is appealing for situations where the computation of propagators, modified by the boundary, 1

Introduction

In the last decades, there has been vast interest in the role of the space boundary in Quantum Field Theory (QFT) [1,2,3]. The main reason is that the presence of a boundary changes the properties of a physical system, originating phenomena like the Casimir Effect [4] or the edge states in the Fractional Quantum Hall Effect [START_REF] Wen | Quantum Field Theory of Many Body Systems: from the Origin of Sound to an Origin of Light and Electrons (Oxford Graduate Texts)[END_REF].

In some cases, the boundary even represents the center of the investigation, as in the case of black holes [START_REF] Bañados | [END_REF], where the horizon of events can be indeed regarded as a boundary. Boundary effects play an important role also in critical phenomena [START_REF] Diehl | Field Theoretical Approach to the Critical Behaviour at Surface[END_REF].

In particular, the effect of a boundary is extremely interesting in topological field theories. Indeed, topological theories are known to have no local observables except when the base manifold has a boundary [2,3]. Our attention will be actually focused on the topological, three dimensional Chern-Simons (CS) model [START_REF] Witten | [END_REF]9].

The introduction of a boundary in CS theory has been investigated in previous works [2,3], with the result that the local observables arising from the presence of the boundary are two-dimensional conserved chiral currents generating the Kač-Moody algebra [10] of the Wess-Zumino-Witten model [11]. In other words,the boundary establishes a strong connection between CS theory and conformal field theories [12].

An analogous result has been obtained for another topological field theory, namely the topological BF model [13] in three dimensions, for which it has been shown as well the existence of chiral currents living on the boundary and satisfying a Kač-Moody algebra with central extension [14].

The existence of a Kač-Moody algebra with central extension in the CS model with boundary has led to interesting applications due to the connection of the CS theory with several physical systems.

In fact, following the equivalence between CS theory and (2+1) dimensional gravity theory with a cosmological constant [15,16], it has been stressed that the algebra plays a crucial role in understanding the statistical origin of the entropy of a black hole [START_REF] Bañados | [END_REF]. It is thus possible to use the algebra to compute the BTZ black hole (negative cosmological constant) entropy [17] and the Kerr-de Sitter space (positive cosmological constant) entropy [18].

In condensed matter physics, the abelian CS model provides an effective low energy theory for the Fractional Quantum Hall Effect [START_REF] Wen | Quantum Field Theory of Many Body Systems: from the Origin of Sound to an Origin of Light and Electrons (Oxford Graduate Texts)[END_REF]. Further, when a boundary is taken into account, the Kač-Moody algebra describes the boundary chiral currents which are indeed observed on the edge of the Hall bar (edge states) [START_REF] Wen | Quantum Field Theory of Many Body Systems: from the Origin of Sound to an Origin of Light and Electrons (Oxford Graduate Texts)[END_REF].

On the other hand, the coupling of the CS model to other theories also gives rise to interesting results. For instance, it can be coupled to fermion or boson fields to attach magnetic flux to charge density, thus providing an explicit realization of anyons, i.e. particles living only in systems with two spatial dimensions and satisfying a fractional statistics [START_REF] Wilczek | Fractional Statistics and Anyon Superconductivity (Singapore[END_REF].

Among the others, one of the most striking properties of the CS term is that, when added to the three dimensional Yang-Mills action, it originates a topologically massive gauge theory [START_REF] Deser | [END_REF].

Its abelian version, the Maxwell-Chern-Simons (MCS) theory, when coupled with fermions defines a three dimensional modified electrodynamics, in which the "photons" are massive and have a single state of helicity [START_REF] Deser | [END_REF]. The Casimir effect for topologically massive electrodynamics could provide, in principle, a way to "measure" the topological mass of the "photons" [21].

Moreover, the addition of a Maxwell term to the effective low energy theory for the Quantum Hall Effect allows the description of the gap between the ground state and the bulk elementary excitations [START_REF] Wen | Quantum Field Theory of Many Body Systems: from the Origin of Sound to an Origin of Light and Electrons (Oxford Graduate Texts)[END_REF].

The Maxwell theory coupled to the CS action on a manifold with boundary has a further application in (2+1) dimensional quantum gravity: the so called Einstein-Maxwell-Chern-Simons theory [22]. In this framework, the boundary can be regarded as the horizon of the black hole solutions, and the gauge field coupled to gravity describes a topologically massive electromagnetic field which provides the black hole with an electric charge, which, coming from the CS term, is "kind of" topological. The result is thus called "charged black hole".

On the other hand, the Maxwell model is not topological, and therefore the question if its addition spoils the chiral current algebra of the CS theory arises naturally: the aim of this paper is actually to discuss this issue.

To reach this task, we first must face a further problem, i.e. the method. Indeed, the inclusion of a boundary in field theory is a highly non trivial task if one wishes to preserve locality and power counting, the most basic ingredients of QFT.

In 1981 K. Symanzik [1] addressed this question: his key idea was to add to the bulk action a local boundary term which modifies the propagators of the fields in such a way that nothing propagates from one side of the boundary to the other. He called this property "separability" and showed that it requires the realization of a well identified class of boundary conditions that can be implemented by a local bilinear interaction.

These ideas strongly inspired the authors of [23], [24] and [25], who used a closely related approach to compute the chiral current algebra living on the boundary of the three-dimensional CS model. Indeed, the authors of [23] added to the action local boundary terms compatible with power counting, using a covariant gauge fixing. In [24,25], on the other hand, a regularization-free procedure was followed: the equations of motion, rather than the action, were modified by appropriate boundary terms, and a non-covariant axial gauge was preferred.

The main reason for the latter choice was that the main advantage of a covariant gauge, i.e. Lorentz invariance, already fails due to the presence of the boundary. On the other hand, the axial choice does not completely fix the gauge [START_REF] Bassetto | Yang-Mills Theories in Algebraic Non Covariant Gauges : Quantization and Renormalization[END_REF], and a residual gauge invariance exists, implying the existence of a Ward identity which plays a crucial role since, when restricted to the boundary, it might generate a chiral current algebra, as we shall see.

In both these works, the explicit computation of the propagators of the theory seems to be necessary. However, this step could be quite difficult in other theories, like MCS, and another way of investigation is worthy, which avoids the explicit computation of the Green functions of the theory. This is precisely the choice that we make. Our approach is actually more similar to that used in [24,25], since we focus on the effect of the boundary on the equations of motion rather than on the action, and we adopt a noncovariant axial gauge rather than a covariant one.

The basic idea is that, after modifying the equations of motion by means of boundary terms satisfying general basic requirements, we integrate them in proximity of the boundary and use Symanzik's idea of separability to determine, rather than impose, the boundary conditions on the propagators, that can be expressed as boundary conditions on the fields [1]. These, in turn, have an effect on the boundary breaking term of the residual Ward identity that generates the Kač-Moody algebra living on the boundary. For these reasons, our method is suitable also for a different kind of investigation. Indeed, the approach in [START_REF] Park | [END_REF] and [28] is good to describe just a portion of space: there is not a beyond the boundary. This is not a problem for the description of systems like electrodynamics on a disk [29] or a Hall bar in the framework of the Quantum Hall Effect [START_REF] Wen | Quantum Field Theory of Many Body Systems: from the Origin of Sound to an Origin of Light and Electrons (Oxford Graduate Texts)[END_REF], in which one is interested in the dynamics of the internal system only.

However, the description of what there is beyond the boundary is fundamental in the physics of defects [30], and our approach is actually suitable for this task. In the study of semiconductors [START_REF] Drabold | Theory of Defects in Semiconductors[END_REF], for instance, it is important to be able to describe local interactions between the particles and the imperfections of the material, approximated by δ-type interactions.

On the other hand, topological defects are gathering more and more importance also in the framework of astroparticle physics, where they are proposed as an explanation for the formation of cosmic structures and for the genera-tion of extremely high-energy cosmic rays [START_REF] Berezinsky | [END_REF]33].

Even considering an impenetrable boundary, we keep both the left and the right side of spacetime. Indeed, we can fix the parameters of our general description in such a way to decouple the opposite sides of spacetime completely, and therefore in principle we are able to examine just one of the two sides, if we wish. In a certain way, this is actually what we will do by imposing the conservation of the bulk discrete symmetry (2.11) which we called parity.

In other words, according to Symanzik's approach, the boundary is defined by the decoupling condition, which prevents correlations between points belonging to opposite sides of the boundary. A defect could be treated by using the same approach described in this paper, by simply relaxing the decoupling condition. In this way, the resulting modified theory describes interactions between a bulk and a δ-type insertion in the action, i.e. a defect, as it is done in [34].

We organized this paper as follows. In Section 2 we reconsider the abelian CS theory with boundary, and we show that our method leads again to the correct results found in [23] and [24] for both the boundary conditions and the residual Ward identity, and we illustrate how the Ward identity generates the Kač-Moody algebra.

Although the method illustrated in Section 2 cannot be entirely found elsewhere already, the main original results can be found in Section 3, where we introduce the Maxwell term in the CS theory. We follow the same scheme of Section 2 to investigate whether, also in this case, conserved chiral currents exist which satisfy some kind of algebra, like in the pure CS theory. We stress again that the answer to this question is not at all to be taken for granted, due to the non-topological character of the bulk theory.

Our concluding remarks summarize our results and draw some conclusions, with some further suggestions of possible applications and extensions.

"Boundarization": the pure Chern-Simons case

The model

In this Section we recall some known results concerning CS theory with planar boundary [23,24,25]. This will give us the opportunity to illustrate the method we shall follow in what really matters us, that is the MCS theory with boundary, in order to investigate the boundary physics without calculating explicitly the correlators of the theory in presence of the boundary.

In euclidean space, the abelian CS action reads :

S cs = - k 2 d 3 x µνρ A µ ∂ ν A ρ . (2.1)
It will be convenient to work in euclidean light-cone coordinates, defined as:

u = x 2 z = 1 √ 2 (x 1 -ix 0 ) z = 1 √ 2 (x 1 + ix 0 ) , (2.2)
which induce similar definitions in the space of the fields:

A u = A 2 A = 1 √ 2 (A 1 + iA 0 ) Ā = 1 √ 2 (A 1 -iA 0 )
.

(2.

3)

The CS action then reads

S cs = -k dudzdz Ā∂ u A + A u ∂ Ā -A u ∂A . (2.4)
The gauge fixing term is :

S gf = -dudzdz A u b , (2.5)
which corresponds to the axial gauge choice

A u = 0 . (2.6)
The complete action is

Σ CS = S cs + S gf + S ext , (2.7)
where in S ext we coupled, as usual, external sources J Φ to the quantum fields Φ = A u , A, Ā, b :

S ext = -dudzdz Φ J Φ Φ . (2.8)
Besides the canonical mass dimensions, the fields appearing in the action Σ CS are assigned an additional quantum number, called helicity, which encodes the two-dimensional Lorentz invariance of the theory on the planes u = constant.

The dimensions and helicities of the fields and sources are summarized in Table 1 :

T able 1 : CS Quantum Numbers A u A Ā b J u J J J b ∂ u ∂ ∂ u z z dim 1 1 1 2 2 2 2 1 1 1 1 -1 -1 -1 hel 0 1 -1 0 0 1 -1 0 0 1 -1 0 -1 +1
Moreover, as it is well known, the axial gauge choice (2.6) does not completely fix the gauge: a residual local gauge invariance remains, which is expressed by the local Ward identity

∂ J + ∂J + ∂ u J u + ∂ u b = 0 , (2.9)
which, once integrated, gives

du ∂ J + ∂J = 0 . (2.10)
Finally, the set of symmetries of the action Σ CS is completed by the discrete transformation involving at the same time coordinates and fields:

z ↔ z u → -u A ↔ Ā A u → -A u b → -b J ↔ J J u → -J u J b → -J b , (2.11)
which we will refer to by using the term "parity".

The field equations of motion are

k ∂A u -∂ u Ā + J = 0 (2.12) k (∂ u A -∂A u ) + J = 0 (2.13) k ∂ Ā -∂A + 1 k b + J u = 0 (2.14) A u + J b = 0 . (2.15)

The boundary

We now want to introduce a boundary in the theory, and we choose the planar surface u = 0.

The effect of the boundary is to break the original equations of motion, in a way which must respect the following basic constraints.

Locality:

The boundary contribution must be local. This means that all possible breaking terms have the form

δ (n) (u)X(z, z, u) , (2.16)
where δ (n) (u) is the n-order derivative of the Dirac delta function with respect to its argument, and X(z, z, u) is a local functional.

Separability: This constraint, called also decoupling condition, refers to Symanzik's original idea [1] according to which the n-point Green functions which involve two fields computed in points belonging to opposite sides of space must vanish. In particular, the propagators of the theory must satisfy :

uu < 0 ⇒ ∆ AB (x -x ) = T (ϕ A (x)ϕ B (x )) = 0 , (2.17)
where

x = (z, z, u) , (2.18)
and ϕ A (x) is a generic field of the theory. This property is satisfied by propagators which split according to

∆ AB (x -x ) = θ + ∆ AB+ (x -x ) + θ -∆ AB-(x -x ) , (2.19)
where ∆ AB+ (x -x ) and ∆ AB-(x -x ) are respectively the propagators for the right and the left side of spacetime,

θ ± = θ(±u)θ(±u ) ( 2 . 2 0 )
and θ(x) is the step function, defined as usual.

The relation (2.19) is referred to as "decoupling condition", which induces the decomposition of the generating functional W of connected Green functions according to

W = W + + W -, (2.21)
where W + and W -are the generators of the connected Green functions for the right side and the left side of spacetime, respectively.

Linearity: Finally, we require that the boundary contributions to the equations of motion be linear in the fields because, in general, the symmetries of the classical action, if only linearly broken at the classical level, nonetheless remain exact symmetries of the quantum action [START_REF] Weinberg | The Quantum Theory of Fields[END_REF]. Moreover, we are considering a free field theory and therefore nonlinear terms in the field must not occur in the equations of motion.

The most general broken equations of motion which satisfy all the above requirements are

k ∂A u -∂ u Ā + J = δ(u)(c + 1 Ā+ + c - 1 Ā-) ( 2 . 2 2 ) k (∂ u A -∂A u ) + J = δ(u)(c + 2 A + + c - 2 A -) ( 2 . 2 3 ) k ∂ Ā -∂A + 1 k b + J u = δ(u)(c + 3 A u+ + c - 3 A u-) (2.24) A u + J b = 0 , (2.25)
where c ± i are constant parameters, and A ± (Z), Ā± (Z) and A u± (Z) are the boundary fields on the right, respectively on the left, of the boundary:

A ± (Z) = lim u→0 ± A(x) ( 2 . 2 6 ) Ā± (Z) = lim u→0 ± Ā(x) ( 2 . 2 7 ) A u± (Z) = lim u→0 ± A u (x) , (2.28)
with Z = (z, z). Under parity (2.11), the boundary fields transform according to

A ± ↔ Ā∓ (2.29) A u± → -A u∓ .
(2.30)

Imposing the parity constraint (2.11) on the broken equations of motion leads to

c ± 1 = c ∓ 2 (2.31) c + 3 = -c - 3 = c 3 . (2.32)
In addition, the broken equations of motion must satisfy what in [24] has been called compatibility, which is nothing else than the commutation property which equations of motion in general obey. This algebraic constraint further reduces the number of parameters appearing in the r.h.s. of the boundary equations of motion :

c + 1 = c - 1 = c . (2.33)
Finally, the equations of motion, broken by the presence of the planar boundary u = 0, which respect parity and compatibility are

k ∂A u -∂ u Ā + J = cδ(u)( Ā+ + Ā-) ( 2 . 3 4 ) k (∂ u A -∂A u ) + J = cδ(u)(A + + A -) ( 2 . 3 5 ) k ∂ Ā -∂A + 1 k b + J u = c 3 δ(u)(A u+ -A u-) (2.36) A u + J b = 0 . (2.37)

The boundary conditions

In QFT with boundary, a crucial issue is represented by the boundary condition for the quantum fields. This problem is often solved by imposing "by hand" boundary conditions and see what happens. In Symanzik's approach, the only constraint is the decoupling condition, which concerns the Green functions, as we discussed previously. The boundary conditions on the fields, rather than being imposed, are read, once the basic separability condition on propagators is realized.

The price to pay is, of course, the need of an explicit computation of the propagators, which, in presence of a boundary, is not at all an easy task, usually.

For CS theory the problem has been solved in [23] and in [24], by two quite different approaches, and, from the expression of the propagators on the boundary, the boundary conditions on the fields -which turn out to be of the Dirichlet type -are inferred. But, for more complicated situations, like the one this paper is devoted to, this program might be out of reach.

Here, we present an easy way out. That is, we claim to be able to carry out Symanzik's program for QFTs with boundary, and find out the boundary physics, without explicitly computing the Green functions.

In this Section we are treating the pure CS case, which is known, just to illustrate the idea.

Definitely more interesting is the adoption of our simple, but powerful, method, to the MCS theory, what will be done in the next Section.

After setting the sources to zero, let us integrate (2.34) and (2.35) with respect to u from -ε to ε. Taking into account also (2.37), we get

(k -c) Ā-= (k + c) Ā+ (2.38) (k + c) A -= (k -c) A + .
(2.39)

Due to the decoupling condition, each side of the above identities must vanish separately:

(k -c) Ā-= 0 (2.40) (k + c) Ā+ = 0 (2.41) (k + c) A -= 0 (2.42) (k -c) A + = 0 .
(2.43)

The nontrivial solutions are given by

c = k ⇒ Ā+ = 0 = A - (2.44) c = -k ⇒ Ā-= 0 = A + .
(2.45) This is exactly the same result previously found in [24,25,[START_REF] Blasi | [END_REF]: the fields obey Dirichlet boundary conditions on both sides of the dividing plane u = 0.

The curious reader is invited to compare how the same result has ben obtained in [23] and [24], where, like we did, the boundary conditions are not put "by hand" in the theory but derived from the Symanzik's decoupling condition.

It is clear that the choices (2.44) and (2.45) essentially describe the same physics, since they are related by the parity transformation (2.11). Therefore, in what follows we choose the solution with c = k, keeping in mind that a kind of "mirror solution" exists.

The equations of motion thus become

k ∂A u -∂ u Ā + J = kδ(u) Ā- (2.46) k (∂ u A -∂A u ) + J = kδ(u)A + (2.47) k ∂ Ā -∂A + 1 k b + J u = c 3 δ(u)(A u+ -A u-) (2.48) A u + J b = 0 . (2.49)
Correspondingly, the local Ward identity (2.9) acquires a boundary breaking:

∂ J + ∂J + ∂ u J u + ∂ u b = kδ(u) ∂A + + ∂ Ā-+ ∂ u [c 3 δ(u)(A u+ -A u-)] ,
(2.50) which yields the integrated Ward identity

1 k du ∂ J + ∂J = ∂A + + ∂ Ā-.
(2.51)

The boundary algebra

In this Section we recall how the Ward identity (2.51) implies the existence of a Kač-Moody algebra of conserved chiral currents on the boundary.

We can rewrite the Ward identity in a functional way as

1 k du ∂ J (x) + ∂J(x) = ∂ δW + δ J(x) u=0 + + ∂ δW - δJ(x) u=0 - . (2.52)
We then differentiate with respect to J(x ), with x lying on the right side of space next to the boundary, and then set the sources to zero. We thus get

1 k du∂δ 3 (x -x ) =   ∂ δ 2 W + δ J(x )δ J(x) u ,u=0 +   J Φ =0 +   ∂ δ 2 W - δ J(x )δJ(x) u =0 + ,u=0 -   J Φ =0 . (2.53)
Recalling that W ± are the generators of the connected Green functions for the u > 0 and u < 0 sides of spacetime respectively, the right-hand side of this expression involves propagators, the second of which vanishes due to the decoupling condition. Therefore, we are left with

1 k ∂δ 2 (Z -Z ) = ∂ T (A + (Z )A + (Z)) . (2.54)
Keeping in mind the definition of the time-ordering operator T , we have to specify the role of time in light-cone variables. This issue has already been extensively studied in the literature [37]. The outcome is that a possibility is to identify the light-cone variable z with time.

We can now explicitly compute the right-hand side of (2.54):

∂ T (A + (Z )A + (Z)) = = ∂ θ(z -z)A + (Z )A + (Z) + θ(z -z )A + (Z)A + (Z ) = -δ(z -z)A + (Z )A + (Z) + θ(z -z)A + (Z ) ∂A + (Z) + δ(z -z )A + (Z)A + (Z ) + θ(z -z ) ∂A + (Z)A + (Z ) = θ(z -z)A + (Z ) ∂A + (Z) + θ(z -z ) ∂A + (Z)A + (Z ) +δ(z -z ) [A + (Z), A + (Z )] . (2.55)
On the other hand, after setting the sources to zero, the Ward identity (2.51) and the decoupling condition yield the chirality condition [23,24,25]:

0 = ∂A + ⇒ A + = A + (z) ( 2 . 5 6 ) 0 = ∂ Ā-⇒ Ā-= Ā-(z) ( 2 . 5 7 )
Substituting (2.56) into (2.55) and then into (2.54) we get

1 k ∂δ 2 (Z -Z ) = 1 k δ(z -z ) ∂δ(z -z ) = δ(z -z ) [A + (z), A + (z )] , (2.58)
which finally yields the commutation relation

[A + (z), A + (z )] = 1 k ∂δ(z -z ) . (2.59)
This is the abelian counterpart of the Kač-Moody algebra [10] of the Wess-Zumino-Witten model [11] generated by the chiral currents found in [3,23,24]. Under this respect, the coefficient 1 k can be seen as the central charge of the Kač-Moody algebra. Note that the parity symmetry (2.11) implies the mirror algebra

Ā-(z), Ā-(z ) = 1 k ∂δ(z -z ) ( 2 . 6 0 )
on the opposite side of the boundary.

Moreover, from (2.56) and (2.44), we get

∂A + + ∂ Ā+ = 0 , (2.61)
which is the conservation relation for the planar boundary field, written in light-cone coordinates [25]. Indeed, recalling (2.3), we can come back to the euclidean components of the gauge field:

A + = 1 √ 2 (A 1+ + iA 0+ ) Ā+ = 1 √ 2 (A 1+ -iA 0+ ) .
(2.62)

The relation (2.61) then takes the form

∂ 1 A 1+ + ∂ 0 A 0+ = 0 , (2.63)
which is easily identified with a continuity relation involving a density A 0+ and a current A 1+ .

Furthermore, from (2.44) and (2.56) follows

∂ 1 A 0+ -i∂ 0 A 0+ = 0 , (2.64)
which, compared to (2.63), implies the identification of A 0+ and A 1+

A 1+ = iA 0+ .

(2.65)

Consequently, the algebra (2.59) can be written in terms of the density A 0+ :

[A 0+ (z), A 0+ (z )] = - 1 2k ∂δ(z -z ) . (2.66)
Summarizing, in this Section we recovered the general result [2,3] that a topological field theory acquires local observables only when a boundary is introduced. In the CS case, the observables are conserved chiral currents living on the boundary and satisfying a Kač-Moody algebra, whose central charge is the inverse of the CS coupling constant, as discussed in [23,24,25].

Our approach led us to the same results previously obtained in [23,24,25], but, we stress once again, with the remarkable difference that we have used an algebraic method which has allowed to avoid the explicit computation of the propagators of the theory. This will be extremely useful in cases, like that described in the next Section, in which the explicit computation of correlators is not that easy.

3 One step beyond: adding a Maxwell term

The model

In this Section we study the effect of a planar boundary on the MCS theory.

We stress that adding a Maxwell term to the CS theory is not a mere extension, in particular for what concerns the boundary physics. The Maxwell term, indeed, breaks the topological character of the theory, and, hence, all properties which are peculiar to topological QFTs in principle do not hold anymore. The Maxwell term introduces local degrees of freedom, which are absent in topological CS theory, and what happens when a boundary is introduced is far from being obvious.

From the physical point of view, the MCS theory describes a massive spin 1 particle with a single state of helicity, and which can be coupled with fermions to define a modified electrodynamics of fermions interacting with each other and with topologically massive "photons" [START_REF] Deser | [END_REF]. The introduction of local degrees of freedom renders the MCS theory particularly relevant both in solid state physics (Quantum Hall Effect [START_REF] Wen | Quantum Field Theory of Many Body Systems: from the Origin of Sound to an Origin of Light and Electrons (Oxford Graduate Texts)[END_REF]) and (black holes [START_REF] Bañados | [END_REF]), as explained in the Introduction.

For this reason, it is interesting to investigate wether 1. chiral 2. conserved currents exist, which 3. satisfy an algebra, and of which type, and, 4. if the Kač-Moody structure is preserved, how sensitive is the central charge to the Maxwell term.

The MCS action, in euclidean space, is

S MCS = S CS + S M , (3.1)
where S CS is the CS action (2.1), S M is the Maxwell action

+ α 4 d 3 xF µν F µν , (3.2)
and F µν (x) is the field strength.

Notice that the theory actually depends only on one parameter only, that we identify with k, which represents the topological mass. In fact, the parameter α in front of the Maxwell term can be reabsorbed by a redefinition of the gauge field A µ (x). Nevertheless, we prefer to keep it in order to be able, at a later step, to switch off the Maxwell term and make contact with the pure CS case. Another good reason to keep α, is that in the nonabelian extension it is not possible to reabsorbe it, and therefore in the general case it is a real coupling constant.

The Maxwell action in light-cone coordinates (2.3), is :

S M = - iα 2 dudzdz ∂A ∂A + ∂ Ā∂ Ā -2∂ Ā ∂A +2∂ u A ∂A u + 2∂ u Ā∂A u -2∂ u Ā∂ u A -2∂A u ∂A u . (3.3)
Recalling that the CS theory in euclidean light-cone coordinates is given by S CS (2.4), that the gauge fixing term for the axial gauge A u = 0 is S gf (2.5), and that the external sources are coupled to quantum fields through S ext (2.8), the complete action for the theory that we now consider is

Σ MCS = S CS + S M + S gf + S ext . (3.4)
The presence of the Maxwell term modifies the canonical mass dimensions of fields and sources. Table 2 summarizes dimensions and helicities of MCS theory. Notice also that the CS coupling constant k acquires a mass dimension [k] = 1.

T able 2 : MCS Quantum Numbers

A u A Ā b J u J J J b dim 1/2 1/2 1/2 5/2 5/2 5/2 5/2 1/2 hel 0 1 -1 0 0 1 -1 0
The local, and integrated, Ward identities (2.9), (2.10), expressing the residual gauge invariance, are left unchanged, and also the discrete parity symmetry (2.11) still holds for MCS theory.

The equations of motion induced by the action Σ MCS (3.4) are

k( ∂A u -∂ u Ā) + iα[ ∂(∂ Ā -∂A) + ∂ u (∂ u Ā -∂A u )] + J = 0 (3.5) k(∂ u A -∂A u ) + iα[∂( ∂A -∂ Ā) + ∂ u (∂ u A -∂A u )] + J = 0 (3.6) k(∂ Ā -∂A) + iα[2∂ ∂A u -∂ u ( ∂A + ∂ Ā)] + b + J u = 0 (3.7) A u + J b = 0 . (3.8)

The boundary

We now introduce the planar boundary u = 0. The decoupling condition imposes that the generating functional of the connected Green functions W can be written as

W = W + + W - (3.9)
and, consequently, the propagators of the theory take the form (2.19).

The presence of the boundary induces in the equations of motion boundary terms that must respect all the constraints of locality, linearity in the quantum fields, power counting, conserved quantum numbers and analiticity in the parameters. The most general broken equations of motion satisfying all the requirements are

k( ∂A u -∂ u Ā) + iα[ ∂(∂ Ā -∂A) + ∂ u (∂ u Ā -∂A u )] + J = = δ(u) α + 1 Ā+ + α - 1 Ā-+ α + 2 ∂ u Ā + + α - 2 ∂ u Ā - +δ (u) α + 3 Ā+ + α - 3 Ā-+ δ(u) α + 6 ∂A u+ + α - 6 ∂A u- (3.10) k(∂ u A -∂A u ) + iα[∂( ∂A -∂ Ā) + ∂ u (∂ u A -∂A u )] + J = = δ(u) γ + 1 A + + γ - 1 A -+ γ + 2 (∂ u A) + + γ - 2 (∂ u A) - +δ (u) γ + 3 A + + γ - 3 A -+ δ(u) γ + 4 ∂A u+ + γ - 4 ∂A u- (3.11) k(∂ Ā -∂A) + iα[2∂ ∂A u -∂ u ( ∂A + ∂ Ā)] + b + J u = = δ(u) α + 4 ∂A + + α - 4 ∂A -+ α + 5 ∂ Ā+ + α - 5 ∂ Ā- +δ(u) α + 8 A u+ + α - 8 A u-+ δ(u) α + 9 (∂ u A u ) + + α - 9 (∂ u A u ) - +δ (u) α + 10 A u+ + α - 10 A u- (3.12) A u + J b = 0 . (3.13)
In the above equations, α ± i and γ ± i are constant parameters. Notice that the lower dimension of the gauge field, with respect to the CS case, allows the presence of differentiated delta functions δ (u) in the right-hand sides of the broken equations of motion.

Again, we have used the boundary fields on the left, respectively on the right of the boundary:

Ā± (Z) = lim u→0 ± Ā(x) ( 3 . 1 4 ) A ± (Z) = lim u→0 ± A(x) ( 3 . 1 5 ) A u± (Z) = lim u→0 ± A u (x) ( 3 . 1 6 ) ∂ u Ā ± (Z) = lim u→0 ± ∂ u Ā(x) ( 3 . 1 7 ) (∂ u A) ± (Z) = lim u→0 ± ∂ u A(x) ( 3 . 1 8 ) (∂ u A u ) ± (Z) = lim u→0 ± ∂ u A u (x) , (3.19)
where Z = (z, z).

Under the parity symmetry (2.11), the boundary fields transform according to

A ± ↔ Ā∓ A u± → -A u∓ (∂ u A) ± ↔ -∂ u Ā ∓ (∂ u A u ) ± ↔ (∂ u A u ) ∓ . (3.20)
We now proceed by imposing the constraints that we discussed in the previous Section.

The parity constraint yields the following conditions: We now consider the algebraic constraint of "compatibility" between equations of motion. We already said in the previous Section that this simply corresponds to requiring that the equations of motion, even if broken by boundary terms, commute with each other.

α ± 1 = γ ∓ 1 (3.21) α ± 2 = -γ ∓ 2 (3.22) α ± 3 = -γ ∓ 3 (3.23) α ± 6 = -γ ∓ 4 (3.
After a little algebra, and recalling some basic properties of the δ-functions [START_REF] Schwartz | Introduction to the Theory of Distributions[END_REF], we get

α + 1 = α - 1 = α 1 (3.29) α ± 2 = α ∓ 3 (3.30) α + 6 = -α + 4 (3.31) α - 6 = α + 5 (3.32) α + 9 = -α + 10 . (3.33)
The broken equations of motion thus become

k( ∂A u -∂ u Ā) + iα[ ∂(∂ Ā -∂A) + ∂ u (∂ u Ā -∂A u )] + J = = δ(u) α 1 Ā+ + Ā-+ α - 3 ∂ u Ā + + α + 3 ∂ u Ā - +δ (u) α + 3 Ā+ + α - 3 Ā--δ(u) α + 4 ∂A u+ -α + 5 ∂A u- (3.34) k(∂ u A -∂A u ) + iα[∂( ∂A -∂ Ā) + ∂ u (∂ u A -∂A u )] + J = = δ(u) α 1 (A + + A -) -α + 3 (∂ u A) + -α - 3 (∂ u A) - -δ (u) α - 3 A + + α + 3 A --δ(u) α + 5 ∂A u+ -α + 4 ∂A u- (3.35) k(∂ Ā -∂A) + iα[2∂ ∂A u -∂ u ( ∂A + ∂ Ā)] + b + J u = = δ(u) α + 4 ∂A + -∂ Ā-+ α + 5 ∂ Ā+ -∂A - +δ(u) α + 8 (A u+ + A u-) -α + 10 (∂ u A u ) + -(∂ u A u ) - +δ (u)α + 10 (A u+ -A u-) ( 3 . 3 6 ) A u + J b = 0 . (3.37)
Correspondingly, the local Ward identity (2.9) acquires a boundary breaking:

∂ J + ∂J + ∂ u J u + ∂ u b = δ(u) α 1 ∂ Ā+ + ∂ Ā-+ ∂A + + ∂A - +δ(u) α - 3 ∂ ∂ u Ā + -∂ (∂ u A) -+ α + 3 ∂ ∂ u Ā - -∂ (∂ u A) + -δ(u) α + 4 + α + 5 ∂ ∂A u+ -α + 4 + α + 5 ∂ ∂A u- +δ (u) α + 3 ∂ Ā+ -∂A -+ α - 3 ∂ Ā--∂A + +δ (u) α + 4 ∂A + -∂ Ā-+ α + 5 ∂ Ā+ -∂A - +δ (u) α + 8 (A u+ + A u-) -α + 10 (∂ u A u ) + -(∂ u A u ) - +δ (u)α + 10 (A u+ -A u-) , (3.38)
which, once integrated and taking into account the axial gauge condition A u = 0, yields the residual Ward identity

du ∂ J + ∂J = α 1 ∂ Ā+ + ∂ Ā-+ ∂A + + ∂A -+ +α - 3 ∂ ∂ u Ā + -∂ (∂ u A) -+ α + 3 ∂ ∂ u Ā - -∂ (∂ u A) + . (3.39)
In the previous Section, we have seen that, in CS theory with boundary, it is the integrated residual Ward identity (3.39) which implies the existence of an algebra of chiral conserved currents on the boundary. Our aim will now be to study the change of the boundary conditions induced by the additional Maxwell term on the fields, and the consequences on the algebra: indeed, it is not obvious that there will be still an algebra of boundary currents.

The boundary conditions I: the set up

We now infer the boundary conditions on the fields, following the method illustrated in the previous Section for the pure CS case.

The first step is to set the sources to zero and integrate the broken equations of motion between the two sides of the boundary, thus getting the conditions

k Ā--Ā+ + iα ε -ε du ∂ ∂ Ā -∂A -iα ∂ u Ā - -∂ u Ā + = = α 1 Ā+ + Ā-+ α - 3 ∂ u Ā + + α + 3 ∂ u Ā - (3.40) k (A + -A -) + iα ε -ε du ∂ ∂A -∂ Ā -iα (∂ u A) --(∂ u A) + = = α 1 (A + + A -) -α + 3 (∂ u A) + -α - 3 (∂ u A) - (3.41) k ε -ε du ∂ Ā -∂A -iα ∂A + -∂A -+ ∂ Ā+ -∂ Ā-+ ε -ε du b = α + 4 ∂A + -∂ Ā-+ α + 5 ∂ Ā+ -∂A -, (3.42)
where we have also used the axial gauge condition A u = 0. Notice that (3.41) can be obtained from the first one by means of the parity transformation (2.11). Therefore, we can consider only (3.40) and then apply parity to the resulting conditions to get the analogous ones generated by (3.41). Taking the limit ε → 0 in (3.40) and (3.42) we get the two conditions

(α 1 + k) Ā+ + α - 3 -iα ∂ u Ā + = = -(α 1 -k) Ā--α + 3 + iα ∂ u Ā - (3.43) α + 4 + iα ∂ Ā-+ α + 5 + iα ∂A -= = α + 4 + iα ∂A + + α + 5 + iα ∂ Ā+ . (3.44)
Let us focus on (3.44) first. As usual, separability splits it into

α + 4 + iα ∂A + + α + 5 + iα ∂ Ā+ = 0 (3.45) α + 4 + iα ∂ Ā-+ α + 5 + iα ∂A -= 0 . (3.46)
Since the parameters α + 4 and α + 5 do not appear in the Ward identity (3.39), the boundary algebra does not depend on them, and hence we can solve (3.45) and (3.46) by choosing :

α + 4 = α + 5 = -iα , (3.47)
so that (3.45) and (3.46) yield

∂A + + ∂ Ā+ = 0 (3.48) ∂A -+ ∂ Ā-= 0 , (3.49)
which, as in the pure CS case, can be interpreted as conservation relations for currents living on the right, respectively on the left of the boundary.

On the other hand, the request for separability splits (3.43) into

(α 1 + k) Ā+ + α - 3 -iα ∂ u Ā + = 0 (3.50) (α 1 -k) Ā-+ α + 3 + iα ∂ u Ā - = 0 . (3.51)
Using the parity transformation (2.11) on these conditions we get the analogous ones generated by (3.41):

(α 1 + k) A --α - 3 -iα (∂ u A) -= 0 (3.52) (α 1 -k) A + -α + 3 + iα (∂ u A) + = 0 . (3.53)
Since the broken equations of motion contain second-order derivatives, the second step towards the determination of the boundary conditions on the quantum fields, consists into integrating them twice.

Let us set the sources to zero and then integrate (3.34):

u -∞ du k( ∂A u -∂ u Ā) + iα[ ∂(∂ Ā -∂A) + ∂ u (∂ u Ā -∂A u )] + J = = u -∞ du δ(u) α 1 Ā+ + Ā-+ α - 3 ∂ u Ā + + α + 3 ∂ u Ā - + u -∞ du δ (u) α + 3 Ā+ + α - 3 Ā-; ( 3 . 5 4 )
from this we will get also the conditions generated by (3.35) by means of the parity transformation (2.11), like we have done above. From (3.54) we get

-k Ā + iα u -∞ du ∂(∂ Ā -∂A) + iα∂ u Ā = = θ(u) α 1 Ā+ + Ā-+ α - 3 ∂ u Ā + + α + 3 ∂ u Ā - +δ(u) α + 3 Ā+ + α - 3 Ā-, (3.55)
where we have assumed that the fields and their derivatives vanish at infinity. We then integrate (3.55) between the two sides of the (infinitesimal) boundary:

- ε -ε du k Ā + iα ε -ε du u -∞ du ∂(∂ Ā -∂A) + iα ε -ε du ∂ u Ā = = ε -ε du θ(u) α 1 Ā+ + Ā-+ α - 3 ∂ u Ā + + α + 3 ∂ u Ā - + ε -ε du δ(u) α + 3 Ā+ + α - 3 Ā-; ( 3 . 5 6 )
taking the limit ε → 0 we finally get

-iα Ā-+ iα Ā+ = α + 3 Ā+ + α - 3 Ā-, (3.57)
which can be rearranged as

-iα + α - 3 Ā-= -iα + α + 3 Ā+ . (3.58)
The request for separability again splits this condition into two separate ones for the opposite sides of the boundary:

iα + α - 3 Ā-= 0 (3.59) -iα + α + 3 Ā+ = 0 ; (3.60)
Applying the parity transformation (2.11) to these conditions we get those generated by (3.35)

iα + α - 3 A + = 0 (3.61) -iα + α + 3 A -= 0 . (3.62)
At the end, we have obtained a set of eight boundary conditions involving both the fields and their first derivatives, which must be simultaneously satisfied.

Summarizing:

(α 1 + k) Ā+ + α - 3 -iα ∂ u Ā + = 0 (3.63) (α 1 -k) Ā-+ α + 3 + iα ∂ u Ā - = 0 (3.64) (α 1 + k) A --α - 3 -iα (∂ u A) -= 0 (3.65) (α 1 -k) A + -α + 3 + iα (∂ u A) + = 0 (3.66) iα + α - 3 Ā-= 0 (3.67) -iα + α + 3 Ā+ = 0 (3.68) iα + α - 3 A + = 0 (3.69) -iα + α + 3 A -= 0 , (3.70)
which must be considered together with the conservation condition (3.47):

α + 4 = α + 5 = -iα (3.71) ∂A + + ∂ Ā+ = 0 (3.72) ∂A -+ ∂ Ā-= 0 . (3.73)
We are now left with the task of solving the above constraints, thus identifying all the possible choices for the parameters and their consequences on the boundary fields and the residual Ward identity.

The boundary conditions II: solution(s) and algebra(s)

The first natural request is to recover the CS result in the limit α → 0. This implies a dependence of the parameter α 1 on α of the type

α 1 = ±k (1 + 2f (α)) , lim
∂ u Ā ± → ∂ u Ā ∓ (3.78) (∂ u A) ± → (∂ u A) ∓ (3.79) k → -k (3.80) α ± 3 → -α ∓ 3 , (3.81)
as it can be directly checked by comparison with (3.63)-(3.70). Therefore, we can focus on the eight cases with α 1 = k. Reminding that the boundary algebra is determined by the residual Ward identity (3.39), for each solution we write the boundary conditions on the fields and the corresponding Ward identity:

1.

     α 1 = k α + 3 = -iα α - 3 = +iα ⇒ A ± = Ā± = 0 du ∂ J + ∂J = iα∂ ∂ u Ā + -∂ u Ā - + iα ∂ (∂ u A) + -(∂ u A) -; (3.82) 2.      α 1 = k α + 3 = -iα α - 3 = -iα ⇒    Ā+ = 0 = A - ∂ u Ā + = 0 = (∂ u A) - du ∂ J + ∂J = k ∂ Ā-+ ∂A + + iα ∂ (∂ u A) + -∂ ∂ u Ā - ; (3.83) 3.      α 1 = k α + 3 = +iα α - 3 = -iα ⇒          (∂ u A) + = ∂ u Ā - = 0 k Ā+ -iα ∂ u Ā + = 0 kA -+ iα (∂ u A) -= 0 du ∂ J + ∂J = k ∂ Ā-+ ∂A + ; ( 3 . 8 4 ) 4.      α 1 = k α + 3 = -iα α - 3 = ∓iα ⇒    Ā± = 0 = A ± ∂ u Ā + = 0 = (∂ u A) - du ∂ J + ∂J = iα ∂ (∂ u A) + -∂ ∂ u Ā - ;
(3.85) 5.

     α 1 = k α + 3 = -iα α - 3 = +iα ⇒    Ā± = 0 = A ± (∂ u A) + = 0 = ∂ u Ā - du ∂ J + ∂J = iα ∂ ∂ u Ā + -∂ (∂ u A) -; (3.86) 6.
     α 1 = k α + 3 = +iα α - 3 = ∓iα ⇒          A + = Ā-= (∂ u A) + = ∂ u Ā - = 0 2k Ā+ + -iα + α - 3 ∂ u Ā + = 0 2kA ---iα + α - 3 (∂ u A) -= 0 du ∂ J + ∂J = k iα + α - 3 iα -α - 3 ∂ Ā+ + ∂A -; (3.87) 7.      α 1 = k α + 3 = ∓iα α - 3 = -iα ⇒    Ā+ = A -= 0 ∂ u Ā ± = (∂ u A) ± = 0 du ∂ J + ∂J = k ∂ Ā-+ ∂A + ; ( 3 . 8 8 ) 8.      α 1 = k α + 3 = ∓iα α - 3 = ∓iα ⇒    A ± = Ā± = 0 (∂ u A) ± = ∂ u Ā ± = 0 du ∂ J + ∂J = 0 .
(3.89)

However, not all these solutions of the system (3.63)-(3.70) are acceptable. Indeed, we must still keep in mind that taking the limit α → 0 we must recover the result found for the CS theory.

This condition eliminates the 1., 4., 5. and 8. Furthermore, the conservation relations (3.72) and (3.73), together with the boundary conditions, imply that the Ward identity for the cases 6.) and 7.) becomes du ∂ J + ∂J = 0 , (3.90) while for the case 2.) it takes the form

du ∂ J + ∂J = +iα ∂ (∂ u A) + -∂ ∂ u Ā - . (3.91)
Recalling the form (2.51) ofthe Ward identity for pure CS theory with boundary, these cases do not have the correct limit α → 0 either, and therefore they are forbidden as well.

At the end, we are left only with the case 3.), which corresponds to Neumann and Robin boundary conditions

         α 1 = k α + 3 = +iα α - 3 = -iα α + 4 = α + 5 = -ik (3.92)
which yields on the rhs of the boundary

       (∂ u A) + = 0 k Ā+ -iα ∂ u Ā + = 0 ∂A + + ∂ Ā+ = 0 (3.93) du ∂ J + ∂J = k ∂ Ā-+ ∂A + .
(3.94)

The Ward identity (3.94) is the same as in the CS case, and it implies both the chirality ∂A + = 0 (3.95) and the boundary Kač-Moody algebra

[A + (z), A + (z )] = 1 k ∂δ(z -z ) ( 3 . 9 6 )
involving the conserved chiral current A + (z).

The corresponding solution on the lhs of the boundary can be obtained by parity, and read

       ∂ u Ā - = 0 kA -+ iα (∂ u A) -= 0 ∂A -+ ∂ Ā-= 0 (3.97) ∂ Ā-= 0 (3.98) Ā-(z), Ā-(z ) = 1 k ∂δ(z -z ) ( 3 . 9 9 )
involving the conserved chiral current Ā-(z).

This is a remarkable result, since we have found not only that the boundary terms are uniquely fixed, but also that there is still a Kač-Moody algebra, which is the same as in the CS case. In other words, the Maxwell term does not affect the boundary physics of the light-cone currents.

This result is in agreement with the outcomes of [START_REF] Park | [END_REF] and [28], where different approaches have been adopted, and a disk instead of a plane has been considered.

More recent approaches, like [START_REF] Ghoshal | [END_REF] just define one boundary (say on the right) and formulate the local boundary interaction by means of a lagrangian or an algebraic way. The introduction of two boundaries or defects/impurities can be done as in [40].

The claim of [START_REF] Park | [END_REF] and [28] is that the boundary is characterized by Kač-Moody algebras of conserved charges with the same structure of the CS model, thus concluding that the algebra is a consequence of the CS term rather than of the nature of the entire model. We refer to our conclusions for a more precise comparison.

We conclude this Chapter by illustrating the reason why the parameter α 1 must be set to the value k. As we previously pointed out, the request that the limit α → 0 lead to the CS result implies (3.74). In such a case, it can be checked that the only acceptable solution of the system (3.63)-(3.70) is

         α 1 = k (1 + 2f (α)) α + 3 = +iα α - 3 = -iα α + 4 = α + 5 = -iα ⇒              k (1 + f (α)) Ā+ -iα ∂ u Ā + = 0 k (1 + f (α)) A -+ iα (∂ u A) -= 0 kf (α)A + -iα (∂ u A) + = 0 kf (α) Ā-+ iα ∂ u Ā - = 0 ,
which corresponds to the broken residual Ward identity

du ∂ J + ∂J = kf (α) ∂ Ā+ + ∂A -+ k (1 + f (α)) ∂ Ā-+ ∂A + .
(3.100) Equation (3.100), with the external sources set to zero, together with separability, implies the relations shows also that the request that the boundary currents be conserved automatically implies their chirality.

kf (α)∂ Ā+ + k (1 + f (α)) ∂A + = 0 (3.101) kf (α) ∂A -+ k (1 + f (α)) ∂ Ā-= 0 , ( 3

Conclusions

In this paper the three dimensional Maxwell-Chern-Simons (MCS) theory with planar boundary has been considered, following the pioneering work of Symanzik [1].

We adopted a method similar to the one introduced in [24], which avoids the problem of regularizing the boundary action, which is necessarily δ-function dependent, but we modified it in a significant manner. In fact, one of the main steps towards the realization of the Symanzik's constraint of separability, which implements the introduction of a boundary in quantum field theory, is the direct computation of the propagators of the theory, taking into account, of course, the boundary interactions. While this is quite feasible in CS theory, in other more complicated, and physically more relevant cases -of which the MCS model is an example -this could be a much more difficult task. In fact, we have been able to reach the same goals, as in the simpler case of pure CS theory, without a direct computation of the propagators of the theory.

As already pointed out, we stress that the addition of the Maxwell term to the CS action spoils its topological character, and thus it is not at all granted that the same results can be obtained.

Indeed, we found that, under certain conditions, 1. chiral conserved currents living on the boundary exist, 2. which satisfy, like in the CS case, a Kač-Moody algebra, whose central charge coincides with the inverse of the CS coupling constant.

In particular, we have found that there are many possible boundary terms compatible with Symanzik's constraint of separability, each associated with different boundary conditions on the fields. However, the requirement to recover the CS case in the limit of vanishing Maxwell term, uniquely sets the boundary parameters to values which imply Neumann and Robin boundary conditions on the fields on both sides of the boundary. In this situation, the same residual Ward identity of the CS case holds, thus implying the results 1. and 2. above mentioned.

This result leads us to conclude that the boundary physics is independent of the Maxwell term which breaks the topological character of the CS model, even though the Maxwell boundary contributions to the equations of motion are nontrivial. This is a remarkable result, since it suggests that the existence of a boundary Kač-Moody algebra of conserved chiral currents depends only on the presence of a CS term in the theory, rather than on the particular theory itself.

We point out that our main results hold also for the nonabelian model, namely the Yang-Mills-Chern-Simons action. Indeed, the conservation of boundary currents, their chirality and the central charge of their Kač-Moody algebra are entirely determined by the quadratic part of the action only, and therefore are shared with the nonabelian extension, which affects the vertices of the theory.

Our result, concerning the existence of a Kač-Moody boundary algebra with the same structure of the CS model, somehow agrees with Deser's recent statement that, for spin one and in three spacetime dimensions, "everything is CS" [START_REF] Deser | Fundamental Interactions[END_REF]. In Ref. [START_REF] Deser | Fundamental Interactions[END_REF], in fact, he shows how a Yang-Mills-Chern-Simons (YMCS) action can be rewritten in the form of a pure CS action in terms of a new field.

But, even in the earlier work [START_REF] Lemes | [END_REF], three dimensional Yang-Mills gauge theories in the presence of the Chern-Simons action were shown to be generated by the pure topological Chern-Simons term through nonlinear covariant redefinitions of the gauge field. And indeed these claims are supported by our results, which do not depend on the presence of a bulk Maxwell term.

Another point of contact is with the outcome of [22] for the (2+1) dimensional black hole coupled with electrodynamics, where it is shown that the black hole coupled to the pure CS theory does not change configuration when a Maxwell term is "turned on".

In the framework of the Fractional Quantum Hall Effect, our result justifies the fact that an additional Maxwell term to the pure CS effective low energy model describes the gap of the elementary bulk excitations, without affecting the properties of the edge states [START_REF] Wen | Quantum Field Theory of Many Body Systems: from the Origin of Sound to an Origin of Light and Electrons (Oxford Graduate Texts)[END_REF].

Finally, we are in agreement also with [START_REF] Park | [END_REF] and [28], where the nonabelian YMCS theory defined on a manifold with boundary is considered in the Weyl gauge. In Ref. [START_REF] Park | [END_REF], a (2+1) dimensional disk is considered, and the Dirac's procedure is followed. The boundary Kač-Moody algebra is then obtained as the projection of the Dirac's brackets on the boundary of the disk.

In [28] the discussion is instead carried out in terms of the Gauss law generators, and the algebra in terms of the fields is just a consequence. Nevertheless, they both find that also in the YMCS theory there is a boundary algebra, which is the same as that of the CS model.

However, our approach is not only simpler, but also stays in a more general framework. In fact, in both [START_REF] Park | [END_REF] and [28] the boundary conditions are chosen a priori, provided that they satify some general requirements, while we find all possible boundary conditions compatible with the simple and fundamental requests of locality and separability, without requiring any other constraint.

Moreover, in our description the boundary is the most general one compatible with very general principles -like locality, power counting and helicity conservation -and with the algebraic structure of the theory.

We conclude our discussion by indicating other possible further developments.

In fact, the three dimensional CS theory is not the only possible topological field theory. The other important Schwarz type topological field theory [START_REF] Witten | [END_REF] is represented by the BF models [13], which, contrarily to the CS theory, which is intrinsically three dimensional, exist in an arbitrary number of dimensions.

In three dimensions, the BF model with boundary has been studied in [14], where a richer algebraic structure than the CS case has been found, always of the Kač-Moody type.

A natural possible extension could therefore be the study of the effect of a (d-1) dimensional boundary in the d-dimensional BF theory. The investigation of the possible boundary algebra in these cases is a challenge.

24 ) α ± 4 = -α ∓ 5 ( 3

 24453

  we will show later, the case f (α) = 0 is incompatible with the conditions (3.72) and (3.73) expressing the conservation condition. Therefore, the choice α 1 = ±k (3.75) is compulsory. Moreover, the case α 1 = -k can be obtained from α 1 = k by means of the changes Ā± → Ā∓ (3.76) A ± → A ∓ (3.77)

  .102) which are incompatible with the existence of conserved quantities as expressed by (3.72) and (3.73) unless f (α) = 0. Notice that this argument