This is in the context of the Riemann-Roch Theorem.We begin with its connection with Physics for the theorem will be most clear to Physics oriented people in this context only. The connection is in the context of a concept in theoretical Particle Physics. This is the concept of supersymmetry.
This story begins in the paper
E. Witten, Constraints on Supersymmetry Breaking, Nucl. Phys. B202 (1982) 253.
As per the INSPIRE HEP database this article has been cited by 1539 papers. This means, among other things, that one can not track anymore the diverse ideas that have used this paper. This papers is about non-perturbative constraints on supersymmetry breaking.
In this paper, by a monumental survey of known supersymmetry models, Edward Witten came to conclusion that dynamical supersymmetry breaking does occur in anyone of these models. This survey is impressive both by techniques employed and the length, breadth and comprehensiveness of the models surveyed. because of this the negative result is rather discouraging.
In summary it means that once you have supersymmetry then you can not break it. This is a disaster when it comes to applications of the beautiful concept of supersymmetry to real Physics. Supersymmetry is the symmetry between fermions and bosons.
Our best theory of Particle Physics is the Standard Model. It has both fermions and bosons. But it has no supersymmetry. Clearly at the practical level, empirical level, in real life, at the experimental level supersymmetry is broken.
In fact the latest searches for supersymmetry at the Large Hadron Collider of CERN in Geneva very extensive experimental survey has yielded so far absolutely no evidence for supersymmetry.
Anyway even if supersymmetry is found the fact will remain that at energies that are available up till now there the world of Particle Physics is not supersymmetric and that means it, supersymmetry, is broken.
This should give us a perspective on the puzzle that we face today about supersymmetry. Experimentally supersymmetry is not found and hence it must be broken while theoretically we do not know even a single model where it is dynamically broken.
One may inquire here in what way dynamical breaking is different from just breaking. The hint in this subtlety of expression is the following. Explicit breaking of supersymmetry means that we begin with such a Lagrangian for Physics that has terms that break supersymmetry. These terms, be definition, do not possess supersymmetry. They are not invariant under supersymmetry transformations. In this case we are starting with a theory that simply does not possess supersymmetry. This is the case of explicit supersymmetry breaking.
Dynamical supersymmetry breaking is more interesting. Here we begin with a Lagrangian, hence a theory, that has supersymmetry to begin with. Then after dynamical effects set in, when dynamics shows its effects, then we end up in a theory that breaks supersymmetry. This process of supersymmetry breaking is more elegant and physically more attractive and acceptable.
Finally we must also explain the meaning of word perturbative. This is a technical term from quantum mechanics and is applicable to every application that uses quantum mechanics. Exactly solvable problems in quantum mechanics are very few. Free particle, particle in a box, particle hitting a potential barrier, particle facing a potential step or a finite potential well, particle in an infinite vertical potential well, particle in harmonic oscillator potential and hydrogen atom problem are practically the exhaustive set of exactly solvable problems.
For every other problem of Physics that uses quantum mechanics we must use some approximation method to get the useful and practical information. Amongst these approximation methods perturbation theory is the most effective and useful technique. This method gives a series expansion for physical quantities like energy.
This series is mostly infinite and this method can not always be applied. It is only applicable when the given problem is close to some exactly solvable problem. By Closeness we mean that the Hamiltonian or the Lagrangian of the theory differs by a term, which we call the perturbation, that is small in some sense. This sense of smallness is not difficult to understand. For example in case of energies the corrections engendered by the new term, the perturbation, should be small as compared to the energy in case of exactly solvable problem.
This much is very well known to every one who has learned basics of quantum mechanics. The perturbation Hamiltonian or Lagrangian in most cases is the interaction term and contains a constant that gives the strength of the interaction and it is called the coupling constant. When the coupling constant is small then the perturbation theory is useful because perturbation series is convergent and hence useful. This is called the perturbative regime or the weak coupling regime or the weak coupling limit.
The other case is of strong coupling and hence of large interaction and in this case perturbation theory is not useful because perturbation series is not convergent. In this case perturbation theory has failed and we can not use it. In such cases we must use other methods to get useful information about the physical system. The unimaginative collective name for such methods is non-perturbative methods.
Apart from this distinction of weak and strong couplings and corresponding respective perturbative and non-perturbative regimes there is another distinct division of physical regimes of quantum field theories. These are the regimes of low and high energies.
These two classifications get entangled with each other because of the following considerations. In quantum field theories, theories that we use to describe Particle Physics, couplings change with changing energies. This means that the perturbation term in the Lagrangian or the Hamiltonian changes with changing energy. In other words the coupling constants are not constants but become the parameters that change (with energy).
This is the case of running coupling constants. A constant is not supposed to change but in case of field theory that is what is an inevitable conclusion. This oxymoron expression, this contradiction in terms, is a very profound and fundamental one. This observation bears the same depth of insight as the first time realization of wave particle duality or uncertainty relations. Unfortunately the wave-particle duality and the uncertainty principle are common knowledge but running of coupling constant is not. The reason is not difficult to understand. quantum field theory is removed from our experience in quantum mechanics by one further level along the same line where quantum mechanics is removed from our classical experience.
Now let is once again summarize what we have discussed above. We would like to talk about the Riemann-Roch Theorem and we have began with its connection with Physics and this connection is through Witten Index that was defined in above cited paper.
To investigate theoretical models that begin with supersymmetry but ultimately break it Witten needed tools to analyze the models of supersymmetry available. In Section 2 of above cited paper he defined the now famous Witten Index. Symbolically he defined it as the operator in which the number minus one is raised to the power F. This is a notation for $\exp(2i\pi J_z)$. This later operator gives plus one eigenvalue for a bosonic state and minus one for a fermionic state.
This might look terribly abstract, obtuse and opaque - just to begin with. Luckily Witten is at his pedagogical best in this Section, as well as the whole paper itself. In this Section he has also exhibits his hallmark simplicity of the genius in statements like - "Supersymmetry breaking just means that the ground-state energy is positive".
By a very simple argument we can show that states with non-zero energy are paired by the action of the supersymmetry generator Q. One implication of this is that at the level of excited states we shall never have supersymmetry breaking - fermionic and bosonic states are paired.
In case of zero energy states this pairing is absent. Thus at zero energy there can be an arbitrary number of bosonic states and another arbitrary number of fermionic states.
As the parameters of the theory, masses and couplings, are varied fermionic and bosonic states might jump up from the zero energy to non-zero or vice-a-versa. In this process the difference between the number of bosonic and fermionic states at the zero energy will remain constant because the states jump up or down in pairs to maintain pairing at non-zero energies.
Clearly the difference of bosonic and fermionic states of zero energy is parameter independent and hence it can be calculated in most convenient limit for us to do calculations.
If this difference is non-zero then there is at least one bosonic or fermionic zero energy state. In this case supersymmetry is unbroken.
This story begins in the paper
E. Witten, Constraints on Supersymmetry Breaking, Nucl. Phys. B202 (1982) 253.
As per the INSPIRE HEP database this article has been cited by 1539 papers. This means, among other things, that one can not track anymore the diverse ideas that have used this paper. This papers is about non-perturbative constraints on supersymmetry breaking.
In this paper, by a monumental survey of known supersymmetry models, Edward Witten came to conclusion that dynamical supersymmetry breaking does occur in anyone of these models. This survey is impressive both by techniques employed and the length, breadth and comprehensiveness of the models surveyed. because of this the negative result is rather discouraging.
In summary it means that once you have supersymmetry then you can not break it. This is a disaster when it comes to applications of the beautiful concept of supersymmetry to real Physics. Supersymmetry is the symmetry between fermions and bosons.
Our best theory of Particle Physics is the Standard Model. It has both fermions and bosons. But it has no supersymmetry. Clearly at the practical level, empirical level, in real life, at the experimental level supersymmetry is broken.
In fact the latest searches for supersymmetry at the Large Hadron Collider of CERN in Geneva very extensive experimental survey has yielded so far absolutely no evidence for supersymmetry.
Anyway even if supersymmetry is found the fact will remain that at energies that are available up till now there the world of Particle Physics is not supersymmetric and that means it, supersymmetry, is broken.
This should give us a perspective on the puzzle that we face today about supersymmetry. Experimentally supersymmetry is not found and hence it must be broken while theoretically we do not know even a single model where it is dynamically broken.
One may inquire here in what way dynamical breaking is different from just breaking. The hint in this subtlety of expression is the following. Explicit breaking of supersymmetry means that we begin with such a Lagrangian for Physics that has terms that break supersymmetry. These terms, be definition, do not possess supersymmetry. They are not invariant under supersymmetry transformations. In this case we are starting with a theory that simply does not possess supersymmetry. This is the case of explicit supersymmetry breaking.
Dynamical supersymmetry breaking is more interesting. Here we begin with a Lagrangian, hence a theory, that has supersymmetry to begin with. Then after dynamical effects set in, when dynamics shows its effects, then we end up in a theory that breaks supersymmetry. This process of supersymmetry breaking is more elegant and physically more attractive and acceptable.
Finally we must also explain the meaning of word perturbative. This is a technical term from quantum mechanics and is applicable to every application that uses quantum mechanics. Exactly solvable problems in quantum mechanics are very few. Free particle, particle in a box, particle hitting a potential barrier, particle facing a potential step or a finite potential well, particle in an infinite vertical potential well, particle in harmonic oscillator potential and hydrogen atom problem are practically the exhaustive set of exactly solvable problems.
For every other problem of Physics that uses quantum mechanics we must use some approximation method to get the useful and practical information. Amongst these approximation methods perturbation theory is the most effective and useful technique. This method gives a series expansion for physical quantities like energy.
This series is mostly infinite and this method can not always be applied. It is only applicable when the given problem is close to some exactly solvable problem. By Closeness we mean that the Hamiltonian or the Lagrangian of the theory differs by a term, which we call the perturbation, that is small in some sense. This sense of smallness is not difficult to understand. For example in case of energies the corrections engendered by the new term, the perturbation, should be small as compared to the energy in case of exactly solvable problem.
This much is very well known to every one who has learned basics of quantum mechanics. The perturbation Hamiltonian or Lagrangian in most cases is the interaction term and contains a constant that gives the strength of the interaction and it is called the coupling constant. When the coupling constant is small then the perturbation theory is useful because perturbation series is convergent and hence useful. This is called the perturbative regime or the weak coupling regime or the weak coupling limit.
The other case is of strong coupling and hence of large interaction and in this case perturbation theory is not useful because perturbation series is not convergent. In this case perturbation theory has failed and we can not use it. In such cases we must use other methods to get useful information about the physical system. The unimaginative collective name for such methods is non-perturbative methods.
Apart from this distinction of weak and strong couplings and corresponding respective perturbative and non-perturbative regimes there is another distinct division of physical regimes of quantum field theories. These are the regimes of low and high energies.
These two classifications get entangled with each other because of the following considerations. In quantum field theories, theories that we use to describe Particle Physics, couplings change with changing energies. This means that the perturbation term in the Lagrangian or the Hamiltonian changes with changing energy. In other words the coupling constants are not constants but become the parameters that change (with energy).
This is the case of running coupling constants. A constant is not supposed to change but in case of field theory that is what is an inevitable conclusion. This oxymoron expression, this contradiction in terms, is a very profound and fundamental one. This observation bears the same depth of insight as the first time realization of wave particle duality or uncertainty relations. Unfortunately the wave-particle duality and the uncertainty principle are common knowledge but running of coupling constant is not. The reason is not difficult to understand. quantum field theory is removed from our experience in quantum mechanics by one further level along the same line where quantum mechanics is removed from our classical experience.
Now let is once again summarize what we have discussed above. We would like to talk about the Riemann-Roch Theorem and we have began with its connection with Physics and this connection is through Witten Index that was defined in above cited paper.
To investigate theoretical models that begin with supersymmetry but ultimately break it Witten needed tools to analyze the models of supersymmetry available. In Section 2 of above cited paper he defined the now famous Witten Index. Symbolically he defined it as the operator in which the number minus one is raised to the power F. This is a notation for $\exp(2i\pi J_z)$. This later operator gives plus one eigenvalue for a bosonic state and minus one for a fermionic state.
This might look terribly abstract, obtuse and opaque - just to begin with. Luckily Witten is at his pedagogical best in this Section, as well as the whole paper itself. In this Section he has also exhibits his hallmark simplicity of the genius in statements like - "Supersymmetry breaking just means that the ground-state energy is positive".
By a very simple argument we can show that states with non-zero energy are paired by the action of the supersymmetry generator Q. One implication of this is that at the level of excited states we shall never have supersymmetry breaking - fermionic and bosonic states are paired.
In case of zero energy states this pairing is absent. Thus at zero energy there can be an arbitrary number of bosonic states and another arbitrary number of fermionic states.
As the parameters of the theory, masses and couplings, are varied fermionic and bosonic states might jump up from the zero energy to non-zero or vice-a-versa. In this process the difference between the number of bosonic and fermionic states at the zero energy will remain constant because the states jump up or down in pairs to maintain pairing at non-zero energies.
Clearly the difference of bosonic and fermionic states of zero energy is parameter independent and hence it can be calculated in most convenient limit for us to do calculations.
If this difference is non-zero then there is at least one bosonic or fermionic zero energy state. In this case supersymmetry is unbroken.