User jeff harvey - MathOverflow

Answer by Jeff Harvey for Example for non equivalent rational full CFTs with same modular invariant (partition function)

2013-05-26T20:37:39Z

I have no idea what a ``symmetric haploid Frobenius algebra" is, so my answer may be terribly naive, but given an even-self dual lattice of rank $d$ there is a well known lattice construction of a rational (actually holomorphic) conformal field theory with central charge $c=d$. There are two even self-dual lattices of rank $16$ ($E_8 \times E_8$ and $Spin(32)/(Z/2)$) and these give rise to two inequivalent holomorphic Conformal Field Theories with $c=16$. The partition function of these theories is $\Theta_{\Gamma}/\eta^{16}$ where $\eta$ is the Dedekind eta function and $\Theta_\Gamma$ is the theta function of the associated lattice. The theta functions for these two lattices are the same because they are modular forms of weight $8$ and there is a unique such modular form. This is related to the famous "can one hear the shape of a drum" problem. Thus this pair of lattices gives an example of two inequivalent CFT's with the same partition function.

Answer by Jeff Harvey for Is there a "right" proof of Riemann's Theta Relation?

2013-04-13T17:32:51Z

Let me consider one of the Riemann theta relations in Mumford, since as you say, once you know one of them it is straightforward to derive the rest. Relation (R5) is $$ \theta_{00}(x) \theta_{00}(y) \theta_{00}(u) \theta_{00}(v) - \theta_{01}(x) \theta_{01}(y) \theta_{01}(u) \theta_{01}(v) - \theta_{10}(x) \theta_{10}(y) \theta_{10}(u) \theta_{10}(v) $$ $$+ \theta_{11}(x) \theta_{11}(y) \theta_{11}(u) \theta_{11}(v) = 2 \theta_{11}(x_1) \theta_{11}(y_1) \theta_{11}(u_1) \theta_{11}(v_1) $$ where the notation is that of Mumford with $\theta_{11}$ the odd theta function and the argument $\tau$ of the theta functions has been suppressed. $x_1, y_1, u_1, v_1$ are defined in terms of $x,y,u,v$ as in the question.

First, note that if we specialize to $x=y=u=v=0$ we obtain Jacobi's abstruse identity $$ (\theta_{00}(0))^4 - (\theta_{01}(0))^4 - (\theta_{10}(0))^4=0 $$ This equation can be interpreted in terms of characters of level one characters for affine $D_4$. There are four irreducible representations, basic, vector and two spinor representations. The above equation is the statement that the vector and spinor characters are equal, $ \chi_{vec}(\tau) = \chi_{sp}(\tau) $ and this is a corollary of the existence of triality for the $D_4$ Dynkin diagram. In open string theory $D_4=SO(8)$ appears as the transverse part of the ten-dimensional Lorentz group, space-time bosons are associate to the vector representation and space-time fermions to one of the spinor representations and the above is the statement that there are equal numbers of fermions and bosons at each mass level as required by supersymmetry.

There are actually two free fermion constructions of level one affine $D_4$. One of these goes by the name of the RNS construction in string theory while the other is known as the GS construction, they two constructions are related by triality in $D_4$. For details see section 5.2 of Vol.1 of "Superstring Theory" by Green, Schwarz and Witten. The Riemann relation can be interpreted as the statement that the RNS and GS formalisms give the same answer for the quantity $\chi_{vec}(\tau,x,y,u,v)-\chi_{sp}(\tau,x,y,u,v)$ where $$ \chi_r(\tau,x,y,u,v}=Tr_{V_r} q^{L_0-c/24} \exp(2 \pi i (x H_1 + y H_2 _ u H_3 + v H_4)) $$ and the $H_i$ are the zero modes of currents in the Cartan sub algebra of $SO(8)$ The left-hand side of the Riemann relation above is the result of computing this difference of characters in the RNS formalism while the right side is the result of the computation in the GS formalism. Apologies for my physics accent. I'm sure there are others who can translate this into more mathematically precise language if desired.

Answer by Jeff Harvey for Applications of n-dimensional crystallographic groups

2013-03-11T07:09:56Z

They are used in string theory to construct Conformal Field Theories which describe orbifold limits of Calabi-Yau spaces. See for example Dixon, Harvey, Vafa and Witten, "Strings on Orbifolds I,II" Nucl. Phys. B274 (19860 285 and Nucl. Phys. B261 (1985) 678 for an early application in string theory and Miles Reid in http://arxiv.org/pdf/math/9911165v1.pdf for a more mathematical take on related material.

Answer by Jeff Harvey for Relation between TQFT and Wilson lines, boundary conditions, surface defects etc

2013-01-16T00:42:43Z

Greg Moore recently gave the Felix Klein lectures and a draft of notes for his lectures is available at

http://www.physics.rutgers.edu/~gmoore/FelixKleinLectureNotes.pdf

You will find in the first few pages a discussion of (extended) TQFT, defects, Wilson lines and so on in a language which I imagine is more suitable to mathematicians than to (most) physicists.

Answer by Jeff Harvey for Meaning of a phrase from "The algebra of grand unified theories".

2013-01-08T15:00:42Z

I suspect part of the confusion is due to the fact that the $SU(2)$ appearing in the Standard Model gauge group $U(1)\times SU(2) \times SU(3)$ is different from the $SU(2)$ of the Cassen-Condon paper. The latter is usually called isospin and is an approximate global symmetry of nuclear interactions. It is only an exact symmetry in the limit that one ignores electromagnetic interactions and the mass difference between the up and down quarks. The $SU(2)$ of the standard model gauge group on the other hand is a (local) gauge symmetry. I'm assuming here that you understand the difference between global and local symmetries as there phrases are used in the physics literature. If not, please consult any book on quantum field theory.

The particular phrase you are asking about is simply the statement that the isospin part of the nucleon (i.e. (neutron, proton)) Hilbert space is an $SU(2)$-module with $SU(2)$ the approximate isospin symmetry of the nuclear interactions.

Answer by Jeff Harvey for What do correlation functions compute in CFT?

2012-12-28T04:40:47Z

I'm not sure exactly what kind of information you want, and CFT is an enormous subject, but here is some information on the physical interpretation of the complex coordinates and correlation functions along with an example of their mathematical interpretation in a special CFT.

An ordinary refrigerator magnet contains a ferromagnetic material. The atoms in such a material have electrons which act as tiny magnets and they have an interaction between them (of quantum mechanical origin) which makes the spin/magnetic moments of the electrons have lower energy when they are aligned. This alignment produces a macroscopic magnetic field. I am simplifying here, because in real materials this interaction only operates over short distances and one actually forms domains of aligned spins with the domains oriented randomly and a net magnetization is produced only by subjecting the material to an external magnetic field which aligns the domains.

If one heats up such a ferromagnet then at some temperature the random thermal motion overcomes the tendency to align and the macroscopic magnetic field goes to zero. The transition point between the state with net magnetization and the state with zero magnetization is known as a second order critical point. The behavior of phase transitions between different states (magnetized vs. unmagnetized, or water vs. ice or liquid vs. gas etc. ) has been one of the central topics in condensed matter physics for many years. Various simplified models have been invented to try to understand such behavior analytically. The simplest of these is the Ising model consisting of spins which take the value $\pm 1$ living on a two-dimensional lattice with nearest neighbor interactions between spins. There are many more complicated models, one of these is known as the Gaussian model because of the Gaussian weight used to define the probability distribution for spins. These models at the critical point have fluctuations on all length scales and one can take a continuum limit. This continuum limit is a conformal field theory. If the model is defined in two spatial dimensions then the continuum limit is a two-dimensional conformal field theory. The continuum limit of the Gaussian model is a conformal field theory with $c=1$ equivalent to the $c=1$ conformal field theory you have defined. The continuum limit of the Ising model is a $c=1/2$ conformal field theory consisting of a free Majorna fermion. The physical interpretation of the complex coordinates is simply that they are complex coordinates in the real two-dimensional plane describing the physical space on which the model is defined. The correlation functions measure the correlations between the microscopic spins at different spatial points at the critical point.

In two-dimensional conformal field theory the dependence on the coordinates of both the two-point and three-point correlation functions is completely determined by conformal invariance, so the only interesting information is in the numerical coefficients appearing in these correlation functions. For $c=1$ CFT there isn't any terribly interesting information in these correlation functions. However for other CFT's there is more interesting information, including some of purely mathematical interest. For example, Frenkel, Lepowsky and Meurman constructed a $c=24$ CFT which has the Monster sporadic group as its automorphism group. This CFT has 196884 dimension 2 fields and the three point correlation function of these dimension 2 fields can be used to compute the structure constants of the Griess algebra which was used in the original construction of the Monster.

Answer by Jeff Harvey for Explanation for E_8's torsion

2012-12-18T01:27:17Z

Take a look at "Finite H-spaces and Lie Groups" by Frank Adams, particularly the letter from E8 and the appendix which follows it.

Answer by Jeff Harvey for Mathematician trying to learn string theory

2012-12-13T18:33:01Z

Many string theorists would like to know more algebraic geometry. There are a few of us who know algebraic geometry at a pretty high level (not me) but many more who would like to learn more and feel it would help with their research but find the literature very difficult. I think the optimal solution would be to find such a string theorist and agree that you will teach them algebraic geometry if they will teach you string theory.

Answer by Jeff Harvey for Does Physics need non-analytic smooth functions?

2012-11-26T18:04:36Z

As a physicist "in nature" perhaps I can give a few examples that illustrate how non-analytic functions can appear in physics and counter the idea that physicists do not worry about the justification of these procedures.

Example 1 involves one of the most precise comparisons between experiment and theory known to physics, namely the g factor of the electron. The quantity g is a proportionality factor between the spin of the electron and its magnetic moment. Perturbation theory in QED gives a formula $$g-2= c_1 \alpha + c_2 \alpha^2 + c_3 \alpha^3 + \cdots $$ where the coefficients $c_i$ can be computed from i-loop Feynman diagrams and $\alpha=e^2/\hbar c \simeq 1/137$ is the fine structure constant. Including up to four loop diagrams gives an expression for $g$ which agrees to one part in $10^{8}$ with experiment. Yet it is known that that this perturbative series has zero radius of convergence. This is true quite generally in quantum field theory. Physicists do not ignore this, rather they regard it as evidence that QFT's are not defined by their perturbation series but must also include non-perturbative effects, generally of the form $e^{-c/g^2}$ with $g$ a dimensionless coupling constant. Much effort has gone into understanding these non-perturbative effects in a variety of quantum field theories. Instanton effects in non-Abelian gauge theory are an important example of non-perturbative phenomena.

Example 2 involves the Hydrogen atom in an electric field of magnitude $E$, aka the Stark effect. One can compute the shift in the energy eigenvalues of the Hydrogen atom Hamiltonian due to the applied electric field as a power series in $E$ using perturbation theory and again one finds excellent agreement with experiment. One can also prove that this series has zero radius of convergence. In fact, the Hamiltonian is not bounded from below and does not have any normalizable energy eigenstates. The physics of this situation explains what is going on. The electron can tunnel through the potential barrier and escape from being bound to the nucleus of the Hydrogen atom, but for reasonable size electric fields the lifetime of these states exceeds the age of the universe. The perturbation theory does not converge because there are no energy eigenstates to converge to, but it still provides an excellent approximation to the energy eigenstates measured experimentally because the experiments are done on a time scale which is very short compared to the lifetime of the metastable state.

So I would say that at least in these examples there is a very nice interplay between the physics and the mathematics. The lack of analyticity has a clear physical interpretation and this is something that is understood by physicists. Of course I'm sure there are other example where such approximations are made without a clear physical justification, but this just means that one should understand the physics better.

Answer by Jeff Harvey for The use of Hall algebras in physics

2012-11-10T15:02:05Z

In supersymmetric field theories and string theories there are special states called BPS states which are annihilated by some of the supercharges and whose mass is determined in terms of their charges by the supersymmetry algebra. The study of these states and how they behave as various moduli are varied has been one of the main tools physicists have used to find evidence for various kinds of dualities, particularly S-duality which relates weakly coupled theories to strongly coupled theories.

One particularly rich example of an S-duality involves a duality between the heterotic string on $K3 \times E$ with a particular choice of $E_8 \times E_8$ gauge bundle where $E$ is an elliptic curve and Type II string theory on Calabi-Yau manifolds which have the form of K3 surfaces fibered over rational curves. In the first paper mentioned by Alexander Chervov, Greg Moore and I computed certain one-loop integrals on the heterotic string side of this story and found two interesting facts. First, that these integrals were determined purely by the spectrum of BPS states, and second that the answers involved denominator formulae for Generalized Kac Moody algebras of the type studied previously by Borcherds. Given this fact it was natural to think that there was an algebraic structure that one could define on the BPS states that would ``explain'' why were getting the denominator formula for a GKM algebra. This was the physics motivation for the introduction of the algebra of BPS states. However we did not achieve our original goal in that we were not able to find a direct connection between the BPS algebra and the GKM denominator formulae. In spite of this failure the idea that there should be an algebraic structure on the space of BPS states seems to have some merit.

If you want to look at more recent developments you might have a look at arXiv:1102.1821 which finds a more direct relation between a particular Borcherds algebra and one-loop integrals. On the mathematical side there is arXiv:1006.2706 by Kontsevich and Soibelman where Hall algebras appear. They mention the idea of an algebra of BPS states as motivation for their construction.
I must confess though that I do not have the level of mathematical sophistication needed to understand this paper and unfortunately my colleague Greg Moore, who does, is not on MO.

Answer by Jeff Harvey for Higgs mechanism from a deformation quantization point of view

2012-10-03T16:15:58Z

The Higgs mechanism in the Standard Model doesn't have much to do with deformation quantization as other people have explained. However there is a version of the Higgs mechanism in string theory which involves stable D-branes arising via the Higgs mechanism from unstable D-branes. This is very hard to study directly in string field theory where one has to resort to approximate numerical techniques, but if one deforms the theory by turning on a B field as discussed in Seiberg and Witten, hep-th/9908142 one obtains a noncommutative version and in the "large B limit" one can obtain exact results. For details see hep-th/0005031 and references therein.

Answer by Jeff Harvey for photon propagator

2012-10-01T12:16:10Z

Theo and David are perfectly correct. To add a bit more of a physical explanation which might help with the why part, a massive spin one particle has 3 physical degrees of freedom so there must be some condition on the four components $A_\mu$. The equation of motion for $A_\mu$ is equivalent to saying that each component of $A_\mu $ satisfies the massive Klein-Gordon equation and that in addition $\partial^\mu A_\mu=0$. This latter condition in momentum space implies that $k^\mu D_{\mu \nu}(k)=0$. So one can understand the $1/(k^2-m^2)$ from each component obeying the massive KG equation and the factor in the numerator as ensuring that $k^\mu D_{\mu \nu}(k)=0$.

Elliptic genus for manifolds with boundary

2011-12-02T20:31:48Z

Let M be a closed spin manifold of dimension $d$. One form of the elliptic genus of $M$ is $$ F(q)=q^{-d/8} \hat A(M) {\rm ch} \otimes_{k=1/2,3/2,\cdots} \Lambda_{q^k}T \otimes_{\ell=1}^\infty S_{q^\ell}T [M] $$ where the notation follows that of E. Witten, ``The Index of the Dirac Operator in Loop Space." The coefficient of $q^{n/2-d/8}$ is the index of a Dirac operator $D_n$ which acts on sections of $S \otimes T_{R_n}$ where $S$ is the spinor bundle and $T_{R_n}$ is the bundle associated to a representation $R_n$ of $Spin(d)$ with the first few representations being $$ R_0=1, \qquad R_1=T, \qquad R_2=\Lambda^2 T \oplus T $$ where $T$ is the fundamental (vector) representation.

I'm interested in the generalization of the elliptic genus to manifolds with boundary. In the actual application I'm interested in one eventually takes the boundary to infinity to obtain a noncompact manifold, but I'd be happy to understand the situation for a compact manifold with boundary first. The index of the Dirac operator in such a situation acquires boundary corrections of the form $$ CS[ \partial M] - \frac{1}{2}(\eta(0)+h) $$ where $h$ is the number of zero modes of the Dirac operator on $\partial M$ and $\eta(0)$ is the $\eta$ invariant. In the examples I'm interested in I believe the Chern-Simons contributions $CS[\partial M]$ vanish.

Summing up these boundary contributions to the index of $D_n$ weighted by $q^{n/2-d/8}$ leads to a "boundary" contribution to the elliptic genus on manifolds with boundary with the "bulk" contribution given by $F(q)$. My questions are whether this variant of the elliptic genus has been studied and if so where, whether this leads to interesting invariants of manifolds with boundary, and whether the modular properties of the bulk and boundary contributions are known.

Answer by Jeff Harvey for On the periods in the periodic table (or Why is a noble gas stable?)

2011-11-18T19:07:44Z

You are completely correct in your analysis of the structure one obtains by considering the Schrodinger equation for Z electrons in a central potential due to a nucleus of charge Ze when the Coulomb interaction between electrons as well as relativistic effects such as spin-orbit coupling are ignored. In such a model the periods would indeed be 2, 12, 18 etc. for exactly the reasons you have described. In physics jargon the energy in this model depends only on the principal quantum number $n \in {\mathbb Z}, n>0$ and the allowed $\ell$ values are $\ell\le n-1$. Orbitals with $\ell=0,1,2,3,4, \cdots$ are labelled by $s,p,d,f \cdots$ for historical reasons. Thus at $n=1$ one can have one or two states in the $(1s)$ orbital (accounting for spin), at $n=2$ one has the $(2s)$ and $(2p)$ orbitals with $2(1+3)=8$ states. And at $n=3$ one should have $18$ states by filling the $(3s),(3p),(3d)$ orbitals. But in the real world this is not what happens. This simple model does not give a correct description of atoms in the real world once you get past Argon. I believe the main effect leading to the breakdown is the Coulomb interaction between the electrons.

So there is no simple mathematical model based say just on the representation theory of $SU(2)$ and simple solutions to the Schrodinger equation which will account for the structure of the periodic table past Argon. However one could ask whether including Coulomb interactions between electrons does gives a model which correctly reproduces the next few rows of the periodic table past Argon. I am not an expert on this, but since I doubt there are physical chemists on MO I'll just give my rough sense of things.

To approach this problem with some level of rigor probably requires difficult numerical work and my impression is that this is beyond the current state of the art. However there are rough models which try to approximate what is going on by assuming that the interactions between electrons can be replaced by a spherically symmetric potential which is no longer of the $1/r$ form. This leaves the shell structure as is, but can change the ordering of which shells are filled first. In such a model instead of filling the $(3d)$ shell after Argon one starts to fill the $(4s)$ and $(3d)$ shells in a somewhat complicated order. Eventually one fills the $(4s),(3d),(4p)$ shells and this leads to the line of the periodic table starting at K and ending at Krypton.

Added note: There is one nice piece of mathematics associated with this problem that I should have mentioned, even if it doesn't by itself explain the detailed structure of the periodic table. When Coulomb interactions between electrons and relativistic effects are ignored the energy levels of the Schrodinger equation with a central $1/r$ potential depend only the quantum number $n$, but not on the quantum number $\ell$ which determines the representation $V_\ell$ of $SO(3)$ referred to above. When screening is included this is no longer the case and the energies depend on both $n$ and $\ell$. Why is this? With a $1/r$ central potential there is an additional vector $\vec D$ which commutes with the Hamiltonian. Classically this vector is the Runge-Lenz vector and its conservation explains why the perihelion of elliptical orbits in a $1/r$ potential do not precess. Quantum mechanically the commutation relations of the operators $\vec D$ along with the angular momentum operators $\vec L$ are those of the Lie algebra of $SO(4)$ (for bound states with negative energy). There are two Casimir invariants, one vanishes and the other is proportional to the energy. As a result the energy spectrum depends only on $n$ and can be computed using group theory without ever solving the Schrodinger equation explicitly. Perturbations due to screening, that is from some averaged effect of the Coulomb interactions between electrons, change the $1/r$ potential to some more general function of $r$ and break the symmetry generated by the Runge-Lenz vector $\vec D$. As a result the energy levels depend on both $n$ and $\ell$.

Answer by Jeff Harvey for Is the Mendeleev table explained in quantum mechanics?

2011-11-09T22:52:35Z

I am not offended by the suggestion that physicists should follow the standards of mathematical proof, but I think this suggestion and the phrasing of the question demonstrate a lack of understanding of how physicists think about such things and more importantly why they put such little emphasis on axioms.

In my view it is rarely useful to think of physics as an axiomatic system, and I think this question reflects the difficulty with thinking of it as such. A different question, which is much more in tune with a physicist's point of view, would be to ask what physical description is required to explain various features of the structure of atoms as reflected in the periodic table at a prescribed level of accuracy. Until you specify what features you want to understand, and at what level of accuracy, you don't even know what the correct starting point should be. If you want just the crudest structure of the periodic table, then indeed non-relativistic quantum mechanics along with the Pauli exclusion principle will give you the rough structure as described in any standard QM textbook. If you want to understand the detailed quantum numbers of large atoms then you have to start including relativistic effects. Spin-orbit coupling is one of the most important and its effects are often summarized by a set of Hund's rules which are described in many QM textbooks or physical chemistry textbooks. If you want very accurate numerical values for ionization energies or the detailed structure of wave functions then one must do hard numerical work which probably becomes impossibly difficult for large atoms. As you ask for greater and greater precision you should eventually use a fully relativistic description. This is even harder. The Dirac equation is not sufficient, one cannot restrict to a Hilbert space with a finite number of particles in a relativistic quantum theory, and bound state problems in Quantum Field Theory are notoriously difficult. So as one asks more detailed and more precise questions, one has to keep changing the mathematical framework used to formulate the theory. Of course this process could end and there could be an axiomatic formulation of some ultimate theory of physics, but even if this were the case this would undoubtedly not be the most useful formulation for most problems of practical interest.

Answer by Jeff Harvey for What exactly is the relation between string theory and conformal field theory?

2011-10-10T15:34:03Z

One must distinguish between quantum/classical on the string world-sheet and in spacetime. Both of your statements are basically correct, but should read something like "CFT theory is the space of classical solutions to the spacetime equations of string theory" and "Quantization of the the world-sheet sigma model of a string theory gives rise to a CFT."

In a little more detail, the sigma-model describing string theory propagation on some manifold M is a 2-dimensional quantum field theory which in order to describe a consistent string theory must be a conformal field theory. The "classical limit" of this 2-dimensional field theory is a limit in which some measure of the curvature of M is small in units of the string tension. To construct a CFT one must solve the sigma-model exactly, including world-sheet quantum effects.

The coupling constants of the sigma-model are fields in spacetime such as the metric $g_{\mu \nu}(X(\sigma))$ on $M$ where $X: \Sigma \rightarrow M$ define the embedding of the string world-sheet $\Sigma$ into $M$. Now there is also a spacetime theory of these fields. You can think of it as a ``string field theory". At low-energies it can sometimes be usefully approximated by a theory of gravity coupled to some finite number of quantum fields, but in full generality it is a theory of an infinite number of quantum fields. Roughly speaking, each operator in the CFT gives rise to a field in spacetime. The spacetime string field theory lives in 10 dimensions for the superstring or 26 dimensions for the bosonic string and it also has a classical limit. The classical limit is $g_s \rightarrow 0$ where $g_s$ is a dimensionless coupling constant. It appears in perturbative string theory as a factor which weights the contribution of a Riemann surface by the Euler number of the surface. It can also be thought of as the constant (in spacetime) mode of a scalar spacetime field known as the dilaton.

The main point is that there are two notions of classical/quantum in string theory, one involving the world-sheet theory, the other the spacetime theory. In order to avoid confusion one must be clear which is being discussed. Unfortunately string theorists often assume it is clear from the context.

In response to the further question about the space of string fields, I would suggest that you have a look at the introductory material in http://arXiv.org/pdf/hep-th/9305026. You may also find http://arXiv.org/pdf/hep-th/0509129 useful. I should add that while string field theory has had some success recently in the description of D-brane states, it is not widely thought to be a completely satisfactory definition of non-perturbative string theory.

Answer by Jeff Harvey for Is there a relation between 4-dimentional general relativity and exotic smooth structures on $\mathbb{R}^4$?

2011-09-21T01:23:05Z

Regarding the first part of this question, in four spacetime dimensions there are no known generic violations of the cosmic censorship hypothesis while above four dimensions there is good evidence that cosmic censorship is violated without fine tuning of initial conditions. The best evidence that I know of for the latter statement comes from the analysis of the Gregory-Laflamme instability in http://arxiv.org/pdf/1006.5960 .

What is the modern understanding of the order of a mock theta function?

2011-09-08T15:17:29Z

Ramanujan introduced mock theta functions and described them by an "order" which he did not define. As a result of the work of Zwegers and others we now have a better understanding of mock theta functions. They appear as the holomorphic projection of weight 1/2 harmonic Maass forms and in the theta expansions of meromorphic Jacobi forms. Given this modern understanding one wonders if there is a natural definition of the "order" which agrees with Ramanujan's. On the Wikipedia page on mock modular forms one finds the statement "Ramanujan's notion of order later turned out to correspond to the conductor of the Nebentypus character of the weight 1/2 harmonic Maass forms which admit Ramanujan's mock theta functions as their holomorphic projections." I can check that this is true in a few specific cases (e.g. the order 3 mock theta functions studied by Bringmann and Ono) but have not been able to find this statement in the literature, hence my questions. First, does the definition of the order in the Wikipedia article agree with the orders (2,3,5,6,7,8,10) of the mock theta functions given later in the same article? Second, is there a reference to the literature for this definition?

Answer by Jeff Harvey for Bimonster and Heterotic String Theory

2011-08-18T13:19:19Z

The bimonster acts as the automorphism group of a particular bosonic closed string theory and D-brane states in this theory that preserve a chiral subalgebra transform in representations of the bimonster. See http://arxiv.org/pdf/hep-th/0202074 for more details. This does not involve $Y555$ or the heterotic string, but might be useful in thinking about such connections.

Answer by Jeff Harvey for What is the motivation for a vertex algebra?

2011-02-01T23:39:32Z

The answers here have focused on the mathematical aspects of VOAs and the motivation coming from QFT, the specialization to Conformal Field Theory, and then the further specialization to two-dimensional holomorphic CFT. Two-dimensional CFT's did arise from string theory where the fields on the 2d world-sheet of the string define a CFT. However there is another important part of the story which has not been mentioned and is more directly tied to physical phenomenon and that is the theory of critical phenomenon. Many systems, such as water-ice, magnetic systems and so on undergo phase transitions as a thermodynamic parameter such as temperature is varied. Typically these are first order transitions, meaning that there is a latent heat associated to the transition. Sometimes one can vary an additional parameter and find a line of first order transitions which terminates at a second order transition. The second order transition is characterized by fluctuations on all scales: the theory becomes scale and conformally invariant at that point. It also turns out that the behavior of thermodynamic quantities as one approaches the critical point are characterized by numbers called critical exponents which are universal for systems with the same symmetry structure. These exponents are related to what are called the conformal dimensions of operators in CFT and they are directly measurable in the lab for a variety of systems. One important tool which was used in the study of critical phenomenon is the operator product expansion or operator algebra of K. Wilson and L. Kadanoff. There is a huge literature on this. Here is a reference to an early paper on the operator algebra for the Ising model: http://prb.aps.org/abstract/PRB/v3/i11/p3918_1 . VOA's are a rigorous mathematical formalization of this kind of algebraic structure. For someone who wants to learn about CFT starting from a particular physical system (or at least a mathematical idealization of a physical system) the Ising Model is a good place to start.

Answer by Jeff Harvey for Mathematical "urban legends"

2011-01-24T23:40:26Z

Since the OP gave a physics example, here is another one, also at Princeton. Why are they always at Princeton? Student finishes his presentation on very mathematical aspects of string theory. An experimentalist on the committee asks him what he knows about the Higgs boson. He hems and haws and finally says "well, it was discovered a few years ago at Fermilab", Experimentalist: "Can you tell me the mass?" Student: "I think around 40 GeV."

This was more than 20 years ago and actually happened. I was there. The student passed, but the next year all Ph.D students working on string theory were required to take a course on the phenomenology of particle physics.

Question in complex analysis arising from large $N$ gauge theory

2011-01-24T14:51:57Z

This is a question in complex analysis that comes up in the treatment of large $N$ gauge theory with gauge group $SU(N)$ in the 't Hooft limit where $N$ is taken to infinity with $\lambda=g^2 N$ fixed with $g$ the gauge coupling constant. In the physics literature one studies two-point correlation functions as a function of four-momentum $q$ and argues indirectly that the leading large $N$ behavior is given by a sum of poles, $\Pi(q)= \sum_i \frac{r_i}{q^2-m_i^2}$. On the other hand one can compute directly at large $Q^2=-q^2$ and finds the asymptotic behavior $\Pi(q) \sim c \log(Q^2)$ where $c$ is a computable constant. It is then claimed that these two results can be consistent only if the sum on $i$ is infinite. My question is whether the sum can be countable as is implicitly assumed in the physics literature. I realize this is probably not a research level question in mathematics, but standard texts on complex analysis that I am familiar with don't discuss such issues and I'm not sure where else to look.

Answer by Jeff Harvey for Topology of black holes

2011-01-19T14:20:05Z

The previous answers dealt with the physically relevant case of $d=4$ spacetime dimensions. One of the surprising discoveries in recent years is that in higher dimensions the possible topologies are much richer. I believe this started with the discovery by Emparan and Reall of a black hole in $d=5$ with horizon topology $S^2 \times S^1$ (hep-th/0110260). The recent paper arXiv:1002.0490 by Hollands et. al. surveys the situation and discusses restrictions on the possible topologies of the horizon for $d=5$ black holes.

Answer by Jeff Harvey for How are the classifying space of $E_8$ and $K(\mathbb{Z},4)$ related?

2011-01-17T04:36:48Z

In addition to the work mentioned in David's very useful answer I would suggest that you take a look at http://arXiv.org/pdf/hep-th/0312069 by Diaconescu, Moore and Freed. They give a mathematically precise definition of the M-theory 3-form in terms of the Chern-Simons term of $E_8$ gauge theory and apply the formalism to study M-theory on manifolds with boundary. My understanding is that the formalism "works" in the sense of giving mathematically well defined answers which agree with various physical constraints, but whether the $E_8$ gauge field is fundamental or not remains elusive.

Answer by Jeff Harvey for What makes four dimensions special?

2011-01-06T23:54:12Z

$4=11-7$ and $11$ is the maximal dimension for supersymmetry with spins $\le 2$ while $7$ is the first dimension in which there exist compact manifolds of exceptional holonomy.

Degree of Transcendentality and Feynman Diagrams

2010-12-27T02:55:58Z

Physicists computing multiloop Feynman diagrams have introduced various techniques and conjectures that involve the notion of Degree of Transcendentality (DoT). From what I understand one defines

1) $DoT(r)=0$, r rational

2) $DoT(\pi^k)=k$, $k \in {\mathbb N}$,

3) $DoT(\zeta(k))=k$,

4) $DoT( a \cdot b)= DoT(a)+DoT(b)$

One then proves for example that the $\ell$-loop contribution to a certain scaling function in $N=4$ Supersymmetric gauge theory consists of a sum of terms all of which have DoT equal to $2 \ell-2$.

This can't be rigorous mathematically, since it is not even known that $\zeta(2n+1)$ is transcendental, but is there some circle of ideas, or conjecture in mathematics that if true would give a precise definition to DoT?

The influence of string theory on mathematics for philosophers.

2010-12-17T14:26:28Z

I've agreed, perhaps unwisely, to give a talk to Philosophers about string theory.

I'd like to give the philosophers an overview of the status and influence of string theory in physics, which I feel competent to do, but I would also like to say something about the influence it has had in mathematics where I am on less familiar ground. I've read the Jaffe-Quinn manifesto and the responses in http://arxiv.org/abs/math.HO/9404229. What I would like from MO are pointers to more recent discussions of this issue in the mathematical community so that I can get a sense of where things stand 16 years later.

Answer by Jeff Harvey for mirror symmetry with algebraic geometry?

2010-12-11T18:44:39Z

Part of the physics motivation for mirror symmetry involves properties of the chiral ring of N=2 superconformal field theories. Some of these have a description in terms of the polynomials appearing in algebraic geometry. One of the earliest references on this is Algebraic Geometry and Effective Lagrangians, Emil J. Martinec, Phys.Lett.B217:431,1989. There are many papers discussing the relation between these "Landau-Ginzburg" models and mirror symmetry. See for example the paper by Berglund and Katz, http://arXiv.org/pdf/hep-th/9406008.

Answer by Jeff Harvey for Examples of non-rigorous but efficient mathematical methods in physics

2010-12-08T23:06:45Z

Perhaps it would not be out of place to quote Miles Reid's Bourbaki seminar on the McKay correspondence here:

"The physicists want to do path integrals, that is, they want to integrate some "Action Man functional" over the space of all paths or loops $ \gamma : [0; 1] \rightarrow Y $. This impossibly large integral is one of the major schisms between math and fizz. The physicists learn a number of computations in finite terms that approximate their path integrals, and when sufficiently skilled and imaginative, can use these to derive marvellous consequences; whereas the mathematicians give up on making sense of the space of paths, and not infrequently derive satisfaction or a misplaced sense of superiority from pointing out that the physicists' calculations can equally well be used (or abused!) to prove 0 = 1. Maybe it's time some of us also evolved some skill and imagination. The motivic integration treated in the next section builds a miniature model of the physicists' path integral,..."

Answer by Jeff Harvey for String theory "computation" for math undergrad audience

2010-11-30T14:31:39Z

I agree that computing partition functions has many pretty applications. My favorite is the use of Jacobi's abstruse identity between theta functions, $\theta_3^4-\theta_4^4=\theta_2^4$, to show the equality between the number of bosons and fermions in open superstring theory as required by supersymmetry. This is explained in sec 4.3 of "Superstring Theory" by Green, Schwarz and Witten.

Another short calculation which quickly gets to the heart of the connection between string theory and gravity is the demonstration that bosonic string theory contains a massless spin two excitation. One way to do this requires regularizing a divergent zero point energy via $\sum_{n=1}^\infty n \rightarrow \sum_{n=1}^\infty n^{-s}$ and then analytically continuing to $s=-1$ to obtain $\zeta(-1)=-1/12$. See sec 2.3 of GSW.

Comment by Jeff Harvey

2013-06-07T14:10:21Z

I think it depends on what you mean by "exist." I do understand Dirac's argument and construction, but monopoles in pure electromagnetism are singular objects and there is no way to compute their mass or study any other properties they might have. For example scattering of charged particles off of magnetic monopoles is sensitive to the short distance description of the monopole. So to embed monopoles into a well defined, predictive physics framework you need to do something like what I described (or variants thereof).

Comment by Jeff Harvey

2013-06-06T14:12:32Z

I understand this is a math question on a math site, but it might be worth mentioning that the question has little to do with how physicists actually understand the possible existence of magnetic monopoles. The simplest construction of well defined magnetic monopoles involves embedding the Standard Model group $H=SU(3) \times SU(2) \times U(1)$ into a simple, compact Lie group $G$ like $SU(5)$ or $SO(10)$ and in that situation the symmetry breaking of $G$ to $H$ by the Higgs mechanism leads to a classification of magnetic monopoles by $\pi_2(G/H)$ and the equations of motion are not Maxwell's.

Comment by Jeff Harvey

2013-05-31T13:41:10Z

I would think that by any reasonable standard Edward Witten is both a professional physicist and a professional mathematician.

Comment by Jeff Harvey

2013-05-26T22:58:35Z

@Marcel can you define what you mean by a "full rational CFT?" I suspect I must know what this is but am not familiar with this terminology.

Comment by Jeff Harvey

2013-05-17T16:09:25Z

Not a comment on the math, but if you really want to reformulate Quantum Mechanics in terms of classical fluid dynamics and want to be taken seriously rather than viewed as a crank then you are first obligated to understand how Quantum Mechanics is currently formulated and used in some detail.

Comment by Jeff Harvey

2013-05-14T15:23:36Z

@user6818 I think you need to read the paper you are citing more carefully. They do specify the $SU(N_c)$ representations on the bottom of p. 9 and top of p.10. They consider four cases and in each case they specify the $SU(N_c)$ representation content (the $R_i$ in their notation).

Comment by Jeff Harvey

2013-05-14T00:05:59Z

@user6818 As in the question is not well posed to start with and second it is not written in language that most mathematicians will understand. I partially understand what you are asking because I happen to be a physicist. I'd suggest that you either ask the question on physics stack exchange or make the effort to translate your question into a precise mathematical question framed in language that mathematicians will understand. Otherwise your question will be and should be closed since this is a site for research level math questions.

Comment by Jeff Harvey

2013-05-13T22:48:09Z

Your question doesn't contain enough information for a sensible answer until you also specify the $SU(N_c)$ representation of the fields and also their statistics (bosons or fermions).

Comment by Jeff Harvey

2013-05-01T23:43:00Z

It is so annoying not to be able to edit comments! Please in the above read $t A A$ to be the transpose of $A$ times $A$ and interpret the $\frac{1}{2}$ as a prefactor in front of a $4 \times 4$ matrix.

Comment by Jeff Harvey

2013-05-01T23:40:06Z

Mumford explains that the relation depends on a matrix $A$ satisfying $tAA=I$ with $I$ the identity matrix ( I have chosen $m=2$ in his notation on p.14 of Tata I). I think you must have a typo because your matrix does not obey this identity. I believe you want $$ A= \frac{1}{2} \begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 1 & -1 \\ 1 & -1 & -1 & 1 \end{matrix} $$ What is special about this matrix is that it acts as a triality transformation on the weight space of so(8)$, transforming vector weights into spinor weights. This is related to my answer to this question.

Comment by Jeff Harvey

2013-04-13T22:59:17Z

Isn't the usual $\theta$ function as defined in Mumford's Tata lectures in the first chapter $\theta: {\mathbb C} \times {\mathbb H} \rightarrow {\mathbb C}$ with ${\mathbb H}$ the upper half plane? What is your "usual" theta function?

Comment by Jeff Harvey

2013-04-13T01:52:10Z

There is a nice interpretation in terms of characters of affine $Spin(8)$ that involves triality and is related to supersymmetry in string theory. I can write up some details tomorrow unless someone else beats me to it.

Comment by Jeff Harvey

2013-04-10T21:25:29Z

To a physicist it is strange to see a current in a Lorentz invariant theory written as a 2-form in space rather than as a 3-form in space-time.

Comment by Jeff Harvey

2013-03-27T17:47:09Z

Dear Jonas, To expand slightly on Emerton's comment, $E_4$ and $E_6$ are modular forms, so in particular $E_4(-1/\tau)=\tau^4 E_4(\tau)$ and $E_6(-1/\tau)= \tau^6 E_6(\tau)$. On the other hand $E_2$ is not a modular form, it is only quasimodular. It obeys $E_2(-1/\tau)= \tau^2 E_2(\tau)- 6 i \tau/\pi$. No algebraic combination of $E_4,E_6$ can transform this way under $\tau \rightarrow -1/\tau$ so $E_2$ is algebraically independent of $E_4,E_6$.

Comment by Jeff Harvey

2013-03-13T01:03:33Z

In many cases no invariant measure exists. There are anomalies. The passage from finite dimensional integrals to path integrals is more subtle than you indicate here.