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- TITLE AND ABSTRACT BY SNARXIV.ORG
- "(P,q) Instantons" by O. H. Silverstein
- # Abstract
- Anomaly constraints in type IIB strings deformed by Chern-Simons terms are transverse. Consequently, over the last decade, Planck predicted that a formulation of Clebsch-Gordon decomposition in an extremal QFT is alternative. Our calculation of type I strings on hyper-Kahler Ricci-flat hyperbolic Enriques surfaces gives rise to fragmentation functions. Bounding is made easier by solving a check of the halo Hilbert space via a test of Tomita-Takesaki theory in twisted Matrix Models in the presence of n-dimensional black holes formed from collapse. When clarifying clarifying general relativity, we derive that, in the limit that squarks can compute the reduction of type IIB strings on a harmonic manifold, quintessence at 7 loops is subleading.
- # Contents
- * Introduction
- * The Argument
- * Conclusion
- # Introduction
- A hyper-Kahler Calabi-Yau orbifold compactification of the type IIB theory on a hyperbolic Enriques surface admits two deformations that remain exact. The first is a standard deformation by the Chern-Simons term $\int \sqrt{g} B \wedge \bar F$ on the Ricci-flat part of the manifold. Our framework computes a specific integral of this term on the part of the manifold that is not Ricci-flat. The deformation can be derived in various ways including $\alpha'$ expansions of the tree-level worldsheet theory, and by $\beta$-deformations on the background field. We computed the one-loop quantum effective action by series expanding the deformation by the background field and subsequently integrating the operator insertions, and use a matching factor to extract the Chern-Simons deformation.
- A second deformation is obtained by the following Higgsing of the theory with an O-plane to a Higgsed theory that is integrable. From the CFT on the orbifold we compute the field theory of the worldsheet, which is a 2d Conformal Quantum Mechanics. The problem of a priori computing the exact vertex operator for a non-linear sigma model is that the correlators need to be evaluated in the target space. This introduces a second new integrand in the correlator over the target space. The alternative to finding a conventional perturbative expansion is to use the integral in the target space as a test to check on the worldsheet theory. The natural target space has four dimensions, but it has a two-dimensional surface that has two distinct conformal boundaries. These boundaries represent the presence of an O-plane. Our calculation shows that the field theory correlator on the worldsheet reproduces the two-dimensional effective action for the Higgsed theory on the boundary.
- Consequently, we believe that the theory of BPS particles in Higgsed theory on a boundary of a non-linear sigma model can be used to understand the spectrum of black holes. This is our second task. In a departure from the standard method, we use the representation theory of the symmetry group of a two-dimensional surface to solve for the Hilbert space of the theory on the boundary.
- Our solutions to the constraints of the one-loop partition function show that the Hilbert space has a simple structure that is constrained by the conjecture that the amplitudes in the scattering of particles are given by the partition function of n-dimensional black holes formed from collapse. We find that the halo states in the space do not include halo states that are singular on the boundary of the black hole. The halo states on the boundary have vanishing energies. When we check the halo Hilbert space against known facts, we find that the model has the expected number of perturbative states.
- The most interesting feature of our solution to the constraints is that the halo Hilbert space is a $Q$-system. This leads us to propose a new formula for the Clebsch-Gordon decomposition. Our formula is the closest analog of the formula for the Clebsch-Gordon decomposition in Euclidean string theory and the Clebsch-Gordon formula in classical mechanics. Our formula does not account for what is missing in the perturbative state. This formulation can be used to derive other things, such as a test of the charge-halo correspondence in the context of Seiberg-Witten curves.
- A computation of a single particle in the Higgsed theory is a scalar propagator on the worldsheet that solves the equations of motion of the Higgsed theory. We compute the propagator from the Feynman rules of the worldsheet theory.
- The worldsheet theory is a $\mathcal{N}=(4,4)$ SYM coupled to the scalar propagator. The Kinematic symmetry is the global symmetry $\text{SO}(4) \times \text{SO}(4)$.
- Our solution uses the holography of a five-dimensional gravitational theory to compute the three-dimensional scalar propagator on a boundary of a five-dimensional gravitational theory. In our model, the boundary is a space of non-linear sigma models that has four dimensions. This theory is the theory of gravity coupled to a scalar field.
- Our solution to the constraints of the three-dimensional scalar propagator is consistent with the known facts about two-dimensional boundary sigma models coupled to gravity and also gives the first rigorous proof of the conformal invariance of the scalar propagator in two dimensions.
- By computing the three-point function in the field theory of the worldsheet, we can compute the higher-point functions of the boundary theory.
- In the context of the path integral for quantum gravity, the worldsheet theory provides an alternative to the Feynman diagrammatic expansion for the perturbative contributions. This is due to the fact that, as shown by ’t Hooft, the integrals in the perturbative expansion are path integrals that converge on non-linear sigma models that are equivalent to the field theory of gravity in two dimensions.
- Because our worldsheet theory is a Q-system, we can compute the asymptotic correlation functions. We can determine the contact term of the three-point function and we are able to show that it is the same as the contact term of the three-point function in two-dimensional quantum gravity.
- By computing the three-point function of the boundary theory, we can compute the 3-loop contribution to the 4-point function. In our computation, we find that the contact term of the 4-point function has a different coefficient than the contact term of the three-point function. This is the first step in computing the exact 4-point function in 2d quantum gravity.
- The most important consequence of our result for the 3-loop contribution to the 4-point function is that it gives a check of the holography of the five-dimensional gravitational theory in the context of the two-dimensional theory of quantum gravity. The three-loop contribution gives the holomorphic factor of the generating function for the correlator.
- We find the holomorphic factor of the 3-loop contribution for the four-point function. The calculation proceeds by computing a matrix model, but it is a test of the holography of the five-dimensional gravitational theory in the context of two-dimensional quantum gravity.
- The conjectured form of the quantum corrections to the three-point function is given by the holomorphic factor of the three-loop contribution. This conjecture is checked using the result that the 3-loop contribution can be computed by the matrix model that we study in the calculation.
- When we compute the holomorphic factor of the three-point function, we find a disagreement with the sum of 1-loop, 2-loop and 3-loop contributions. This means that there is a problem with the formulation of the summation of quantum corrections in a conventional perturbative expansion.
- In the conventional formulation of quantum corrections, the corrections to the holomorphic factor of the four-point function are given by $\sum_n a_n L_n$, where the sum over $n$ is over the first order contributions from the perturbative expansion.
- In our computation, we find the holomorphic factor of the four-point function to be $\sum_n a_n^{(1)} L_n + a_n^{(2)} L_n^{2} + a_n^{(3)} L_n^{3} + a_n^{(4)} L_n^{4}$. We find that $a_n^{(4)}$ is zero.
- Therefore, our result for the three-loop contribution to the four-point function agrees with the holomorphic factor of the 3-loop contribution that we computed.
- # The Argument
- After introducing a formalism to compute fermion and scalar fragmentation functions in a type I string on a compact hyper-Kahler Ricci-flat surface, we begin by going on to, and then briefly stating, a sequel, and thus:
- - The Conjecture,
- - The Check,
- - The Solution
- - The Final Check
- - Concluding
- ## The Conjecture
- The three body problem is solvable, and there exist satisfactory solutions to the matrix model for the Mandelstam-Wilson (MW) twist. Also, there are ripples in the Hesse potential from black holes coming from bubble solutions in string theory.
- ## The Check
- This paper is centered on looking at a very special sort of hyperbolic Enriques surface. Though it is known that a del Pezzo surface with the right Euler characteristic and degree is hyperbolic, the Enriques surfaces has been generally thought of as merely having a singularity at a point. By further imposing the hyper-Kahler condition, the problem becomes solvable.
- ## The Solution
- The gravitational part of the bosonic moduli space for the family of hyper-Kahler Ricci-flat surfaces is a moduli space for hyper-Kahler toric Calabi-Yau manifolds. The twist is the MW twist in type I strings on hyper-Kahler toric manifolds.
- ## The Final Check
- The matrix model is expected to pick out the Ricci-flat sector of the superconformal field theory, and since the dimension of the matrix model is (1-2N+12)N, the dynamics of the matrix model is precisely the dynamics of a type I string on a hyper-Kahler Ricci-flat surface with N hypermultiplets and 24N moduli. This is a contradiction, since the type I string on a Ricci-flat hyper-Kahler surface admits only the N=1 vacua with 4 moduli.
- ## Concluding
- - The singularity at the point $x_0$ is actually a point charge.
- - In the presence of the point charge, the supergravity solution consists of a supergravity solution on the manifold with point charge $Q$, and an other solution on the space with metric $G(x_0,x)$ at points which are away from the singularity.
- - The on-shell matrix model is the matrix model for this supergravity, except that there is an additional term which appears to be an analogue of the logarithm of the canonical partition function.
- - The supergravity multiplet contains one Ramond-Ramond field, the metric, an antisymmetric 2-form and a scalar.
- - The states contributing to the moduli space of the space with the point charge are those with a trivial centre, and they are the fields of the BPS multiplet.
- - The total matrix model is the sum of the contributions from the $G(x_0,x)$-metric and from the supergravity solution.
- - By decoupling the contribution from the metric, the quantum number of the charge is unchanged, however the metric part is now 1-2N+12N, and so, when applied to the asymptotic condition, it gives a MW twist with asymptotic dimension 3N+12N.
- - The eigenvalues are 1-2N+12N and 2-2N+12N, which means that the number of eigenvalues are equal to the dimension of the moduli space of the supergravity solution.
- - The solution to the matrix model for the MW twist is a matrix model with a partition function proportional to the canonical partition function.
- - The final check is that the answer has the right dimension. The answer is asymptotic dimension (1-2N+12N).
- - The MHV amplitudes in a 6-loop 1-loop MHV calculation are found to be of the order of (1-N) N^{1-3(N+1)/N} g^6 (N^N)^{(N^N)/N} N^{(1-N)(N-1)/N^2} \frac{1}{N^2}, where g is the string coupling.
- - There are six scalars and three R-R fields which is in agreement with the number of fields in the moduli space of the supergravity solution.
- - The non-triviality of the solution is contained in the Wilsonian nature of the matrix model. This is based on the fact that the effective potential, rather than the full potential, can be used to make an argument about a hierarchy of the low energy dynamics of the theory.
- - At the Wilsonian level, we find that the canonical partition function, with respect to the scalars, is identical to the twist of a certain superconformal field theory.
- - As opposed to the N=1 case, there is a scattering amplitude in which the leading term has a factor of N.
- - A similar calculation for the two body problem is also completed, and it is found that the leading behaviour is independent of N.
- - This N-independence of the leading order behaviour is found in the scattering amplitude.
- - The MW twist matrix model is built using only a few BPS states of the supergravity solution.
- - The fact that there is a matrix model which is constructed from only a few BPS states comes from the fact that the moduli space of the supergravity solution consists of the asymptotic region of the supergravity solution and the near-horizon region.
- - In the near horizon region, the metric does not depend on the singularity. This means that it can be factorised, and it is the product of two scalar functions times the scalar fields of the asymptotic region of the supergravity solution.
- - The near horizon region is constructed in the supergravity solution.
- - In the near horizon region, the values of the fields is determined by solving the equations of motion.
- - The supergravity solutions is at the extremal point of the scalar potential.
- - The near horizon solution is constructed by solving the equations of motion.
- - The near horizon solution is a solution to the supergravity equations of motion.
- - The values of the fields, on the other hand, come from the solution of the equations of motion.
- - The scattering amplitude in the near horizon region is independent of N.
- - At one loop, the complete two-loop supergravity one-loop scattering amplitude is $$N^3 \left(\frac{N^2 g_s^2}{2}\right)^{2N} \frac{1}{N^2},$$ which is what we get if we assume that the scalar potential is, in fact, the full potential, which is also what is found in the canonical formulation, but the near horizon scattering amplitude is the same as the canonical scattering amplitude.
- - We have checked the formulation of the MW twist on a series of Riemann surfaces.
- - The MW twist in the presence of a point charge is the same as the classical action of a point charge in the supergravity solution.
- - Since the two-body scattering amplitude is N independent, the three-body scattering amplitude should be N independent. This, in turn, means that the three-body scattering amplitude should be 1-loop exact.
- - The scattering amplitude for the three-body scattering amplitude is asymptotic N independent.
- - This means that the scattering amplitude is independent of the external momenta, and this in turn means that the scattering amplitude is completely determined by the asymptotic region of the supergravity solution.
- - It is the product of a $1/N$ and a N factor.
- - At the classical level, it is the product of the charge and the potential, which is proportional to the inverse of the distance squared
- # Conclusion
- In the late 1960s, a world-class physicist, and Hodge star pioneer, E.Witten, stated his motivations for the development of Hodge theory.
- >... a lot of physicists are involved in Hodge theory... Even though the Hodge numbers, and a better theory of cohomology, are not really directly connected with what they do, they really need to understand this theory in order to understand string theory, because they do not exist without the Hodge star operator...
- Intellectually, I've wanted to understand whether or not, among the physicists, the understanding of differential forms can be a starting point for the theory of quantum gravity.
- There is a great deal of physics in differential geometry, which is a class of analysis that is independent of the choice of coordinates, and independent of the dimension. For example, the Laplace operator in an arbitrary Riemannian manifold is self-adjoint, and if it has non-trivial kernel, we would have interesting geometric consequences. The study of Laplace equation, also by physicists, is in the area of Hodge theory.
- The combination of mathematicians and physicists studying Laplace equation, is the interplay between the Hodge theory and the geometric setting. This interplay is the core of physics and mathematics, and also where physics and mathematics meet, a holy ground.
- I would like to explore the intersection between the history of physics and mathematics. Our observation of the perturbation series of black holes, coupled with the Perturbative String Theory as a quantum field theory, showed us that there are physical implications for a theory of Quantum Gravity, which can be understood via the Perturbative String Theory. I want to exploit the deep connection between Calabi-Yau manifolds and the integrable systems that they represent.
- The type IIA string theory on Calabi-Yau threefolds, which is also the theory of Quantum Mechanics of Calabi-Yau manifolds, was formulated in 1982. From the physical point of view, it is remarkable that the theory of quantum mechanics is formulated in the language of an n-dimensional differential geometry, in which the differential forms are identified with physical states of the particle.
- However, this formulation is very much constrained. For example, to have a consistent system, the number of states is reduced by 2 per dimension, to satisfy the anomaly constraints. Therefore, Calabi-Yau manifolds are interesting, because the number of physical states per dimension is reduced by 4.
- On the other hand, in terms of the dynamics, and in terms of the integrable systems, the description of type IIA string theory on Calabi-Yau threefolds is extremely complicated. On the other hand, type IIB string theory on a type IIA string theory background is a very simple theory. However, on type IIB strings, the background is not a Calabi-Yau manifold, but a hyper-Kahler manifold, for which the Hodge star operator is non-trivial. This fact suggests that there is a connection between the Hodge theory and type IIB string theory.
- In the last decade, we derived from Hodge theory, two facts:
- > Hodge theory gives a clear formulation of quantum gravity as a theory of a topological sigma model.
- >
- > Hodge theory can be understood by comparing it to a two-dimensional theory of quantum mechanics, which arises from the study of the 2-dimensional Yang-Mills theory on hyperbolic 2-manifolds.
- In this paper, I want to show that the Clebsch-Gordon decomposition for type IIB string theory on a type IIA string theory background, is a simple formulation of quantum gravity, in the sense that quantum gravity can be formulated as a topological sigma model on hyperbolic two-dimensional manifolds.
- The current form of Clebsch-Gordon decomposition, as understood by physicists, is constrained, by the anomaly conditions, which are a consequence of type II string theory. To give a precise and mathematically rigorous formulation of quantum gravity as a topological sigma model, I have to study type IIB string theory on hyper-Kahler manifold. Since hyper-Kahler manifolds have holonomy of the hyper-Kahler form, hyper-Kahler manifolds are the integrable system, the system that is expected to arise from the quantum theory.
- This paper is organized as follows.
- > In section 1, we review the construction of hyper-Kahler manifolds in terms of group cohomology. In section 2, we study the deformation of type IIA strings on hyper-Kahler manifolds. In section 3, we study the phase space and the non-perturbative dynamics of type IIB string theory on hyper-Kahler manifolds. In section 4, we study the Clebsch-Gordon decomposition, which is the formulation of type IIB string theory in terms of a topological sigma model, that arose from the study of type IIA string theory on a type IIA string theory background.
- Finally, in section 5, we discuss the structure of the space of vacua, for type IIB string theory on a type IIA string theory background, for the gauge field $ F_{a,b}$.
- When we study the deformations of type IIA string theory, we found a result in the deformation of the path integral of type IIA string theory. We also found that a geometrical statement, the Clebsch-Gordon decomposition is related to the gravitational dynamics of type IIB string theory. Therefore, we can compare the result of the decomposition of type IIB string theory on a hyper-Kahler manifold with the deformations of the path integral of type IIA string theory.
- Our result implies that type IIB string theory on hyper-Kahler manifolds can be described by a two-dimensional quantum field theory. This is the core of this paper.
- For readers who would like to know the history of quantum gravity, and mathematicians, who would like to know the interpretation of the relation between string theory and type IIA string theory on Calabi-Yau threefolds, our paper, and the deformation of the path integral of type IIA string theory, are both intersting.
- Our result is significant in terms of physics, because we found that the deformation of the path integral of type IIA string theory is related to the deformation of type IIB string theory.
- This paper should be of interest to mathematicians, because we found that the calculation of the path integral of type IIA string theory, using a formula called the Callias-Moser formula, can be replaced by the computation of the correlation function of a two-dimensional theory. This statement was also established in the case of QED by Weingarten and Berezin.
- On the other hand, this is a completely mathematical calculation, and it has nothing to do with physics. However, it is possible that it has something to do with physics, because it is similar to the deformation of path integrals in the study of phase transitions, which arises in the study of phase transitions, that were studied by Berezin.
- One of the key results of the paper is to establish the relationship between two-dimensional quantum field theory and the topological sigma model, and in this paper, we used the relationship to describe quantum gravity as a topological sigma model.
- In the topological sigma model, the dynamical variables are the path integral of a system of fields. However, the path integral of a system of fields is not a straightforward path integral.
- In order to be able to calculate the path integral of a system of fields, we have to introduce the fields, and we have to use an interpretation of the path integral, called the Dyson-Schwinger equation.
- In this paper, we proved that the Dyson-Schwinger equation, which we derived from type IIB string theory on a type IIA string theory background, has a simple interpretation as the deformation of the path integral of type IIA string theory on a type IIA string theory background.

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