sign in sign up
<<
Home , Arithmetic geometry

PHYSICS

::::owan, C., Keehan, S., et al. (2007). Health spending Modular Forms as Clues to the Emergence of in String 1anges obscure part D's impact. Health Affairs, 26, Theory

Nathan Vander Werf .. , Ayanian, J.Z., Schwabe, A., & Krischer, J.P. Abstract .nd race on early detection of cancer. Journal ofth e 409-1419. In order to be consistent as a quantum theory, string theory requires our universe to have extra . The extra dimensions form a particu­ lar type of , called Calabi-Yau manifolds. A fundamental open ions ofNursi ng in the Community: Community question in string theory over the past two decades has been the problem ;, MO: Mosby. of establishing a direct relation between the physics on the worldsheet and the structure of spacetime. The strategy used in our work is to use h year because of lack of insurance. British Medical methods from geometry to find a link between the geome­ try of spacetime and the physical theory on the string worldsheet . The approach involves identifying modular forms that arise from the Omega Fact Finder: Economic motives with modular forms derived from the underlying conformal letrieved November 20, 2008 from: theory. The aim of this article is to first give a motivation and expla­ nation of string theory to a general audience. Next, we briefly provide some of the needed up to modular forms, so that one has es. (2007). National Center for Health the tools to find a string theoretic interpretation of modular forms de­ : in the Health ofAmericans. Publication rived from K3 surfaces of Brieskorm-Pham type. The problem involves the calculation of theoretic constructions called Jacobi sums, and : 20, 2008 from: the use these to construct the Omega-motivic L-function and q-expansion. us07 .pdf#fig30 Then, one must determine whether the q-expansion can be factored into Hecke-indefinite modular forms that arise from the worldsheet.

1 Introduction

1.1 The Deep Questions in Life :n are all senior nursing students who We live in a wonderfully complex, but elegant universe. Time and time again humans ;472 class, A Multi-System Approach to has asked questions like, What is space and time, and what is the universe made of? We are lucky to live in a time when much progress has been made toward finding some of these answers. String theory is an attempt to answer questions like these unify all of physics as we know it. But before we start on string theory, let's briefly review the developments in theoretical physics over the last one hundred years that have led so many theoretical physicists have taken an interest in string theory.

1.2 Relativity Toward the end of the 19th century, it was thought by many physics professors that the task of physics was more or less complete. The main theories were thought to be already in place, through the technological triumphs of Newton's laws, Maxwell's electromagnetic theory, and thermodynamics. They believed that only a few details needed to be filled here and there by the future generations, and the study of physics would be complete. This view could not have been any more wrong.

In 1905, Albert Einstein submitted a paper that would soon completely overturn the prior notions of space and time, and replace them with something one would never have guessed from common experience. It was known from Maxwell's formulation of

45 PHYSJCS electromagnetism that light always traveled at the same speed, roughly 3 · 108 m/s. 1.3 Quantum Mechanics and Qua As a teenager, Einstein wondered what would happen if one were to chase after light We will now look at how our understand at nearly the speed of light. Intuition and Newtonian mechanics tells us that we could has matured. The story starts in 1900, wl catch up to it so that light would appear stationary. On the other hand, Maxwell was studying the black body problem, w tells us that there is no such thing as stationary light. Einstein resolved this paradox radiation depends on the frequency of the with what is called the Special Theory of Relativity. In order for light to travel at the In order to derive a formula from the exper same speed for two observers that are in relative motion to one another, each will have the idea that energy is quantized. In othe1 different perceptions of distance and time. In other words, two identical atomic clocks a finite number of quanta. This is very di with cameras attached in relative motion to each other will tick at different rates, a continuous wave. Physicists soon came disagree on the time that elapsed between two recorded events, and will disagree on world, energy quanta of light has both wa measurements taken during these two events. Einstein's theory showed that concepts Bohr went on to develop a theory on th such as space and time are not separate, but deeply interwoven. From now on, space plained the spectral lines by using the no and time will be called spacetime. He also was able to show that two other physical justification and no one understood why t concepts were not separate: He discovered a deep connection between mass and energy, agreed with experimental results. These 1 2 theory. In 1924, a French physicist nam E = mc , matter by coming to the conclusion that where E is the rest energy of a particle, m is the mass of the particle, and c is the and not just light. Using the ideas of E speed of light. and Max Born developed a matrix mechar mechanics. At about the same time, the With the success of special relativity, Einstein set out to conquer two new problems. de Broglie's ideas to develop a wave me< One was to generalize his idea of relativity to accelerating reference frames, and the Schrodinger also showed that his approa< other to formulate gravity into this framework. In Newton's successful theory of Heisenberg formulated the uncertainty p gravity, two masses seem to instantly know about each other, which goes against quantities such as position and momentu the notion that no information can travel faster than the speed of light. Einstein came time. It is impossible to know both of th to the realization that being in accelerated motion is indistinguishable from being what kind of experiment one uses to de1 in a gravitational field that produces the same force, and called this the equivalence leads to quantum effects that are just not principle. With this, he was able to declare that all observers, regardless of whether the relation !::..E!:lt ::'.". n implies that one c they are in an accelerated state of motion or not, share the same laws of nature. In vacuum energy can be created for a short 1916, he fully developed these thoughts into a mathematical framework that is called the general theory of relativity (GR). This formulation led to a second link in uniting In 1927, British physicist Paul Dirac took gravity and accelerated motion, the curvature of space and time. He found that since into quantum mechanics. In 1928, he deve accelerated motion warps spacetime, gravity also warps spacetime. Further, strength tion for the electron. This new formulat of gravity, or this spacetime warp, is proportion to the produce of the two products. intrinsic spin. Also, it predicted the exis As an example of what this means, consider a bowling ball placed on a trampoline. matter known as anti-matter that can le~ Think of the bowling ball as the sun and the trampoline fabric to be a 2-dimensional regular matter. In the 1930's, some physi< version of our 3-dimensional space. When the bowling ball is absent, any other smaller relationships that these formulations wer ball on the trampoline will just move in a straight . But with the presence of the formulated a rigorous mathematical basis bowling ball, other smaller balls will be forced to move in a curved path. In the right ear operators on Hilbert spaces. conditions, the smaller ball will go into an orbit around the bowling ball. This idea is analogous to why the planets orbit the sun, except the 2-dimensional trampoline In 1927, physicists began to take on the p fabric is replaced with our three dimensional space, and the warping of space is what number of particles. The way to do this " a gravitational field is (due to a mass). Even the path of light will bend due to the theories are what now we call (relativisti presence of a large mass. Later that year, a physicist Karl Schwarzschild use the the rest of the 1930s many researchers we equations of GR to predict the existence of black holes. Also with this framework magnetism on the quantum level. The >1 physicists were able to make accurate models on the past, present, and future of our is now known as quantum electrodynam: universe, giving birth to modern cosmology. and the electromagnetic field. It is argua nomena ever. By this time, it was knm microscopic level, the strong and weak 1 constructed for the strong force known as

46 PHYSICS d at the same speed, roughly 3 · 108 m/s. 1.3 Quantum Mechanics and Quantum Field Theory 11ld happen if one were to chase after light We will now look at how our understanding of the universe at the microscopic level lewtonian mechanics tells us that we could has matured. The story starts in 1900, when a German physicist named Max Planck stationary. On the other hand, Maxwell was studying the black body problem, which can be stated as how the intensity of mary light. Einstein resolved this paradox radiation depends on the frequency of the radiation and the temperature of the body. elativity. In order for light to travel at the In order to derive a formula from the experimental data at the time, Planck introduced :i.tive motion to one another, each will have the idea that energy is quantized. In other words, the light energy can be divided into [n other words, two identical atomic clocks a finite number of quanta. This is very different from the prior notion that light was to each other will tick at different rates, a continuous wave. Physicists soon came to the acceptance that in the microscopic two recorded events, and will disagree on world, energy quanta of light has both wave-like and particle-like properties. In 1913, ts. Einstein's theory showed that concepts Bohr went on to develop a theory on the structure of the hydrogen atom that ex­ Jt deeply interwoven. From now on, space plained the spectral lines by using the notion of quantization. Though there was no was able to show that two other physical justification and no one understood why this theory of quantization should be true, it deep connection between mass and energy, agreed with experimental results. These theories became known as the old quantum me2 , theory. In 1924, a French physicist named Louis de Broglie developed a theory of matter by coming to the conclusion that matter also has this wave-particle duality, i is the mass of the particle, and c is the and not just light. Using the ideas of Bohr in 1925, physicists Werner Heisenberg and Max Born developed a matrix mechanics formulation to explain modern quantum mechanics. At about the same time, the Austrian physicist Erwin Schri:idinger used ;ein set out to conquer two new problems. de Broglie's ideas to develop a wave mechanics formulation of quantum mechanics. · to accelerating reference frames, and the Schri:idinger also showed that his approach was equivalent to Heisenberg's. In 1927, 2work. In Newton's successful theory of Heisenberg formulated the uncertainty principle, which states that certain physical )W about each other, which goes against quantities such as position and momentum can't both have precise values at a given ster than the speed of light. Einstein came time. It is impossible to know both of these features with total precision, no matter :d motion is indistinguishable from being what kind of experiment one uses to detect these values. The uncertainty relation ame force, and called this the equivalence leads to quantum effects that are just not seen in the world we perceive. For example, e that all observers, regardless of whether the relation 6.E6.t 2: n implies that one or more particles with energy 6.E above the or not, share the same laws of nature. In vacuum energy can be created for a short time 6.t, and are called virtual particles. o a mathematical framework that is called formulation led to a second link in uniting In 1927, British physicist Paul Dirac took on the problem of putting special relativity Jre of space and time. He found that since into quantum mechanics. In 1928, he developed a wave equation called the Dirac equa­ ;y also warps spacetime. Further, strength tion for the electron. This new formulation predicted the existence of an electron's irtion to the produce of the two products. intrinsic spin. Also, it predicted the existence of a positron, which is a new type of er a bowling ball placed on a trampoline. matter known as anti-matter that can lead to a huge explosion when it collides with 1e trampoline fabric to be a 2-dimensional regular matter. In the 1930's, some physicists began to realize the deep mathematical he bowling ball is absent, any other smaller relationships that these formulations were using. For example, John von Neumann 3traight line. But with the presence of the formulated a rigorous mathematical basis for quantum mechanics as the theory of lin­ ced to move in a curved path. In the right ear operators on Hilbert spaces. orbit around the bowling ball. This idea sun, except the 2-dimensional trampoline In 1927, physicists began to take on the problem of applying quantum mechanics to a al space, and the warping of space is what number of particles. The way to do this was to create a relativistic field theory. These ven the path of light will bend due to the theories are what now we call (relativistic) quantum field theories(QFT) today. For r, a physicist Karl Schwarzschild use the the rest of the 1930s many researchers worked to understand the dynamics of electro­ of black holes. Also with this framework magnetism on the quantum level. The work of many physicists culminated in what els on the past, present, and future of our is now known as quantum electrodynamics (QED), a theory of electrons, positrons, and the electromagnetic field. It is arguably the most precise theory of natural phe­ nomena ever. By this time, it was known that there were two other forces on the microscopic level, the strong and weak nuclear forces. In early 1960s, a QFT was constructed for the strong force known as quantum chromodynamics (QCD), and was

47 PHYSICS

completed around 1975. Also in the 1960's, a few physicists began to realize that the feel that if we really understood how the uni weak force looks very similar to the electromagnetic force when the energies being then the universe can be described by a logi used to probe it are above what is called the "unification energy" of roughly 100 GeV. consistent and describes all four forces on £ This development is now called the electroweak theory,which describes what is now modifying either GR or QM have met with l called the electroweak interaction in an electroweak field. These theories of the three a few proposed quantum theories of gravit) non-gravitational forces became known as the standard model of particle physics. theory being string theory. The standard model describes three messenger particles for the electromagnetic, strong and weak forces that are called the photon, gluon, and the weak gauge bosons respec­ tively. These are the smallest possible units of energy each force can transmit. In this 2 String Theory framework, the particles are not thought to have any internal structure; instead they are elementary and can be thought of as points. The standard model also consists 2 .1 History of String Theory of 6 quarks, 6 leptons, their associated antiparticles, and the Higgs Boson. After the The story of string theory can be traced ba success of the electroweak theory, people have worked to unify the electroweak theory named Gabriel Veneziano was trying underE and QCD into a QCD-electroweak interaction. These similar theories have come to ties of the strong nuclear force. He discover be known as grand unification theories (GUT)s. It predicts that at extremely high the Euler beta-function, seemed to describ 14 energies of about 10 GeV, the electromagnetic, weak, and strong forces are fused into In the year 1970, physicists Yoichiro Namb a single unified QCD-electroweak field. Up untill now, energies can't be made high showed that if one modeled elementary "objE enough to confirm these GUTs. By applying other symmetric principles, physicists then the nuclear force could be described t have come up with extensions to the standard model such as supersymmetry, which not long before the string description was : proposes a whole new list of particles that have never been directly observed due to the 1970's showed that the string model made . fact that they are to massive to be produced in a particle accelerator. The standard and a particle theory called Quantum model falls short of being a complete theory of fundamental interactions due to the successful at describing the nuclear force. , fact that gravity is not included and a host of other problems. Also, in many places was dead. The electroweak force was estal constants are put into place to match data from experiments, and are not derived from high enough energies, the electroweak woul< the theory itself. come one force. Through the success of qm of data from high energy experiments, phy~ 1.4 The Incompatibility of General Relativity and Quantum Field The- standard model. Most people had lost inter ory of the standard model, but the early pion that string theory must be right. At the ti1 So far we have not talked about gravity in quantum mechanics. The equations of GR additional messenger-like particles that did are applied mainly to distances of astronomical , although there are some ex­ rent experiments. A predicted particle calle< ceptions. GR says nothing about what the microscopic properties of space would look speed of light. It seemed that the more the) like, making it a classical theory. The strength of the gravitation force is some 10- 4o were found. times as strong as the strength of the electromagnetic field. In almost all cases, this implies that the gravitational force in quantum systems is negligible, which is why the In 1970s, physicists John Schwarz and Joel standard model was able to get away with totally ignoring the gravitational force. GR these additional particles, but kept failing. demands that space is smooth, and that it stays smooth no matter what lengths we are equations. Then one day while brainstormi1 looking at. However, quantum mechanics says that everything is subjected to quan­ cerning one of these messenger-like particleE tum fluctuations due to the uncertainty principle. The smaller the region of space we vibration, they came to the realization that probe into, the more the gravitational field will fluctuate. These quantum fluctuations than this massless hypothetical particle had cause violent distortions to the surrounding space, sometimes called quantum foam. have. String theory was not just a theory When one tries to incorporate the equations of GR into those of quantum mechanics, theory of gravity! String theory was able t probabilities that are supposed to be between 0 and 1 become infinity. Getting infinity missing from the standard model. Schwarz < in a probability is nature's way of telling us that something is very wrong in our frame­ tion. However, it was shown during this tir work. In order for the effects of gravity to become apparent on the quantum scale, one and their discovery was given almost no att would have to look at distances as small as the plank , which is roughly 10-34 some time, Michael Greene joined Schwarz m. Physicists have tried again and again to incorporat~ gravity into the framework for over a decade. In 1984, Schwarz and Gr• of the standard model with no luck. This incompatibility between the theories of the work was culminated in the summer of 19E big and small points to some kind of flaw in the understanding of our universe. Due a single calculation. They succeeded and to the successful unification of the weak and electromagnetic force, many physicists

48 PHYSICS

3, a few physicists began to realize that the feel that if we really understood how the universe works at its most fundamental level, ;romagnetic force when the energies being then the universe can be described by a logically sound theory that is mathematically ie " unification energy" of roughly 100 GeV. consistent and describes all four forces on any scale. However, numerous attempt of ·oweak theory,which describes what is now modifying either GR or QM have met with failure after failure. Today, there are quite ~ ctroweak field. These theories of the three a few proposed quantum theories of gravity, with the most successful and promising the standard model of particle physics. theory being string theory. iger particles for the electromagnetic, strong , gluon, and the weak gauge bosons respec­ .s of energy each force can transmit. In this 2 String Theory .o have any internal structure; instead they 2.1 History of String Theory points. The standard model also consists tiparticles, and the Higgs Boson. After the The story of string theory can be traced back to the late 1960's. In 1968, a physicist ave worked to unify the electroweak theory named Gabriel Veneziano was trying understand the experimentally observed proper­ :tion. These similar theories have come to ties of the strong nuclear force. He discovered that a two hundred year old equation, jUT)s. It predicts that at extremely high the Euler beta-function, seemed to describe various properties of the nuclear force. netic, weak, and strong forces are fused into In the year 1970, physicists Yoichiro Nambu, Roiger Nielsen, and Leonard Susskind ·p untill now, energies can't be made high showed that if one modeled elementary "objects" as vibrating, one-dimensional strings, ting other symmetric principles, physicists then the nuclear force could be described by Euler's function. Unfortunately, it was dard model such as supersymmetry, which not long before the string description was rejected. High energy experiments in the tave never been directly observed due to the 1970's showed that the string model made a number of predictions that were wrong, 3ed in a particle accelerator. The standard and a point particle theory called Quantum Chromodynamics (QCD) was much more )ry of fundamental interactions due to the successful at describing the nuclear force. At that point, it looked like string theory 3t of other problems. Also, in many places was dead. The electroweak force was established and physicists believed that given from experiments, and are not derived from high enough energies, the electroweak would have united with the strong force to be­ come one force. Through the success of quantum field theories and the accumulation of data from high energy experiments, physicists were able develop a QFT called the tl Relativity and Quantum Field The- standard model. Most people had lost interest in string theory because of the success of the standard model, but the early pioneers of string theory were still convinced that string theory must be right. At the time, their string theory has problems with quantum mechanics. The equations of GR additional messenger-like particles that did not appear to have any relevance to cur­ mica! lengths, although there are some ex­ rent experiments. A predicted particle called the tachyon that traveled faster than the microscopic properties of space would look 40 speed of light. It seemed that the more they studied string theory, the more problems ngth of the gravitation force is some 10- were found. tromagnetic field. In almost all cases, this tum systems is negligible, which is why the In 1970s, physicists John Schwarz and Joel Scherk tried all sorts of ways to get rid of totally ignoring the gravitational force. GR these additional particles, but kept failing. For four years, they tried to tame those ;tays smooth no matter what lengths we are equations. Then one day while brainstorming, they made the startling discovery con­ says that everything is subjected to quan­ cerning one of these messenger-like particles. While studying the patterns of a strings inciple. The smaller the region of space we vibration, they came to the realization that if they were to make the strings smaller, will fluctuate. These quantum fluctuations than this massless hypothetical particle had the same properties that a graviton would ig space, sometimes called quantum foam. have. String theory was not just a theory of the nuclear force, but also a quantum 1s of GR into those of quantum mechanics, theory of gravity! String theory was able to produce a piece of the puzzle that was en 0 and 1 become infinity. Getting infinity missing from the standard model. Schwarz quickly submitted his research for publica­ that something is very wrong in our frame­ tion. However, it was shown during this time that string theory still had fatal flaws, >ecome apparent on the quantum scale, one 34 and their discovery was given almost no attention from the physics community. After s the plank length, which is roughly 10- some time, Michael Greene joined Schwarz on his quest to conqueror these anomalies to incorporate gravity into the framework for over a decade. In 1984, Schwarz and Green finally made a breakthrough and their mcompatibility between the theories of the work was culminated in the summer of 1984. Amazingly, everything boiled down to in the understanding of our universe. Due a single calculation. They succeeded and the reactions to their success was heard .nd electromagnetic force, many physicists

49 PHYSICS throughout the worldwide physics community. This new string theory had the mathe­ Another consequence extremely large tensi matical depth to encompass all four forces, and all known matter. In less than a year, ber of different vibrational patterns. Thi hundreds of particle physicists dropped their projects to either join in the quest to experimental data. However, the large te either develop string theory or show that it can't possible be true. This period from tional patterns will correspond to particle: 1984 to around 1986 is now known as the "first superstring revolution". [l].Thousands is likely that the heavier particles that st of research papers were published during this time on string theory. They showed that lighter particles. numerous features of the standard model that took decades of experimental research to understand seemed to emerge naturally from the structure of string theory. Many of The answer to the apparent contradiction the features of the standard model had a more satisfying explanation in string theory ory is now one that is conceptually unders than what could be found in the standard model. It also became apparent that in between QM and GR is that general relat order for string theory to work, it needed to have six extra dimensions. where the six fabric to be smooth while QM's uncertain1 dimensions are thought to be very small. In order to produce the four dimensional have very violent distortions. Due to the spacetime we know, these extra dimensions must be of a particular geometry, known lot of energy and a very short length. Wi1 as Calabi-Yau manifolds, which are small and compact. One of the few examples of microscopic discreteness and fluctuations o Calabi-Yau manifolds that can be visualized is a torus. GR and QM can work together. These flu in string theory. They only exist in a form1 However, time and time again, string theorist found new anomalies that had to be as point particles. tamed, only to slowly figure out a way around them. By the 1990's it was shown that there was not just one, but five different string theories that explained the universe as we know it. This was frustrating, since theorists felt as if there should only be one 3 String Theory R esearch at Il theory of everything. In 1995, theoretical physicist Edward Witten gave a talk where he suggested the existence of M-theory which has one more to explain 3.1 Motivation a number of observed dualities between the different string theories. This marked the birth of what is now called the "second superstring revolution" . Here some of Here at IUSB, the two string theorists Roi: the versions of string theory were unified by new equivalences known as S-duality, ing to understand the structure of spacetim T-duality, U-duality and mirror . on the worldsheet [2]. The aim here is to understand the problem in the context of • are formulating a framework that combin 2. 2 Basics of String Theory and arithmetic geometry to Calabi-Yau h As complicated as the mathematical formalism of string theory is, it can be understood to show that the basic building blocks of 1 in a very elegant and simple way conceptually. Consider the strings on a violin. derived from the geometry of these Calabi· When plucked, each sting can go through a number of different vibrational patterns. the modular forms derived in elliptic cur Just as our ears can here the different musical notes different resonant vibrations, forms derived from the underlying supercc the different vibrational patterns of a fundamental string can give rise to a host of this result to higher genus is difficu different masses and forces charges. So one vibrational pattern can describe an electron theorem is known, even conjecturally [3]. while another vibrational pattern may describe a photon. The amplitude of a string higher genus surfaces to determine wheth vibration correspond to a larger energy with a shorter wavelength. Due to special they can be written in modular forms deri relativity, we can confer that the mass of an elementary particle is determined by already been shown in [4] that some of tl the energy of the vibrational pattern of its internal string. Similarly although a bit be written in terms of string partition fun more complicated, the electric, weak, and strong charge carried by a particular string be write various computer programs that c is determined precisely by the way the string vibrates. Instead of the notion in the than a day. standard model that different particles are made of different "stuff'', string theory states that all matter and all the forces really come from the same constituent, a string. 4 R elevant Math em atics a nd D The universe is nothing but a cosmic symphony. Notice that in the violin example, the string has tension, which causes it to be able to plucked. The fundamental string's This next section assumes that the reader tension that would produce a graviton-like particle has been computed to be about matics to read definitions and fill in some 8.9 * 1042 N. Very detailed calculations have showed th~t with this tension, the string length is about the same as the Plank length,

lp = ~ = 1.61 * 10-35m.

50 PHYSICS nity. This new string theory had the mathe­ Another consequence extremely large tension is that a string can be in a infinite num­ l, and all known matter. In less than a year, ber of different vibrational patterns. This at first may seem to be in conflict with ;heir projects to either join in the quest to experimental data. However, the large tension means that only a few of the vibra­ it can't possible be true. This period from tional patterns will correspond to particles at masses of the particles we observe. It 'first superstring revolution" . [l].Thousands is likely that the heavier particles that string theory predicts would decay into the his time on string theory. They showed that lighter particles. that took decades of experimental research from the structure of string theory. Many of The answer to the apparent contradiction between general relativity and string the­ more satisfying explanation in string theory ory is now one that is conceptually understandable. As a reminder, the main conflict rd model. It also became apparent that in between QM and GR is that general relativity demands the properties of the spatial to have six extra dimensions, where the six fabric to be smooth while QM's uncertainty principle gives a picture of spatial fabric In order to produce the four dimensional have very violent distortions. Due to the large tension on a string, the string has a as must be of a particular geometry, known lot of energy and a very short length. With this property, it simply smears outs this and compact. One of the few examples of microscopic discreteness and fluctuations of the gravitational field just enough so that ~ed is a torus. GR and QM can work together. These fluctuations then, in some sense, do not arise in string theory. They only exist in a formulation that thinks of fundamental particles ~ orist found new anomalies that had to be as point particles. mnd them. By the 1990's it was shown that string theories that explained the universe theorists felt as if there should only be one 3 String Theory Research at IUSB physicist Edward Witten gave a talk where which has one more dimension to explain 3 .1 Motivation the different string theories. This marked )nd superstring revolution". Here some of Here at IUSB, the two string theorists Rolf Schimmrigk and Monika Lynker are work­ :I by new equivalences known as S-duality, ing to understand the structure of spacetime from first principles in terms of the physics on the worldsheet [2]. The aim here is to use methods from arithmetic geometry to r. understand the problem in the context of exactly solvable Calabi-Yau varieties. They are formulating a framework that combines methods from algebraic and arithmetic geometry to Calabi-Yau hypersurfaces of n dimensions. The goal is .!ism of string theory is, it can be understood to show that the basic building blocks of underlying string partition functions can be ptually. Consider the strings on a violin. derived from the geometry of these Calabi-Yau hypersurfaces. It has been shown that a number of different vibrational patterns. the modular forms derived in elliptic curves can be expressed in terms of modular nusical notes different resonant vibrations, forms derived from the underlying superconformal field theory. The generalization of 1damental string can give rise to a host of this result to higher genus curves is difficult since no analog of the elliptic modularity vibrational pattern can describe an electron theorem is known, even conjecturally [3]. It then suffices to look at special classes of scribe a photon. The amplitude of a string higher genus surfaces to determine whether it has modular forms and then whether with a shorter wavelength. Due to special they can be written in modular forms derived from the conformal field theory. It has >f an elementary particle is determined by already been shown in [4] that some of these surfaces have modular forms that can ts internal string. Similarly although a bit be written in terms of string partition functions. To help in this work, I will have to strong charge carried by a particular string be write various computer programs that can compute very large computations in less ring vibrates. Instead of the notion in the than a day. .re made of different "stuff', string theory lly come from the same constituent, a string. 4 Relevant Mathematics and Data Collected 1phony. Notice that in the violin example, e able to plucked. The fundamental string's This next section assumes that the reader has had enough prior knowledge in mathe­ :e particle has been computed to be about matics to read definitions and fill in some of the spaces between the blanks. ve showed that with this tension, the string gth,

1.61 * 10- 35m.

51 PHYSICS

4.1 Calabi-Yau Hypersurfaces CY hypersurfaces with n = 3 are a specia I am to investigate this class of CY hypen Definition 4.1.1. Projective Space hypersurfaces with n = 3: A complex projective space IP'n ={the set of complex lines in icn+i through the origin (0, · · · , 0) , more precisely List of CY hy JP>141 _ 4 JP>161 1,1,1,l) - 8 (1,1,1,3) IP'n = {(z1,· · · ,zn+1) E iCn+l } (1) JP> f8] _ 58 JP>[lO] (1,1,2,4) - (1,2,2,5) with an equivalence relation ~ defined by: JP>[12] _ 512A JP>[12] (1,1,4,6) - (1,2,3,6) IP'[18] - 518 IP'[20] (z1, · · · , Zn+i) c::: >.(z1, · · · , Zn+1) = (.Az1, · · · , AZn+1), A E IC*, IC* =IC - {O} (2) (1,2,6,9) - (1,4,5,20)

Note if A= l /z1, where z1i= 0 , then (z1, · · · ,Zn+i) c::: l /z1(z1, · · · ,zn+1) 4.2 Zeta Function = (1 ~ · · · Zn+l) = (1 Yl · · · Y ) ' z 1 ' z 1 ' ' ' n In other words, Definition 4.2.1. lF P is the field of dim !P'n= n (3) Let Definition 4.1.2. We define the degree of a polynomial p(x) E IP'n to be the largest exponent in which a variable is raised to. denote the number of solutions manifold x • One way to find properties of some mathemE Definition 4.1.3. Weighted Projective Space arithmetic) that may reveal some informat A projective space is called a weighted projective space if: function made for this use is known as the Artin as,

Z(X/lFp, t) =exp ( with ~ being an equivalence relation defined by:

(4) oo n wit·hx e = '\"~ I.x The congruence ze tfa ur n. Definition 4.1.4. Let x E IP'(ki, .... ,kn+i)· Then x has degreed if: n=O function. Grothendieck showed in [5] that tl as (5) Z(X/lFp, t) = Then (6) w/ deg P~(t) =dim W(x) = bi(x) given by Next is a theorem conjectured by Andre W. Definition 4.1.5. Hypersurfaces in Weighted Projected Spaces A polynomial x E IP'ck,, ..... ,kn+il is called a hypersurface in a weighted projective space Theorem 4.1. The congruence zeta functic if deg x = d for some d E z+

1 Z(X/lFp, t Let IP'~~ 1 , .... ,kn+i) denote the set of hypersurfaces with deg din IP'(ki, .... ,kn+il Definition 4.1.6. A Calabi-Yau manifold is a Kahler manifold with a vanishing first The definition of Z needs apparently a Chern class Q are polynomials, it must be the case tha it. This is important because it means we ; Example 1. In one complex dimension, a Calabi-Yau (CY) manifold is a complex a finite amount of information as opposed 1 elliptic . The only compact example is a torus. A K3 surface is an example of a case of K3 surfaces can be shown that Calabi-Yau manifold with complex dimension two. Z(X/lFp, t) = -(- 1- Definition 4.1.7. Let x E IP'~~1 , .... ,kn+il Then x is ·a Calabi-Yau Hypersurface if n+l which leads us to the Hasse Weil L-functi01 d=I>i i=l

52 PHYSICS

CY hypersurfaces with n = 3 are a special type of K3 surfaces that are algebraic. I am to investigate this class of CY hypersurfaces. Here is a list of all possible CY hypersurfaces with n = 3: .f complex lines in cn+i through the origin

List of CY hypersurfaces IP' l4J _ s4 . IP'l6J _ s6A IP'l6J _ s 6s (1,1,1,1 ) - 11,1,1 ,3) - (1,1 ,2,2) - Zn+1) E cn+l} (1) IP'(8] - IP' 10] - sio IP'(12] - x12 (1,1,2,4) - ss (1,2, 2,5) - (2,3,3,4) - 2 IP'112J _ s12A IP'1121 _ s12s IP'1121 _ s12c (1,1 ,4,6) - 11,2,3,6) - 11,3,4,4) - IP'1 1s1 _ sis IP' 201 _ s20 IP' 241 _ s24 , · · · , >. zn+1), >. EC*, C* = C- {O} (2) (1,2,6,9} - (1,4,5,20) - (1 ,3,8,12} -

· ,Zn+1)'.::::' l /z1(z1,··· , Zn+i) I 4.2 Congruence Zeta Function

D efinition 4.2.1. IFp is the field of {O, 1, · · · ,p - 1} modulo p. (3) 1=n Let ·a polynomial p(x) E IP'n to be the largest Nr,p(X) = #(X/IFpr) (7) denote the number of solutions manifold x over the field Zi E IF pr . One way to find properties of some mathematical object is to define a function (usually ace arithmetic) that may reveal some information about the object itself. A generating ective space if: function made for this use is known as the congruence zeta function defined by Emil Artin as, cn+1 - {O} )/ ~ Z(X/IFp, =exp(~ Nr , ~(X) (8) d by: t) tr)

= (>.k'z1, · · · , >.k n+ 1 zn+1), >. EC* (4) with e"' = f ~~. The congruence zeta function uses Nr,p(X) to define a generating ['hen x has degree d if: n=O function. Grothendieck showed in [5] that the congruence zeta function can be written

I· ~ ,kd d· Q as I = L_, /\ I l zi I = (5) 11 p;i+l(t) Z(X/IFp, t) = . (9) 11 ppl(t)2 I or d = k;d; E z+'v'i (6) w/ deg P~(t) =dim W(x) = bi(x) given by the ith betti number. Next is a theorem conjectured by Andre Weil and proven by Bernard Dworkin 1960. ted Projected Spaces ypersurface in a weighted projective space Theorem 4.1. The congruence zeta function can be written as a ratio of polynomials,

P(t) Z(X/IFp, t) = Q(t) (10) :faces with deg d in IP'(ki. .... ,kn+il

:s a Kahler manifold with a vanishing first The definition of Z needs apparently an infinite # of Nr,p(x) . But since P and Qare polynomials, it must be the case that only a finite# of Nr,p(x) will determine it. This is important because it means we are more likely to extract out of an object . Calabi-Yau (CY) manifold is a complex a finite amount of information as opposed to infinite amount of information. For the a torus. A K3 surface is an example of a case of K3 surfaces can be shown that m two. Z(X/IFp, t) = 1 (11) Then x is a Calabi-Yau Hypersurface if which leads us to the Hasse Weil L-function.

53 PHYSICS

4.3 L-functions With these ingredients, one can solve fo1 Jacobi sum variables, Definition 4.3.1. a Dirichlet L-series is a infinite series in the form p~- l(t) = II c1 - c L(s, x) = x(~), (12) f n aEAi- n=l where where xis the Dirichlet character ands is a complex variables with Re(s) > 1. when the function is extended to a meromorphic function on the whole complex by analytic continuation, it is called a Dirichlet L-function and denoted L(X, s).

The factorization of (3) leads to the definition of the cohomological L-functions and f is determined via (pf - 1) =QI (i) _ 1 L (X/lFp, s) - II pi ( -s) (13) p prime P p For my purposes, s=3 and f=l. Here are so the Calabi-Yau hypersurfaces mentioned ab associated to the ith cohomology group H i(X). Sometimes the L-functions of a variety the Jacobi sum results that has been collect X can lead to a nice complex function that has lots of . This was proven to be true by for elliptic curves. For higher dimensional surfaces, a program called the has made a series of conjectures. The hope is that the L-functions can be used to find modular forms, and that some of these modular forms are the same as the string modular forms derived from conformal field theories This paper will focus on the computation of a special part of a L-function that has lots of nice symmetries known as the 0-motivic L-function, which will be defined later.

4.4 Jacobi Sums For weighted Fermat varieties there is a second method for obtaining both the cardi­ nalities and the L-function.

Theorem 4.2. For a smooth weighted projective surface with degree vector '.11 = (no, ... 'ns), 0 ) X'l = {z; + z~' + · · · + z;• = O} C lP' (ko ,ki.·. , k 8 (14) defined over the lF q, define the set A~ ·Tic of rational vectors a = (ao, ai, · · · , as) as

5 1 A~ · Tic = {a E Q + IO< a; < 1, d; = gcd(ni, q - 1), d;ai = 0 mod 1, t ai = 0 mod 1} 1=0 (15) For each (s + 1)-tuple a define the Jacobi sum

1 jq(ao, ai, · · · , as)= -- X<>o(uo)x" 1 (u1) ··· x". (us), (16) q- 1 L u iEIFq uo+u1 +···+us=O mod p

2 where x"•( u ; ) = e "i"•m• with integers m; determined via u; = g=;, where g E lF q is a generator. Then the cardinality X Tic is given by

#(Xrl/JFq) = Ni,q(Xrl) = 1 + q + · · · + qs-l + L jq(a). (17) aEAi'n

54 P HYS ICS

With these ingredients, one can solve for the polynomials p~- 1 (t) in terms of the is a infinite series in the form Jacobi sum variables,

= x(n) p~- 1 (t) = (1- (-1)5-1jpr(a)tf)1/f f, (12) IT (18) n=l ns ' aEAi-

: is a complex variables with Re(s) > 1. when where phic function on the whole complex plane by 1 :hlet L-function and denoted L(X, s). A;rr = {a E 'Ct+ 10 < ai < 1, niai = 0 mod 1, ~ ai = 0 mod 1} (19) definition of the cohomological L-functions and f is determined via I1 ~ (13) (pr - 1) = 0 mod 1, 'Vi. (20) P prime Pp For my purposes, s=3 and f=l. Here are some of the Jacobi sum results for some of the Calabi-Yau hypersurfaces mentioned above. The following will describe some of li(X). Sometimes the L-functions of a variety the Jacobi sum results that has been collected .hat has lots of symmetries. This was proven ; curves. For higher dimensional surfaces, a has made a series of conjectures. The hope find modular forms, and that some of these g modular forms derived from conformal field .putation of a special part of a L-function that ~ 0-motivic L-function, which will be defined

second method for obtaining both the cardi- projective surface with degree vector !!: -

+ z~ · = O} C 1P'(ko,k1 , ··· ,k, ) (14) et A'.;·!1 of rational vectors a = ( ao, a1, · · · , a,) li,q -1),diai = 0 mod 1, ~ai = 0 mod l} (15) ·i sum

Xa 0 (uo)xa 1 (u1) · · · Xa , (u,), (16) 'o 8 =0 mod p i; determined via u; = gmi, where g E lF q is ~iven by

- q + ... + q s-1 + '"°'~ Jq. (a ). (17) oEA~ · !!

55 ~ ~ =~ ~ n -~

[8] IP\1,1,2,4) 41 p 17 . (1 1 Jp B' B' , 3 + 8v'2 + ( -12 + 2v'2)i 15 + 16v'2 + ( -12 + 20v'2)i

jp(~,~,, 3 - 8v'2 + (12 + 2v'2)i 15 - 16v'2 + (12 + 20v'2)i

jp(~,~,, 3 - 8v'2 - (12 + 2v'2)i 15 - 16v'2 - (12 + 20v'2)i jp(!. '!_ 3 + 8v'2 - ( -12 + 2v'2)i 15 + 16v'2 - ( -12 + 20v'2)i

[12] IP(1,1,4,6) p 7 13 19 31 37

'!_ - 7../3i 37 37i../3 13j3)i + 19~../3i Jp( ~,~, ~, ~) 2 2 -¥ - ¥ -31 -2- -2- . (5 1 1 1) 7 7v'3. ~ 19iv'3 i _ _:rr 37iv'3 2 + -2 - i -~ + 13v'3)i + ]p ~·3, 3, 2 2 2 2 2 -31 2 2 Jp( 12, ~,~, ~) 13i 37 1 1 1) 9 3 ]p·(1 11",cg, 3, 2 ¥ +3V3+i(-5+ f) ¥-9V3+i(-33- f) jp(12,4,~,~) 13i 37 . (5 5 2 1) 9 3 ]p 11",12, 3, 2 ¥ - 3V3 + i( -5 - f) ¥ +9V3+i(-33+ f)

jp(12,~'~'~) - 13i 37 9 3 jp( fi, fi, ~,~) ¥ -3V3-i(-5- f) ¥ + 9V3- i(-33 + f)

jp ( H, ~,~, ~) -13i 37 9 3 jp(H,H.~.~) !.§. + 3V3 - i(-5 + -{3) .!! - gyl3 - i(-33 - -{3)

[12] IP(1,2,3,6) p 13 37

jp(f2",%,~.~)13i 37 · (1111) 2 + 6V3 + (3 - 4V3) 5 + 12V3 + ( -30 + 2V3)i ]p V' ~·~ ~· ]p ( 12, 3, 4, 2) 13i 37 "(5511) 2 - 6V3 + (3 + 4V3) 5 - 12V3 + ( -30 - 2V3)i ]p V' ~·4, 2 jp(12,3,~,~)-13i 37 · (7131) 2 - 6V3 - (3 + 4V3) 5 - 12V3 - ( -30 - 2V3)i ]p Il' ~·i' ~ ]p(12,3,4,2) - 13i 37 ,;(.!!§.;Jl\ ') ...L ~.IQ- (Q _ A, /Q\ r; ...L i 0,n _ (_Qn ...L 9,/Q\; p lJ HI ;:n 37 !_ - 7v'3i 13j3)i .!_g + 19iv'3i -31 _;)_'!_ - 37iv'3 Jp(~,~, ~) ~, 2 2 -¥ - 2 2 2 -2- 7 7./3. .!_g _;)_?_ + 37i./3 '(5111) + -2~i _g + 13./3)i - 19i./3i -3 1 )p ~,3, 3> 2 2 2 2 2 2 2 2

Jp(12, ~,~, ~) 13i 37 . (1 1 1 1) 9 gy'3 + i( -33 - 3 Jp ~,V' 3, 2 ¥ +3v'3+i(-5+ f) ¥ - f) Jp(12,4,~,~) 13i 37 5 2 1) 9 3 Jp'(5 -q, 12, 3, 2 ¥ - 3y'3 + i( -5 - f) ¥ + gy'3 + i(-33 + f)

Jp( 12, ~,~, ~) -13i 37 9 3 Jp(fi, fi, ~,~) ¥ - 3v'3- i(-5- f) ¥ + 9v'3- i(-33 + f) jp(H,~'~'~) -13i 37 9 3 Jp(f-*,f-*,~,~).!_§_+ 3v'3-i(-5 + f) l! - gy'3 - i(-33 - f)

[12] JP>(1,2,3,6) p 13 37

jp(Y2,1,~,~) 13i 37 '(1111) 2 + 6v'3 + (3 - 4v'3) 5 + 12y'3 + ( -30 + 2v'3)i )p l?' ~, 1~, Jp(12,3,4,2) 13i 37 '(5511) 2 - 6v'3 + (3 + 4v'3) 5 - 12y'3 + (-30 - 2v'3)i Jp V'~'t'1 )p ( 12, 3, 4, 2) -13i 37 . ( 7 1 3 1) 2 - 6v'3 - (3 + 4v'3) 5 - 12y'3 - ( -30 - 2v'3)i )p Il' ~,t' 1 Jp(12,3,4,2) -13i 37

Jp(f-*,*'~'~)2 + 6v'3 - (3 - 4v'3) 5 + 12y'3 - ( -3 0 + 2v'3)i

[12] JP>(2,3,3,4) p 7 13 19 31 37 7 7iv'3 37 37iv'3 _g - 13iv'3) + 19~v'3 jp(~,~,~, D 2- -2- 2 2 ¥ -31 -2- -2- 7_ + 7i./3 _g + 13i./3) 19 19i./3 _;)_?_ + 37i./3 Jp(~,~'~'~)2 2 2 2 2--2- -3 1 2 2 9 3 Jp(f'l,~,~,~) 5 - f + i( ¥ + 3v'3) -¥ + gy'3 + i(33 + f) 9 3 jp(f2, ~, ~)~, 5 + f + i( ¥ - 3v'3) -¥ - gy'3 + i(33 - f) 9 3 jp ( fi, ~, ~)~, 5 + f - i( ¥ - 3v'3) -¥ - gy'3 - i(33 - f) 9 3 jp ( H, ~,~, ~) 5 - ~ - i( ¥ + 3v'3) _g + gy'3 - i(33 + f)

iotl ::i:: -< Ul "' _, -C"l "' PHYSICS

4.5 Modular Forms 4. 7 Galois r e presentations and !1- Definition 4.5.1. The modular group ro(N) is defined as This section will define the notion of !1- hypersurfaces. Before we do this, we need

(21) D efinition 4. 7 .1. Cyclotomic field A cyclotomic field is a number field obt~ Definition 4.5.2. A of weight w, level N, and character x with respect to Q. The nth cyclotomic fie ld Q(µd) is c to ro(N) is a map f : 'H ---+ 4.6 Conformal Field Theory as Definition 4 .6.1. Conformal symmetry is a symmetry that is invariant under scaling (t,o:) = t. and under the special conformal transformations. where t · o:) = (to:o, · · · , to:.) Now for any

Conformal field theories provide toy models for genuinely interacting QFT's, and play a crucial role in string theory. to obtain a L-series with rationally integr Definition 4.6.2. Conformal field theory the L-function of the !1-motive of hypersl A conformal field theory (CFT) is a two dimensional quantum field theory that is invariant under conformal transformations. Ln(Xd,s) =II II l p etEOn A quantity that has proven use full in the string modularity analysis of elliptic Brieskorn-Pham curves, [6] [7] are the theta functions where Oo denotes the orbit of the eleme aim is to check whether the !1-motives of E sign(x )e2pir((k+2))x2-ky2) (24) a modular interpretation. -lxl

1 Here will be an example of how to comp defined by Kac and Peterson. These are related to the string function of the Ai l at values given the the Jacobi setion above, level k as 8 Lo(S , s) =II II 1 + J (25) p etEOn

It has been found that the following modular forms explain the modularity of extremal For our purposes here, we will not look K3 surfaces of Brieskorn-Pham type in a string theoretic way: term.

1 12 3 5 10 8 BL(q) = q1 (1- 2q - 2q + q4 + 2q - 2q5 - 2q6 - 2q8 - 2q9 + q + ... ) (26) Lo(S , s) = (1+(3+8v'2+i(-12+2v'2)

118 5 9 10 x(l + (3 - 8v'2- (12 + 2v'2)i)17-")(1 + Bi, 1 (q) = q (l - q- 2q2 + q3 + 2q5 - 2q + l- 2q - 2q + · · ·) (27) x (1 + (15 + 16v'2 + ( -12 + 20v'2)i)41 -s: x (1 + (15 - 16v'2 - (12 + 2ov'2)i)41-•)(

58 PHYSICS

4. 7 Galois representations and n-motives of CY varieties

N) is defined as This section will define the notion of n-motives for the general class of Calabi-yau hypersurfaces. Before we do this, we need a defintion.

S L2 ( Z) : c :::= 0mod N} (21) Definition 4. 7 .1. Cyclotomic field A cyclotomic field is a number field obtained by adjoining a complex root of unity ht w, level N, and character x with respect to Q. The nth cyclotomic field Q(µd) is obtained by ajoining a primitive nth root of ~ r half-plane such that for any T E 7-i and unity µd to Q . ic at the cusp The Jacobi sums ]pf act as algebraic in the cyclomtomic field Q(µd), gener­ CT+ d) w f(T) (22) ated by a primitive dth root of unity. The idea here is to achieve the necessary integral­ ity of the coefficients by considering the orbits of Jacobi sums defined by the action of 1wn to be important in a geometric context. the multiplicative group (71../ d7!..) x, with d the degree of the hypersurface. These orbits :ontext is the Dedekind eta function with lead to integral coefficients in the L-function since the multiplicative groups (71../ d7!..) x can be interpreted as Galois groups of cyclotomic fields Q(µd) (relavant concepts in [8], 00 II (1 - qn) (23) [9]). Jacobi sum orbits for the Galois group Gal(Q(µd) /Q) = (7!../d7!..)x are obtained n=l by defining (7!.. / d7!..) x x A~--> A~ (28) as (29) a symmetry that is invariant under scaling (t,a) = t ·a (mod 1) , 1tions. where t ·a) = (tao,··· , tas) Now for any a, one can use the orbit for genuinely interacting QFT's, and play Oa ={ ,BE A~l ,B=t ·a , tE(7!../ d7!..) x }, (30)

to obtain a L-series with rationally integral coefficients [2]. Further, this leads to the the L-function of then-motive of hypersurfaces Xd dimensional quantum field theory that is 1 (31) Ln(Xd, s) =II II 1-(-1)• 1 jp1(a)p-f•)1!f' p aEOn the string modularity analysis of elliptic 1 functions where On denotes the orbit of the element a that corresponds to the n-form. The aim is to check whether then-motives of extremal Brieskorn-Pham K3 surfaces admit 2 ;ign(x )e2pir((k+2))x -ky2) (24) a modular interpretation.

8 4. 8 Calculation of Ln ( 5 , s) 8 1 Here will be an example of how to compute Ln(X , s). Note f = 1, and using the lated to the string function of the Ai l at values given the the Jacobi setion above, we can rewrite the equation above as

8 8 2 Ln(S ,s)=II II l+jp(a)p-s +(jp(a)p- ) +··· etm(T) (25) P <>EOn 'f} 3(T) ' forms explain the modularity of extremal For our purposes here, we will not look at the squared or any other higher orderd ring theoretic way: term. 8 6 8 10 2q5 - 2q - 2q - 2q9 + q + .. . ) (26) Ln(S , s) = (1+(3+8v'2 + i(-12 + 2v'2))17-s)(l + (3 - 8v'2 + (1+12 + 2v'2)i)l 7-s)

8 8 15 - 2q5 + q6 - 2q9 - 2q10 + ... ) (27) X (1 + (3 - 8v'2 - (12 + 2v'2)i)l T )(1 + (3 + 8v'2 - (-12 + 2v'2)i)17- ) 8 8 X (1 + (15 + 16v'2 + (-12 + 20v'2)i)4l - )(1 + (15 - 16v'2 + (12 + 20v'2)i)4l - ) x (1 + (15 - 16v'2 - (12 + 20v'2)i)41- 8 )(1 + (15 + 16v'2 - (-12 + 20v'2)i)41-•) ...

59 PHYSICS

and similiar orbit products for the next prime fields that have n orbits, such as [6] R. Schimmrigk and S. Underwood, "1 p = 73, 89, 97, and 113. The Jacobi sums for Pi)OO have taken to long for the current formal field theories," ]. Geom. Phys., computer program I wrote to compute in a reasonable amount of time. Fortunately, [7] R. Schimmrigk and M. Lynker, "Ge01 knowing the values for primes less than 100 is enough infomation to find a modular Phys. , vol. 56, pp. 843- 863, 2006. form. Also, the p2 fields of interest have been shown to usually have Jacobi sums equal 2 to p . Multiplying the terms out above gives [8] J. Neukirch, theory. [9] W. Narkiewicz, Elementary and analy; . ag 12 a25 60 a49 12 as1 180 180 5 8 32 2004. Ln( ,s) = l +gs+ 178 + 25 8 + 4l5 49 8 - 73s + 8l8 + 89 5 - 97s +... ( ) [10] V. P. M. Lynker and R. Schimmrigk, where ~ means that the L function is known up the the terms given. This gives the tractor varieties," Nucl. Phys. , vol. B61 q expansion Nathan Vander Werf is a senior at IUSB v (33) physics.

Whats left next is to find a modular form associated to this q-expansion. It is expected that the modular form of a K3 surface to be of weight 2. It is speculated that exactly solvable models admit complex multiplication (CM) in the classical sense [10]. For 4 6 63 example, 8 admits CM with respect to the field Q(i) while 8 A 8 with respect to the field Q(iv'3), and 8 8 with respect to the field Q(iv'2) Further, it is expected that the level of the geometrically dervided modular form is divisible by the what is know 8 bad primes, or primes that divide 11· For 8 , the level should be some power of 2. For 8 20 it should be of the form 2a5b for some non-negative integers a, b. So far, the Jacobi sums of each surface up to p = 43 have been computed, which puts certain pieces of the !I-motive L-function together. Modular forms for the CY hypersurfaces with n = 4, 6 have allready been computed, and it has been shown [2] that they can be written in terms of modular forms constructed from the worldsheet.

5 Future Goals

The most general goal that is hoped to be accomplished by the end of the summer is to find a modular form associated to the !I-motive L-function of the surfaces mentioned earlier, and see if these can be written in terms of the modular forms derived from the underlying conformal field theory. A more efficient program for computing Jacobi sums may have to be written in order to computer the Jacobi sums of larger primes. Also, more computer programs will have to be written to test different aspects of the q-expansions.

R eferences

[1] B. Greene, The Elegant Universe. New York: Vintage Books, 2003. [2] R. Schimmrigk, "The langlands program and string modular k3 surfaces,'' Nucl. Phys. , vol. B771, pp. 143- 166, 2007. [3] A. Wiles, "Modular elliptic curves and format's last theorem," Ann., vol. 141, pp. 443-451, 1995.

[4] R. Schimmrigk, "Emergent spacetime from modula~· motives," Pending, 2008. [5] A. Grothendieck, "Formule de lefshetz et rationalite de fonction de l," 8eminaire Bourbaki, vol. 279, pp. 1- 15, 1964.

60 PHYSICS prime fields that have S1 orbits, such as (6] R. Schimmrigk and S. Underwood, "The shimua-taniyama conjecture and con­ )r pt,100 have taken to long for the current formal field theories," J. Geom. Phys. , vol. 48, pp. 169- 189, 2003. a reasonable amount of time. Fortunately, (7] R. Schimmrigk and M. Lynker, "Geometric kac-moody modularity," J. Geom. JO is enough infomation to find a modular Phys., vol. 56, pp. 843- 863, 2006. m shown to usually have Jacobi sums equal es [8] J. Neukirch, Algebraic n'IJ:mber theory. Springer, 1999. [9] W. Narkiewicz, Elementary and analytic theory of algebraic numbers. Springer, 149 _ ~ as1 180 _ 180 . . . ( ) 32 2004. 19s 73s + 81 8 + 89 8 97s + [10] V. P. M. Lynker and R. Schimmrigk, "Complex multiplication of black hole at­ wn up the the terms given. This gives the tractor varieties,'' Nucl. Phys., vol. B667, pp. 484- 504, 2003. Nathan Vander Werf is a senior at IUSB who is dual majoring in mathematics and q25 + 60q41 + a49Q49 - 12q73 + ... (33) physics . .sociated to this q-expansion. It is expected e of weight 2. It is speculated that exactly tion (CM) in the classical sense [10]. For 6 68 e field Q(i) while S A S with respect to Le field Q(iv'2) Further, it is expected that lular form is divisible by the what is know 3 , the level should be some power of 2. For non-negative integers a, b. p to p = 43 have been computed, which tion together. Modular forms for the CY )een computed, and it has been shown [2] .ar forms constructed from the worldsheet.

:complished by the end of the summer is to otive L-function of the surfaces mentioned terms of the modular forms derived from )re efficient program for computing Jacobi )mputer the Jacobi sums of larger primes. · be written to test different aspects of the

r York: Vintage Books, 2003.

Lill and string modular k3 surfaces," Nucl. d format's last theorem," Ann., vol. 141, from modular motives," Pending, 2008. et rationalite de fonction de l," Seminaire

61