# Ω Ω /ω. To these, one wants to add a fourth condition that arises from physics, what is known as the anomaly cancellation, namely that

Size: px
Start display at page:

Download "Ω Ω /ω. To these, one wants to add a fourth condition that arises from physics, what is known as the anomaly cancellation, namely that"

Transcription

1 String theory and balanced metrics One of the main motivations for considering balanced metrics, in addition to the considerations already mentioned, has to do with the theory of what are known as heterotic strings with super-symmetry. The physics here is complicated to explain, having to do with accommodating both bosons and fermions simultaneously. But the mathematics involved has a reasonably straight-forward expression, though it is far from obvious how to find interesting examples of the type of system involved. To jump immediately to the description of the structure one would like to find(we shall return later to motivational matters) we would like to find a complex manifold M of complex dimension 3, a vector bundle V M with a Hermitian Yang Mills metric h and associated Hermitian type (1,0) connection with curvature F h (as in our previous notation) and a holonorphic 3-form on M, denoted by Ω, which satisfy several equations: 1 F h,0 = F h,0 =0 (this is equivalent to the bundle being holomorphic and the connection being Hermitian) F h ω =0 (equivalently, c 1 (V)=0) 3 d Ω = 0 where Ω ω ω ω = by definition 3 Ω Ω /ω. (In previous cases, where one had Ricci flatness, one was looking for an Ω with constant norm in condition 3, so that the equation became just that of a balanced metric. ) To these, one wants to add a fourth condition that arises from physics, what is known as the anomaly cancellation, namely that 4 1 ω = trrω Rω trfh Fh Solving equations 1 and amounts just to the Hermitian Yang Mills situation that we have discussed before. Namely, it can be done is ω is a balanced metric and V is holomorphic and stable with respect to ω. In order to deal with equation 3, leaving aside for the time being condition 4, one could 1/ think of introducing a new two-form ω = Ω ω. Then ω has dω = 0. So one can investigate first just the condition that for some (1,1) form ω, dω =0. There are actually several constructions that can be used here. We consider one that involved branch covers. 1

2 Suppose L M is a line bundle with a nontrivial holomorphic section s and let D be the divisor determined by s=0. And suppose further that one has for some reason or another that c 1 (L) is divisible by a positive integer m >1 (in the Z cohomology, that is the quotient is still an integral class). Then for some line bundle L', L is the mth power of L'. This corresponds to a map on the fibres (x,z) (x, z m ). [Note that if L has a topological mth root in this sense, then it has a holomorphic mth root, since the only obstruction to taking the mth roots of the transition functions coherently is topological. Thus the divisibility of the first Chern class of L is the only obstruction.] Let F:L' L be this mth power fibre map. Then in the total space of L', one can consider F -1 (s(m)), where s(m) denotes the image under s in the total space of L of the base manifold M. Here we are interpreting s as a map M L in the usual way. The set F -1 (s(m)) is a branched cover over M with m "sheets" (m preimages of a generic point in M) and m-fold branching over the zeroes of the section s. The importance of this construction for us is that the branch-cover manifold admits a balanced metric(assuming that M does). For this, note that we can regard the branch cover as a submanifold of LxL' by sending a point in the branch cover (x,s), s in the fibre of L' over x to (x, s, s m ) in LxL' over M. We put a metric on this submanifold as follows: write for each open set in a trivializing cover U α, z = hα ( z ). Considering z as a function of all variables, we can consider z. This may not be positive but is positive in fibre directions, if perhaps not in base manifold directions. (In the Calabi Conjecture negative curvature case, this is actually positive in base directions also and is a positive definite metric in all variable, in fact.) Now consider the sum, for some (large positive) constant C : 1 * ( z ) n n + Cπ ω 1 where π is the projection onto M. Since we are working on a compact set(a compact submanifold, the image of M), we can choose C so large that this is in fact a metric on the subvariety. And similarly on LxL', we can use n 1 n 1 ( z ) ( z ) C π * ω n to get a positive form on the (compact) submanifold L of LxL' already discussed (which is indeed the branched cover). Note here that the concept of positivity of a (p,p) form makes sense without respect to the overall dimension. So we can restrict to F -1 (s(m)), and noting that all forms are closed, we get a balanced metric on the branch cover F -1 (s(m)).

3 Note that in fact we constructed not the Kahler form itself of the balanced metric but rather a positive closed (n-1,n-1) form (on the n dimensional branch cover) directly. But as noted earlier, every such (n-1, n-1) form arises as the (n-1) st wedge product power of a uniquely determined positive (1,1) form, point by point, by linear algebra(cf the earlier discussion of balanced metrics). In this situation if D is nonsingular then the branched covering is also nonsingular. Unfortunately, it often arises that one wishes to deal with singularities in the divisor D and one then needs some resolution work. This should usually be doable, however. Now let us return to the situation of twistor space Tw(M) for a conformally self-dual four manifold M, as discussed earlier. In this case, one has a fibration CP 1 Tw(M) M, Tw(M) is a complex manifold(from the self duality) and Tw(M) is balanced in its natural metric as discussed earlier also. However, in general c 1 (M) is nonzero so that one cannot find nontrivial holomorphic 3- forms without zeroes. (Recall that we wanted to use the norm of a holomorphic form as a multiplication factor to get a new metric so one needs to have a holomorphic form and have that the form is nowhere 0, too.) To deal with this, we might try to find a divisor in twistor space that would function to give a branched cover with respect to which the construction can be carried out. In short, we hope to "kill" (trivialize) the canonical bundle by a branched cover. This is related to an idea in Riemann surface theory. By forming suitable branch covers, one can introduce holomorphic forms where none existed before. Recall the usual form of this idea: Suppose N is a compact Riemann surface and let K -1 be the anti-canonical bundle (of local sections of the holomorphic tangent bundle. In higher dimensions this would be the n times wedge product of the holomorphic tangent space, n= complex dimension). Then if s is a meromorphic section of K -1, then it has locally the form f (z) ( / z) where f is a meromorphic function(similar in higher dimensions). So s -1 has a pole along the divisor D, the anticanonical divisor defined by the vanishing locus of s, the section of the anitcanonical bundle. We want to get rid of the pole. And the thing to do is to pass to a branch cover over the poles of s -1. 3

4 But even for Riemann surfaces, this situation is more subtle than at first appears. Imagine a branch covering of order m, say defined in local coordinates by u m =z, where z is the base, u is in the cover. Then the coordinate vector field / u pushed down (at a given point where u is not 0) becomes a multiple of / z by the complex number mu m-1. This goes to 0 as u goes to 0 but it has "fractional order" as a function of (the magnitude) of z, 1 (1/ m) namely it has order of magnitude z. So one picks up some advantage but only in terms of fractional order. In bundle terms, one is really gaining only in a bundle a power of which is the given anti-canonical bundle. This somewhat 19 th century description actually works. Think of the anti-canonical bundle of CP 1. A holomorphic section of this(and there are some) has total degree, that is the sum of the orders of the zeroes is. If one is looking at two -fold branch covers, then according to the above "heurisitic", each (two -fold) branch point account for a gain of 1-(1/)= ½. So one expects to need four such branch points to get a branched cover with a holomorphic nowhere zero (1,0) form. This is the right answer! By the Riemann Hurwitz formula, such a cover has Euler characteristic x 4x1=0, where the terms are respectively (Euler characteristic of the base) and sum of branching order -1 over all branch points. So the branched cover is a torus and the required holomorphic (1,0) form exists. Similarly, a triple (three-sheet) covering with three triple branch point will give a branched cover with Euler characteristic 0 and thus with a trivial canonical bundle. Here in the heuristic above each branch gives an advantage of 1-(1/3) so that three of such branch points gets rid of 3(/3) of the total order of zero for the holomorphic tangent bundle section, and again one sees that one is in the trivial canonical bundle situation in the branched cover. Clearly, this is a quite subtle matter to deal with in general cases. But the general principle applies that one gets advantage passing to branched covers but in fractional amounts depending on the branching order, the fractions corresponding in effect to "roots" (locally) of the bundle. In higher dimensions, an interesting set of examples are the K3 surfaces. These Kahler surfaces all admit Ricci-flat metrics by Yau's solution of the Calabi Conjecture in case of first Chern class 0. These manifolds are anti-self-dual automatically with respect to this metric(a Kahler metric is anti-self-dual if and only if its scalar curvature vanishes). By the Bochner technique argument discussed earlier, any holomorphic (,0) form in this case is actually parallel with respect to the Kahler metric connection. And such a form Ω can be chosen to have Ω Ω= ω. This shows the important observation that the holonomy reduces from U() to SU()! since the form Ω is parallel from which one sees that the holonomy action on the volume form is not just of absolute value 1(as is inevitable from parallel translation being an isometry) but is actually equal exactly to 1. 4

5 Now in this case, one can find additional complex structures as follows: The choice of T 0,1 determines an almost complex structure. For a form A not zero at a point and of type (,0), one has that T 0,1 is determined via {X: i X A=0}. Now we are in possession of three two-forms, namely Ω, Ω, and ω. So we can form new almost complex structures(which are actually integrable) by chosing various possibilities for A. Namely, we can build in this situation a family of integrable almost complex structures by taking A= aω+ bω + cω where a +b +c =1, a,b,c, constants. This is still closed, and because all three are parallel, this form is never 0 if it is nonzero at one point. This gives a new integrable complex structure so one has a sphere of integrable almost structures, corresponding to the sphere fibre at one point in the twistor space over M, translated to every point by parallelism. And again by parallelism and the construction, all these are compatible with the metric g. Note that in this situation, one has that Tw(M) is smoothly(not biholomorphically) the product of M and CP 1, where the coordinates a,b,c are the coordinates on CP 1 identified as usual with the unit sphere in R 3. Now there are three sections here,thinking of Tw(M) as a complex bundle over M with CP 1 fibres. So as we have been discussing, we can form a branch covering over the divisors corresponding to the images of these sections in Tw(M). This produces an elliptic fibre space over the K3 surface M: the fibres are elliptic curves that arise from the branching over the three points in each CP 1 fibre over a point in M. (As we observed earlier, a three-sheeted cover of CP 1 branched over three points, each with branching order 3, is a torus as a Riemann surface, that is, an elliptic curve). This new space, the elliptic curve fibration over the K3 surface admits a holomorphic 3-form. Thus one obtains a large class of compact complex manifolds of dimension 3 with c 1 =0 and with a holomorphic 3-form. This is an interesting situation to consider. There are other interesting ideas for constructing examples of the type we are interested in beyond the idea we have been discussing, of branched coverings arising from twistor spaces. One of these is a generalization of the Calabi-Eckmann construction. Recall that in the classical Calabi-Eckmann construction, one considers, for example, the fibration S 1 x S 1 S 3 x S 3 CP 1 x CP 1 and the complex structure on the product of the 3-spheres is built from that on the product of the projective spaces and the complex structure on the torus fibres. Now Goldstein and some physicists suggested that one could look at the following generalization: 5

6 Choose two line bundles(which could be the same one) L 1, L over a K3 surface M, or an arbitrary Calabi-Yau manifold. We want to have here that the (first) Chern form c ( ) 0 1 Li ω = so as to have an anti-self-dual Chern form. There is in fact a large set of line bundles for which this will happen. This follows from dimension considerations. The (complex coefficient) cohomology of the K3 surface is dimension (over C) =. Only three of these dimensions are generated by the Kahler form and the holomorphic (,0) parallel form and its complex conjugate. The rest of these are generated by forms which are type (1,1) and have wedge product with ω=0. Now take the principal circle bundles associated to two line bundles(possibly equal) of the sort indicated. Then the torus bundle over M attached to this situation has a natural complex structure(details of this will be covered later). Robert E. Greene 6

### As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing).

An Interlude on Curvature and Hermitian Yang Mills As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing). Suppose we wanted

### Hermitian vs. Riemannian Geometry

Hermitian vs. Riemannian Geometry Gabe Khan 1 1 Department of Mathematics The Ohio State University GSCAGT, May 2016 Outline of the talk Complex curves Background definitions What happens if the metric

### The Strominger Yau Zaslow conjecture

The Strominger Yau Zaslow conjecture Paul Hacking 10/16/09 1 Background 1.1 Kähler metrics Let X be a complex manifold of dimension n, and M the underlying smooth manifold with (integrable) almost complex

### Some new torsional local models for heterotic strings

Some new torsional local models for heterotic strings Teng Fei Columbia University VT Workshop October 8, 2016 Teng Fei (Columbia University) Strominger system 10/08/2016 1 / 30 Overview 1 Background and

### Linear connections on Lie groups

Linear connections on Lie groups The affine space of linear connections on a compact Lie group G contains a distinguished line segment with endpoints the connections L and R which make left (resp. right)

### Two simple ideas from calculus applied to Riemannian geometry

Calibrated Geometries and Special Holonomy p. 1/29 Two simple ideas from calculus applied to Riemannian geometry Spiro Karigiannis karigiannis@math.uwaterloo.ca Department of Pure Mathematics, University

### Pietro Fre' SISSA-Trieste. Paolo Soriani University degli Studi di Milano. From Calabi-Yau manifolds to topological field theories

From Calabi-Yau manifolds to topological field theories Pietro Fre' SISSA-Trieste Paolo Soriani University degli Studi di Milano World Scientific Singapore New Jersey London Hong Kong CONTENTS 1 AN INTRODUCTION

### Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture)

Building Geometric Structures: Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture) A geometric structure on a manifold is a cover

### Constructing compact 8-manifolds with holonomy Spin(7)

Constructing compact 8-manifolds with holonomy Spin(7) Dominic Joyce, Oxford University Simons Collaboration meeting, Imperial College, June 2017. Based on Invent. math. 123 (1996), 507 552; J. Diff. Geom.

### HYPERKÄHLER MANIFOLDS

HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly

### Vanishing theorems and holomorphic forms

Vanishing theorems and holomorphic forms Mihnea Popa Northwestern AMS Meeting, Lansing March 14, 2015 Holomorphic one-forms and geometry X compact complex manifold, dim C X = n. Holomorphic one-forms and

### Geometry of Twistor Spaces

Geometry of Twistor Spaces Claude LeBrun Simons Workshop Lecture, 7/30/04 Lecture Notes by Jill McGowan 1 Twistor Spaces Twistor spaces are certain complex 3-manifolds which are associated with special

### Stable bundles on CP 3 and special holonomies

Stable bundles on CP 3 and special holonomies Misha Verbitsky Géométrie des variétés complexes IV CIRM, Luminy, Oct 26, 2010 1 Hyperkähler manifolds DEFINITION: A hyperkähler structure on a manifold M

### The geometry of Landau-Ginzburg models

Motivation Toric degeneration Hodge theory CY3s The Geometry of Landau-Ginzburg Models January 19, 2016 Motivation Toric degeneration Hodge theory CY3s Plan of talk 1. Landau-Ginzburg models and mirror

### Holomorphic line bundles

Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

### ABEL S THEOREM BEN DRIBUS

ABEL S THEOREM BEN DRIBUS Abstract. Abel s Theorem is a classical result in the theory of Riemann surfaces. Important in its own right, Abel s Theorem and related ideas generalize to shed light on subjects

### Definition We say that a topological manifold X is C p if there is an atlas such that the transition functions are C p.

13. Riemann surfaces Definition 13.1. Let X be a topological space. We say that X is a topological manifold, if (1) X is Hausdorff, (2) X is 2nd countable (that is, there is a base for the topology which

### LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS

LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS JON WOLFSON Abstract. We show that there is an integral homology class in a Kähler-Einstein surface that can be represented by a lagrangian twosphere

### The Canonical Sheaf. Stefano Filipazzi. September 14, 2015

The Canonical Sheaf Stefano Filipazzi September 14, 015 These notes are supposed to be a handout for the student seminar in algebraic geometry at the University of Utah. In this seminar, we will go over

### K-stability and Kähler metrics, I

K-stability and Kähler metrics, I Gang Tian Beijing University and Princeton University Let M be a Kähler manifold. This means that M be a complex manifold together with a Kähler metric ω. In local coordinates

### Structure theorems for compact Kähler manifolds

Structure theorems for compact Kähler manifolds Jean-Pierre Demailly joint work with Frédéric Campana & Thomas Peternell Institut Fourier, Université de Grenoble I, France & Académie des Sciences de Paris

### Algebraic geometry over quaternions

Algebraic geometry over quaternions Misha Verbitsky November 26, 2007 Durham University 1 History of algebraic geometry. 1. XIX centrury: Riemann, Klein, Poincaré. Study of elliptic integrals and elliptic

### INTRODUCTION TO THE HODGE CONJECTURE

INTRODUCTION TO THE HODGE CONJECTURE JIM BROWN Abstract. These are notes from a talk introducing the Hodge conjecture to undergraduates attending the 2009 Clemson REU. The description given here of the

### Holonomy groups. Thomas Leistner. School of Mathematical Sciences Colloquium University of Adelaide, May 7, /15

Holonomy groups Thomas Leistner School of Mathematical Sciences Colloquium University of Adelaide, May 7, 2010 1/15 The notion of holonomy groups is based on Parallel translation Let γ : [0, 1] R 2 be

### η = (e 1 (e 2 φ)) # = e 3

Research Statement My research interests lie in differential geometry and geometric analysis. My work has concentrated according to two themes. The first is the study of submanifolds of spaces with riemannian

### INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD

INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:

### RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY

RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY NICK MCCLEEREY 0. Complex Differential Forms Consider a complex manifold X n (of complex dimension n) 1, and consider its complexified tangent bundle T C

### CANONICAL METRICS AND STABILITY OF PROJECTIVE VARIETIES

CANONICAL METRICS AND STABILITY OF PROJECTIVE VARIETIES JULIUS ROSS This short survey aims to introduce some of the ideas and conjectures relating stability of projective varieties to the existence of

### Hyperkähler geometry lecture 3

Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1 Broom Bridge Here as he walked by on the 16th of October 1843

### Heterotic Geometry and Fluxes

Heterotic Geometry and Fluxes Li-Sheng Tseng Abstract. We begin by discussing the question, What is string geometry? We then proceed to discuss six-dimensional compactification geometry in heterotic string

### Homological mirror symmetry via families of Lagrangians

Homological mirror symmetry via families of Lagrangians String-Math 2018 Mohammed Abouzaid Columbia University June 17, 2018 Mirror symmetry Three facets of mirror symmetry: 1 Enumerative: GW invariants

### New Aspects of Heterotic Geometry and Phenomenology

New Aspects of Heterotic Geometry and Phenomenology Lara B. Anderson Harvard University Work done in collaboration with: J. Gray, A. Lukas, and E. Palti: arxiv: 1106.4804, 1202.1757 J. Gray, A. Lukas and

### Intro to Geometry and Topology via G Physics and G 2 -manifolds. Bobby Samir Acharya. King s College London. and ICTP Trieste Ψ(1 γ 5 )Ψ

Intro to Geometry and Topology via G 2 10.07.2014 Physics and G 2 -manifolds Bobby Samir Acharya King s College London. µf µν = j ν dϕ = d ϕ = 0 and ICTP Trieste Ψ(1 γ 5 )Ψ The Rich Physics-Mathematics

### UNIVERSAL UNFOLDINGS OF LAURENT POLYNOMIALS AND TT STRUCTURES. Claude Sabbah

UNIVERSAL UNFOLDINGS OF LAURENT POLYNOMIALS AND TT STRUCTURES AUGSBURG, MAY 2007 Claude Sabbah Introduction Let f : (C ) n C be a Laurent polynomial, that I assume to be convenient and non-degenerate,

### Kähler manifolds and variations of Hodge structures

Kähler manifolds and variations of Hodge structures October 21, 2013 1 Some amazing facts about Kähler manifolds The best source for this is Claire Voisin s wonderful book Hodge Theory and Complex Algebraic

### Conjectures in Kahler geometry

Conjectures in Kahler geometry S.K. Donaldson Abstract. We state a general conjecture about the existence of Kahler metrics of constant scalar curvature, and discuss the background to the conjecture 1.

### Higgs Bundles and Character Varieties

Higgs Bundles and Character Varieties David Baraglia The University of Adelaide Adelaide, Australia 29 May 2014 GEAR Junior Retreat, University of Michigan David Baraglia (ADL) Higgs Bundles and Character

### Knot Homology from Refined Chern-Simons Theory

Knot Homology from Refined Chern-Simons Theory Mina Aganagic UC Berkeley Based on work with Shamil Shakirov arxiv: 1105.5117 1 the knot invariant Witten explained in 88 that J(K, q) constructed by Jones

### Topological reduction of supersymmetric gauge theories and S-duality

Topological reduction of supersymmetric gauge theories and S-duality Anton Kapustin California Institute of Technology Topological reduction of supersymmetric gauge theories and S-duality p. 1/2 Outline

### Perspective on Geometric Analysis: Geometric Structures III

Perspective on Geometric Analysis: Geometric Structures III Shing-Tung Yau Harvard University Talk in UCLA 1 When the topological method of surgery and gluing fails, we have to find a method that does

### Lecture XI: The non-kähler world

Lecture XI: The non-kähler world Jonathan Evans 2nd December 2010 Jonathan Evans () Lecture XI: The non-kähler world 2nd December 2010 1 / 21 We ve spent most of the course so far discussing examples of

### Calabi-Yau Fourfolds with non-trivial Three-Form Cohomology

Calabi-Yau Fourfolds with non-trivial Three-Form Cohomology Sebastian Greiner arxiv: 1512.04859, 1702.03217 (T. Grimm, SG) Max-Planck-Institut für Physik and ITP Utrecht String Pheno 2017 Sebastian Greiner

### Riemann Surfaces and Algebraic Curves

Riemann Surfaces and Algebraic Curves JWR Tuesday December 11, 2001, 9:03 AM We describe the relation between algebraic curves and Riemann surfaces. An elementary reference for this material is [1]. 1

### COMPLEX ALGEBRAIC SURFACES CLASS 4

COMPLEX ALGEBRAIC SURFACES CLASS 4 RAVI VAKIL CONTENTS 1. Serre duality and Riemann-Roch; back to curves 2 2. Applications of Riemann-Roch 2 2.1. Classification of genus 2 curves 3 2.2. A numerical criterion

### Topics in Geometry: Mirror Symmetry

MIT OpenCourseWare http://ocw.mit.edu 18.969 Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIRROR SYMMETRY:

### Lecture VI: Projective varieties

Lecture VI: Projective varieties Jonathan Evans 28th October 2010 Jonathan Evans () Lecture VI: Projective varieties 28th October 2010 1 / 24 I will begin by proving the adjunction formula which we still

### Mathematical Research Letters 2, (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS. Shuguang Wang

Mathematical Research Letters 2, 305 310 (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS Shuguang Wang Abstract. It is shown that the quotients of Kähler surfaces under free anti-holomorphic involutions

### Math. Res. Lett. 13 (2006), no. 1, c International Press 2006 ENERGY IDENTITY FOR ANTI-SELF-DUAL INSTANTONS ON C Σ.

Math. Res. Lett. 3 (2006), no., 6 66 c International Press 2006 ENERY IDENTITY FOR ANTI-SELF-DUAL INSTANTONS ON C Σ Katrin Wehrheim Abstract. We establish an energy identity for anti-self-dual connections

### Heterotic Vector Bundles, Deformations and Geometric Transitions

Heterotic Vector Bundles, Deformations and Geometric Transitions Lara B. Anderson Harvard University Work done in collaboration with: J. Gray, A. Lukas and B. Ovrut arxiv: 1010.0255, 1102.0011, 1107.5076,

### arxiv:hep-th/ v3 4 Nov 2006

Imperial/TP/2006/JG/02 New Supersymmetric AdS 3 Solutions arxiv:hep-th/0608055v3 4 Nov 2006 Jerome P. Gauntlett, Oisín A. P. Mac Conamhna, Toni Mateos and Daniel Waldram Theoretical Physics Group, Blackett

### The Calabi Conjecture

The Calabi Conjecture notes by Aleksander Doan These are notes to the talk given on 9th March 2012 at the Graduate Topology and Geometry Seminar at the University of Warsaw. They are based almost entirely

### CALIBRATED FIBRATIONS ON NONCOMPACT MANIFOLDS VIA GROUP ACTIONS

DUKE MATHEMATICAL JOURNAL Vol. 110, No. 2, c 2001 CALIBRATED FIBRATIONS ON NONCOMPACT MANIFOLDS VIA GROUP ACTIONS EDWARD GOLDSTEIN Abstract In this paper we use Lie group actions on noncompact Riemannian

### Poles of Instantons and Jumping Lines of Algebraic Vector Bundles on P

No. 5] Proc. Japan Acad., 5, Ser. A (1979) 185 Poles of Instantons and Jumping Lines of Algebraic Vector Bundles on P By Motohico ]V[ULASE Research Institute for Mathematical Science.s, Kyoto University

### Compact 8-manifolds with holonomy Spin(7)

Compact 8-manifolds with holonomy Spin(7) D.D. Joyce The Mathematical Institute, 24-29 St. Giles, Oxford, OX1 3LB. Published as Invent. math. 123 (1996), 507 552. 1 Introduction In Berger s classification

### arxiv:alg-geom/ v1 29 Jul 1993

Hyperkähler embeddings and holomorphic symplectic geometry. Mikhail Verbitsky, verbit@math.harvard.edu arxiv:alg-geom/9307009v1 29 Jul 1993 0. ntroduction. n this paper we are studying complex analytic

### LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS

LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS Contents 1. Almost complex manifolds 1. Complex manifolds 5 3. Kähler manifolds 9 4. Dolbeault cohomology 11 1. Almost complex manifolds Almost complex structures.

### Knots and Mirror Symmetry. Mina Aganagic UC Berkeley

Knots and Mirror Symmetry Mina Aganagic UC Berkeley 1 Quantum physics has played a central role in answering the basic question in knot theory: When are two knots distinct? 2 Witten explained in 88, that

### Generalized Global Symmetries

Generalized Global Symmetries Anton Kapustin Simons Center for Geometry and Physics, Stony Brook April 9, 2015 Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries

### Lecture 8: More characteristic classes and the Thom isomorphism

Lecture 8: More characteristic classes and the Thom isomorphism We begin this lecture by carrying out a few of the exercises in Lecture 1. We take advantage of the fact that the Chern classes are stable

### Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

### Useful theorems in complex geometry

Useful theorems in complex geometry Diego Matessi April 30, 2003 Abstract This is a list of main theorems in complex geometry that I will use throughout the course on Calabi-Yau manifolds and Mirror Symmetry.

### Hodge Theory of Maps

Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1 - Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar

### THE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS

THE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS AILANA FRASER AND JON WOLFSON Abstract. In this paper we study the topology of compact manifolds of positive isotropic

### Theorem 2. Let n 0 3 be a given integer. is rigid in the sense of Guillemin, so are all the spaces ḠR n,n, with n n 0.

This monograph is motivated by a fundamental rigidity problem in Riemannian geometry: determine whether the metric of a given Riemannian symmetric space of compact type can be characterized by means of

### Minimal surfaces in quaternionic symmetric spaces

From: "Geometry of low-dimensional manifolds: 1", C.U.P. (1990), pp. 231--235 Minimal surfaces in quaternionic symmetric spaces F.E. BURSTALL University of Bath We describe some birational correspondences

### Techniques for exact calculations in 4D SUSY gauge theories

Techniques for exact calculations in 4D SUSY gauge theories Takuya Okuda University of Tokyo, Komaba 6th Asian Winter School on Strings, Particles and Cosmology 1 First lecture Motivations for studying

### Topics on Geometric Analysis

Topics on Geometric Analysis Shing-Tung Yau Harvard University C.C. Hsiung Lecture Lehigh University May 28, 2010 I am proud to give the first C.C. Hsiung Lecture in memory of an old friend, who had many

### Mirror Symmetry: Introduction to the B Model

Mirror Symmetry: Introduction to the B Model Kyler Siegel February 23, 2014 1 Introduction Recall that mirror symmetry predicts the existence of pairs X, ˇX of Calabi-Yau manifolds whose Hodge diamonds

### Let X be a topological space. We want it to look locally like C. So we make the following definition.

February 17, 2010 1 Riemann surfaces 1.1 Definitions and examples Let X be a topological space. We want it to look locally like C. So we make the following definition. Definition 1. A complex chart on

### BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS

BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS SHOO SETO Abstract. These are the notes to an expository talk I plan to give at MGSC on Kähler Geometry aimed for beginning graduate students in hopes to motivate

### Holonomy groups. Thomas Leistner. Mathematics Colloquium School of Mathematics and Physics The University of Queensland. October 31, 2011 May 28, 2012

Holonomy groups Thomas Leistner Mathematics Colloquium School of Mathematics and Physics The University of Queensland October 31, 2011 May 28, 2012 1/17 The notion of holonomy groups is based on Parallel

### Non-Kähler Calabi-Yau Manifolds

Non-Kähler Calabi-Yau Manifolds Shing-Tung Yau Harvard University String-Math 2011 University of Pennsylvania June 6, 2011 String and math have had a very close interaction over the past thirty years.

### Riemannian Curvature Functionals: Lecture III

Riemannian Curvature Functionals: Lecture III Jeff Viaclovsky Park City Mathematics Institute July 18, 2013 Lecture Outline Today we will discuss the following: Complete the local description of the moduli

### A Joint Adventure in Sasakian and Kähler Geometry

A Joint Adventure in Sasakian and Kähler Geometry Charles Boyer and Christina Tønnesen-Friedman Geometry Seminar, University of Bath March, 2015 2 Kähler Geometry Let N be a smooth compact manifold of

### Conjectures on counting associative 3-folds in G 2 -manifolds

in G 2 -manifolds Dominic Joyce, Oxford University Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics, First Annual Meeting, New York City, September 2017. Based on arxiv:1610.09836.

### APPROXIMATE YANG MILLS HIGGS METRICS ON FLAT HIGGS BUNDLES OVER AN AFFINE MANIFOLD. 1. Introduction

APPROXIMATE YANG MILLS HIGGS METRICS ON FLAT HIGGS BUNDLES OVER AN AFFINE MANIFOLD INDRANIL BISWAS, JOHN LOFTIN, AND MATTHIAS STEMMLER Abstract. Given a flat Higgs vector bundle (E,, ϕ) over a compact

### Topic: First Chern classes of Kähler manifolds Mitchell Faulk Last updated: April 23, 2016

Topic: First Chern classes of Kähler manifolds itchell Faulk Last updated: April 23, 2016 We study the first Chern class of various Kähler manifolds. We only consider two sources of examples: Riemann surfaces

### LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS

LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS WEIMIN CHEN, UMASS, SPRING 07 1. Basic elements of J-holomorphic curve theory Let (M, ω) be a symplectic manifold of dimension 2n, and let J J (M, ω) be

### RICCI CURVATURE: SOME RECENT PROGRESS A PROGRESS AND OPEN QUESTIONS

RICCI CURVATURE: SOME RECENT PROGRESS AND OPEN QUESTIONS Courant Institute September 9, 2016 1 / 27 Introduction. We will survey some recent (and less recent) progress on Ricci curvature and mention some

### arxiv: v2 [math.dg] 23 Dec 2016

ON NON-KÄHLER COMPACT COMPLE MANIFOLDS WITH BALANCED AND ASTHENO-KÄHLER METRICS ADELA LATORRE AND LUIS UGARTE ariv:1608.06744v2 [math.dg] 23 Dec 2016 Abstract. In this note we construct, for every n 4,

### Compact Riemannian manifolds with exceptional holonomy

Unspecified Book Proceedings Series Compact Riemannian manifolds with exceptional holonomy Dominic Joyce published as pages 39 65 in C. LeBrun and M. Wang, editors, Essays on Einstein Manifolds, Surveys

### Complex manifolds, Kahler metrics, differential and harmonic forms

Complex manifolds, Kahler metrics, differential and harmonic forms Cattani June 16, 2010 1 Lecture 1 Definition 1.1 (Complex Manifold). A complex manifold is a manifold with coordinates holomorphic on

### Transparent connections

The abelian case A definition (M, g) is a closed Riemannian manifold, d = dim M. E M is a rank n complex vector bundle with a Hermitian metric (i.e. a U(n)-bundle). is a Hermitian (i.e. metric) connection

### k=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula

20 VASILY PESTUN 3. Lecture: Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim

### Metrics and Holonomy

Metrics and Holonomy Jonathan Herman The goal of this paper is to understand the following definitions of Kähler and Calabi-Yau manifolds: Definition. A Riemannian manifold is Kähler if and only if it

### The structure of algebraic varieties

The structure of algebraic varieties János Kollár Princeton University ICM, August, 2014, Seoul with the assistance of Jennifer M. Johnson and Sándor J. Kovács (Written comments added for clarity that

### Lecture 1: Introduction

Lecture 1: Introduction Jonathan Evans 20th September 2011 Jonathan Evans () Lecture 1: Introduction 20th September 2011 1 / 12 Jonathan Evans () Lecture 1: Introduction 20th September 2011 2 / 12 Essentially

### ERRATUM TO AFFINE MANIFOLDS, SYZ GEOMETRY AND THE Y VERTEX

ERRATUM TO AFFINE MANIFOLDS, SYZ GEOMETRY AND THE Y VERTEX JOHN LOFTIN, SHING-TUNG YAU, AND ERIC ZASLOW 1. Main result The purpose of this erratum is to correct an error in the proof of the main result

### Modern Geometric Structures and Fields

Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island Preface

### An Algorithmic Approach to Heterotic Compactification

An Algorithmic Approach to Heterotic Compactification Lara B. Anderson Department of Physics, University of Pennsylvania, and Institute for Advanced Study, Princeton Work done in collaboration with: LBA,

### Each is equal to CP 1 minus one point, which is the origin of the other: (C =) U 1 = CP 1 the line λ (1, 0) U 0

Algebraic Curves/Fall 2015 Aaron Bertram 1. Introduction. What is a complex curve? (Geometry) It s a Riemann surface, that is, a compact oriented twodimensional real manifold Σ with a complex structure.

### Geometric Structures in Mathematical Physics Non-existence of almost complex structures on quaternion-kähler manifolds of positive type

Geometric Structures in Mathematical Physics Non-existence of almost complex structures on quaternion-kähler manifolds of positive type Paul Gauduchon Golden Sands, Bulgaria September, 19 26, 2011 1 Joint

### ALF spaces and collapsing Ricci-flat metrics on the K3 surface

ALF spaces and collapsing Ricci-flat metrics on the K3 surface Lorenzo Foscolo Stony Brook University Simons Collaboration on Special Holonomy Workshop, SCGP, Stony Brook, September 8 2016 The Kummer construction

### The exceptional holonomy groups and calibrated geometry

Proceedings of 12 th Gökova Geometry-Topology Conference pp. 110 139 Published online at GokovaGT.org The exceptional holonomy groups and calibrated geometry Dominic Joyce Dedicated to the memory of Raoul

### David R. Morrison. String Phenomenology 2008 University of Pennsylvania 31 May 2008

: : University of California, Santa Barbara String Phenomenology 2008 University of Pennsylvania 31 May 2008 engineering has been a very successful approach to studying string vacua, and has been essential

### Twistor Strings, Gauge Theory and Gravity. Abou Zeid, Hull and Mason hep-th/

Twistor Strings, Gauge Theory and Gravity Abou Zeid, Hull and Mason hep-th/0606272 Amplitudes for YM, Gravity have elegant twistor space structure: Twistor Geometry Amplitudes for YM, Gravity have elegant

### Porteous s Formula for Maps between Coherent Sheaves

Michigan Math. J. 52 (2004) Porteous s Formula for Maps between Coherent Sheaves Steven P. Diaz 1. Introduction Recall what the Thom Porteous formula for vector bundles tells us (see [2, Sec. 14.4] for

### How to recognize a conformally Kähler metric

How to recognize a conformally Kähler metric Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge MD, Paul Tod arxiv:0901.2261, Mathematical Proceedings of