Kähler Potential of Moduli Space. of Calabi-Yau d-fold embedded in CP d+1

Size: px
Start display at page:

Download "Kähler Potential of Moduli Space. of Calabi-Yau d-fold embedded in CP d+1"

Transcription

1 March 2000 hep-th/ Kähler Potential of Moduli Space arxiv:hep-th/000364v 20 Mar 2000 of Calabi-Yau d-fold embedded in CP d+ Katsuyuki Sugiyama Department of Fundamental Sciences Faculty of Integrated Human Studies, Kyoto University Yoshida-ihon-Matsu cho, Sakyo-ku, Kyoto , Japan ABSTRACT We study a Kähler potential K of a one parameter family of Calabi-Yau d-fold embedded in CP d+. By comparing results of the topological B-model and the data of the CFT calculation at Gepner point, the K is determined unambiguously. It has a moduli parameter ψ that describes a deformation of the CFT by a marginal operator. Also the metric, curvature and hermitian two-point functions in the neighborhood of the Gepner point are analyzed. We use a recipe of tt fusion and develop a method to determine the K from the point of view of topological sigma model. It is not restricted to this specific model and can be applied to other Calabi-Yau cases. sugiyama@phys.h.kyoto-u.ac.jp

2 Introduction There are great progresses in studying worldsheet instanton corrections by the discovery of the mirror symmetry [],[2]. When we study topological sigma model [3], physical observables are constructed by combining couplings, correlation functions and metrics of operators. Typical constituent blocks are three-point and two-point functions (metrics). Their behaviors are controlled by moduli parameters and these observables contain information about moduli space of the sigma model [4]-[0]. In this paper, we investigate a Kähler potential of the B-model moduli space from the point of view of the topological sigma model together with results by CFT at Gepner point. 2 Calabi-Yau d-fold We concentrate on a one-parameter family of Calabi-Yau d-fold M realized as a zero locus of a hypersurface in a projective space CP d+ W ; {p = 0}/Z ( ), p = X +X 2 + +X ψx X 2 X = 0, where we introduce a number = d+2. Hodge numbers h p,q of this d fold are calculated as [] h p,q = δ p+q.d, (0 p d,0 q d,p q), ( ) ( ) p d+2 (p+ m)(d+)+p h p,p = δ 2p,d + ( ) m, m d+ m=0 and deformation of complex structure is parametrized by a complex parameter ψ. In order to describe the complex structure moduli space near Gepner point (ψ 0), we uses a basis of periods They behave around ψ 0 [ ( )] k (ψ) k k(ψ) = Γ Γ(k) G k(ψ), G k := + Γ( k +n) n= Γ ( Γ(k) k Γ(n+k) (ψ)n. [ ( k k(ψ) = Γ )] (ψ) k Γ(k) [ +O(ψ ) ].

3 This set is a projective coordinate of the B-model moduli space. Also we use a canonical basis { (0) k } of the coordinates at the Gepner point as [ ( )] k k = (0) k Γ, and take ratios to construct normalized coordinates ω k := (0) k / (0) = (ψ)k (k )! Gk G (k =,2,, ). Some part of the cohomology classes of H d (W) is associated with the complex structure deformation with Hodge numbers h l,d l = (l = 0,,2,,d). Then associated operators are represented as O (k) := (X X 2 X ) k (k = 0,,2,, 2), and each O (k) is related to a canonical period ω k. Also a coupling ψ := ψ is associated with an operator O () = X X 2 X. When we choose a canonical set of periods, these operators are normalized appropriately and fusion couplingsκ l are evaluated at ψ = 0 ω := t (ω 0 ω ω 2 ), 0 κ 0 0 κ 0 κ 2 K := κ d ψ ω = K ω, O () O (l) =κ l O (l+) (l = 0,,,d ). κ 0 =, κ = ω = +O(ψ ), κ l = κ l κ ω l 2 κ l = +O(ψ ), (l 2). 0 In this canonical set of periods, the three-point couplings are normalized to units up to terms with order O(ψ ). 3 Kähler potential ext let us consider a Kähler potential K of the moduli space. We can construct K as a quadratic form of periods e K = i d2 M Ω Ω = 2 I m,n m n. m,n=

4 The matrix I = {I m,n } is determined by properties of intersection numbers of homology cycles and does not change under any infinitesimal deformation of continuous moduli parameters. That is, the I m,n s cannot depend on any moduli parameters and turns out to be constant numbers. On the other hand, global structures of the moduli space are encoded in monodromies and it is important to see property of K under a global monodromy transformation. Our basis { k} diagonalizes a cyclic Z monodromy ψ αψ (α = e 2πi/ ) at ψ = 0 k(αψ) = α k k(ψ) (k =,2,, ). α = e 2πi/. But this transformation induces a change of the I m,n I m,n I m,n α m+n (m,n =,2,, ). Because the Kähler potential is physical quantity and should not depend on monodromies, it is invariant under the transformation. Only invariant parts of this transformation are diagonal ones I m,m (m =,2,, ) and we find that the matrix I is diagonal one: e K = k= I k k k. () 4 Determination of I k ext, in order to determine the Kähler potential, all we have to do is to fix the diagonal matrix. We use a method of the tt fusion in fixing the I k (k =,2,, ). First note that the Kähler potential is related to a moduli space metric g ψ ψ K = g ψ ψ, = ψ, = ψ. When we introduce a set of hermitian two-point functions Ō(l) O (l) = e q l (0 l 2), (2) the Kähler potential and Zamolodchikov metric g ψ ψ are expressed as e q 0 = e K, g ψ ψ = e q q 0, q 0 = e q q 0. (3) These two-point functions are different from topological metricsη lm = O (l) O (m) = δ l+m,d. 3

5 The equation Eq.(3) is a part of the tt -fusion equation [2, 9] of the Calabi-Yau d-fold with fusion couplingsκ n q 0 + κ 0 2 e q q 0 = 0, q l + κ l 2 e q l+ q l κ l 2 e q l q l = 0 ( l d ), q d κ d 2 e q d q d = 0. (4) ThissetofequationsEq.(4)representsanA d typetodasystem. Byintroducingnewvariables q 0 = q 0, q l = q l + l n=0 log κ n 2 (l ), and noting relations q l = q l, we reexpress the Toda system into a formula l U l = (l+)u 0 + (l n)dlog( U n ) ( l d ), U 0 = Dq 0 = Dlog n=0 [ + l=2 I l I U d = 0, n U n = Dq m ( n d), m=0 l 2], D :=, κ n 2 e q n+ q n = U n (0 n d ) () The number of diagonal components I k in the Kähler potential is (d+) and the above Toda system with (d+) equations gives us consistency conditions of the I k s. ow recall that the components of the matrix I are numerical constants. We may truncate terms of the periods ks in the series expansions up to order O(ψ ) for the purpose of determination of I k(ψ) = f k (ψ) [ k +O(ψ ) ] (k =,2,, ), [ ] U 0 = Dlog [ ( k f k := Γ + l=2 a l ( ψ ψ) l + )] Γ(k), a n := I n I. f 2 n. (6) f We calculate the U n concretely for n 0, and propose a conjecture for the {U n }s at the Gepner point ψ = 0 U 0 = a 2, U d = 0, U n = (n+) 2a n+2 a n+ ( n d ). 4

6 When we use definitions of a n and f n, the U n s are expressed as ) U 0 = a 2 = I 2 Γ( 2 2 I Γ ( U n = (n+) 2a n+2 a n+ = I n+2 I n+, Γ( n+2 Γ ( n+ ) 2 ( n d ). Also we can obtain expressions of hermitian two-point functions by remarking that the threepoint couplings become constantsκ n = (n = 0,,,d) at the ψ = 0 e q n+ q n = I n+2 I n+ Γ( n+2 Γ ( n+ ) 2 (0 n d ). Equivalently, normalized two-point functions are represented as ) Ō(m) O (m) Ō(0) O (0) = eqm q 0 = ( ) m Im+ Γ( 2 m+ I Γ ( (0 m d). (7) ow let us compare these results with those of CFT calculations [3, 2] for minimal models associated with the d-fold M e qn = e qn q 0 = Γ( n+ Γ ( ) Γ( n+ Γ ( (n = 0,,2,, 2), n+ ) ( ) Γ ) Γ ( n+. (8) By comparing two results Eq.(7) and Eq.(8), we can obtain the components of the matrix I up to one numerical constant c I m = c ( ) m ( sin πm In this case, the e K in Eq.() is given as e K = m= c ( ) m π ) (m =,2,, ). ) Γ( m Γ ( m (2 ψ ψ) m [Γ(m)] 2 G m (ψ) 2. Because the Kähler potential is not a function but a section of a line bundle, there is an arbitrariness of multiplication of arbitrary (anti-)holomorphic functions. We choose the normalization factor as c = π +2,

7 then the Kähler potential is written as e K = (ψ ψ) ( ) m Γ ( ) m m= Γ ( ( 2 ψ ψ) m G m [Γ(m)] 2 m (ψ) 2. (9) In order to confirm the validity of this convention, we restrict ourselves to the 3-fold ( = ) case and consider a 3-point functionκof the operator O () in the B-model κ :=κ ψψψ = 3 ψ 2 ψ. (0) (In considering the tt equation, we use normalized periods ω k by taking ratios. In that case, the e K in Eq.(9) and the three-point function Eq.(0) are divided by a factor ψ 2. ) This coupling is a section of holomorphic line bundle and changes with a normalization of the Kähler potential. But there is an invariant 3-point coupling (g ψ ψ) 3/2 e K κ, and its value is evaluated at the ψ = 0 in our normalization ) (g ψ ψ) 3/2 e K κ = Γ( 3 Γ ( 2 /2 ) Γ( Γ ( 4 This result coincides with that of the calculation of CFT [3, 2]. /2 = Metric and Curvature ow we have an exact formula of the K with a moduli parameter ψ and can evaluate the corrections in the physical observables by the marginal deformation of the CFT. For simplicity, we will consider leading corrections of the Kähler potential, metric and scalar curvature in the moduli space. When we define a function A m A m = ( 2 ) m we can obtain these formulae ( A m Γ(n) = ( 2 ) m n A n Γ(m) e K = Γ ( ) Γ ( [Γ(m)] 2 ) 2 Γ ( ) m Γ ( m ) ( ) Γ( m Γ n Γ ( ) ( m Γ n ) (ψ ψ) 2 6 Γ( 2 ( m ), (m,n ), ) ( ) Γ Γ ( ) ( 2 Γ ) (ψ ψ)+ ( 3),

8 ) ( ) g ψ ψ = 2 Γ( 2 Γ [ ( ) 2 ( ) ] Γ ( A2 A3 ) ( 2 Γ +(ψ ψ) ( 4), A ) A Γ ( ( ) ) R = 4+2 ) Γ 3 Γ ( ) ( Γ( 2 2 Γ 3 Γ ( 2 [ ( ) ( )( ) 2 ( )( ) ] A3 A A3 A A4 +(ψ ψ) ( ). A 2 A 2 A 2 A 2 A 2 The Ricci tensor of the moduli space is represented as R ψ ψ = 2 g ψ ψr. For the torus ( = 3) case, its metric is a standard one of the upper-half plane and the scalar curvature is constant R = 4 except for points ψ = e 2πil/3 (l = 0,,2). Also the K3 ( = 4) metric is given by the above formula and related scalar curvature is a negative constant number R = 2 except for ψ = e πil/2 (l = 0,,2,3). We plot this scalar curvature at the ψ = 0 in Fig.. It increases monotonically with the dimension d. But the derivative with respect to ψ ψ is negative for cases at ψ = 0 as shown in Fig Figure : Leading part R 0 of Scalar Curvature R. This part depends on the dimension = d + 2 of the Calabi-Yau d-fold. It increases monotonically with the d. For the torus ( = 3) and K3 ( = 4) cases, associated curvatures are negative. But the other cases have positive curvatures at ψ = 0. Let us return to the hermitian two-point functions. They are represented as some combinations of these K, g ψ ψ and R e q 0 = e K, e q q 0 = 2g ψ ψ, e q 2 q = ( ) R 2g ψ ψ 2 +2, e q 3 q 2 = [ ( ) R 2g ψ ψ g ψ ψ ( )] R ψ ψlog 2 +2, 7

9 Figure 2: Subleading part R of Scalar Curvature R. This part is a coefficient of ψ ψ in the R and depends on the dimension of the CY d-fold. For the 0 cases, its value is positive. But the R is negative for the other cases. e q 4 q 3 = [ 2g ψ ψ 4+3R 2g ψ ψ ( ) R ψ ψlog g ψ ψ [ ( ) R ψ ψlog g ψ ψ ( ) ] R ] ψ ψlog 2 +2, ow we know the formula of the K, R, and g ψ ψ and evaluate moduli dependences of these correlators. In this paper we develop a method to determine the Kähler potential unambiguously by comparing the result of CFT with that of topological sigma model. We calculate the metric and curvature in the neighborhood of the Gepner point, which have dependences of moduli parameter ψ. The result represents a marginal deformation of the CFT. But the formula of the K is exact and we can study it at all points in the moduli space by analytic continuation. The method we developed here is not restricted to the specific model and can be applied to any other Calabi-Yau spaces. Acknowledgement This work is supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture

10 References [] B. Greene and M. Plesser, ucl. Phys. B338 (990). P. Candelas, M. Lynker and R. Schimmrigk, ucl. Phys. B34 (990) 383. [2] P. Candelas, X. de la Ossa, P. Green and L. Parkes, Phys. Lett. 28B (99) 8; ucl. Phys. B39 (99) 2. [3] E. Witten, Mirror Manifolds and Topological Field Theory, in Essays on Mirror Manifolds, ed. S.-T. Yau, (Int. Press, Hong Kong, 992), pp P. Aspinwall and D. Morrison, Commun. Math. Phys. (993) 24. [4] B. Greene, D. Morrison and M. Plesser, Commun. Math. Phys. 73 (99) 9, hepth/ [] M. Jinzenji and M. agura, Int. J. Mod. Phys. A (996) 27, hep-th/ [6] M. agura and K. Sugiyama, Int. J. Mod. Phys. A0 (99) 233, hep-th/9329. [7] K. Sugiyama, Ann. Phys. 247 (996) 06, hep-th/ [8] K. Sugiyama, Int. J. Mod. Phys. A (996) 229, hep-th/9044. [9] K. Sugiyama, ucl. Phys. B49 (996) 693, hep-th/904. [0] K. Sugiyama, ucl. Phys. B37 (999) 99, hep-th/ [] V. Batyrev and D. Dais, Topology 3 (996) 90. [2] S. Cecotti and C. Vafa, ucl. Phys. B367 (99) 39. [3] S. Cecotti, Int. J. Mod. Phys. A6 (99) 749, S. Cecotti, ucl. Phys. B3 (99) 7. 9

On the BCOV Conjecture

On the BCOV Conjecture Department of Mathematics University of California, Irvine December 14, 2007 Mirror Symmetry The objects to study By Mirror Symmetry, for any CY threefold, there should be another CY threefold X, called

More information

Geometry of the Calabi-Yau Moduli

Geometry of the Calabi-Yau Moduli Geometry of the Calabi-Yau Moduli Zhiqin Lu 2012 AMS Hawaii Meeting Department of Mathematics, UC Irvine, Irvine CA 92697 March 4, 2012 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 1/51

More information

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

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

More information

Topics in Geometry: Mirror Symmetry

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:

More information

Mirror Symmetry: Introduction to the B Model

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

More information

Mirror Maps and Instanton Sums for Complete Intersections in Weighted Projective Space

Mirror Maps and Instanton Sums for Complete Intersections in Weighted Projective Space LMU-TPW 93-08 Mirror Maps and Instanton Sums for Complete Intersections in Weighted Projective Space arxiv:hep-th/9304034v 8 Apr 993 Albrecht Klemm and Stefan Theisen Sektion Physik Ludwig-Maximilians

More information

Topological Mirrors and Quantum Rings

Topological Mirrors and Quantum Rings HUTP-91/A059 Topological Mirrors and Quantum Rings arxiv:hep-th/9111017v1 7 Nov 1991 Cumrun Vafa 1 Lyman Laboratory of Physics Harvard University Cambridge, Ma 02138, USA Aspects of duality and mirror

More information

Overview of classical mirror symmetry

Overview of classical mirror symmetry Overview of classical mirror symmetry David Cox (notes by Paul Hacking) 9/8/09 () Physics (2) Quintic 3-fold (3) Math String theory is a N = 2 superconformal field theory (SCFT) which models elementary

More information

An introduction to heterotic mirror symmetry. Eric Sharpe Virginia Tech

An introduction to heterotic mirror symmetry. Eric Sharpe Virginia Tech An introduction to heterotic mirror symmetry Eric Sharpe Virginia Tech I ll begin today by reminding us all of ordinary mirror symmetry. Most basic incarnation: String theory on a Calabi-Yau X = String

More information

D-Branes and Vanishing Cycles in Higher Dimensions.

D-Branes and Vanishing Cycles in Higher Dimensions. Preprint typeset in JHEP style. - HYPER VERSION D-Branes and Vanishing Cycles in Higher Dimensions. Mark Raugas Department of Mathematics, Columbia University New York, NY 10027, USA raugasm@math.columbia.edu

More information

Mirror symmetry for G 2 manifolds

Mirror symmetry for G 2 manifolds Mirror symmetry for G 2 manifolds based on [1602.03521] [1701.05202]+[1706.xxxxx] with Michele del Zotto (Stony Brook) 1 Strings, T-duality & Mirror Symmetry 2 Type II String Theories and T-duality Superstring

More information

The geometry of Landau-Ginzburg models

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

More information

A Two Parameter Family of the Calabi-Yau d-fold

A Two Parameter Family of the Calabi-Yau d-fold arxiv:hep-th/9410184v1 25 Oct 1994 A Two Parameter Family of the Calabi-Yau d-fold Katsuyuki Sugiyama 1 Department of Physics, University of Tokyo Bunkyo-ku, Tokyo 113, Japan ABSTRACT We study a two parameter

More information

An Introduction to Quantum Sheaf Cohomology

An Introduction to Quantum Sheaf Cohomology An Introduction to Quantum Sheaf Cohomology Eric Sharpe Physics Dep t, Virginia Tech w/ J Guffin, S Katz, R Donagi Also: A Adams, A Basu, J Distler, M Ernebjerg, I Melnikov, J McOrist, S Sethi,... arxiv:

More information

GENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY. Sampei Usui

GENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY. Sampei Usui GENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY Sampei Usui Abstract. This article is a geometric application of polarized logarithmic Hodge theory of Kazuya Kato and Sampei Usui. We prove generic Torelli

More information

Departement de Physique Theorique, Universite de Geneve, Abstract

Departement de Physique Theorique, Universite de Geneve, Abstract hep-th/950604 A NOTE ON THE STRING ANALOG OF N = SUPER-SYMMETRIC YANG-MILLS Cesar Gomez 1 and Esperanza Lopez 1 Departement de Physique Theorique, Universite de Geneve, Geneve 6, Switzerland Abstract A

More information

Topological String Partition Functions as Polynomials

Topological String Partition Functions as Polynomials hep-th/0406078 Topological String Partition Functions as Polynomials arxiv:hep-th/0406078v1 9 Jun 2004 Satoshi Yamaguchi and Shing-Tung Yau yamaguch@math.harvard.edu, yau@math.harvard.edu Harvard University,

More information

Some new torsional local models for heterotic strings

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

More information

Heterotic Torsional Backgrounds, from Supergravity to CFT

Heterotic Torsional Backgrounds, from Supergravity to CFT Heterotic Torsional Backgrounds, from Supergravity to CFT IAP, Université Pierre et Marie Curie Eurostrings@Madrid, June 2010 L.Carlevaro, D.I. and M. Petropoulos, arxiv:0812.3391 L.Carlevaro and D.I.,

More information

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

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

More information

G 2 manifolds and mirror symmetry

G 2 manifolds and mirror symmetry G 2 manifolds and mirror symmetry Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics First Annual Meeting, New York, 9/14/2017 Andreas Braun University of Oxford based on [1602.03521]

More information

arxiv:hep-th/ v3 25 Aug 1999

arxiv:hep-th/ v3 25 Aug 1999 TIT/HEP-411 arxiv:hep-th/9904155v3 25 Aug 1999 Puzzles on the Duality between Heterotic and Type IIA Strings Mitsuko Abe and Masamichi Sato Department of Physics Tokyo Institute of Technology Oh-okayama,

More information

Heterotic Mirror Symmetry

Heterotic Mirror Symmetry Heterotic Mirror Symmetry Eric Sharpe Physics Dep t, Virginia Tech Drexel University Workshop on Topology and Physics September 8-9, 2008 This will be a talk about string theory, so lemme motivate it...

More information

arxiv:hep-th/ v1 22 Aug 1995

arxiv:hep-th/ v1 22 Aug 1995 CLNS-95/1356 IASSNS-HEP-95/64 hep-th/9508107 TOWARDS MIRROR SYMMETRY AS DUALITY FOR TWO DIMENSIONAL ABELIAN GAUGE THEORIES arxiv:hep-th/9508107 v1 Aug 1995 DAVID R. MORRISON Department of Mathematics,

More information

Topics in Geometry: Mirror Symmetry

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:

More information

Mirrored K3 automorphisms and non-geometric compactifications

Mirrored K3 automorphisms and non-geometric compactifications Mirrored K3 automorphisms and non-geometric compactifications Dan Israe l, LPTHE New Frontiers in String Theory, Kyoto 2018 Based on: D.I. and Vincent Thie ry, arxiv:1310.4116 D.I., arxiv:1503.01552 Chris

More information

2 Type IIA String Theory with Background Fluxes in d=2

2 Type IIA String Theory with Background Fluxes in d=2 2 Type IIA String Theory with Background Fluxes in d=2 We consider compactifications of type IIA string theory on Calabi-Yau fourfolds. Consistency of a generic compactification requires switching on a

More information

HOMOLOGICAL GEOMETRY AND MIRROR SYMMETRY

HOMOLOGICAL GEOMETRY AND MIRROR SYMMETRY HOMOLOGICAL GEOMETRY AND MIRROR SYMMETRY Alexander B. GIVENTAL Dept. of Math., UC Berkeley Berkeley CA 94720, USA 0. A popular example. A homogeneous degree 5 polynomial equation in 5 variables determines

More information

Calabi-Yau Geometry and Mirror Symmetry Conference. Cheol-Hyun Cho (Seoul National Univ.) (based on a joint work with Hansol Hong and Siu-Cheong Lau)

Calabi-Yau Geometry and Mirror Symmetry Conference. Cheol-Hyun Cho (Seoul National Univ.) (based on a joint work with Hansol Hong and Siu-Cheong Lau) Calabi-Yau Geometry and Mirror Symmetry Conference Cheol-Hyun Cho (Seoul National Univ.) (based on a joint work with Hansol Hong and Siu-Cheong Lau) Mirror Symmetry between two spaces Mirror symmetry explains

More information

Arithmetic Mirror Symmetry

Arithmetic Mirror Symmetry Arithmetic Mirror Symmetry Daqing Wan April 15, 2005 Institute of Mathematics, Chinese Academy of Sciences, Beijing, P.R. China Department of Mathematics, University of California, Irvine, CA 92697-3875

More information

Generalized Tian-Todorov theorems

Generalized Tian-Todorov theorems Generalized Tian-Todorov theorems M.Kontsevich 1 The classical Tian-Todorov theorem Recall the classical Tian-Todorov theorem (see [4],[5]) about the smoothness of the moduli spaces of Calabi-Yau manifolds:

More information

Topological reduction of supersymmetric gauge theories and S-duality

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

More information

Aspects of (0,2) theories

Aspects of (0,2) theories Aspects of (0,2) theories Ilarion V. Melnikov Harvard University FRG workshop at Brandeis, March 6, 2015 1 / 22 A progress report on d=2 QFT with (0,2) supersymmetry Gross, Harvey, Martinec & Rohm, Heterotic

More information

String-Theory: Open-closed String Moduli Spaces

String-Theory: Open-closed String Moduli Spaces String-Theory: Open-closed String Moduli Spaces Heidelberg, 13.10.2014 History of the Universe particular: Epoch of cosmic inflation in the early Universe Inflation and Inflaton φ, potential V (φ) Possible

More information

Compactifications of F-Theory on Calabi Yau Threefolds II arxiv:hep-th/ v2 31 May 1996

Compactifications of F-Theory on Calabi Yau Threefolds II arxiv:hep-th/ v2 31 May 1996 DUKE-TH-96-107 HUTP-96/A012 hep-th/9603161 March, 1996 (Revised 5/96) Compactifications of F-Theory on Calabi Yau Threefolds II arxiv:hep-th/9603161v2 31 May 1996 David R. Morrison Department of Mathematics,

More information

Constructing compact 8-manifolds with holonomy Spin(7)

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.

More information

Generalized N = 1 orientifold compactifications

Generalized N = 1 orientifold compactifications Generalized N = 1 orientifold compactifications Thomas W. Grimm University of Wisconsin, Madison based on: [hep-th/0602241] Iman Benmachiche, TWG [hep-th/0507153] TWG Madison, Wisconsin, November 2006

More information

D-branes and Normal Functions

D-branes and Normal Functions arxiv:0709.4028 D-branes and Normal Functions David R. Morrison a,b and Johannes Walcher c arxiv:0709.4028v1 [hep-th] 26 Sep 2007 a Center for Geometry and Theoretical Physics, Duke University, Durham,

More information

Pseudoholomorphic Curves and Mirror Symmetry

Pseudoholomorphic Curves and Mirror Symmetry Pseudoholomorphic Curves and Mirror Symmetry Santiago Canez February 14, 2006 Abstract This survey article was written for Prof. Alan Weinstein s Symplectic Geometry (Math 242) course at UC Berkeley in

More information

Magdalena Larfors

Magdalena Larfors Uppsala University, Dept. of Theoretical Physics Based on D. Chialva, U. Danielsson, N. Johansson, M.L. and M. Vonk, hep-th/0710.0620 U. Danielsson, N. Johansson and M.L., hep-th/0612222 2008-01-18 String

More information

Enumerative Geometry: from Classical to Modern

Enumerative Geometry: from Classical to Modern : from Classical to Modern February 28, 2008 Summary Classical enumerative geometry: examples Modern tools: Gromov-Witten invariants counts of holomorphic maps Insights from string theory: quantum cohomology:

More information

Topological DBI actions and nonlinear instantons

Topological DBI actions and nonlinear instantons 8 November 00 Physics Letters B 50 00) 70 7 www.elsevier.com/locate/npe Topological DBI actions and nonlinear instantons A. Imaanpur Department of Physics, School of Sciences, Tarbiat Modares University,

More information

* +, ...! Einstein. [1] Calabi-Yau [2] Calabi-Yau. :;æåø!! :; õ ø!!

* +, ...! Einstein. [1] Calabi-Yau [2] Calabi-Yau. :;æåø!! :; õ ø!! !"#$%$%&! '()*+,-./01+,-.234!!"#$%&'()! * +, -! 56789:;?@ABC

More information

Sphere Partition Functions, Topology, the Zamolodchikov Metric

Sphere Partition Functions, Topology, the Zamolodchikov Metric Sphere Partition Functions, Topology, the Zamolodchikov Metric, and Extremal Correlators Weizmann Institute of Science Efrat Gerchkovitz, Jaume Gomis, ZK [1405.7271] Jaume Gomis, Po-Shen Hsin, ZK, Adam

More information

Generalized complex geometry and topological sigma-models

Generalized complex geometry and topological sigma-models Generalized complex geometry and topological sigma-models Anton Kapustin California Institute of Technology Generalized complex geometry and topological sigma-models p. 1/3 Outline Review of N = 2 sigma-models

More information

Heterotic Flux Compactifications

Heterotic Flux Compactifications Heterotic Flux Compactifications Mario Garcia-Fernandez Instituto de Ciencias Matemáticas, Madrid String Pheno 2017 Virginia Tech, 7 July 2017 Based on arxiv:1611.08926, and joint work with Rubio, Tipler,

More information

Two simple ideas from calculus applied to Riemannian geometry

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

More information

An introduction to mirror symmetry. Eric Sharpe Virginia Tech

An introduction to mirror symmetry. Eric Sharpe Virginia Tech An introduction to mirror symmetry Eric Sharpe Virginia Tech This will be a talk about string theory, so let me discuss the motivation. Twentieth-century physics saw two foundational advances: General

More information

The Strominger Yau Zaslow conjecture

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

More information

Introduction To K3 Surfaces (Part 2)

Introduction To K3 Surfaces (Part 2) Introduction To K3 Surfaces (Part 2) James Smith Calf 26th May 2005 Abstract In this second introductory talk, we shall take a look at moduli spaces for certain families of K3 surfaces. We introduce the

More information

HMS Seminar - Talk 1. Netanel Blaier (Brandeis) September 26, 2016

HMS Seminar - Talk 1. Netanel Blaier (Brandeis) September 26, 2016 HMS Seminar - Talk 1 Netanel Blaier (Brandeis) September 26, 2016 Overview Fukaya categories : (naive) Lagrangian Floer homology, A -structures Introduction : what is mirror symmetry? The physical story

More information

Counting black hole microstates as open string flux vacua

Counting black hole microstates as open string flux vacua Counting black hole microstates as open string flux vacua Frederik Denef KITP, November 23, 2005 F. Denef and G. Moore, to appear Outline Setting and formulation of the problem Black hole microstates and

More information

The Pfaffian Calabi Yau, its Mirror, and their link to the Grassmannian G(2,7)

The Pfaffian Calabi Yau, its Mirror, and their link to the Grassmannian G(2,7) arxiv:math/9801092v1 [math.ag] 20 Jan 1998 The Pfaffian Calabi Yau, its Mirror, and their link to the Grassmannian G(2,7 Einar Andreas Rødland January 20, 1998 Abstract The rank 4 locus of a general skew-symmetric

More information

Conformal Sigma Models in Three Dimensions

Conformal Sigma Models in Three Dimensions Conformal Sigma Models in Three Dimensions Takeshi Higashi, Kiyoshi Higashijima,, Etsuko Itou, Muneto Nitta 3 Department of Physics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043,

More information

THE QUANTUM CONNECTION

THE QUANTUM CONNECTION THE QUANTUM CONNECTION MICHAEL VISCARDI Review of quantum cohomology Genus 0 Gromov-Witten invariants Let X be a smooth projective variety over C, and H 2 (X, Z) an effective curve class Let M 0,n (X,

More information

Vanishing theorems and holomorphic forms

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

More information

arxiv:hep-th/ v3 13 Oct 1995

arxiv:hep-th/ v3 13 Oct 1995 TIFR/TH/95-15 hep-th/9504003 Revised version August 8, 1995 arxiv:hep-th/9504003v3 13 Oct 1995 Covariantising the Beltrami equation in W-gravity Suresh Govindarajan 1 Theoretical Physics Group Tata Institute

More information

The exact quantum corrected moduli space for the universal hypermultiplet

The exact quantum corrected moduli space for the universal hypermultiplet The exact quantum corrected moduli space for the universal hypermultiplet Bengt E.W. Nilsson Chalmers University of Technology, Göteborg Talk at "Miami 2009" Fort Lauderdale, December 15-20, 2009 Talk

More information

Non-Geometric Calabi- Yau Backgrounds

Non-Geometric Calabi- Yau Backgrounds Non-Geometric Calabi- Yau Backgrounds CH, Israel and Sarti 1710.00853 A Dabolkar and CH, 2002 Duality Symmetries Supergravities: continuous classical symmetry, broken in quantum theory, and by gauging

More information

N = 2 String Amplitudes *

N = 2 String Amplitudes * LBL-37660 August 23, 1995 UCB-PTH-95/30 N = 2 String Amplitudes * Hirosi Oogurit Theoretical Physics Group Lawrence Berkeley Laboratory University of California Berkeley, California 94 720 To appear in

More information

Riemannian Curvature Functionals: Lecture III

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

More information

Asymptotic of Enumerative Invariants in CP 2

Asymptotic of Enumerative Invariants in CP 2 Peking Mathematical Journal https://doi.org/.7/s4543-8-4-4 ORIGINAL ARTICLE Asymptotic of Enumerative Invariants in CP Gang Tian Dongyi Wei Received: 8 March 8 / Revised: 3 July 8 / Accepted: 5 July 8

More information

Takao Akahori. z i In this paper, if f is a homogeneous polynomial, the correspondence between the Kodaira-Spencer class and C[z 1,...

Takao Akahori. z i In this paper, if f is a homogeneous polynomial, the correspondence between the Kodaira-Spencer class and C[z 1,... J. Korean Math. Soc. 40 (2003), No. 4, pp. 667 680 HOMOGENEOUS POLYNOMIAL HYPERSURFACE ISOLATED SINGULARITIES Takao Akahori Abstract. The mirror conjecture means originally the deep relation between complex

More information

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

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 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

More information

ENOKI S INJECTIVITY THEOREM (PRIVATE NOTE) Contents 1. Preliminaries 1 2. Enoki s injectivity theorem 2 References 5

ENOKI S INJECTIVITY THEOREM (PRIVATE NOTE) Contents 1. Preliminaries 1 2. Enoki s injectivity theorem 2 References 5 ENOKI S INJECTIVITY THEOREM (PRIVATE NOTE) OSAMU FUJINO Contents 1. Preliminaries 1 2. Enoki s injectivity theorem 2 References 5 1. Preliminaries Let us recall the basic notion of the complex geometry.

More information

Some half-bps solutions of M-theory

Some half-bps solutions of M-theory Preprint typeset in JHEP style - PAPER VERSION arxiv:hep-th/0506247v2 10 Feb 2006 Some half-bps solutions of M-theory Micha l Spaliński So ltan Institute for Nuclear Studies ul. Hoża 69, 00-681 Warszawa,

More information

Chern-Simons Theory & Topological Strings

Chern-Simons Theory & Topological Strings 1 Chern-Simons Theory & Topological Strings Bonn August 6, 2009 Albrecht Klemm 2 Topological String Theory on Calabi-Yau manifolds A-model & Integrality Conjectures B-model Mirror Duality & Large N-Duality

More information

Monodromy of the Dwork family, following Shepherd-Barron X n+1. P 1 λ. ζ i = 1}/ (µ n+1 ) H.

Monodromy of the Dwork family, following Shepherd-Barron X n+1. P 1 λ. ζ i = 1}/ (µ n+1 ) H. Monodromy of the Dwork family, following Shepherd-Barron 1. The Dwork family. Consider the equation (f λ ) f λ (X 0, X 1,..., X n ) = λ(x n+1 0 + + X n+1 n ) (n + 1)X 0... X n = 0, where λ is a free parameter.

More information

Mathematical Research Letters 1, (1994) A MATHEMATICAL THEORY OF QUANTUM COHOMOLOGY. Yongbin Ruan and Gang Tian

Mathematical Research Letters 1, (1994) A MATHEMATICAL THEORY OF QUANTUM COHOMOLOGY. Yongbin Ruan and Gang Tian Mathematical Research Letters 1, 269 278 (1994) A MATHEMATICAL THEORY OF QUANTUM COHOMOLOGY Yongbin Ruan and Gang Tian 1. Introduction The notion of quantum cohomology was first proposed by the physicist

More information

Mathematical Results Inspired by Physics

Mathematical Results Inspired by Physics ICM 2002 Vol. III 1 3 Mathematical Results Inspired by Physics Kefeng Liu Abstract I will discuss results of three different types in geometry and topology. (1) General vanishing and rigidity theorems

More information

arxiv:math/ v2 [math.dg] 11 Nov 2004

arxiv:math/ v2 [math.dg] 11 Nov 2004 arxiv:math/990402v2 [math.dg] Nov 2004 Lagrangian torus fibration of quintic Calabi-Yau hypersurfaces I: Fermat quintic case Wei-Dong Ruan Department of mathematics Columbia University New York, NY 0027

More information

Critical metrics on connected sums of Einstein four-manifolds

Critical metrics on connected sums of Einstein four-manifolds Critical metrics on connected sums of Einstein four-manifolds University of Wisconsin, Madison March 22, 2013 Kyoto Einstein manifolds Einstein-Hilbert functional in dimension 4: R(g) = V ol(g) 1/2 R g

More information

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

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.

More information

LANDAU-GINZBURG MODEL FOR A CRITICAL TOPOLOGICAL STRING

LANDAU-GINZBURG MODEL FOR A CRITICAL TOPOLOGICAL STRING LANDAU-GINZBURG MODEL FOR A CRITICAL TOPOLOGICAL STRING Debashis Ghoshal Mehta Research Institute of Mathematics & Mathematical Physics 10, Kasturba Gandhi Marg, Allahabad 211 002, India and Sunil Mukhi

More information

The Yau-Tian-Donaldson Conjectuture for general polarizations

The Yau-Tian-Donaldson Conjectuture for general polarizations The Yau-Tian-Donaldson Conjectuture for general polarizations Toshiki Mabuchi, Osaka University 2015 Taipei Conference on Complex Geometry December 22, 2015 1. Introduction 2. Background materials Table

More information

Exact Results in D=2 Supersymmetric Gauge Theories And Applications

Exact Results in D=2 Supersymmetric Gauge Theories And Applications Exact Results in D=2 Supersymmetric Gauge Theories And Applications Jaume Gomis Miami 2012 Conference arxiv:1206.2606 with Doroud, Le Floch and Lee arxiv:1210.6022 with Lee N = (2, 2) supersymmetry on

More information

Heterotic Geometry and Fluxes

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

More information

Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current

Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current Author, F., and S. Author. (2015) Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current, International Mathematics Research Notices, Vol. 2015, Article ID rnn999, 7 pages. doi:10.1093/imrn/rnn999

More information

Topological Strings, D-Model, and Knot Contact Homology

Topological Strings, D-Model, and Knot Contact Homology arxiv:submit/0699618 [hep-th] 21 Apr 2013 Topological Strings, D-Model, and Knot Contact Homology Mina Aganagic 1,2, Tobias Ekholm 3,4, Lenhard Ng 5 and Cumrun Vafa 6 1 Center for Theoretical Physics,

More information

2d-4d wall-crossing and hyperholomorphic bundles

2d-4d wall-crossing and hyperholomorphic bundles 2d-4d wall-crossing and hyperholomorphic bundles Andrew Neitzke, UT Austin (work in progress with Davide Gaiotto, Greg Moore) DESY, December 2010 Preface Wall-crossing is an annoying/beautiful phenomenon

More information

Lecture 1: Introduction

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

More information

Mirror symmetry. Mark Gross. July 24, University of Cambridge

Mirror symmetry. Mark Gross. July 24, University of Cambridge University of Cambridge July 24, 2015 : A very brief and biased history. A search for examples of compact Calabi-Yau three-folds by Candelas, Lynker and Schimmrigk (1990) as crepant resolutions of hypersurfaces

More information

HODGE NUMBERS OF COMPLETE INTERSECTIONS

HODGE NUMBERS OF COMPLETE INTERSECTIONS HODGE NUMBERS OF COMPLETE INTERSECTIONS LIVIU I. NICOLAESCU 1. Holomorphic Euler characteristics Suppose X is a compact Kähler manifold of dimension n and E is a holomorphic vector bundle. For every p

More information

SOME ASPECTS ON CIRCLES AND HELICES IN A COMPLEX PROJECTIVE SPACE. Toshiaki Adachi* and Sadahiro Maeda

SOME ASPECTS ON CIRCLES AND HELICES IN A COMPLEX PROJECTIVE SPACE. Toshiaki Adachi* and Sadahiro Maeda Mem. Fac. Sci. Eng. Shimane Univ. Series B: Mathematical Science 32 (1999), pp. 1 8 SOME ASPECTS ON CIRCLES AND HELICES IN A COMPLEX PROJECTIVE SPACE Toshiaki Adachi* and Sadahiro Maeda (Received December

More information

EXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION

EXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION EXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION JAN CHRISTIAN ROHDE Introduction By string theoretical considerations one is interested in Calabi-Yau manifolds since Calabi-Yau 3-manifolds

More information

arxiv:hep-th/ v1 24 Sep 1998

arxiv:hep-th/ v1 24 Sep 1998 ICTP/IR/98/19 SISSA/EP/98/101 Quantum Integrability of Certain Boundary Conditions 1 arxiv:hep-th/9809178v1 24 Sep 1998 M. Moriconi 2 The Abdus Salam International Centre for Theoretical Physics Strada

More information

On the Geometry of the String Landscape. and the Swampland

On the Geometry of the String Landscape. and the Swampland CALT-68-2600, HUTP-06/A017 On the Geometry of the String Landscape arxiv:hep-th/0605264v1 26 May 2006 and the Swampland Hirosi Ooguri 1 and Cumrun Vafa 2 1 California Institute of Technology Pasadena,

More information

Looking Beyond Complete Intersection Calabi-Yau Manifolds. Work in progress with Hans Jockers, Joshua M. Lapan, Maurico Romo and David R.

Looking Beyond Complete Intersection Calabi-Yau Manifolds. Work in progress with Hans Jockers, Joshua M. Lapan, Maurico Romo and David R. Looking Beyond Complete Intersection Calabi-Yau Manifolds Work in progress with Hans Jockers, Joshua M. Lapan, Maurico Romo and David R. Morrison Who and Why Def: X is Calabi-Yau (CY) if X is a Ricci-flat,

More information

BPS states, Wall-crossing and Quivers

BPS states, Wall-crossing and Quivers Iberian Strings 2012 Bilbao BPS states, Wall-crossing and Quivers IST, Lisboa Michele Cirafici M.C.& A.Sincovics & R.J. Szabo: 0803.4188, 1012.2725, 1108.3922 and M.C. to appear BPS States in String theory

More information

Classifying complex surfaces and symplectic 4-manifolds

Classifying complex surfaces and symplectic 4-manifolds Classifying complex surfaces and symplectic 4-manifolds UT Austin, September 18, 2012 First Cut Seminar Basics Symplectic 4-manifolds Definition A symplectic 4-manifold (X, ω) is an oriented, smooth, 4-dimensional

More information

F- 理論におけるフラックスコンパクト化. Hajime Otsuka( 大塚啓 ) (Waseda U.) Physics Lett. B. 774 (2017) 225 with Y. Honma (National Tsing-Hua U.) Sangyo U.

F- 理論におけるフラックスコンパクト化. Hajime Otsuka( 大塚啓 ) (Waseda U.) Physics Lett. B. 774 (2017) 225 with Y. Honma (National Tsing-Hua U.) Sangyo U. F- 理論におけるフラックスコンパクト化 Hajime Otsuka( 大塚啓 ) (Waseda U.) Physics Lett. B. 774 (2017) 225 with Y. Honma (National Tsing-Hua U.) 2018/1/29@Kyoto Sangyo U. Outline Introduction Flux compactification in type

More information

Differential equations concerned with mirror symmetry of toric K3 hypersurfaces with arithmetic properties. Atsuhira Nagano (University of Tokyo)

Differential equations concerned with mirror symmetry of toric K3 hypersurfaces with arithmetic properties. Atsuhira Nagano (University of Tokyo) Differential equations concerned with mirror symmetry of toric K3 hypersurfaces with arithmetic properties Atsuhira Nagano (University of Tokyo) 1 Contents Section 1 : Introduction 1: Hypergeometric differential

More information

N = 2 supersymmetric gauge theory and Mock theta functions

N = 2 supersymmetric gauge theory and Mock theta functions N = 2 supersymmetric gauge theory and Mock theta functions Andreas Malmendier GTP Seminar (joint work with Ken Ono) November 7, 2008 q-series in differential topology Theorem (M-Ono) The following q-series

More information

Homological mirror symmetry via families of Lagrangians

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

More information

Sigma-models with complex homogeneous target spaces

Sigma-models with complex homogeneous target spaces EPJ Web of onferences 15, 0500 (016) DOI: 10.1051/ epjconf/016150500 QUARKS-016 Sigma-models with complex homogeneous target spaces Dmitri Bykov 1,, 1 Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut,

More information

arxiv: v2 [hep-th] 25 Aug 2017

arxiv: v2 [hep-th] 25 Aug 2017 Topological Chern-Simons/Matter Theories arxiv:1706.09977v2 [hep-th] 25 Aug 2017 Mina Aganagic a, Kevin Costello b, Jacob McNamara c and Cumrun Vafa c a Center for Theoretical Physics, University of California,

More information

Singlet Couplings and (0,2) Models arxiv:hep-th/ v1 14 Jun 1994

Singlet Couplings and (0,2) Models arxiv:hep-th/ v1 14 Jun 1994 PUPT 1465 hep-th@xxx/9406090 Singlet Couplings and (0,2) Models arxiv:hep-th/9406090v1 14 Jun 1994 Jacques Distler and Shamit Kachru Joseph Henry Laboratories Princeton University Princeton, NJ 08544 USA

More information

Refined Donaldson-Thomas theory and Nekrasov s formula

Refined Donaldson-Thomas theory and Nekrasov s formula Refined Donaldson-Thomas theory and Nekrasov s formula Balázs Szendrői, University of Oxford Maths of String and Gauge Theory, City University and King s College London 3-5 May 2012 Geometric engineering

More information

Statistics of supersymmetric vacua in string/m theory

Statistics of supersymmetric vacua in string/m theory Statistics of supersymmetric vacua in string/m theory Steve Zelditch Department of Mathematics Johns Hopkins University Joint Work with Bernard Shiffman Mike Douglas 1 Supergravity theory An effective

More information

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

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

More information