Refined ChernSimons Theory, Topological Strings and Knot Homology


 Milton Richardson
 3 years ago
 Views:
Transcription
1 Refined ChernSimons Theory, Topological Strings and Knot Homology Based on work with Shamil Shakirov, and followup work with Kevin Scheaffer arxiv: arxiv:
2 ChernSimons theory played a central role in mathematics and physics ever since Witten s discovery in 88 that the knot invariant J K (q) constructed by Jones is computed by SU(2) ChernSimons theory on S 3.
3 ChernSimons theory with gauge group G on a threemanifold M is given in terms of the path integral where is a connection valued in the Lie algebra of G is an integer, and the action is The theory is independent of the metric on M, so is automatically topologically invariant.
4 The natural observable in the theory is the Wilson loop, associated to a knot K in M taken in various representations of the gauge group. To any three manifold with a collection of knots in it, the ChernSimons path integral associates a knot and three manifold invariant.
5 ChernSimons theory is solvable explicitly, so it provides a physical way to not only interpret knot polynomials, but to calculate them as well.
6 ChernSimons theory gained new relevance when it emerged as a part of string theory and the topological string.
7 Topological string is counting holomorphic maps to a CalabiYau threefold X. φ : Σ g X φ = 0
8 This can be extended to open topological string theory, counting maps from Riemann surfaces with boundaries to X, where the boundaries fall on a Lagrangian submanifold M of X. M φ : Σ g X φ = 0 Since X is a CalabiYau threefold, M is three dimensional.
9 Contact with knot theory emerges if we take open topological string on X = T * M Witten 92 where the boundaries map to the zero section. We take N copies of M, and distinguish which copy a boundary maps to.
10 Open topological string in this case is the same as SU(N) ChernSimons theory on M. Z CS (M,q) = Z Top (T * M,λ) The parameter that keeps track of the genus in topological string is related to q of ChernSimons theory by 2πi k + q = e N = e λ 2g 2+# holes λ # edges # vertices λ
11 We can introduce knots in the theory by adding Lagrangian branes to X. For a knot K in S^3, we associate a Lagrangian L_K K S 3 T * S 3 L K S 3 = K Now, we must allow boundaries on L_K as well.
12 As explained by Gukov, Schwarz and Vafa, categorified knot invariants are related to the refined topological string.
13 Categorification program, initiated by Khovanov in 99, interprets ChernSimons knot invariants as Euler characteristics of a knot homology theory. Categorifying the Jones polynomial, one associates to a knot K bigraded cohomology groups in such a way that the Jones polynomial arises as the Euler characteristic, taken with respect to one of the gradings.
14 In this way, one makes integrality of the knot polynomials manifest. This also leads to refinement, where one computes Poincare polynomials of the knot homology theory instead.
15 String theory allows one to relate topological string, and hence also ChernSimons partition functions, to computing certain indices in Mtheory on the CalabiYau. In the Mtheory formulation, contact with knot homology can be made.
16 Ordinary topological string on a CalabiYau X is related to Mtheory on (X 2 S 1 ) q where as one goes around the 2 S 1 rotate by, the complex coordinates of
17 The supersymmetric partition function of Mtheory in this geometry Z M (X,q) = Tr( 1) F q S 1 S 2 equals the closed topological string partition function of X Gopakumar, Vafa Dijkgraaf, Vafa,Verlinde Z M (X,q) = Z top (X,λ) Here, around the S 1,S 2 z 1, z 2 are generators of rotations planes, and q = e λ The Mtheory trace corresponds to trace around the factor in S 1 (X 2 S 1 ) q
18 Open topological string, with boundaries on arizes by putting M5 branes on L X L S 1 in X 2 S 1
19 In particular, if we take X = T * M SU(N) ChernSimons theory on M corresponds Mtheory on (T * M 2 S 1 ) q with N M5 branes on where corresponds to one of the two factors in 2
20 The partition function of N M5 branes in this geometry Z M 5 (M,q) = Tr ( 1) F q S 1 S 2 OoguriVafa equals the SU(N) ChernSimons partition function on M Z M 5 (M,q) = Z CS (M ) S 1,S 2 2πi k + N Here q = e is related to the level k of ChernSimons and are generators of rotations around the z 1,z 2 planes.
21 The refined topological string theory is defined via Mtheory on a slightly more general background, Nekrasov Nekrasov, Okounkov... (X 2 S 1 ) q,t where as one goes around the, the complex coordinates of 2 S 1 rotate by
22 To preserve supersymmetry of the trace, in the refined theory we need an additional U(1) symmetry that acts on the supersymmetry generators Q defining the trace
23 When the theory on N M5 branes on preserves supersymmetry, its partition function Z M 5 (M,q,t) = Tr ( 1) F q S 1 S R t S R S 2 defines the partition function of the refined SU(N) ChernSimons theory on M. For q=t, this reduces to the index that computes ordinary ChernSimons partition function.
24 The extra U(1) symmetry exists when M is a Seifert manifold S 1 M Σ The extra U(1) symmetry arizes as follows: consider a rank 2 subbundle of T^*M, corresponding to cotangent vectors orthogonal to the. Rotations of the fiber of this is the U(1) we need. S 1
25 We can introduce knots in the theory by introducing, for each knot K an additional Lagrangian and M5 branes wrapping L K S 1. The L K preserves the U(1) symmetry if K wraps the S 1 fibers. L K
26 Thus, when M is a Seifert three manifold, with Seifert knots, we can define refined ChernSimons theory as a supersymmetric partition function of M5 brane theory in Nekrasov background.
27 Moreover, just like ordinary ChernSimons theory, the refined ChernSimons theory is solvable exactly. Recall that, in a topological theory in three dimensions, all amplitudes, corresponding to any threemanifold and collection of knots in it, can be written in terms of three building blocks, the S and T matrices and the braiding matrix B.
28 In refined ChernSimons theory, only a subset of amplitudes generated by S and T enter, since only those preserve the U(1) symmetry.
29 S The and T matrices of a 3d topological QFT, provide a unitary representation of the action of the modular group SL(2, Z) on the Hilbert space of the theory on T. 2
30 A basis of Hilbert space is obtained by considering the path integral on a solid torus, obtained by shrinking the (1,0) cycle of the boundary, with Wilson loops in representations inserted in its interior, along the (0,1) cycle of the torus. R i
31 An element of SL(2,Z) acting on the boundary of the torus, permutes the states in The matrix element has interpretation as the path integral on a three manifold obtained by gluing two solid tori, with knots colored by and inside, R i R j up to the SL(2,Z) transformation of their boundaries.
32 From S and T, by cutting and gluing rules of a 3d TQFT one can compute any amplitude for arbitrary Seifert knots in Seifert manifolds.
33 In ChernSimons theory with gauge group G and level k, the Hilbert space is the space of conformal blocks of the corresponding WZW model. The S, T are the S, T matrices of the G k WZW model This allows one solve the theory explicitly, and in particular obtain the Jones and other knot polynomials.
34 Explicitly, for G=SU(N), the S and the T matrices are given by where s R (x) = Tr R x are Schur functions. For q = e 2πi k + N this generates a unitary representation of SL(2,Z) in the Hilbert space labeled by SU(N) representations at level k.
35 The S and T matrices of refined ChernSimons theory are computable explicitly from Mtheory, for any A, D, E gauge group. They depend on q and one additional parameter, t = q β
36 In the refined SU(N) ChernSimons theory, t = q β S 00 = c N 1 n=0 i=1 (q n/2 t i/2 q n/2 t i/2 ) N i (q n/2 t (i+1)/2 q n/2 t (i+1)/2 ) N i where M R (x i,q,t) are the Macdonald polynomials. For q = e 2πi k +βn this gives a unitary representation of SL(2,Z) in the Hilbert space is labeled by SU(N) representations at level k.
37 Similar formulas hold for any ADE group where and where y is the dual Coxeter number of G.
38 The fact that these provide a unitary representation of SL(2,Z) was proven in the 90s by Ivan Cherednik.
39 The relation of refined topological string to knot homologies, leads us to conjecture the following.
40 Let M be a Seifert threemanifold with a collection of (linked) knots wrapping the S^1 fiber. 1) For any G=A,D, E, the colored knot homology should admit and additional grade H ij = k H ijk which allows one to define a refined index ( 1) k q i t j k dim H ijk i,j,k such that setting t = 1 one gets ChernSimons invariants back. 2) The index is computable explicitly in terms of S and T matrices of the refined ChernSimons theory with the corresponding group.
41 The idea of the derivation is to use the M theory definition and topological invariance to solve the theory on the simplest piece, the solid torus M L = D S 1 where D is a disk. S,T and everything else can be recovered from this by gluing.
42 For separated M5 branes, the Mtheory index Z M 5 (M,q,t) = Tr ( 1) F q S 1 S R t S R S 2 can be computed by counting cohomologies of the moduli of holomorphic curves embedded in Y, endowed with a flat U(1) bundle, and with boundaries on the M5 branes. Gopakumar and Vafa Ooguri and Vafa
43 In the present case, all the holomorphic curves are annuli, which are isolated. The moduli of the flat bundle is an S 1 One gets precisely two cohomology classes. From the Mtheory perspective, the simplicity of the spectrum is the key to solvability of the theory.
44 The partition function on the solid torus, D S 1 is a wavefunction, depending on the holonomy around the S 1. x I 0 x = Δ q,t (x) = 1 I < J N n=1 (q n/2 e (x J x I )/2 q n/2 e (x I x J )/2 ) (q n/2 t 1/2 e (x J x I )/2 q n/2 t 1/2 e (x I x J )/2 )
45 This implies that line operators inserting wilson loops in a solid torus x R i = x O Ri 0 = Δ q,t (x)m Ri (e x ) R i and giving a basis for are given by Macdonald polynomials. In this basis, the identity operator acting on has the natural matrix elements corresponding to Z(S 2 S 1, R i, R j ) R i R j = d N x Δ q, t (x) 2 M Ri (e x )M Rj (e x ) = 1 ij This equals 1 or 0 depending on whether R i, j are equal or not.
46 Obtaining the S and T matrices from solving the theory on two solid tori M L = D S 1, and gluing the pieces back together, lifts in Mtheory to solving the theory on T * M L = 2 * with N M5 branes on M L and gluing.
47 One can give the S 1 fibration over S 2 first Chern class, by multiplying one of two wave functions by exp( ptr x 2 / 2ε 1 ) before gluing. p For example, viewed this way, is the Hopf fibration, with. S 3 p = 1
48 The Mtheory partition function on is computing the TST matrix element corresponding to refined ChernSimons theory on a framed S^3. R i TST R j = d N From Mtheory, we obtain T * S 3 The fact that this takes the explicit form we gave before 1 x Δ q,t (x) 2 e Tr x 2 2ε 1 M Ri (e x )M Rj (e x ) was proven in the 90 s in the context of proving Macdonald Conjectures Cherednik, Kirillov, Etingof
49 The theory also generalizes Verlinde algebra. The algebra of line operators k O Ri O Rj = N ij k has coefficients O Rk N ijk =
50 They satisfy the Verlinde formula, as required by threedimensional topological invariance.
51 The refined ChernSimons theory connects to the recent work of Witten on obtaining knot homologies from string theory. Witten studies M5 branes in Mtheory on the same geometry, M S 1 inside T * M 2 S 1, but for a general three manifold M, for which the refined index need not exist.
52 The index Z M 5 (M,q,t) = Tr H ( 1) F q S 1 S R t S R S 2 is simple to compute, when it exists: as a partition function, it is computable by cutting and gluing. This is the case when the theory has an extra U(1) symmetry, and the Hilbert space has an extra grade S R associated to it.
53 By contrast, the Poincare polynomial of the theory of N M5 branes wrapping M in, X = T * M Tr H Q q S 1 t S 2 is defined for any three manifold, and any collection of knots, but it is hard to compute in general, since one needs to know the cohomology the supersymmetry operator Q. H Q of
54 The refined SU(N) ChernSimons theory has a large N dual description in terms of the refined topological string on a different manifold. To begin with, recall how this works in the case of the ordinary ChernSimons theory.
55 The SU(N) ChernSimons theory on S 3, or equivalently, the ordinary open topological string on X = T * S 3 is dual to closed topological string theory on X V = O( 1) O( 1) P 1
56 The large N duality is a geometric transition that shrinks the S 3 and grows the P 1 while eliminating boundaries SU(N) SU(N) ChernSimons theory on S 3, or open topological string on GromovWitten theory on X V = O( 1) O( 1) P 1 X = T * S 3 with P 1 of size Nλ
57 OoguriVafa Adding a knot on K S 3 corresponds to adding a Lagrangian in, and geometric transition relates this to a Lagrangian in. L K X L K X V X = T * S 3 X V = O( 1) O( 1) P 1
58 The large N duality of ChernSimons theory implies topological string on OoguriVafa 99 X V = O( 1) O( 1) P 1 with boundaries on L_K computes colored HOMFLY polynomial. Relating this to Mtheory on this space and the Mtheory index explains the integrality of the HOMFLY polynomial.
59 The large N duality extends to the refined case: e.g. the partition function of the refined SU(N) ChernSimons theory on equals the partition function of the refined topological string on S 3 X Z CS (q,t, N) = Z rtop (X,q,t,a) a = t N t 1/2 q 1/2
60 The duality relates computing knot invariants of the refined ChernSimons theory on to computing the refined index Z M 5 (L K, X,q,t) = Tr ( 1) F q S 1 S R t S R S 2 S 3 of M5 branes wrapping (L K S 1 ) q,t after the transition, in Mtheory on (X 2 S 1 ) q,t
61 Gukov, Vafa and Schwarz proposed that keeping track of the two gradings S_1, S_2 in Mtheory instead, corresponds to categorification of the colored HOMFLY polynomial.
62 A triply graded knot homology theory categorifying the HOMFLY polynomial was constructed by Dunfield, Rassmussen and Gukov, (the colored versions were studied recently by Gukov et. al.) The Poincare polynomial of this theory is the superpolynomial Setting t=1, this reduces to the HOMFLY polynomial H(K,q,a). Taking the cohomology with respect to a certain differential d_n and setting a=q^n, this reduces to knot homology theory of Khovanov and Rozansky, categorifying the SL(N) knot invariants.
63 While refined ChernSimons theory is computing an index, in many cases this index agrees with the Poincare polynomial of the theory DGR constructed.
64 The invariant of an (n, m) torus knot in S 3 is computed from refined ChernSimons by the TQFT rules where K represents an element of SL(2, ) that takes the (0,1) cycle into (n, m) cycle
65 There is strong evidence that, for all torus knots K, the refined SU(N) ChernSimons invariant colored by the fundamental representation, computes the superpolynomial of the knot homology theory categorifying the HOMFLY polynomial when written in terms of q = t, t = q / t and a = q N / t
66 There also seems to persist, for all representations colored by knots in symmetric or antisymmetric representations.
67 Our results provide support for the conjecture of Gukov, Vafa and Schwarz, that the knot homologies of a triply graded homology theory categorifying the HOMFLY polynomial are, up to regrading, the spaces of supersymmetric ground states in Mtheory, with M5 branes on X = O( 1) O( 1) P 1 L K graded by the spins S 1,S and their charge in H 2 2 (X, ).
68 Some structural properties of the answer follow from its Mtheory definition.
69 Topological string is computing a particular combination of ChernSimons knot observables det(1 U V ) 1 = Tr SU (N ) R (U) q Tr SU (N ) R (V ) R
70 In the refined topological string, the story is richer we have to specify which branes we are studying: whether they wrap or plane q t 1 in Mtheory on (X 2 S 1 ) q,t We will call these the q and the tbranes.
71 Depending on the kinds of branes, the topological string is computing different refined ChernSimons amplitudes. n=0 det(1 q n U V ) det(1 tq n U V ) det(1 U V ) SU ( N ) t 1 = M R (U) SU (N )q M R (V ) R G R = ˆM R (U) ( 1) R M SU ( N ) R T (V ) t q q t q R q q
72 a = t N t / q a = t N t / q 1 M R (U) SU (N )q M R (V ) R G R ˆM R T (U) ( 1) R M SU ( N ) R (V ) t R a However, at large N, the branes on the hence we get a prediction S 3 disappear, Z R1 R n (K 1, K n,a,q,t) / G R1 G Rn = ( 1) R 1+ +R n Z R T 1 R T n (K 1, K n,a,t 1,q 1 )
73 The refined topological string on a large class of toric CalabiYau manifolds is captured by the combinatorics of the refined topological vertex, analogously to the ordinary topological string. Cozacs, Iqbal, Vafa Gukov, Iqbal, Vafa X = O( 1) O( 1) P 1 Cozacs, Iqbal It was proven recently that the refined topological vertex predictions for the Hopflink colored by arbitrary representations, S ij ee with the Smatrix amplitude of the refined ChernSimons theory, at large N, as predicted by large N duality.
74 One can show that similarly, the refined topological vertex Aganagic, Shakirov follows from refined ChernSimons theory, just as the original derivation of the ordinary topological vertex came from ChernSimons theory, using large N dualities. Aganagic, Klemm, Marino, Vafa 03
75
Knot Homology from Refined ChernSimons Theory
Knot Homology from Refined ChernSimons 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
More informationKnots 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
More informationKnot invariants, Apolynomials, BPS states
Knot invariants, Apolynomials, BPS states Satoshi Nawata Fudan University collaboration with Rama, Zodin, Gukov, Saberi, Stosic, Sulkowski ChernSimons theory Witten M A compact 3manifold gauge field
More informationRefined ChernSimons Theory. and. Topological String
Refined ChernSimons Theory and Topological String arxiv:1210.2733v1 [hepth] 9 Oct 2012 Mina Aganagic 1,2 and Shamil Shakirov 1,2,3 1 Center for Theoretical Physics, University of California, Berkeley,
More informationKnot polynomials, homological invariants & topological strings
Knot polynomials, homological invariants & topological strings Ramadevi Pichai Department of Physics, IIT Bombay Talk at STRINGS 2015 Plan: (i) Knot polynomials from ChernSimons gauge theory (ii) Our
More informationTopological Strings, DModel, and Knot Contact Homology
arxiv:submit/0699618 [hepth] 21 Apr 2013 Topological Strings, DModel, 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 informationVolume Conjecture: Refined and Categorified
Volume Conjecture: Refined and Categorified Sergei Gukov based on: hepth/0306165 (generalized volume conjecture) with T.Dimofte, arxiv:1003.4808 (review/survey) with H.Fuji and P.Sulkowski, arxiv:1203.2182
More informationKnot Contact Homology, ChernSimons Theory, and Topological Strings
Knot Contact Homology, ChernSimons Theory, and Topological Strings Tobias Ekholm Uppsala University and Institute MittagLeffler, Sweden Symplectic Topology, Oxford, Fri Oct 3, 2014 Plan Reports on joint
More informationElements of Topological MTheory
Elements of Topological MTheory (with R. Dijkgraaf, S. Gukov, C. Vafa) Andrew Neitzke March 2005 Preface The topological string on a CalabiYau threefold X is (loosely speaking) an integrable spine of
More information(Quantum) SuperApolynomial
Piotr Suªkowski UvA (Amsterdam), Caltech (Pasadena), UW (Warsaw) String Math, Bonn, July 2012 1 / 27 Familiar curves (SeibergWitten, mirror, Apolynomial)......typically carry some of the following information:
More informationUC Santa Barbara UC Santa Barbara Previously Published Works
UC Santa Barbara UC Santa Barbara Previously Published Works Title Link Homologies and the Refined Topological Vertex Permalink https://escholarship.org/uc/item/075890 Journal Communications in Mathematical
More informationTopological String Theory
Topological String Theory Hirosi Ooguri (Caltech) Strings 2005, Toronto, Canada If String Theory is an answer, what is the question? What is String Theory? If Topological String Theory is an answer, what
More informationFree fermion and wallcrossing
Free fermion and wallcrossing Jie Yang School of Mathematical Sciences, Capital Normal University yangjie@cnu.edu.cn March 4, 202 Motivation For string theorists It is called BPS states counting which
More informationEnumerative Invariants in Algebraic Geometry and String Theory
Dan Abramovich . Marcos Marino Michael Thaddeus Ravi Vakil Enumerative Invariants in Algebraic Geometry and String Theory Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy June 611,
More informationChernSimons Theory & Topological Strings
1 ChernSimons Theory & Topological Strings Bonn August 6, 2009 Albrecht Klemm 2 Topological String Theory on CalabiYau manifolds Amodel & Integrality Conjectures Bmodel Mirror Duality & Large NDuality
More informationTopological Strings and DonaldsonThomas invariants
Topological Strings and DonaldsonThomas invariants University of Patras Πανɛπιστήµιo Πατρών RTN07 Valencia  Short Presentation work in progress with A. Sinkovics and R.J. Szabo Topological Strings on
More informationCHERNSIMONS THEORY AND LINK INVARIANTS! Dung Nguyen! U of Chicago!
CHERNSIMONS THEORY AND LINK INVARIANTS! Dung Nguyen! U of Chicago! Content! Introduction to Knot, Link and Jones Polynomial! Surgery theory! Axioms of Topological Quantum Field Theory! Wilson loopsʼ expectation
More informationRefined DonaldsonThomas theory and Nekrasov s formula
Refined DonaldsonThomas theory and Nekrasov s formula Balázs Szendrői, University of Oxford Maths of String and Gauge Theory, City University and King s College London 35 May 2012 Geometric engineering
More informationChernSimons gauge theory The ChernSimons (CS) gauge theory in three dimensions is defined by the action,
Lecture A3 ChernSimons gauge theory The ChernSimons (CS) gauge theory in three dimensions is defined by the action, S CS = k tr (AdA+ 3 ) 4π A3, = k ( ǫ µνρ tr A µ ( ν A ρ ρ A ν )+ ) 8π 3 A µ[a ν,a ρ
More informationG 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 informationTopological Matter, Strings, Ktheory and related areas September 2016
Topological Matter, Strings, Ktheory and related areas 26 30 September 2016 This talk is based on joint work with from Caltech. Outline 1. A string theorist s view of 2. Mixed Hodge polynomials associated
More informationDuality ChernSimons CalabiYau and Integrals on Moduli Spaces. Kefeng Liu
Duality ChernSimons CalabiYau and Integrals on Moduli Spaces Kefeng Liu Beijing, October 24, 2003 A. String Theory should be the final theory of the world, and should be unique. But now there are Five
More informationDualities and Topological Strings
Dualities and Topological Strings Strings 2006, Beijing  RD, C. Vafa, E.Verlinde, hepth/0602087  work in progress w/ C. Vafa & C. Beasley, L. Hollands Robbert Dijkgraaf University of Amsterdam Topological
More informationTHE MASTER SPACE OF N=1 GAUGE THEORIES
THE MASTER SPACE OF N=1 GAUGE THEORIES Alberto Zaffaroni CAQCD 2008 Butti, Forcella, Zaffaroni hepth/0611229 Forcella, Hanany, Zaffaroni hepth/0701236 Butti,Forcella,Hanany,Vegh, Zaffaroni, arxiv 0705.2771
More informationSuperApolynomials of Twist Knots
SuperApolynomials of Twist Knots joint work with Ramadevi and Zodinmawia to appear soon Satoshi Nawata Perimeter Institute for Theoretical Physics Aug 28 2012 Satoshi Nawata (Perimeter) SuperApoly
More informationString Duality and Localization. Kefeng Liu UCLA and Zhejiang University
String Duality and Localization Kefeng Liu UCLA and Zhejiang University Yamabe Conference University of Minnesota September 18, 2004 String Theory should be the final theory of the world, and should be
More informationTopological Holography and Chiral Algebras. Work in progress with Kevin Costello
Topological Holography and Chiral Algebras Work in progress with Kevin Costello General Motivations Topological twist of sugra/superstrings as holographic dual of twisted SQFT (Costello) Topological Holography:
More informationRemarks on ChernSimons Theory. Dan Freed University of Texas at Austin
Remarks on ChernSimons Theory Dan Freed University of Texas at Austin 1 MSRI: 1982 2 Classical ChernSimons 3 Quantum ChernSimons Witten (1989): Integrate over space of connections obtain a topological
More informationLocalization and Conjectures from String Duality
Localization and Conjectures from String Duality Kefeng Liu June 22, 2006 Beijing String Duality is to identify different theories in string theory. Such identifications have produced many surprisingly
More informationCOUNTING BPS STATES IN CONFORMAL GAUGE THEORIES
COUNTING BPS STATES IN CONFORMAL GAUGE THEORIES Alberto Zaffaroni PISA, MiniWorkshop 2007 Butti, Forcella, Zaffaroni hepth/0611229 Forcella, Hanany, Zaffaroni hepth/0701236 Butti,Forcella,Hanany,Vegh,
More informationString Duality and Moduli Spaces
String Duality and Moduli Spaces Kefeng Liu July 28, 2007 CMS and UCLA String Theory, as the unified theory of all fundamental forces, should be unique. But now there are Five different looking string
More informationLinear 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)
More information1 Electrons on a lattice, with noisy electric field
IHESP/05/34 XXIII Solvay Conference Mathematical structures: On string theory applications in condensed matter physics. Topological strings and two dimensional electrons Prepared comment by Nikita Nekrasov
More informationTopics 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 information5d SCFTs and instanton partition functions
5d SCFTs and instanton partition functions Hirotaka Hayashi (IFT UAMCSIC) HeeCheol Kim and Takahiro Nishinaka [arxiv:1310.3854] Gianluca Zoccarato [arxiv:1409.0571] Yuji Tachikawa and Kazuya Yonekura
More informationTwo 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 informationFactorization Algebras Associated to the (2, 0) Theory IV. Kevin Costello Notes by Qiaochu Yuan
Factorization Algebras Associated to the (2, 0) Theory IV Kevin Costello Notes by Qiaochu Yuan December 12, 2014 Last time we saw that 5d N = 2 SYM has a twist that looks like which has a further Atwist
More informationGeometry, Quantum Topology and Asymptotics
Geometry, Quantum Topology and Asymptotics List of Abstracts Jørgen Ellegaard Andersen, Aarhus University The Hitchin connection, degenerations in Teichmüller space and SYZmirror symmetry In this talk
More informationTowards a modular functor from quantum higher Teichmüller theory
Towards a modular functor from quantum higher Teichmüller theory Gus Schrader University of California, Berkeley ld Theory and Subfactors November 18, 2016 Talk based on joint work with Alexander Shapiro
More informationGeneralized 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
More informationKnot invariants, Hilbert schemes and Macdonald polynomials (joint with A. Neguț, J. Rasmussen)
Knot invariants, Hilbert schemes and Macdonald polynomials (joint with A. Neguț, J. Rasmussen) Eugene Gorsky University of California, Davis University of Southern California March 8, 2017 Outline Knot
More informationTopological reduction of supersymmetric gauge theories and Sduality
Topological reduction of supersymmetric gauge theories and Sduality Anton Kapustin California Institute of Technology Topological reduction of supersymmetric gauge theories and Sduality p. 1/2 Outline
More informationΩdeformation and quantization
. Ωdeformation and quantization Junya Yagi SISSA & INFN, Trieste July 8, 2014 at Kavli IPMU Based on arxiv:1405.6714 Overview Motivation Intriguing phenomena in 4d N = 2 supserymmetric gauge theories:
More informationDisk Instantons, Mirror Symmetry and the Duality Web
HUTP01/A023 HUEP01/21 hepth/0105045 Disk Instantons, Mirror Symmetry and the Duality Web Mina Aganagic 1, Albrecht Klemm 2 and Cumrun Vafa 1 1 Jefferson Physical Laboratory Harvard University, Cambridge,
More informationNew worlds for Lie Theory. math.columbia.edu/~okounkov/icm.pdf
New worlds for Lie Theory math.columbia.edu/~okounkov/icm.pdf Lie groups, continuous symmetries, etc. are among the main building blocks of mathematics and mathematical physics Since its birth, Lie theory
More informationQuantum Entanglement of Baby Universes
CALT686 NSFKITP06118 Quantum Entanglement of Baby Universes arxiv:hepth/061067 v1 7 Dec 006 Mina Aganagic, 1 Hirosi Ooguri, and Takuya Okuda 3 1 University of California, Berkeley, CA 9470, USA California
More informationDavid 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
More informationFtheory effective physics via Mtheory. Thomas W. Grimm!! Max Planck Institute for Physics (WernerHeisenbergInstitut)! Munich
Ftheory effective physics via Mtheory Thomas W. Grimm Max Planck Institute for Physics (WernerHeisenbergInstitut) Munich Ahrenshoop conference, July 2014 1 Introduction In recent years there has been
More informationThe 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 informationInstanton effective action in  background and D3/D(1)brane system in RR background
Instanton effective action in  background and D3/D(1)brane system in RR background Speaker : Takuya Saka (Tokyo Tech.) Collaboration with Katsushi Ito, Shin Sasaki (Tokyo Tech.) And Hiroaki Nakajima
More informationOn the Virtual Fundamental Class
On the Virtual Fundamental Class Kai Behrend The University of British Columbia Seoul, August 14, 2014 http://www.math.ubc.ca/~behrend/talks/seoul14.pdf Overview DonaldsonThomas theory: counting invariants
More informationInstantons and Donaldson invariants
Instantons and Donaldson invariants George Korpas Trinity College Dublin IFT, November 20, 2015 A problem in mathematics A problem in mathematics Important probem: classify dmanifolds up to diffeomorphisms.
More informationIntegrable spin systems and fourdimensional gauge theory
Integrable spin systems and fourdimensional gauge theory Based on 1303.2632 and joint work with Robbert Dijkgraaf, Edward Witten and Masahito Yamizaki Perimeter Institute of theoretical physics Waterloo,
More informationTwo Dimensional YangMills, Black Holes. and Topological Strings
hepth/0406058 HUTP04/A026 Two Dimensional YangMills, arxiv:hepth/0406058v2 7 Jul 2004 Black Holes and Topological Strings Cumrun Vafa Jefferson Physical Laboratory, Harvard University, Cambridge, MA
More informationarxiv:hepth/ v4 9 May 2005
ChernSimons Theory and Topological Strings Marcos Mariño Department of Physics, CERN, Geneva 3, Switzerland We review the relation between ChernSimons gauge theory and topological string theory on noncompact
More informationNUMBER THEORY IN STRING THEORY: SKEW, MOCK, FALSE, AND QUANTUM
NUMBER THEORY IN STRING THEORY: SKEW, MOCK, FALSE, AND QUANTUM Sarah M. Harrison based on work with Cheng, Duncan, Harvey, Kachru, Rayhaun ( 17) and Cheng, Chun, Ferrari, Gukov (to appear) INTRODUCTION
More informationLarge N Duality, Lagrangian Cycles, and Algebraic Knots
Large N Duality, Lagrangian Cycles, and Algebraic Knots The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher
More informationTorus Knots and q, tcatalan Numbers
Torus Knots and q, tcatalan Numbers Eugene Gorsky Stony Brook University Simons Center For Geometry and Physics April 11, 2012 Outline q, tcatalan numbers Compactified Jacobians Arc spaces on singular
More informationFour Lectures on Web Formalism and Categorical WallCrossing. collaboration with Davide Gaiotto & Edward Witten
Four Lectures on Web Formalism and Categorical WallCrossing Lyon, September 35, 2014 Gregory Moore, Rutgers University collaboration with Davide Gaiotto & Edward Witten draft is ``nearly finished Plan
More informationDisk Instantons, Mirror Symmetry and the Duality Web
Disk Instantons, Mirror Symmetry and the Duality Web Mina Aganagic, Albrecht Klemm a, and Cumrun Vafa Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA a Institut für Physik,
More informationOn 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 informationA wallcrossing formula for 2d4d DT invariants
A wallcrossing formula for 2d4d DT invariants Andrew Neitzke, UT Austin (joint work with Davide Gaiotto, Greg Moore) Cetraro, July 2011 Preface In the last few years there has been a lot of progress
More informationarxiv: v2 [hepth] 25 Aug 2017
Topological ChernSimons/Matter Theories arxiv:1706.09977v2 [hepth] 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 informationLocalization and Conjectures from String Duality
Localization and Conjectures from String Duality Kefeng Liu January 31, 2006 Harvard University Dedicated to the Memory of Raoul Bott My master thesis is about Bott residue formula and equivariant cohomology.
More informationGeometry and Physics. Amer Iqbal. March 4, 2010
March 4, 2010 Many uses of Mathematics in Physics The language of the physical world is mathematics. Quantitative understanding of the world around us requires the precise language of mathematics. Symmetries
More informationOn Flux Quantization in FTheory
On Flux Quantization in FTheory Raffaele Savelli MPI  Munich Bad Honnef, March 2011 Based on work with A. Collinucci, arxiv: 1011.6388 Motivations Motivations The recent attempts to find UVcompletions
More informationHiggs 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
More informationGauge Theory and Mirror Symmetry
Gauge Theory and Mirror Symmetry Constantin Teleman UC Berkeley ICM 2014, Seoul C. Teleman (Berkeley) Gauge theory, Mirror symmetry ICM Seoul, 2014 1 / 14 Character space for SO(3) and Toda foliation Support
More informationAlgebraic structure of the IR limit of massive d=2 N=(2,2) theories. collaboration with Davide Gaiotto & Edward Witten
Algebraic structure of the IR limit of massive d=2 N=(2,2) theories IAS, October 13, 2014 Gregory Moore, Rutgers University collaboration with Davide Gaiotto & Edward Witten draft is ``nearly finished
More information2d4d wallcrossing and hyperholomorphic bundles
2d4d wallcrossing and hyperholomorphic bundles Andrew Neitzke, UT Austin (work in progress with Davide Gaiotto, Greg Moore) DESY, December 2010 Preface Wallcrossing is an annoying/beautiful phenomenon
More informationSpectral Networks and Their Applications. Caltech, March, 2012
Spectral Networks and Their Applications Caltech, March, 2012 Gregory Moore, Rutgers University Davide Gaiotto, o, G.M., Andy Neitzke e Spectral Networks and Snakes, pretty much finished Spectral Networks,
More informationPossible Advanced Topics Course
Preprint typeset in JHEP style  HYPER VERSION Possible Advanced Topics Course Gregory W. Moore Abstract: Potential List of Topics for an Advanced Topics version of Physics 695, Fall 2013 September 2,
More informationPolynomial Structure of the (Open) Topological String Partition Function
LMUASC 57/07 Polynomial Structure of the (Open) Topological String Partition Function Murad Alim a and Jean Dominique Länge a arxiv:0708.2886v2 [hepth] 9 Oct 2007 a Arnold Sommerfeld Center for Theoretical
More informationarxiv: v1 [math.gt] 4 May 2018
The action of full twist on the superpolynomial for torus knots Keita Nakagane Abstract. We show, using Mellit s recent results, that Kálmán s full twist formula for the HOMFLY polynomial can be generalized
More informationKhovanov Homology And Gauge Theory
hepth/yymm.nnnn Khovanov Homology And Gauge Theory Edward Witten School of Natural Sciences, Institute for Advanced Study Einstein Drive, Princeton, NJ 08540 USA Abstract In these notes, I will sketch
More informationI. Why Quantum Ktheory?
Quantum groups and Quantum Ktheory Andrei Okounkov in collaboration with M. Aganagic, D. Maulik, N. Nekrasov, A. Smirnov,... I. Why Quantum Ktheory? mathematical physics mathematics algebraic geometry
More informationTechniques 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
More informationCounting surfaces of any topology, with Topological Recursion
Counting surfaces of any topology, with Topological Recursion 1... g 2... 3 n+1 = 1 g h h I 1 + J/I g 1 2 n+1 Quatum Gravity, Orsay, March 2013 Contents Outline 1. Introduction counting surfaces, discrete
More informationTopological Field Theory Amplitudes for A N 1 Fibration
Topological Field Theory Amplitudes for A N 1 Fibration arxiv:1507.066v [hepth] 5 Oct 015 Amer Iqbal, a,b,c Ahsan Z. Khan a Babar A. Qureshi a Khurram Shabbir d Muhammad A. Shehper a a Department of Physics,
More informationChernSimons Theory and Its Applications. The 10 th Summer Institute for Theoretical Physics KiMyeong Lee
ChernSimons Theory and Its Applications The 10 th Summer Institute for Theoretical Physics KiMyeong Lee Maxwell Theory Maxwell Theory: Gauge Transformation and Invariance Gauss Law Charge Degrees of
More informationEnumerative Geometry: from Classical to Modern
: from Classical to Modern February 28, 2008 Summary Classical enumerative geometry: examples Modern tools: GromovWitten invariants counts of holomorphic maps Insights from string theory: quantum cohomology:
More informationHomological mirror symmetry via families of Lagrangians
Homological mirror symmetry via families of Lagrangians StringMath 2018 Mohammed Abouzaid Columbia University June 17, 2018 Mirror symmetry Three facets of mirror symmetry: 1 Enumerative: GW invariants
More informationApplications of Macdonald Ensembles. Shamil Shakirov. A dissertation submitted in partial satisfaction of the. requirements for the degree of
Applications of Macdonald Ensembles by Shamil Shakirov A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division
More informationTopological vertex and quantum mirror curves
Topological vertex and quantum mirror curves Kanehisa Takasaki, Kinki University Osaka City University, November 6, 2015 Contents 1. Topological vertex 2. Onstrip geometry 3. Closed topological vertex
More informationarxiv:hepth/ v2 4 Dec 2006
Noncompact Topological Branes on Conifold Seungjoon Hyun and SangHeon Yi Department of Physics, College of Science, Yonsei University, Seoul 120749, Korea arxiv:hepth/0609037v2 4 Dec 2006 Abstract
More informationDisk Instantons, Mirror Symmetry and the Duality Web
HUTP01/A023 HUEP01/21 hepth/0105045 arxiv:hepth/0105045v1 4 May 2001 Disk Instantons, Mirror Symmetry and the Duality Web Mina Aganagic 1, Albrecht Klemm 2 and Cumrun Vafa 1 1 Jefferson Physical Laboratory
More informationHYPERKÄHLER 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
More informationAnomalies and SPT phases
Anomalies and SPT phases Kazuya Yonekura, Kavli IPMU Based on A review of [1508.04715] by Witten [1607.01873] KY [1609.?????] Yuji Tachikawa and KY Introduction One of the motivations: What is the most
More informationConstructing compact 8manifolds with holonomy Spin(7)
Constructing compact 8manifolds 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 informationThink Globally, Act Locally
Think Globally, Act Locally Nathan Seiberg Institute for Advanced Study Quantum Fields beyond Perturbation Theory, KITP 2014 Ofer Aharony, NS, Yuji Tachikawa, arxiv:1305.0318 Anton Kapustin, Ryan Thorngren,
More informationCHARACTERISTIC CLASSES
1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact
More informationBobby Samir Acharya Kickoff Meeting for Simons Collaboration on Special Holonomy
Particle Physics and G 2 manifolds Bobby Samir Acharya Kickoff Meeting for Simons Collaboration on Special Holonomy King s College London µf µν = j ν dϕ = d ϕ = 0 and ICTP Trieste Ψ(1 γ 5 )Ψ THANK YOU
More informationIntroduction Curves Surfaces Curves on surfaces. Curves and surfaces. Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway
Curves and surfaces Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway What is algebraic geometry? IMA, April 13, 2007 Outline Introduction Curves Surfaces Curves on surfaces
More informationHeterotic 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 informationOpen GromovWitten Invariants from the Augmentation Polynomial
S S symmetry Article Open GromovWitten Invariants from the Augmentation Polynomial Matthe Mahoald Department of Mathematics, Northestern University, Evanston, IL 60208, USA; m.mahoald@u.northestern.edu
More informationEric Perlmutter, DAMTP, Cambridge
Eric Perlmutter, DAMTP, Cambridge Based on work with: P. Kraus; T. Prochazka, J. Raeymaekers ; E. Hijano, P. Kraus; M. Gaberdiel, K. Jin TAMU Workshop, Holography and its applications, April 10, 2013 1.
More informationBlack Hole Microstate Counting using Pure Dbrane Systems
Black Hole Microstate Counting using Pure Dbrane Systems HRI, Allahabad, India 11.19.2015 UC Davis, Davis based on JHEP10(2014)186 [arxiv:1405.0412] and upcoming paper with Abhishek Chowdhury, Richard
More informationSurface Defects and the BPS Spectrum of 4d N=2 Theories
Surface Defects and the BPS Spectrum of 4d N=2 Theories Solvay Conference, May 19, 2011 Gregory Moore, Rutgers University Davide Gaiotto, G.M., Andy Neitzke Wallcrossing in Coupled 2d4d Systems: 1103.2598
More informationKnots and Physics. Louis H. Kauffman
Knots and Physics Louis H. Kauffman http://front.math.ucdavis.edu/author/l.kauffman Figure 1  A knot diagram. I II III Figure 2  The Reidemeister Moves. From Feynman s Nobel Lecture The character of
More informationA Landscape of Field Theories
A Landscape of Field Theories Travis Maxfield Enrico Fermi Institute, University of Chicago October 30, 2015 Based on arxiv: 1511.xxxxx w/ D. Robbins and S. Sethi Summary Despite the recent proliferation
More information