The structure of 2D semi-simple field theories

Teleman, Constantin

doi:10.1007/s00222-011-0352-5

Cited by 209 publications

(320 citation statements)

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“…According to the results of Givental and Teleman [13,35], a semisimple CohFT can be obtained via the action of an R-matrix on a Topological Field Theory; the result is an expression for the CohFT as a summation over graphs. A similar procedure works for R ct g,A , and it can be used to write Hain's formula as a graph sum.…”

Section: Pixton's Conjectural Formulamentioning

confidence: 99%

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Pixton’s double ramification cycle relations

Clader

Janda

2018

Geom. Topol.

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Section: Pixton's Conjectural Formulamentioning

confidence: 99%

“…The work of Givental and Teleman [12,35] implies that a semisimple CohFT can be expressed as Ω = R · ω,…”

Section: The Cohftsmentioning

confidence: 99%

Pixton’s double ramification cycle relations

Clader

Janda

2018

Geom. Topol.

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“…Examples of manifolds with semisimple quantum cohomology include Grassmanians and Fano toric manifolds. It was conjectured by Givental [56] and proved by Teleman [100] that if the quantum multiplication is semi-simple, then D X is given by a formula of the following type:…”

Section: N I=1mentioning

confidence: 99%

-constraints for the total descendant potential of a simple singularity

2013

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Abstract. Simple, or Kleinian, singularities are classified by Dynkin diagrams of type ADE. Let g be the corresponding finitedimensional Lie algebra, and W its Weyl group. The set of ginvariants in the basic representation of the affine Kac-Moody algebraĝ is known as a W-algebra and is a subalgebra of the Heisenberg vertex algebra F . Using period integrals, we construct an analytic continuation of the twisted representation of F . Our construction yields a global object, which may be called a W -twisted representation of F . Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest weight vector for the W-algebra.1. Introduction 1.1. Motivation from Gromov-Witten theory. Recall that the Gromov -Witten (GW) invariants of a projective manifold X consist of correlators (1.1) τ k 1 (v 1 ), . . . , τ kn (v n ) g,n,dwhere v 1 , . . . , v n ∈ H * (X; C) are cohomology classes and the enumerative meaning of the correlator is the following. Let C 1 , . . . , C n be n cycles in X in a sufficiently generic position that are Poincaré dual to v 1 , . . . , v n , respectively. Then the GW invariant (1.1) counts the number of genus-g, degree-d holomorphic curves in X that are tangent (in an appropriate sense) to the cycles C i with multiplicities k i . For the precise definition we refer to [103,75,8,83]. After A. Givental [57], we organize the GW invariants in a generating series D X called the total descendant potential of X and defined as follows. Choose a basis {v i } N i=1of the vector (super)space H = H * (X; C) and let t k = τ k 1 (t k 1 ), . . . , τ kn (t kn ) g,n,d , where t = (t 0 , t 1 , . . . ) = (t i k ) and the definition of the correlator is extended multi-linearly in its arguments. The function D X is interpreted as a formal power series in the variables t i k with coefficients formal Laurent series in whose coefficients are elements of the Novikov ring C [Q].When X is a point and hence d = 0, the potential D pt (also known as the partition function of pure gravity) is a generating function for certain intersection numbers on the Deligne-Mumford moduli space of Riemann surfaces M g,n . It was conjectured by Witten [103] and proved by Kontsevich [74] that D pt is a tau-function for the KdV hierarchy of soliton equations. (We refer to [19,102] for excellent introductions to soliton equations.) In addition, D pt satisfies one more constraint called the string equation, which together with the KdV hierarchy determines uniquely D pt (see [103]). It was observed in [22,54,69] that the taufunction of KdV satisfying the string equation is characterized as the unique solution of L n D pt = 0 for n ≥ −1, where L n are certain differential operators representing the Virasoro algebra. This means that D pt is a highest-weight vector for the Virasoro algebra and in addition satisfies the string equation L −1 D pt = 0.One of the fundamental open questions in Gromov-Witten theory is the Virasoro conjecture suggested by S. Katz and the physicists Eguchi, Hori, Xiong, and...

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“…It would be interesting to relate both expressions explicitly (see also [12,15,23]). Alternatively, one could combine Teleman's classification of semi simple cohomological field theories ( [37]) with Givental's results to deduce that this is the right expression for the solution. Nevertheless our result can be applied on other frameworks, as it will mentioned in §3.6).…”

Section: If the Matrix (D Ij ) Is Diagonal Thenmentioning

confidence: 99%

“…On the other hand, one also wants to know if the generating function is given by (the logarithm of) a particular tau-function of an integrable hierarchy. Nowadays, the case of varieties with semisimple quantum cohomology is well understood and the answer to both questions is affirmative ( [37] and [7]; see also [6,13,14,26,30,31]). …”

Section: Introductionmentioning

confidence: 99%

Virasoro and KdV

Martı́n¹,

Prieto²

2016

Lett Math Phys

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Abstract. We investigate the structure of representations of the (positive half of the) Virasoro algebra and situations in which they decompose as a tensor product of Lie algebra representations. As an illustration, we apply these results to the differential operators defined by the Virasoro conjecture and obtain some factorization properties of the solutions as well as a link to the multicomponent KP hierarchy.

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The structure of 2D semi-simple field theories

Cited by 209 publications

References 28 publications

Pixton’s double ramification cycle relations

Pixton’s double ramification cycle relations

-constraints for the total descendant potential of a simple singularity

Virasoro and KdV

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