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Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions
Abstract
We show that various holomorphic quantities in supersymmetric gauge theories
can be conveniently computed by configurations of D4-branes and D6-branes.
These D-branes intersect along a Riemann surface that is described by a
holomorphic curve in a complex surface. The resulting I-brane carries
two-dimensional chiral fermions on its world-volume. This system can be mapped
directly to the topological string on a large class of non-compact Calabi-Yau
manifolds. Inclusion of the string coupling constant corresponds to turning on
a constant B-field on the complex surface, which makes this space
non-commutative. Including all string loop corrections the free fermion theory
is elegantly formulated in terms of holonomic D-modules that replace the
classical holomorphic curve in the quantum case.
... In what follows, it will also be convenient to exploit the fact that there exists a natural collection of n line bundles L 1 , . . . , L n on TN n (see [62,63] for an explicit construction) with anti-selfdual curvature α i such that ...
... Chiral fermions. M5 branes on the spacetime T 2 × TN n are known to possess BPS degrees of freedom that are localized at the Taub-NUT center and give rise to chiral fermionic particles on T 2 [63]. These degrees of freedom can also be understood within the present Type IIB frame. ...
... In other words, let us look at the rank r SCFT at the superconformal fixed point, with no B-field turned on at the singularity. As detailed in [63], the fermions furnish a u(n r) 1 current algebra, and the symmetries supported on the two stacks of NS5 branes are visible in the conformal embedding ...
We compute the equivariant partition function of the six-dimensional M-string SCFTs on a background with the topology of a product of a two-dimensional torus and an ALE singularity. We determine the result by exploiting BPS strings probing the singularity, whose worldvolume theories we determine via a chain of string dualities. A distinguished feature we observe is that for this class of background the BPS strings’ worldsheet theories become relative field theories that are sensitive to finer discrete data generalizing to 6d the familiar choices of flat connections at infinity for instantons on ALE spaces. We test our proposal against a conjectural 6d \( \mathcal{N} \) = (1, 0) generalization of the Nekrasov master formula, as well as against known results on ALE partition functions in four dimensions.
... As will be derived in section 9, the long distance limit of this holographic system will involve a Chern-Simons theory in the AdS bulk direction. Constructions similar to this work can be found in [16,17]. Differences between our construction and those in [16,17] will be explained in section 9. ...
... Constructions similar to this work can be found in [16,17]. Differences between our construction and those in [16,17] will be explained in section 9. ...
... The second formulation of the problem uses a hint from representation theory, and leads to the expression of Ehr Qg (z) in one full sweep. The second approach is inspired from a duality relation constructed from string theory (see section 9 and ref [16,17]), without which it is not obvious how one can make a connection of Ehr Qg (z) to being solved by representation theory. ...
A bstract
The Hilbert space of level q Chern-Simons theory of gauge group G of the ADE type quantized on T ² can be represented by points that lie on the weight lattice of the Lie algebra g up to some discrete identifications. Of special significance are the points that also lie on the root lattice. The generating functions that count the number of such points are quasi-periodic Ehrhart polynomials which coincide with the generating functions of SU( q ) representation of the ADE subgroups of SU(2) given by the McKay correspondence. This coincidence has roots in a string/M theory construction where D3(M5)-branes are put along an ADE singularity. Finally, a new perspective on the McKay correspondence that involves the inverse of the Cartan matrices is proposed.
... , c N are diagonal matrices. These matrices correspond to finite order difference operators of the form (2). As we show in the next section, the partially factorized forms (39) and (41)-equivalently (1) as difference operators-of L andL −1 yield a reduction of the Toda hierarchy to the generalized Ablowitz-Ladik hierarchy. In this sense, (40) are initial conditions for a special solution of the generalized Ablowitz-Ladik hierarchy. ...
... . Thus (39) and (41) do not imply the trivial situation wherē L (0) is proportional to L (0) . ...
... Let us first confirm that the modified Lax equations (43) imply the Lax equations (24) of the Toda hierarchy for matrices L andL −1 defined as (39) and (41) suggest. ...
This paper addresses the issue of integrable structures in topological string theory on generalized conifolds. Open string amplitudes of this theory can be expressed as the matrix elements of an operator on the Fock space of 2D charged free fermion fields. The generating function of these amplitudes with respect to the product of two independent Schur functions becomes a tau function of the 2D Toda hierarchy. The associated Lax operators turn out to have a particular factorized form. This factorized form of the Lax operators characterizes a generalization of the Ablowitz–Ladik hierarchy embedded in the 2D Toda hierarchy. The generalized Ablowitz–Ladik hierarchy is thus identified as a fundamental integrable structure of topological string theory on the generalized conifolds.
... We review the M-theory framework that motivates our paper [13,16]. We begin with an overview of the general structure, and then discuss the special class of backgrounds that give rise to the examples we consider in this paper. ...
... Alternatively we can parametrize the T 3 -action in terms of three independent (global) ( 1 , 2 , 3 ) vtx 1 1 , 2 , 3 2 1 , − 2 , 3 + 2 2 3 − 1 , 2 , 3 + 2 1 4 − 1 , − 2 , 3 + 2 1 + 2 2 (6.15) If we denote by H v (v = 1, 2, 3, 4) the Hamiltonian at the fixed points we have 16) which are expressed in terms of global ( 1 , 2 , 3 ). These shifts are uniquely fixed by the compact P 1 's. ...
... (A. 16) We are interested in the toric geometry of the resolved A n−1 space, which can be realized as Kähler quotient C n+1 //U (1) n−1 with charge matrix Eq. (A.14) and moment maps |z α | 2 − 2|z α+1 | 2 + |z α+2 | 2 = t α , α = 1, 2, . . . , n − 1, (A. 17) where we have (n − 1) t's. ...
Motivated by M-theory, we study rank n K-theoretic Donaldson–Thomas theory on a toric threefold X . In the presence of compact four-cycles, we discuss how to include the contribution of D4-branes wrapping them. Combining this with a simple assumption on the (in)dependence on Coulomb moduli in the 7d theory, we show that the partition function factorizes and, when X is Calabi–Yau and it admits an ADE ruling, it reproduces the 5d master formula for the geometrically engineered theory on $$A_{n-1}$$ A n - 1 ALE space, thus extending the usual geometric engineering dictionary to $$n>1$$ n > 1 . We finally speculate about implications for instanton counting on Taub-NUT.
... The precise dictionary between the two descriptions is obtained by identifying the topological string partition function on the CY 3 with the supersymmetric index of the gauge theory, which is conjectured to capture the exact BPS content of its 5d SCFT completion [2]. More generally, the supersymmetric index of the gauge theory with surface defects is matched with the corresponding D-brane open topological string wave function [3,4]. The coupling constants and the moduli of the QFT arise from the geometric engineering as CY moduli parameters (Kähler and complex in the A and B-model picture, respectively). ...
... We see that the mutations for x 1 , x 2 do not involve the variables x 3 , x 4 , so that no limit is actually necessary, and in fact they are already in the form of mutations for the Kronecker subquiver. Further, these mutations preserve the limiting value of x 3 , x 4 . In fact, the choice of the subquiver is completely arbitrary: by this limiting procedure we can consider any of the Kronecker subquivers of the quiver in Fig. 2. The limit is less trivial on the q-Painlevé flow: ...
... It is possible to study the autonomous limit of this object, following the discussion of [19] for the tau functions. One sets s = e η/ , q= e , u = e a , (C. 4) sending → 0. In this limit, one can write the partition function as Z(a, , t) = exp 1 2 ∞ n=0 2n F n (a, t) , (C.5) so that we want to take the "semi-classical" (actually autonomous) → 0 limit of the dual partition functions, keeping the leading and first subleading term. Let us review the argument: in [7] it was shown that the saddle point of Z is the Seiberg-Witten A-period: let us denote this by a * . ...
We study the discrete flows generated by the symmetry group of the BPS quivers for Calabi–Yau geometries describing five-dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlevé equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with $$\tau $$ τ -functions of the cluster algebra associated to the quiver. We exemplify our construction in the case corresponding to five-dimensional SU (2) pure super Yang–Mills and $$N_f=2$$ N f = 2 on a circle.
... [15,16]). The twisted partition function is well-known to give a character of the 2D CFT [17][18][19], where a string theory interpretation is given in [20]. ...
... As mentioned in the introduction, the generating function (2.4) originates with the U(N ) instanton partition function on C 2 /Z n with a massless adjoint hypermultiplet in the case of 1 + 2 = 0 and pertains to the partition function of an N = 4 twisted Yang-Mills theory on C 2 /Z n [19] (see [20] for a string theory interpretation). ...
... and is considered to be described by N n free fermions (see below (3.22) and e.g. [20]). ...
We propose, following the AGT correspondence, how the \( {\mathcal{W}}_{N,n}^{\mathrm{para}} \) (n-th parafermion \( {\mathcal{W}}_N \) ) minimal model characters are obtained from the U(N ) instanton counting on ℂ2/ℤn with Ω-deformation by imposing specific conditions which remove the minimal model null states.A preprint version of the article is available at ArXiv.
... From the point of view of the string theory, a quantum structure behind the Gromov-Witten theory has been considered in a different way [3,33,34,71]. The string theoretical quantum structure emerges in the higher genus (open) string free energy in the topological A-model. ...
... , p n ) ∈ Z n , where each integer p i specifies how many times the i-th boundary of a world-sheet Riemann surface C g,n wraps around the one-cycle in L. t i ∈ C (i = 1, . . . , dim H 2 (Y )) denote the Kähler moduli parameters of .7) is interpreted as a quantum curve (see Sect. 5.2.3 as an example) [33,34]. We then find that the quantum curve arises from a hidden quantum mechanical system behind the open topological strings. ...
... The brane partition function obeys a q-difference equation known as the Schrödinger equation or the quantum curve 13 [33,34]. For the brane partition function Z Y N −1,n A-brane (x) in (5.15), one finds the q-difference equation (c.f. ...
In this article, a novel description of the hypergeometric differential equation found from Gel'fand-Kapranov-Zelevinsky's system (referred to GKZ equation) for Givental's $J$-function in the Gromov-Witten theory will be proposed. The GKZ equation involves a parameter $\hbar$, and we will reconstruct it as the WKB expansion from the classical limit $\hbar\to 0$ via the topological recursion. In this analysis, the spectral curve (referred to GKZ curve) plays a central role, and it can be defined as the critical point set of the mirror Landau-Ginzburg potential. Our novel description is derived via the duality relations of the string theories, and various physical interpretations suggest that the GKZ equation is identified with the quantum curve for the brane partition function in the cohomological limit. As an application of our novel picture for the GKZ equation, we will discuss the Stokes matrix for the equivariant $\mathbb{C}\textbf{P}^{1}$ model and the wall-crossing formula for the total Stokes matrix will be examined. And as a byproduct of this analysis we will study Dubrovin's conjecture for this equivariant model.
... Indeed, some of the boundary algebras above already appeared in the literature, having been inferred from computations of boundary anomalies and halfindices. Examples include some Neumann algebras of the form (1.10) for boundary conditions supporting 2d free fermions [29,50,51], and the WZW algebras on Dirichlet b.c. [31]. ...
... The bulk theories flow to topological Chern-Simons theories, with trivial algebras V of local operators (in cohomology); and the boundary conditions support chiral WZW models, with trivial centers and Sugawara stress tensors. Other simple examples include the boundary coset models discussed in [29,50,51] and [31,Sec. 7]. ...
We study the holomorphic twist of 3d N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{N}=2$$\end{document} gauge theories in the presence of boundaries, and the algebraic structure of bulk and boundary local operators. In the holomorphic twist, both bulk and boundary local operators form chiral algebras (a.k.a. vertex algebras). The bulk algebra is commutative, endowed with a shifted Poisson bracket and a “higher” stress tensor; while the boundary algebra is a module for the bulk, may not be commutative, and may or may not have a stress tensor. We explicitly construct bulk and boundary algebras for free theories and Landau–Ginzburg models. We construct boundary algebras for gauge theories with matter and/or Chern–Simons couplings, leaving a full description of bulk algebras to future work. We briefly discuss the presence of higher A-infinity like structures.
... Type IIA little strings of A N −1 type possess N = (1, 1) supersymmetry, whereas type IIB little strings of type A N −1 possess N = (2, 0) supersymmetry. The former little string can also arise through the decoupling limit of a stack of N N S5 branes in type IIB string theory and the latter as the decoupling limit [14,18,17,15, 1] of a stack of N NS5 branes in type IIA string theory. This dual description is a result of T-daulity that S 1 -compactified little strings enjoy. ...
... and its generating function can be expressed in terms of the weight 10 automorphic form Φ 10 (ρ, σ, ν) of Sp(2, Z) as [14] Z ...
We compute partition functions of the mass deformed multiple M5-branes theory on $K3\times T^2$ using the refined topological vertex formalism and the Borcherds lift. The seed of the lift is calculated by taking the universal part of the type IIb little string free energy of the CY3-fold $X_{N,1}$. We provide explicit modular covariant expressions, as expansions in the mass parameter $m$, of the genus two Siegel modular forms of the paramodular groups produced by the Borcherds lift of the first few seed functions. We also discuss the relation between genus-one free energy and Ray-Singer Torsion, and the automorphic properties of the latter.
... We are thus led to conclude that supersymmetric localization of the N = (2, 0) free tensor multiplet on R 4 , × C with a topological-holomorphic twist should be equivalent to supersymmetric localization N = 4 U(1) SYM R 4 , with the Donaldson-Witten twist. To arrive at the final result we can appeal to the results in [35] where it was argued that the theory resulting from this supersymmetric localization is the same as that of a chiral fermion which can be thought of as moving on the chiral algebra plane. While it will be nice to derive this result more rigorously, we will take it at face value and proceed to bosonize this chiral fermion and thus find a u(1) Kac-Moody algebra on C. ...
... More generally for N = (2, 0) theories of type g one should find a system of generalized para-fermions of type RCFT[A k−1 , g] ⊕ RCFT[g, A k−1 ]. Indeed, this appears to be in line with the discussion in [35], where the localization of the four-dimensional theory resulting from the abelian N = (2, 0) theory was studied. In section 4.3 we discuss the results in [24] which provide supporting evidence for this conjecture by computing the chiral algebra central charge via an explicit equivariant integration of the anomaly polynomial of the 6d N = (2, 0) SCFT of type g. ...
A bstract
Every six-dimensional $$ \mathcal{N} $$ N = (2 , 0) SCFT on R ⁶ contains a set of protected operators whose correlation functions are controlled by a two-dimensional chiral algebra. We provide an alternative construction of this chiral algebra by performing an Ω-deformation of a topological-holomorphic twist of the $$ \mathcal{N} $$ N = (2 , 0) theory on R ⁶ and restricting to the cohomology of a specific supercharge. In addition, we show that the central charge of the chiral algebra can be obtained by performing equivariant integration of the anomaly polynomial of the six-dimensional theory. Furthermore, we generalize this construction to include orbifolds of the R ⁴ transverse to the chiral algebra plane.
... The precise dictionary between the two descriptions is obtained by identifying the topological string partition function on the CY 3 with the supersymmetric index of the gauge theory, which is conjectured to capture the exact BPS content of its 5d SCFT completion [2]. More generally, the supersymmetric index of the gauge theory with surface defects is matched with the corresponding D-brane open topological string wave function [3,4]. The coupling constants and the moduli of the QFT arise from the geometric engineering as CY moduli parameters (Kähler and complex in the A and B-model picture respectively). ...
... It is possible to study the autonomous limit of this object, following the discussion of [19] for the tau functions. One sets s = e η/ , q = e , u = e a , (C. 4) sending → 0. In this limit one can write the partition function as Z(a, , t) = exp 1 2 ∞ n=0 2n F n (a, t) , (C.5) so that we want to take the "semi-classical" (actually autonomous) → 0 limit of the dual partition functions, keeping the leading and first subleading term. Let us review the argument: in [7] it was shown that the saddle point of Z is the Seiberg-Witten A-period: let us denote this by a * . ...
We study the discrete flows generated by the symmetry group of the BPS quivers for Calabi-Yau geometries describing five dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlev\'e equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with $\tau$-functions of the cluster algebra associated to the quiver. The X-cluster variables describe quantum periods of the underling Calabi-Yau geometry. We exemplify our construction in the case corresponding to five dimensional $SU(2)$ pure Super Yang-Mills and $N_f = 2$ on a circle.
... The operatorĤ t is called quantum curve in the literature [2,11,10,9,18]. ...
... is the so-called quantum curve, which generates a holonomic system determines the B-model partition function [2,10,9,25] and can be viewed as a quantization of the classical spectral curve, and will return to its classical form when taking the classical limit, i.e.Ĥ (x, − ∂ x ) = H(x, y(x)), → 0. ...
We construct the quantum curve for the Baker-Akhiezer function of the orbifold Gromov-Witten theory of the weighted projective line $\mathbb P[r]$. Furthermore, we deduce the explicit bilinear Fermionic formula for the (stationary) Gromov-Witten potential via the lifting operator constructed from the Baker-Akhiezer function.
... An extended discussion of Neumann-like b.c. for pure Yang-Mills-Chern-Simons theory, their IR behavior, and their connection to level-rank duality appeared in [51] and led to N = (0, 2) trialities [52]. Related constructions of WZW and coset models on boundaries in supersymmetric Chern-Simons theory appeared in [68,69]. ...
... The idea that Neumann b.c. for Yang-Mills-Chern-Simons theory leads to coset models is familiar in the literature, cf. [51,68,69], though we believe the statement about Dirichlet boundary conditions is new. We will give evidence of both statements for G = U(N ) and SU(N ) (and, in the case of D b.c., general simple G) by computing half-indices. ...
A bstract
We propose matching pairs of half-BPS boundary conditions related by IR dualities of 3d $$ \mathcal{N}=2 $$ N = 2 gauge theories. From these matching pairs we construct duality interfaces. We test our proposals by anomaly matching and the computation of supersymmetric indices. Examples include basic abelian dualities, level-rank dualities, and Aharony dualities.
... An extended discussion of Neumann-like b.c. for pure Yang-Mills-Chern-Simons theory, their IR behavior, and their connection to level-rank duality appeared in [51] and led to N = (0, 2) trialities [52]. Related constructions of WZW and coset models on boundaries in supersymmetric Chern-Simons theory appeared in [68,69]. ...
... The idea that Neumann b.c. for Yang-Mills-Chern-Simons theory leads to coset models is familiar in the literature, cf. [51,68,69], though we believe the statement about Dirichlet boundary conditions is new. We will give evidence of both statements for G = U (N ) and SU (N ) (and, in the case of D b.c., general simple G) by computing half-indices. ...
We propose matching pairs of half-BPS boundary conditions related by IR dualities of 3d $\mathcal{N}=2$ gauge theories. From these matching pairs we construct duality interfaces. We test our proposals by anomaly matching and the computation of supersymmetric indices. Examples include basic abelian dualities, level-rank dualities, and Aharony dualities.
... Quantum curves are intriguing objects, identified originally in various problems related to string theory and supersymmetric gauge theories [1][2][3][4][5], and analyzed from various mathematical perspectives e.g. in [6][7][8][9][10][11][12][13][14][15][16][17]. In general, quantum curves take form of differential operators A( x, y) imposing Schroedinger-like equations on appropriately defined wave-functions Ψ(x) ...
... where V (2) F (x, z a ) is defined in (6.22). For polynomial potentials, h (0) (x) and f (0) (x) are polynomials of x. ...
A bstract
As we have shown in the previous work, using the formalism of matrix and eigenvalue models, to a given classical algebraic curve one can associate an infinite family of quantum curves, which are in one-to-one correspondence with singular vectors of a certain (e.g. Virasoro or super-Virasoro) underlying algebra. In this paper we reformulate this problem in the language of conformal field theory. Such a reformulation has several advantages: it leads to the identification of quantum curves more efficiently, it proves in full generality that they indeed have the structure of singular vectors, it enables identification of corresponding eigenvalue models. Moreover, this approach can be easily generalized to other underlying algebras. To illustrate these statements we apply the conformal field theory formalism to the case of the Ramond version of the super-Virasoro algebra. We derive two classes of corresponding Ramond super-eigenvalue models, construct Ramond super-quantum curves that have the structure of relevant singular vectors, and identify underlying Ramond super-spectral curves. We also analyze Ramond multi-Penner models and show that they lead to supersymmetric generalizations of BPZ equations.
... Here, the denotes the fact that the D4 branes end on the D6 branes. The D6 branes in table 2 are instead similar to the ones studied in [50] (see also [51]). The corresponding orbifold singularities in the 4d-2d frame reduce to a graviphoton flux in the 3d-3d frame, which is responsible for the Chern-Simons coupling through (2.3). ...
... See also[50,51] for related discussions. ...
Recently, a physical derivation of the Alday-Gaiotto-Tachikawa correspondence has been put forward. A crucial role is played by the complex Chern-Simons theory arising in the 3d-3d correspondence, whose boundary modes lead to Toda theory on a Riemann surface. We explore several features of this derivation and subsequently argue that it can be extended to a generalization of the AGT correspondence. The latter involves codimension two defects in six dimensions that wrap the Riemann surface. We use a purely geometrical description of these defects and find that the generalized AGT setup can be modeled in a pole region using generalized conifolds. Furthermore, we argue that the ordinary conifold clarifies several features of the derivation of the original AGT correspondence.
... The notion of quantum curves was conceived in string theory by Aganagic, Dijkgraaf, Gukov, Hollands, Klemm, Marino, Sulkowski, Vafa, and others [1,5,6,22]. We are far from establishing a complete theory at this moment. ...
... As a consequence, this extension has led to the discovery of the relation between Hitchin spectral curves and Gromov-Witten invariants in few examples (as the one in Section 2.1 and Section 4.1). More precisely, the novelty of this approach is the discovery of the PDE differential recursions of free energies F g,n in [Definition 6.6, [11]] (as well as [Equation 6.5, [10]]) that implies the WKB analysis of the quantization Theorem 4.1. ...
This paper provides an introduction to the mathematical notion of \emph{quantum curves}. We start with a concrete example arising from a graph enumeration problem. We then develop a theory of quantum curves associated with Hitchin spectral curves. A conjecture of Gaiotto, which predicts a new construction of opers from a Hitchin spectral curve, is explained. We give a step-by-step detailed description of the proof of the conjecture for the case of rank $2$ Higgs bundles. Finally, we identify the two concepts of \textit{quantum curve} arising from the topological recursion formalism with the limit oper of Gaiotto's conjecture.
... In contrast, gauge bosonic and fermionic effective or string fields can be expressed as a logistic function raised by some power. Figure 1 claims odd integer spin bosonic or half-spin fermionic string fields possess open strings whose vibratory modes satisfy solely Dirichlet boundary conditions; the strings, called Dbranes, about such fields, have endpoints that are fixed in spacetime [83,86,87]. Thus, GFT implies the odd integer spin bosonic or half-spin fermionic string field's are equal to null. ...
The concealed logic of archetypal thought among our ancestors is most vividly manifested in decorative arts and architecture. Symbols of the Sky and Earth are organically woven into the cultural world—by observing the sky, humanity learned to measure space and time. The essence and purpose of the spiritual process, initially ex-pressed through geometric symbolism, gave rise to numerous variations that serve both symbolic and practical functions.
Keywords: Azerbaijan, Gok Tanry, "Babylon" symbol, cult, tripartite cosmology.
The journal has a certificate of registration at the International Centre in Paris-ISSN 5782-5319. The frequency of publication-12 times per year. Reception of articles in the journal-on the daily basis.
... (2.1). See for example [20,[43][44][45]- [46][47][48][49][50]. It would be interesting to understand the effects of the resolution provided in eq. ...
A bstract
In this paper we holographically study the strongly coupled dynamics of the field theory on I-branes (D5 branes intersecting on a line). In this regime, the field theory becomes (2 + 1) dimensional with 16 supercharges. The dual background has an IR singularity. We resolve this singularity by compactifiying the theory on a circle, preserving 4 supercharges. We study various aspects: confinement, symmetry breaking, Entanglement Entropy, etc. We also discuss a black membrane solution and make some comments on the string σ -model on our backgrounds.
... More generally, on the intersection brane, which is R 1,1 , we have chiral fermions in the bifundamental of U(N ) × U(Q) where in the brane picture there are Q coincident D6 intersecting with N coincident D4 branes. These chiral fermions have a gauge anomaly that cancels the corresponding gauge anomaly of the 5d SYM that lives on the D4 brane [10]. By bosonizing we get a chirally gauged WZW on R 1,1 and again that theory has a gauge anomaly that cancels the corresponding gauge anomaly of the D4 brane [5]. ...
A bstract
We study M5 branes on ℝ 1 , 1 × Taub-NUT that we view as a singular fibration. Reducing the M5 branes along the fiber gives 5d SYM. Due to the singularity, this 5d theory has a gauge anomaly and it is not supersymmetric. To cure these problems we add a supersymmetric gauged chiral WZW theory on the 2d submanifold where the circle fiber vanishes. In addition we add a mass term for the five scalar fields located on this submanifold. With all this, we obtain a fully supersymmetric and gauge invariant theory.
... This is the dimensional reduction of the two-dimensional chiral fermions that arise from an intersection of D4-branes and D6-branes [65,66]. ...
We present a correspondence between two-dimensional \mathcal{N} = (2,2) 𝒩 = ( 2 , 2 ) supersymmetric gauge theories and rational integrable \mathfrak{gl}(m|n) 𝔤 𝔩 ( m | n ) spin chains with spin variables taking values in Verma modules. To explain this correspondence, we realize the gauge theories as configurations of branes in string theory and map them by dualities to brane configurations that realize line defects in four-dimensional Chern–Simons theory with gauge group GL(m|n) G L ( m | n ) . The latter configurations embed the superspin chains into superstring theory. We also provide a string theory derivation of a similar correspondence, proposed by Nekrasov, for rational \mathfrak{gl}(m|n) 𝔤 𝔩 ( m | n ) spin chains with spins valued in finite-dimensional representations.
... After normalisation by the gauge theory perturbative part, the partition function Z (1,1) (τ, ρ, m) can be written as [11,15,43] Z (1,1) ( τ, ρ, m) where c(4kl − p 2 ) are the Fourier coefficients of the elliptic genus of K 3 ...
In this article we study certain degenerations of the mirror curves associated with the Calabi–Yau threefolds XN,M, and the effect of these degenerations on the refined topological string partition function of XN,M. We show that when the mirror curve degenerates and become the union of the lower genus curves the corresponding partition function factorizes into pieces corresponding to the components of the degenerate mirror curve. Moreover we show that using degeneration of a generalised mirror curve it is possible to obtain the partition function corresponding to XN,M-1 from XN,M.
... 6. It would be interesting if we can have a better understanding of the free field representation of the instanton partition function, generalizing the discussion in [32]. 7. We may consider the tetrahedron instantons with supergroups by adding negative branes [31,117,125] in our construction. ...
We introduce and study tetrahedron instantons, which can be realized in string theory by $$\hbox {D}1$$ D 1 -branes probing a configuration of intersecting $$\hbox {D}7$$ D 7 -branes in flat spacetime with a proper constant B -field. Physically they capture instantons on $$\mathbb {C}^{3}$$ C 3 in the presence of the most general intersecting real codimension-two supersymmetric defects. Moreover, we construct the tetrahedron instantons as particular solutions of general instanton equations in noncommutative field theory. We analyze the moduli space of tetrahedron instantons and discuss the geometric interpretations. We compute the instanton partition function both via the equivariant localization on the moduli space of tetrahedron instantons and via the elliptic genus of the worldvolume theory on the $$\hbox {D}1$$ D 1 -branes probing the intersecting $$\hbox {D}7$$ D 7 -branes, obtaining the same result. The instanton partition function of the tetrahedron instantons lies between the higher-rank Donaldson–Thomas invariants on $$\mathbb {C}^{3}$$ C 3 and the partition function of the magnificent four model, which is conjectured to be the mother of all instanton partition functions. Finally, we show that the instanton partition function admits a free field representation, suggesting the existence of a novel kind of symmetry which acts on the cohomology of the moduli spaces of tetrahedron instantons.
... On the other hand, aspects of special curve theory provide certain correlations between Frenet vectors and curvatures of curve pairs; such as Bertrand, Mannheim and involute-evolute curves [23,25,28,32]. Quantum curves, in modern mathematical physics occurs in various fields such as quantization of spectral curves in matrix models [19,1], in topological string theory as quantization of mirror curves [2], in intersecting membrane systems and Seiberg-Witten theory [13,12], generalizations and physical realizations in quantum polynomials and knot theory [14,15]. In all these fields, it is generally thought that a quantum curve can be uniquely assigned to a particular classical curve. ...
Quaternions are widely used in physics. Quaternions, an extension of complex numbers, are closely related to many fundamental concepts (e.g. Pauli matrices) in physics. The aim of this study is the geometric structure underlying the quaternions used in physics. In this paper, we have investigated a new structure of unit speed associated curves, such as spatial quaternionic and quaternionic osculating direction curves. For this, we have assumed that the vector fields
χ(%) = υ1(%)t(%) + υ2(%)n(%)
where υ21(%) + υ22(%) = 1 for the spatial quaternionic curve and
χ(%) = λ1(%)>(%) + λ2(%)η(%) + λ3β2(%);
where λ21(%) + λ2 2(%) + λ2 3(%) = 1 for the quaternionic curve φ. Then, we have given the relationship between (spatial) quaternionic (OD)-curves and Mannheim curve pair. Moreover, we have examined in which cases the (spatial) quaternionic (OD)-curve can be helix or slant helix. So, considering that helices also take place in electron physics, it is thought that this study will create a bridge between physics and geometry. Finally, we have given the examples and draw the figures of curves in the examples.
... The difference equations we consider also arise in a different context, that of A model topological strings on a Calabi-Yau threefold X with a D-brane placed on a Lagrangian submanifold L ⊂ X, e.g. [12,15,64]. In this language Σ is the mirror curve of X. ...
We propose a connection between 3d-5d exponential networks and exact WKB for difference equations associated to five dimensional Seiberg-Witten curves, or equivalently, to quantum mirror curves to toric Calabi-Yau threefolds $X$: the singularities in the Borel planes of local solutions to such difference equations correspond to central charges of 3d-5d BPS KK-modes, and the central charges of 5d BPS KK-modes are related to the singularities in the Borel planes of the closed topological string free energy on $X$. It follows that there should be distinguished local solutions of the difference equation in each domain of the complement of the exponential network, and these solutions jump at the walls of the network. We verify this picture in detail in two simple examples of 3d-5d systems, corresponding to taking the toric Calabi-Yau $X$ to be either $\mathbb{C}^3$ or the resolved conifold. We provide the full list of local solutions in each sector of the Borel plane and in each domain of the complement of the exponential network, and find that local solutions in disconnected domains correspond to non-perturbative open topological string amplitudes on $X$ with insertions of branes at different positions of the toric diagram. We also study the Borel summation of the closed refined topological string free energy on $X$ and the corresponding non-perturbative effects.
... The branes D3 α , α ≤ K, and D5 i , i > m, intersect along the time axis R, and from strings stretched between them we get fermions ϕ 10 ∈ Hom(C K , C n ) , (4.49) ϕ 10 ∈ Hom(C n , C K ) . This is the dimensional reduction of the two-dimensional chiral fermions that arise from an intersection of D4-branes and D6-branes [65,66]. ...
We present a correspondence between two-dimensional $\mathcal{N} = (2,2)$ supersymmetric gauge theories and rational integrable $\mathfrak{gl}(m|n)$ spin chains with spin variables taking values in Verma modules. To explain this correspondence, we realize the gauge theories as configurations of branes in string theory and map them by dualities to brane configurations that realize line defects in four-dimensional Chern-Simons theory with gauge group $\mathrm{GL}(m|n)$. The latter configurations embed the superspin chains into superstring theory. We also provide a string theory derivation of a similar correspondence, proposed by Nekrasov, for rational $\mathfrak{gl}(m|n)$ spin chains with spins valued in finite-dimensional representations.
... Since we are now studying the boundary dynamics on D6', we may treat D6 heavy and only freeze the associated gauge field. [16,40] also discussed such an assignment of the flavor and gauge symmetry in an analogous intersecting D4-D6 brane system. We have already seen a candidate for the gauge invariant algebra of operators in §3.2 ...
We study a four-dimensional domain wall in twisted M-theory. The domain wall
is engineered by intersecting D6 branes in the type IIA frame. We identify the
classical algebra of operators on the domain wall in terms of a higher vertex
operator algebra, which describes the holomorphic subsector of a 4d
$\mathcal{N}=1$ supersymmetric field theory, and compute the associated mode
algebra. We conjecture that the quantum deformation of the classical algebra is
isomorphic to the bulk algebra of operators from which we establish twisted
holography of the domain wall.
... 5. It would be interesting if we can have a better understanding of the free field representation of the instanton partition function, generalizing the discussion in [122]. 6. ...
We introduce and study tetrahedron instantons, which can be realized in string theory by D$1$-branes probing a configuration of intersecting D$7$-branes in flat spacetime with a nonzero constant background $B$-field. Physically they capture instantons on $\mathbb{C}^{3}$ in the presence of the most general intersecting codimention-two supersymmetric defects. Moreover, we construct the tetrahedron instantons as particular solutions of general instanton equations in noncommutative field theory. We analyze the moduli space of tetrahedron instantons and discuss the geometric interpretations. We compute the instanton partition function both via the equivariant localization on the moduli space of tetrahedron instantons and via the elliptic genus of the worldvolume theory on the D$1$-branes probing the intersecting D$7$-branes, obtaining the same result. The instanton partition function of the tetrahedron instantons lies between the higher-rank Donaldson-Thomas invariants on $\mathbb{C}^{3}$ and the partition function of the magnificent four model, which is conjectured to be the mother of all instanton partition functions. Finally, we show that the instanton partition function admits a free field representation.
... This generalizes to the SL(N , C) case by considering representation of W N algebra, rather than Virasoro, as in [21,22]. In turn, this makes contact, rather than with the usual AGT correspondence [23], with a four-dimensional limit of topological strings, that are more naturally connected to free fermions [29,31,34,59]. Due to the extra Fock space, the system that is needed in the end is that of N -component complex free fermions, which we define in this section without introducing degenerate fields of W N . ...
We study the relation between class $$\mathcal {S}$$ S theories on punctured tori and isomonodromic deformations of flat SL ( N ) connections on the two-dimensional torus with punctures. Turning on the self-dual $$\Omega $$ Ω -background corresponds to a deautonomization of the Seiberg–Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding $$\tau $$ τ -function is proportional to the dual gauge theory partition function, the proportionality factor being a nontrivial function of the solution of the deautonomized Seiberg–Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to $$W_N$$ W N free fermion correlators on the torus.
... nN Fermi multiplets in 2d N = (0, 2) language -that transform in a bifundamental representation of U(n) × U(N ), cf. [26]. ...
A bstract
While the study of bordered (pseudo-)holomorphic curves with boundary on Lagrangian submanifolds has a long history, a similar problem that involves (special) Lagrangian submanifolds with boundary on complex surfaces appears to be largely overlooked in both physics and math literature. We relate this problem to geometry of coassociative submanifolds in G 2 holonomy spaces and to Spin(7) metrics on 8-manifolds with T ² fibrations. As an application to physics, we propose a large class of brane models in type IIA string theory that generalize brane brick models on the one hand and 2d theories T [ M 4 ] on the other.
... After normalisation by the gauge theory perturbative part , the partition function Z (1,1) (τ, ρ, m) can be written as [13] Z ...
In this paper we study certain degenerations of the mirror curves, associated with Calabi-Yau threefolds $X_{N,M}$, and the effect of these degenerations on the topological string partition function of $X_{N,M}$. We show that when the mirror curve degenerates and become the union of the lower genus curves the corresponding partition function factorizes into pieces corresponding to the components of the degenerate mirror curve. Moreoever we show that using degeneration of a generalised mirror curve it is possible to obtain the partition function corresponding to $X_{N,M-1}$ from $X_{N,M}$.
... In order to impose the Chern relations (2.8), for each Y ∈ C s σ , define k σ (Y ) 8 When p = N , the central charge c(W para N,n ) = 0 in (3.3). 9 Without the Burge conditions, the generating functions correspond to the partition functions of N = 4 topologically twisted U (N ) supersymmetric gauge theories [36,42]. A description of the generating functions/WZW characters in terms of 'orbifold partitions', and a realization in terms of intersecting D4 and D6-branes can be found in [43,44]. 10 In fact, this solution is unique except in the case where σ 1 = σ 2 = · · · = σ N . ...
Generalizations of the AGT correspondence between 4D $\mathcal{N}=2$ $SU(2)$ supersymmetric gauge theory on ${\mathbb {C}}^2$ with $\Omega$-deformation and 2D Liouville conformal field theory include a correspondence between 4D $\mathcal{N}=2$ $SU(N)$ supersymmetric gauge theories, $N = 2, 3, \ldots$, on ${\mathbb {C}}^2/{\mathbb {Z}}_n$, $n = 2, 3, \ldots$, with $\Omega$-deformation and 2D conformal field theories with $\mathcal{W}^{\ para}_{N, n}$ ($n$-th parafermion $\mathcal{W}_N$) symmetry and $\widehat{\mathfrak{sl}}(n)_N$ symmetry. In this work, we trivialize the factor with $\mathcal{W}^{\ para}_{N, n}$ symmetry in the 4D $SU(N)$ instanton partition functions on ${\mathbb {C}}^2/{\mathbb {Z}}_n$ (by using specific choices of parameters and imposing specific conditions on the $N$-tuples of Young diagrams that label the states), and extract the 2D $\widehat{\mathfrak{sl}}(n)_N$ WZW conformal blocks, $n = 2, 3, \ldots$, $N = 1, 2, \ldots\, .
... We might, for instance, consider IIB on spacetimes of the form C 2 /Γ 1 × C 2 /Γ 2 × T 2 . As argued in [117], such a configuration leads to a chiral WZW model (with algebra determined by Γ 1 and Γ 2 ) living on T 2 . It is natural to conjecture that a careful analysis of the boundary conditions of IIB in this background should reproduce the structure of conformal blocks of the chiral WZW model, and in particular give a geometric picture for the Verlinde formula for these theories [118][119][120][121][122] (and relatedly, a direct string theory interpretation of the work by Nakajima [123]). ...
A bstract
We discuss the origin of the choice of global structure for six dimensional (2 , 0) theories and their compactifications in terms of their realization from IIB string theory on ALE spaces. We find that the ambiguity in the choice of global structure on the field theory side can be traced back to a subtle effect that needs to be taken into account when specifying boundary conditions at infinity in the IIB orbifold, namely the known non-commutativity of RR fluxes in spaces with torsion. As an example, we show how the classification of $$ \mathcal{N} $$ N = 4 theories by Aharony, Seiberg and Tachikawa can be understood in terms of choices of boundary conditions for RR fields in IIB. Along the way we encounter a formula for the fractional instanton number of $$ \mathcal{N} $$ N = 4 ADE theories in terms of the torsional linking pairing for rational homology spheres. We also consider six-dimensional (1 , 0) theories, clarifying the rules for determining commutators of flux operators for discrete 2-form symmetries. Finally, we analyze the issue of global structure for four dimensional theories in the presence of duality defects.
... nN Fermi multiplets in 2d N = (0, 2) language -that transform in a bifundamental representation of U (n) × U (N ), cf. [23]. ...
While the study of bordered (pseudo-)holomorphic curves with boundary on Lagrangian submanifolds has a long history, a similar problem that involves (special) Lagrangian submanifolds with boundary on complex surfaces appears to be largely overlooked in both physics and math literature. We relate this problem to geometry of coassociative submanifolds in $G_2$ holonomy spaces and to $Spin(7)$ metrics on 8-manifolds with $T^2$ fibrations. As an application to physics, we propose a large class of brane models in type IIA string theory that generalize brane brick models on the one hand and 2d theories $T[M_4]$ on the other.
... This generalizes to the SL(N, C) case by considering representation of W N algebra, rather than Virasoro, as in [21,22]. In turn, this makes contact, rather than with the usual AGT correspondence [23], with a four-dimensional limit of topological strings, that are more naturally connected to free fermions [29,31,34,59]. Due to the extra Fock space, the system that is needed in the end is that of N -component complex free fermions, which we define in this section without introducing degenerate fields of W N . ...
We study the relation between class S theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two dimensional torus with punctures. Turning on the self dual $\Omega$-background corresponds to a deautonomization of the Seiberg-Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding $\tau$-function is proportional to the dual gauge theory partition function, the proportionality factor being a non trivial function of the solution of the deautonomized Seiberg-Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to $W_N$ free fermion correlators on the torus.
... We might, for instance, consider IIB on spacetimes of the form C 2 /Γ 1 × C 2 /Γ 2 × T 2 . As argued in [110], such a configuration leads to a chiral WZW model (with algebra determined by Γ 1 and Γ 2 ) living on T 2 . It is natural to conjecture that a careful analysis of the boundary conditions of IIB in this background should reproduce the structure of conformal blocks of the chiral WZW model, and in particular give a geometric picture for the Verlinde formula for these theories [111][112][113][114][115] (and relatedly, a direct string theory interpretation of the work by Nakajima [116]). ...
We discuss the origin of the choice of global structure for six dimensional $(2,0)$ theories and their compactifications in terms of their realization from IIB string theory on ALE spaces. We find that the ambiguity in the choice of global structure on the field theory side can be traced back to a subtle effect that needs to be taken into account when specifying boundary conditions at infinity in the IIB orbifold, namely the known non-commutativity of RR fluxes in spaces with torsion. As an example, we show how the classification of $\mathcal{N}=4$ theories by Aharony, Seiberg and Tachikawa can be understood in terms of choices of boundary conditions for RR fields in IIB. Along the way we encounter a formula for the fractional instanton number of $\mathcal{N}=4$ ADE theories in terms of the torsional linking pairing for rational homology spheres. We also consider six-dimensional $(1,0)$ theories, clarifying the rules for determining commutators of flux operators for discrete 2-form symmetries. Finally, we analyze the issue of global structure for four dimensional theories in the presence of duality defects.
... Examples of quantum curves have been found from several seemingly unrelated perspectives. First, they were identified in the system of intersecting branes, which encode Seiberg-Witten theory and provide quantization of Seiberg-Witten curves [1,2]. Second, it was realized that partition functions of branes in topological string theory satisfy difference equations, which quantize mirror curves [3]. ...
We show that quantum curves arise in infinite families and have the structure of singular vectors of a relevant symmetry algebra. We analyze in detail the case of the hermitian one-matrix model with the underlying Virasoro algebra, and the super-eigenvalue model with the underlying super-Virasoro algebra. In the Virasoro case we relate singular vector structure of quantum curves to the topological recursion, and in the super-Virasoro case we introduce the notion of super-quantum curves. We also discuss the double quantum structure of the quantum curves and analyze specific examples of Gaussian and multi-Penner models.
... Theresee sect. 4.7 and further elaborated in [65,66] -it is also suggested that for higher genus Riemann surfaces the most natural framework is given by considering the free-fermion grand-canonical partition function leading precisely to the Fourier basis and thus to the Nekrasov-Okounkov partition function. Indeed this latter can be regarded as a character of W 1+∞ -algebra as in the topological B-brane setting of [64]. ...
In this paper we study the extension of Painlev\'e/gauge theory correspondence to circular quivers by focusing on the special case of $SU(2)$ $\mathcal{N}=2^*$ theory. We show that the Nekrasov-Okounkov partition function of this gauge theory provides an explicit combinatorial expression and a Fredholm determinant formula for the tau-function describing isomonodromic deformations of $SL_2$ flat connections on the one-punctured torus. This is achieved by reformulating the Riemann-Hilbert problem associated to the latter in terms of chiral conformal blocks of a free-fermionic algebra. This viewpoint provides the exact solution of the renormalization group flow of the $SU(2)$ $\mathcal{N}=2^*$ theory on self-dual $\Omega$-background and, in the Seiberg-Witten limit, an elegant relation between the IR and UV gauge couplings.
... which is the quantum curve related to Z q t U . As discussed below, (7.4) is a qt-version of the quantum curve of 3 in string theory [1,16,15,25]. ...
We show, in a number of simple examples, that Macdonald-type $qt$-deformations of topological string partition functions are equivalent to topological string partition functions that are without $qt$-deformations but with brane condensates, and that these brane condensates lead to geometric transitions.
... Examples of quantum curves have been found from several seemingly unrelated perspectives. First, they were identified in the system of intersecting branes, which encode Seiberg-Witten theory and provide quantization of Seiberg-Witten curves [1,2]. Second, it was realized that partition functions of branes in topological string theory satisfy difference equations, which quantize mirror curves [3]. ...
We show that quantum curves arise in infinite families and have the structure of singular vectors of a relevant symmetry algebra. We analyze in detail the case of the hermitian one-matrix model with the underlying Virasoro algebra, and the super-eigenvalue model with the underlying super-Virasoro algebra. In the Virasoro case we relate singular vector structure of quantum curves to the topological recursion, and in the super-Virasoro case we introduce the notion of super-quantum curves. We also discuss the double quantum structure of the quantum curves and analyze specific examples of Gaussian and multi-Penner models.
... The simplest such theory fitting to this context is the notion of quantum curves. The relation between quantum curves and B-model geometry is considered in [2,19,17,27,28,30,49,50] and others. The key point for this idea to work is when the holomorphic geometry Y corresponding to the Gromov-Witten theory of X is captured by an algebraic or analytic curve. ...
This article consists of two parts. In Part 1, we present a formulation of two-dimensional topological quantum field theories in terms of a functor from a category of Ribbon graphs to the endofuntor category of a monoidal category. The key point is that the category of ribbon graphs produces all Frobenius objects. Necessary backgrounds from Frobenius algebras, topological quantum field theories, and cohomological field theories are reviewed. A result on Frobenius algebra twisted topological recursion is included at the end of Part 1. In Part 2, we explain a geometric theory of quantum curves. The focus is placed on the process of quantization as a passage from families of Hitchin spectral curves to families of opers. To make the presentation simpler, we unfold the story using SL_2(\mathbb{C})-opers and rank 2 Higgs bundles defined on a compact Riemann surface $C$ of genus greater than $1$. In this case, quantum curves, opers, and projective structures in $C$ all become the same notion. Background materials on projective coordinate systems, Higgs bundles, opers, and non-Abelian Hodge correspondence are explained.
... The purpose of this paper is to construct a geometric theory of quantum curves. The notion of quantum curves was introduced in the physics literature (see for example, [1,16,17,37,38,43,54,65]). A quantum curve is supposed to compactly capture topological invariants, such as certain Gromov-Witten invariants, Seiberg-Witten invariants, and quantum knot polynomials. ...
Quantum curves were introduced in the physics literature. We develop a mathematical framework for the case associated with Hitchin spectral curves. In this context, a quantum curve is a Rees $\mathcal{D}$-module on a smooth projective algebraic curve, whose semi-classical limit produces the Hitchin spectral curve of a Higgs bundle. We give a method of quantization of Hitchin spectral curves by concretely constructing one-parameter deformation families of opers. We propose a generalization of the topological recursion of Eynard-Orantin and Mirzakhani for the context of singular Hitchin spectral curves. We show a surprising result that a PDE version of the topological recursion provides all-order WKB analysis for the Rees $\mathcal{D}$-modules, defined as the quantization of Hitchin spectral curves associated with meromorphic $SL(2,\mathbb{C})$-Higgs bundles. Topological recursion is thus identified as a process of quantization of Hitchin spectral curves. We prove that these two quantizations, one via the construction of families of opers, and the other via the PDE topological recursion, agree for holomorphic and meromorphic $SL(2,\mathbb{C})$-Higgs bundles. Classical differential equations such as the Airy differential equation provides a typical example. Through these classical examples, we see that quantum curves relate Higgs bundles, opers, a conjecture of Gaiotto, and quantum invariants, such as Gromov-Witten invariants.
... A similar expression also appeared as the partition function for the I-brane system of[52][53][54]. ...
We elucidate the relation between Painlev\'e equations and four-dimensional rank one ${\cal N= 2}$ theories by identifying the connection associated to Painlev\'e isomonodromic problems with the oper limit of the flat connection of the Hitchin system associated to gauge theories and by studying the corresponding renormalisation group flow. Based on this correspondence we provide long-distance expansions at various canonical rays for all Painlev\'e functions in terms of magnetic and dyonic Nekrasov partition functions for ${\cal N= 2}$ SQCD and Argyres-Douglas theories at self-dual Omega background $\epsilon_1+\epsilon_2= 0$, or equivalently in terms of $c= 1$ irregular conformal blocks.
По периодической двудольной плоской димерной модели мы строим некоммутативную кривую. Эта кривая позволяет реконструировать обратную матрицу Кастелейна и, вместе с ней, все корреляционные функции. Ее можно рассматривать как квантование конструкции предельной формы, предложенной Кеньоном и автором. Мы также обсуждаем различные возможные направления для обобщений.
We construct the quantum curve for the Baker–Akhiezer function of the orbifold Gromov–Witten theory of the weighted projective line ℙ[r]. Furthermore, we deduce the explicit bilinear Fermionic formula for the (stationary) Gromov–Witten potential via the lifting operator contructed from the Baker–Akhiezer function.
We demonstrate that twisted equivariant differential K-theory of transverse complex curves accommodates exotic charges of the form expected of codimension=2 defect branes, such as of D7-branes in IIB/F-theory on 𝔸-type orbifold singularities, but also of their dual 3-brane defects of class-S theories on M5-branes. These branes have been argued, within F-theory and the AGT correspondence, to carry special SL(2)-monodromy charges not seen for other branes, at least partially reflected in conformal blocks of the su2-WZW model over their transverse punctured complex curve. Indeed, it has been argued that all “exotic” branes of string theory are defect branes carrying such U-duality monodromy charges — but none of these had previously been identified in the expected brane charge quantization law given by K-theory.
Here we observe that it is the subtle (and previously somewhat neglected) twisting of equivariant K-theory by flat complex line bundles appearing inside orbi-singularities (“inner local systems”) that makes the secondary Chern character on a punctured plane inside an 𝔸-type singularity evaluate to the twisted holomorphic de Rham cohomology which Feigin, Schechtman and Varchenko showed realizes 𝔰𝔩2̂k-conformal blocks, here in degree 1 — in fact it gives the direct sum of these over all admissible fractional levels k=−2+κ/r. The remaining higher-degree 𝔰𝔩2̂k-conformal blocks appear similarly if we assume our previously discussed “Hypothesis H” about brane charge quantization in M-theory. Since conformal blocks — and hence these twisted equivariant secondary Chern characters — solve the Knizhnik–Zamolodchikov equation and thus constitute representations of the braid group of motions of defect branes inside their transverse space, this provides a concrete first-principles realization of anyon statistics of — and hence of topological quantum computation on — defect branes in string/M-theory.
This paper describes the reconstruction of the topological string partition function for certain local Calabi–Yau (CY) manifolds from the quantum curve, an ordinary differential equation obtained by quantising their defining equations. Quantum curves are characterised as solutions to a Riemann–Hilbert problem. The isomonodromic tau-functions associated to these Riemann–Hilbert problems admit a family of natural normalisations labelled by the chambers in the extended Kähler moduli space of the local CY under consideration. The corresponding isomonodromic tau-functions admit a series expansion of generalised theta series type from which one can extract the topological string partition functions for each chamber.
The low energy effective behavior of N=2 gauge theory has the geometric characterization due to the Seiberg–Witten theory. In particular, the algebraic curve, called the Seiberg–Witten curve, geometrically encodes the information about the prepotential of N=2 theory. In this Chapter, we show how to obtain such an algebraic object from the microscopic path integral formalism together with the instanton counting described in Part I.
In this paper we study the extension of Painlevé/gauge theory correspondence to circular quivers by focusing on the special case of SU(2) \({\mathcal {N}}=2^*\) theory. We show that the Nekrasov–Okounkov partition function of this gauge theory provides an explicit combinatorial expression and a Fredholm determinant formula for the tau-function describing isomonodromic deformations of \(SL_2\) flat connections on the one-punctured torus. This is achieved by reformulating the Riemann–Hilbert problem associated to the latter in terms of chiral conformal blocks of a free-fermionic algebra. This viewpoint provides the exact solution of the renormalization group flow of the SU(2) \({\mathcal {N}}=2^*\) theory on self-dual \(\Omega \)-background and, in the Seiberg–Witten limit, an elegant relation between the IR and UV gauge couplings.
A bstract
We study M5-branes wrapped on a multi-centred Taub-NUT space. Reducing to String Theory on the S ¹ fibration leads to D4-branes intersecting with D6-branes. D-braneology shows that there are additional charged chiral fermions from the open strings which stretch between the D4-branes and D6-branes. From the M-theory point of view the appearance of these charged states is mysterious as the M5-branes are wrapped on a smooth manifold. In this paper we show how these states arise in the M5-brane worldvolume theory and argue that are governed by a WZWN-like model where the topological term is five-dimensional.
We discuss the Gromov-Witten/Donaldson-Thomas correspondence for 3-folds in both the absolute and relative cases. Descendents in Gromov-Witten theory are conjectured to be equivalent to Chern characters of the universal sheaf in Donaldson-Thomas theory. Relative constraints in Gromov-Witten theory are conjectured to correspond in Donaldson-Thomas theory to cohomology classes of the Hilbert scheme of points of the relative divisor. Independent of the conjectural framework, we prove degree 0 formulas for the absolute and relative Donaldson-Thomas theories of toric varieties.
We conjecture an equivalence between the GromovSimons theory of Donaldson and Thomas. For CalabiWitten theory and $q$ is the Euler characteristic parameter of DonaldsonYau toric surfaces.
Recently, a D-brane construction in type IIA string theory was shown to yield the electric/magnetic duality of four-dimensional N = 1 supersymmetric U(Nc) gauge theories with Nf flavours of quark. We present here an extension of that construction which yields the electric/magnetic duality for the SO(Nc) and USp(Nc) gauge theories with Nf quarks, by adding an orientifold plane which is consistent with the supersymmetry. Due to the intersection of the orientifold plane with the NS-NS fivebranes already present, new features arise which are crucial in determining the correct final structure of the dualities.
We construct metastable configurations of branes and anti-branes wrapping 2-spheres inside local Calabi–Yau manifolds and study their large N duals. These duals are Calabi–Yau manifolds in which the wrapped 2-spheres have been replaced by 3-spheres with flux through them, and supersymmetry is spontaneously broken. The geometry of the non-supersymmetric vacuum is exactly calculable to all orders of the 't Hooft parameter, and to the leading order in 1/N. The computation utilizes the same matrix model techniques that were used in the supersymmetric context. This provides a novel mechanism for breaking supersymmetry in the context of flux compactifications.
We find the exact spectrum of a class of quarter BPS dyons in a generic N = 4 supersymmetric Z(N) orbifold of type IIA string theory on K3 x T-2 or T-6. We also find the asymptotic expansion of the statistical entropy to first non-leading order in inverse power of charges and show that it agrees with the entropy of a black hole carrying same set of charges after taking into account the effect of the four derivative Gauss-Bonnet term in the effective action of the theory.
A recently discovered relation between 4D and 5D black holes is used to derive weighted BPS black hole degeneracies for 4D N = 4 string theory from the well-known 5D degeneracies. They are found to be given by the Fourier coefficients of the unique weight 10 automorphic form of the modular group Sp( 2, Z). This result agrees exactly with a conjecture made some years ago by Dijkgraaf, Verlinde and Verlinde.
In this note we give a direct proof using the theory of modular forms of a beautiful fact explained in the preceding paper by Robbert Dijkgraaf [1, Theorem 2 and Corollary]. Let \(
{\tilde M_*}({\Gamma _1})
\) denote the graded ring of quasi-modular forms on the full modular group Γ= PSL(2, ℤ). This is the ring generated by G2, G4, G6, and graded by assigning to each Gk
the weight where \(
{G_k} = - \frac{{{B_k}}}{{2k}} + \sum\limits_{n = 1}^\infty {\left( {{{\sum\limits_{d|n} d }^{k - 1}}} \right)} {q^n}\left( {k \geqslant 2,{B_k} = kth Bernoulli number} \right)
\) are the classical Eisenstein series, all of which except G
2 are modular.
We consider the realization of four-dimensional theories with N=2
supersymmetry as M-theory configurations including a five-brane. Our
emphasis is on the spectrum of massive states that are realized as
two-branes ending on the five-brane. We start with a determination of
the supersymmetries that are left unbroken by the background metric and
five-brane. We then show how the central charge of the N=2 algebra
arises from the central charge associated with the M-theory two-brane.
This determines the condition for a two-brane configuration to be BPS
saturated in the four-dimensional sense. By imposing certain conditions
on the moduli, we can give concrete examples of such two-branes. This
leads us to conjecture that vector multiplet and hypermultiplet
BPS-saturated states correspond to two-branes with the topology of a
cylinder and a disk, respectively.
A recently
discovered relation between 4D and 5D black holes is used to
derive weighted BPS black hole degeneracies for 4D = 4 string
theory from the well-known 5D degeneracies. They are found to be
given by the Fourier coefficients of the unique weight 10
automorphic form of the modular group SP(2,). This
result agrees exactly with a conjecture made some years ago by
Dijkgraaf, Verlinde and Verlinde.
The two-loop path integral for the closed bosonic string is given by a genus-two modular form: the product of the even theta functions. Thus the two-loop vacuum amplitude, scattering amplitudes, and the one-loop free energy may be expressed simply in terms of theta functions.
By taking the critical limit of Penner's matrix model we obtain a continuum theory whose free energy at genus-g is the Euler characteristic of moduli space of Riemann surfaces of genus-g. The exponents, and the appearance of logarithmic corrections suggest that we are dealing with a theory at c=1.
The equivalence of NS5-branes and ALF spaces under T-duality is well known. However, a naive application of T-duality transforms the ALF space into a smeared NS5-brane, de-localized on the dual, transverse, circle. In this paper we re-examine this duality, starting from a two-dimensional Script N = (4, 4) gauged linear sigma model describing Taub-NUT space. After dualizing the S1 fiber, we find that the smeared NS5-brane target space metric receives corrections from multi-worldsheet instantons. These instantons are identified as Nielsen-Olesen vortices. We show that their effect is to break the isometry of the target space, localizing the NS5-brane at a point. The contribution from the k-instanton sector is shown to be proportional to the weighted integral of the Euler form over the k-vortex moduli space. The duality also predicts the, previously unknown, asymptotic exponential decay coefficient of the BPS vortex solution.
We find the exact spectrum of a class of quarter BPS dyons in a generic
Script N = 4 supersymmetric Bbb ZN orbifold of type IIA
string theory on K3 × T2 or T6. We also find
the asymptotic expansion of the statistical entropy to first non-leading
order in inverse power of charges and show that it agrees with the
entropy of a black hole carrying same set of charges after taking into
account the effect of the four derivative Gauss-Bonnet term in the
effective action of the theory.
We derive a compact and explicit expression for the generating functional of all correlation functions of tachyon operators in two-dimensional string theory. This expression makes manifest relations of the c = 1 system to KP flow nd W1 + ∞ constraints. Moreover we derive a Kontsevich-Penner integral representation of this generating functional.
The matrix model representation of c = 1 matter coupled to two-dimensional quantum gravity is analyzed as a string theory. It is shown that a natural representation of the string field theory is that of free, two-dimensional Dirac fermions. One of the dimensions is that of the target space, the other arises from the space of the eigenvalues of the matrices and is identified with the zero mode of the Liouville field. The higher-order contributions to the correlation functions of the observables arise from translationally non-invariant corrections to the fermion kinetic energy. We discuss the integrability of this model and present some calculations of correlation functions using this formalism.
It is shown that Sugawara's energy momentum tensor, bilinear in fermionic currents associated with a group G, equals the energy momentum tensor for free fermions if there exists a symmetric space G′/G with the symmetric space generators transforming under G as the fermions do. This result provides a list of chiral field theories with Wess-Zumino term that are equivalent to free fermion theories and specifies which representations of Kac-Moody algebras bilinear in fermions are finitely reducible.
As a step towards development of an efficient method for separation of cephalosporin C from fermentation broth, reactive extraction in supported liquid membrane has been investigated using simulated feed solution. Experiments were conducted in a diaphragm type cell with Celgard-2400 membrane as the solid support and Aliquat-336 as the carrier dissolved in butyl acetate as the diluent. Cephalosporin C was permeated across the membrane from an alkaline feed phase of carbonate buffer into an acidic strip solution of acetate buffer via a coupled transport phenomenon involving Cl ions. The permeation process was analyzed from the hypothesis of an overall transport dependance on diffusion across the liquid membrane and total re-extracting in the stripping phase.
We study metastable nonsupersymmetric configurations in type IIA string theory, obtained by suspending D4-branes and -branes between holomorphically curved NS5's, which are related to those of hep-th/0610249 by T-duality. When the numbers of branes and antibranes are the same, we are able to obtain an exact M theory lift which can be used to reliably describe the vacuum configuration as a curved NS5 with dissolved RR flux for gs≪1 and as a curved M5 for gs≫1. When our weakly coupled description is reliable, it is related by T-duality to the deformed IIB geometry with flux of hep-th/0610249 with moduli exactly minimizing the potential derived therein using special geometry. Moreover, we can use a direct analysis of the action to argue that this agreement must also hold for the more general brane/antibrane configurations of hep-th/0610249. On the other hand, when our strongly coupled description is reliable, the M5 wraps a nonholomorphic minimal area curve that can exhibit quite different properties, suggesting that the residual structure remaining after spontaneous breaking of supersymmetry at tree level can be further broken by the effects of string interactions. Finally, we discuss the boundary condition issues raised in hep-th/0608157 for nonsupersymmetric IIA configurations, their implications for our setup, and their realization on the type IIB side.
We exhibit a duality of primary fields, conformal dimensions, braid matrices, and modular transformation matrix elements S0,λ/S0,0 (so also Weyl characters) for sequences of dual pairs (SU (N)K, SU(K)N), (SO(N)K, (SO(N)K, SO(K)N) and Sp(N)K, Sp(K)N. As a consequence an analogous duality of Chern-Simons skein relations and Wilson line observables is uncovered.
Various constructions of the affine Lie algebra action on the moduli space of instantons on 4-manifolds are discussed. The analogy between the local-global principle and the role of mass is also explained. The detailed proofs are given in separated papers [16,17].
We describe the explicit construction of Yang-Mills instantons on Asymptotically Locally Euclidean (ALE) spaces, following the work of Kronheimer and Nakajima. For multicenter ALE metrics, we determine the Abelian instanton connections that are needed for the construction in the non-Abelian case. We compute the partition function of Maxwell theories on ALE manifolds and comment on the issue of electromagnetic duality. We discuss the topological characterization of the instanton bundles as well as the identification of their moduli spaces. We generalize the 't Hooft ansatz to SU(2) instantons on ALE spaces and on other hyper-Kähler manifolds. Specializing to the Eguchi-Hanson gravitational background, we explicitly solve the ADHM equations for SU(2) gauge bundles with second Chern class , 1 and .
We derive a Kontsevich-type matrix model for the c = 1 string directly from the W∞ solution of the theory. The model that we obtain is different from previous proposals, which are proven to be incorrect. Our matrix model contains the Penner and Kontsevich cases, and we study its quantum effective action. The simplicity of our model leads to an encouraging interpretation in the context of background-independent non-critical string field theory.
We study the BPS states of the M-fivebrane which correspond to monopoles of N = 2 SU(2) gauge theory. Far away from the centres of the monopoles these states may be viewed as solitons in the Seiberg-Witten effective action. It is argued that these solutions are smooth and some properties of their moduli space are discussed.
We study general perturbations of two-dimensional conformal field theories by holomorphic fields. It is shown that the genus one partition function is controlled by a contact term (pre-Lie) algebra given in terms of the operator product expansion. These models have applications to vertex operator algebras, two-dimensional QCD, topological strings, holomorphic anomaly equations and modular properties of generalized characters of chiral algebras such as the W1+∞ algebra, that is treated in detail.
We present a generic lagrangian, in arbitrary spacetime dimension D, describing the interaction of a dilaton, a graviton and an antisymmetric tensor of arbitrary rank d. For each D and d, we find black solutions where . These solutions display a spacetime singularity surrounded by an event horizon and are characterized by a mass per unit , d̃, and a topological “magnetic” charge . The theory also admits elementary p-brane solutions with “electric” Noether charge ed, obeying the Dirac quantization rule . We then present the lagrangian describing the theory dual to the original theory, whose antisymmetric tensor has rank d̃ and for which the roles of topological and elementary solutions are interchanged. In the extreme limits or , the singularity and event horizon coalesce. In this case, the metics and their duals are mutually nonsingular. For specific values of D and d, these extreme solutions also exhibit supersymmetry and some may be identified with previously classified heterotic, Type IIA and Type IIB super . In particular, within the context of Type II theory, the electric/magnetic duality of Gibbons and Perry in D = 4 is seen to be a consequence of particle/sixbrane duality in D = 10. Among the new solutions is a self-dual superstring in D = 6.
We continue our study of compactifications of F-theory on Calabi-Yau threefolds. We gain more insight into F-theory duals of heterotic strings and provide a recipe for building F-theory duals for arbitrary heterotic compactifications on elliptically fibered manifolds. As a byproduct we find that string/string duality in six dimensions gets mapped to fiber/base exchange in F-theory. We also construct a number of new N = 1, d = 6 examples of F-theory vacua and study transitions among them. We find that some of these transition points correspond upon further compactification to 4 dimensions to transitions through analogues of Argyres-Douglas points of N = 2 moduli. A key idea in these transitions is the notion of classifying (0,4) fivebranes of heterotic strings.
N = 2 supersymmetric gauge theories in four dimensions are studied by formulating them as the quantum field theories derived from configurations of fourbranes, fivebranes, and sixbranes in Type IIA superstrings, and then reinterpreting those configurations in M-theory. This approach leads to explicit solutions for the Coulomb branch of a large family of four-dimensional N = 2 field theories with zero or negative beta function.
We investigate Landau-Ginzburg string theory with the singular superpotential X−1 on arbitrary Riemann surfaces. This theory, which is a topological version of the c = 1 string at the self-dual radius, is solved using results from intersection theory and from the analysis of matter Landau-Ginzburg systems, and consistency requirements. Higher-genus amplitudes decompose as a sum of contributions from the bulk and the boundary of moduli space. These amplitudes generate the W∞ algebra.
We study the vacuum structure and don spectrum of N=2 supersymmetric gauge theory in four dimensions, with gauge group SU(2). The theory turns out to have remarkably rich and physical properties which can nonetheless be described precisely; exact formulas can be obtained, for instance, for electron and dyon masses and the metric on the moduli space of vacua. The description involves a version of Olive-Montonen electric-magnetic duality. The “strongly coupled” vacuum turns out to be a weakly coupled theory of monopoles, and with a suitable perturbation confinement is described by monopole condensation.
We show how the Riemann surface σ of N = 2 Yang-Mills field theory arises in type II string compactifications on Calabi-Yau threefolds. The relevant local geometry is given by fibrations of ALE spaces. The 3-branes that give rise to BPS multiplets in the string descend to self-dual strings on the Riemann surface, with tension determined by a canonically fixed Seiberg-Witten differential λ. This gives, effectively, a dual formulation of Yang-Mills theory in which gauge bosons and monopoles are treated on equal footing, and represents the rigid analog of type II/heterotic string duality. The existence of BPS states is essentially reduced to a geodesic problem on the Riemann surface with metric |λ|2. This allows us, in particular, to easily determine the spectrum of stable BPS states in field theory. Moreover, we identify the six-dimensional space as the world-volume of a five-brane and show that BPS states correspond to two-branes ending on this five-brane.
We show that the non-critical c = 1 string at the self-dual radius is equivalent to topological strings based on the deformation of the conifold singularity of Calabi-Yau threefolds. The Penner sum giving the genus expansion of the free energy of the c = 1 string theory at the self-dual radius therefore gives the universal behaviour of the topological partition function of a Calabi-Yau threefold near a conifold point.
Using the recent advances in our understanding of non-perturbative aspects of type 11 strings we show how non-trivial exact results for N = 2 quantum field theories can be reduced to T-dualities of string theory. This is done by constructing a local geometric realization of quantum field theories together with a local application of mirror symmetry. This construction is not based on any duality conjecture and thus reduces non-trivial quantum field theory results to much better understood T-dualities of type 11 strings. Moreover it can be used in principle to construct in a systematic way the vacuum structure for arbitrary N = 2 gauge theories with matter representations.
We show that a special superconformal coset (with c ^=3) is equivalent to c=1 matter coupled to two-dimensional gravity. This identification allows a direct computation of the correlation functions of the c=1 non-critical string to all genus, and at nonzero cosmological constant, directly from the continuum approach. The results agree with those of the matrix model. Moreover we connect our coset with a twisted version of a euclidean two-dimensional black hole, in which the ghost and matter systems are mixed.
We present a microscopic index formula for the degeneracy of dyons in four-dimensional N = 4 string theory. This counting formula is manifestly symmetric under the duality group, and its asymptotic growth reproduces the macroscopic Bekenstein-Hawking entropy. We give a derivation of this result in terms of the type 11 five-brane compactified on K3, by assuming that its fluctuations are described by a closed string theory on its world-volume. We find that the degeneracies are given in terms of the denominator of a generalized super Kac-Moody algebra. We also discuss the correspondence of this result with the counting of D-brane states.
By studying the partition function of N=4 topologically twisted supersymmetric Yang-Mills on four-manifolds, we make an exact strong coupling test of the Montonen-Olive strong-weak duality conjecture. Unexpected and exciting links are found with two-dimensional rational conformal field theory.
We study SYM gauge theories living on ALE spaces. Using localization formulae we compute the prepotential (and its gravitational corrections) for SU(N) supersymmetric N = 2 , 2* gauge theories on ALE spaces of the An type. Furthermore we derive the Poincaré polynomial describing the homologies of the corresponding moduli spaces of self-dual gauge connections. From these results we extract the N = 4 partition function which is a modular form in agreement with the expectations of SL (2, ℤ) duality.
We study the degenerating limits of superconformal theories for compactifications on singular K3 and Calabi-Yau threefolds. We find that in both cases the degeneration involves creating an Euclidean two-dimensional black hole coupled weakly to the rest of the system. Moreover we find that the conformal theory of An singularities of K3 are the same as that of the symmetric fivebrane. We also find intriguing connections between ADE (1, n) non-critical strings and singular limits of superconformal theories on the corresponding ALE space.
We study compactifications of F-theory on certain Calabi-Yau threefolds. We find that N = 2 dualities of type II/heterotic strings in 4 dimensions get promoted to N = 1 dualities between heterotic string and F-theory in 6 dimensions. The six-dimensional heterotic/heterotic duality becomes a classical geometric symmetry of the Calabi-Yau in the F-theory setup. Moreover the F-theory compactifacation sheds light on the nature of the strong coupling transition and what lies beyond the transition at finite values of heterotic string coupling constant.
We discuss random matrix-model representations of D = 1 string theory, with particular emphasis on the case in which the target space is a circle of finite radius. The duality properties of discretized strings are analyzed and shown to depend on the dynamics of vortices. In the representation in terms of a continuous circle of matrices we find an exact expression for the free energy, neglecting non-singlet states, as a function of the string coupling and the radius which exhibits exact duality. In a second version, based on a discrete chain of matrices, we find that vortices induce, for a finite radius, a Kosterlitz-Thouless phase transition that takes us to a c = 0 theory.
Explicit formulae are obtained for two- and three-loop vacuum amplitudes in the theory of closed oriented bosonic strings at d=26 n terms of the theta constants, with the moduli space being parametrized by period matrices.
We present evidence for the following conjecture: when quantized, the magnetic monopole soliton solutions constructed by 't Hooft and Polyakov, as modified by Prasad, Sommerfield and Bogomolny, form a gauge triplet with the photon, corresponding to a Lagrangian similar to the original Georgi-Glashow one, but with magnetic replacing electric charge.
We study tensor products of the spin modules (i.e. the Fermion Fock space representations) for classical (simple or afiine) Kac-Moody Lie algebras. We find out that there are mutually commutant pairs of classical Kac-Moody algebras acting on the spin modules, and describe the irreducible decompositions in terms of Young diagrams. As applications, we obtain a simple explanation of Jimbo-Miwa's branching rule duality (i.e. isomorphisms between coset Virasoro modules) [JM], generalization thereof and the duality of the modular transformation rules of affine Lie algebra characters. © 1989, Japan Society of Clinical Chemistry. All rights reserved.