Topological Partition Function Research Articles

Topological twists of massive SQCD, Part I

We consider topological twists of four-dimensional 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} supersymmetric QCD with gauge group SU(2) and Nf≤3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_f\\le 3$$\\end{document} fundamental hypermultiplets. The twists are labelled by a choice of background fluxes for the flavour group, which provides an infinite family of topological partition functions. In this Part I, we demonstrate that in the presence of such fluxes the theories can be formulated for arbitrary gauge bundles on a compact four-manifold. Moreover, we consider arbitrary masses for the hypermultiplets, which introduce new intricacies for the evaluation of the low-energy path integral on the Coulomb branch. We develop techniques for the evaluation of these path integrals. In the forthcoming Part II, we will deal with the explicit evaluation.

Topological Recursion and Uncoupled BPS Structures II: Voros Symbols and the tau -Function

We continue our study of the correspondence between BPS structures and topological recursion in the uncoupled case, this time from the viewpoint of quantum curves. For spectral curves of hypergeometric type, we show the Borel-resummed Voros symbols of the corresponding quantum curves solve Bridgeland’s “BPS Riemann–Hilbert problem”. In particular, they satisfy the required jump property in agreement with the generalized definition of BPS indices Omega in our previous work. Furthermore, we observe the Voros coefficients define a closed one-form on the parameter space, and show that (log of) Bridgeland’s tau -function encoding the solution is none other than the corresponding potential, up to a constant. When the quantization parameter is set to a special value, this agrees with the Borel sum of the topological recursion partition function Z_textrm{TR}, up to a simple factor.

2-Parameter $$\tau $$-Function for the First Painlevé Equation: Topological Recursion and Direct Monodromy Problem via Exact WKB Analysis

We show that a 2-parameter family of $\tau$-functions for the first Painlev\'e equation can be constructed by the discrete Fourier transform of the topological recursion partition function for a family of elliptic curves. We also perform an exact WKB theoretic computation of the Stokes multipliers of associated isomonodromy system assuming certain conjectures.

Multi-trace correlators from permutations as moduli space

We study the n-point functions of scalar multi-trace operators in the U(Nc) gauge theory with adjacent scalars, such as mathcal{N} = 4 super Yang-Mills, at tree-level by using finite group methods. We derive a set of formulae of the general n-point functions, valid for general n and to all orders of 1/Nc. In one formula, the sum over Feynman graphs becomes a topological partition function on Σ0,n with a discrete gauge group, which resembles closed string interactions. In another formula, a new skeleton reduction of Feynman graphs generates connected ribbon graphs, which resembles open string interaction. We define the moduli space {mathrm{mathcal{M}}}_{g,n}^{mathrm{gauge}} from the space of skeleton-reduced graphs in the connected n-point function of gauge theory. This moduli space is a proper subset of {mathrm{mathcal{M}}}_{g,n} stratified by the genus, and its top component gives a simple triangulation of Σg,n.

Equivariant U(N) Verlinde algebra from Bethe/gauge correspondence

We compute the topological partition function (twisted index) of mathcal{N} = 2 U(N) Chern-Simons theory with an adjoint chiral multiplet on Σg × S1. The localization technique shows that the underlying Frobenius algebra is the equivariant Verlinde algebra which is obtained from the canonical quantization of the complex Chern-Simons theory regularized by U(1) equivariant parameter t. Our computation relies on a Bethe/Gauge correspondence which allows us to represent the equivariant Verlinde algebra in terms of the Hall-Littlewood polynomials Pλ(xB, t) with a specialization by Bethe roots xB of the q-boson model. We confirm a proposed duality to the Coulomb branch limit of the lens space superconformal index of four dimensional mathcal{N} = 2 theories for SU(2) and SU(3) with lower levels. In SU(2) case we also present more direct computation based on Jeffrey-Kirwan residue operation.

Beyond triality: dual quiver gauge theories and little string theories

The web of dual gauge theories engineered from a class of toric Calabi-Yau threefolds is explored. In previous work, we have argued for a triality structure by compiling evidence for the fact that every such manifold XN,M (for given (N, M)) engineers three a priori different, weakly coupled quiver gauge theories in five dimensions. The strong coupling regime of the latter is in general described by Little String Theories. Furthermore, we also conjectured that the manifold XN,M is dual to XN ′,M ′ if NM = N′M′ and gcd(N, M) = gcd(N′, M′). Combining this result with the triality structure, we currently argue for a large number of dual quiver gauge theories, whose instanton partition functions can be computed explicitly as specific expansions of the topological partition function {mathcal{Z}}_{N,M} of XN,M. We illustrate this web of dual theories by studying explicit examples in detail. We also undertake first steps in further analysing the extended moduli space of XN,M with the goal of finding other dual gauge theories.

Stringy instanton counting and topological strings

We study the stringy instanton partition function of four dimensional $$ \mathcal{N}=2 $$ U(N) supersymmetric gauge theory which was obtained by Bonelli et al. in 2013. In type IIB string theory on $$ {\mathrm{\mathbb{C}}}^2\times {T}^{*}{\mathrm{\mathbb{P}}}^1\times \mathrm{\mathbb{C}} $$ , the stringy U(N) instantons of charge k are described by k D1-branes wrapping around the $$ {\mathrm{\mathbb{P}}}^1 $$ bound to N D5-branes on $$ {\mathrm{\mathbb{C}}}^2\times {\mathrm{\mathbb{P}}}^1 $$ . The KK corrections induced by compactification of the $$ {\mathrm{\mathbb{P}}}^1 $$ give the stringy corrections. We find a relation between the stringy instanton partition function whose quantum stringy corrections have been removed and the K-theoretic instanton partition function, or by geometric engineering, the refined topological A-model partition function on a local toric Calabi-Yau threefold. We also study the quantum stringy corrections in the stringy instanton partition function which is not captured by the refined topological strings.

Quiver matrix model and topological partition function in six dimensions

We consider a topological quiver matrix model which is expected to give a dual description of the instanton dynamics of topological U(N) gauge theory on D6 branes. The model is a higher dimensional analogue of the ADHM matrix model that leads to Nekrasov's partition function. The fixed points of the toric action on the moduli space are labeled by colored plane partitions. Assuming the localization theorem, we compute the partition function as an equivariant index. It turns out that the partition function does not depend on the vacuum expectation values of Higgs fields that break U(N) symmetry to U(1)^N at low energy. We conjecture a general formula of the partition function, which reduces to a power of the MacMahon function, if we impose the Calabi-Yau condition. For non Calabi-Yau case we prove the conjecture up to the third order in the instanton expansion.

𝒩 = 4 SYM on K3 and the AdS3/CFT2correspondence

We study the Fareytail expansion of the topological partition function of N=4 SU(N) super Yang-Mills theory on K3. We argue that this expansion corresponds to a sum over geometries in asymptotically AdS_3 spacetime, which is holographically dual to a large number of coincident fundamental heterotic strings.

Formula omitted] topological amplitudes and string effective action

Certain scattering amplitudes in the gravitational sector of type II string theory on K 3 × T 2 are found to be computed by correlation functions of the N = 4 topological string. This analysis extends the already known results for K3 by Berkovits and Vafa, which correspond to six-dimensional terms in the effective action, involving four Riemann tensors and 4 g − 4 graviphotons, R 4 T 4 g − 4 , at genus g. We find two additional classes of topological amplitudes that use the full internal SCFT of K 3 × T 2 . One of these string amplitudes is mapped to a 1-loop contribution in the heterotic theory, and is studied explicitly. It corresponds to the four-dimensional term R 2 ( d d Φ ) 2 T 2 g − 4 , with Φ a Kaluza–Klein graviscalar from T 2 . Finally, the generalization of the harmonicity relation for its moduli dependent coupling coefficient is obtained and shown to contain an anomaly, generalizing the holomorphic anomaly of the N = 2 topological partition function F g .

Dimensional reduction of Seiberg-Witten monopole equations, N=2 noncommutative supersymmetric field theories and Young diagrams

We investigate the Seiberg-Witten monopole equations on noncommutative(N.C.) R^4 at the large N.C. parameter limit, in terms of the equivariant cohomology. In other words, N}=2 supersymmetric U(1) gauge theories with hypermultiplet on N.C. R}^4 are studied. It is known that after topological twisting partition functions of N}>1 supersymmetric theories on N.C. R^2D are invariant under N.C.parameter shift, then the partition functions can be calculated by its dimensional reduction. At the large N.C. parameter limit, the Seiberg-Witten monopole equations are reduced to ADHM equations with the Dirac equation reduced to 0 dimension. The equations are equivalent to the dimensional reduction of non-Abelian U(N) Seiberg-Witten monopole equations in N -> \infty. The solutions of the equations are also interpreted as a configuration of brane anti-brane system. The theory has global symmetries under torus actions originated in space rotations and gauge symmetries. We investigate the Seiberg-Witten monopole equations reduced to 0 dimension and the fixed point equations of the torus actions. We show that the Dirac equation reduced to 0 dimension is trivial when the fixed point equations and the ADHM equations are satisfied. For finite N, it is known that the fixed points of the ADHM data are isolated and are classified by the Young diagrams. We give a new proof of this statement by solving the ADHM equations and the fixed point equations concretely and by giving graphical interpretations of the field components and these equations.

Open string topological amplitudes and gaugino masses

We discuss the moduli-dependent couplings of the higher derivative F-terms ( Tr W 2 ) h − 1 , where W is the gauge N = 1 chiral superfield. They are determined by the genus zero topological partition function F ( 0 , h ) , on a world-sheet with h boundaries. By string duality, these terms are also related to heterotic topological amplitudes studied in the past, with the topological twist applied only in the left-moving supersymmetric sector of the internal N = ( 2 , 0 ) superconformal field theory. The holomorphic anomaly of these couplings relates them to terms of the form Π n ( Tr W 2 ) h − 2 , where Π's represent chiral projections of non-holomorphic functions of chiral superfields. An important property of these couplings is that they violate R-symmetry for h ⩾ 3 . As a result, once supersymmetry is broken by D-term expectation values, ( Tr W 2 ) 2 generates gaugino masses that can be hierarchically smaller than the scalar masses, behaving as m 1 / 2 ∼ m 0 4 in string units. Similarly, Π Tr W 2 generates Dirac masses for non-chiral brane fermions, of the same order of magnitude. This mechanism can be used, for instance, to obtain fermion masses at the TeV scale for scalar masses as high as m 0 ∼ O ( 10 13 ) GeV . We present explicit examples in toroidal string compactifications with intersecting D-branes.

Topological string amplitudes, complete intersection Calabi-Yau spaces and threshold corrections

We present the most complete list of mirror pairs of Calabi-Yau complete intersections in toric ambient varieties and develop the methods to solve the topological string and to calculate higher genus amplitudes on these compact Calabi-Yau spaces. These symplectic invariants are used to remove redundancies in examples. The construction of the B-model propagators leads to compatibility conditions, which constrain multi-parameter mirror maps. For K3 fibered Calabi-Yau spaces without reducible fibers we find closed formulas for all genus contributions in the fiber direction from the geometry of the fibration. If the heterotic dual to this geometry is known, the higher genus invariants can be identified with the degeneracies of BPS states contributing to gravitational threshold corrections and all genus checks on string duality in the perturbative regime are accomplished. We find, however, that the BPS degeneracies do not uniquely fix the non-perturbative completion of the heterotic string. For these geometries we can write the topological partition function in terms of the Donaldson-Thomas invariants and we perform a non-trivial check of S-duality in topological strings. We further investigate transitions via collapsing D5 del Pezzo surfaces and the occurrence of free Z2 quotients that lead to a new class of heterotic duals.

Anomaly cancellation in K3 orientifolds

We study in detail the pattern of anomaly cancellation in D=6 Type IIB Z N orientifolds, occurring through a generalized Green–Schwarz mechanism involving several RR antisymmetric tensors and scalars fields. The starting point is a direct string theory computation of the inflow of anomaly arising from magnetic interaction of D-branes, O-planes and fixed points, which are encoded in topological one-loop partition functions in the RR odd spin-structure. All the RR anomalous couplings of these objects are then obtained by factorization. They are responsible for a spontaneous breaking of U(1) factors through a Higgs mechanism involving the corresponding hypermultiplets. Some of them are also related by supersymmetry to gauge couplings involving the NSNS scalars sitting in the tensor multiplets. We also comment on the possible occurrence of tensionless strings when these couplings diverge.

Topological amplitudes in heterotic superstring theory

We show that certain heterotic string amplitudes are given in terms of correlators of the twisted topological (2,0) SCFT, corresponding to the internal sector of the N = 1 space-time supersymmetric background. The genus g topological partition function F g corresponds to a term in the effective action of the form W 2 g , where W is the gauge or gravitational superfield. We study also recursion relations related to holomorphic anomalies, showing that, contrary to the type II case, they involve correlators of anti-chiral superfields. The corresponding terms in the effective action are of the form W 2 g Π n , where Π is a chiral superfield obtained by chiral projection of a general superfield. We observe that the structure of the recursion relations is that of N = 1 spacetime supersymmetry Ward identity. We give also a solution of the tree-level recursion relations and discuss orbifold examples.

Topological partition function and string-string duality

Evidence for string/string-duality can be given not only by matching the vector couplings but also by matching the gravitational couplings. In this note this is shown in the rank three example, the closest stringy analog of the Seiberg/Witten-setup, which is related to the Calabi-Yau WP 1,1,2,2,6 4(12). I provide an exact analytical verification of a relation checked by coefficient comparison to fourth order by Kaplunovsky, Louis and Theisen.

N = 2 type II-heterotic duality and higher-derivative F-terms

We test the recently conjectured duality between N = 2 supersymmetric type II and heterotic string models by analysing a class of higher-dimensional interactions in the respective low-energy Lagrangians. These are F-terms of the form F g W 2 g where W is the gravitational superfield. On the type II side these terms are generated at the g-loop level and in fact are given by topological partition functions of the twisted Calabi-Yau sigma model. We show that on the heterotic side these terms arise at the one-loop level. We study in detail a rank-3 example and show that the corresponding couplings F g satisfy the same holomorphic anomaly equations as in the type II case. Moreover we study the leading singularities of F g 's on the heterotic side, near the enhanced symmetry point and show that they are universal poles of order 2 g − 2 with coefficients that are given by the Euler number of the moduli space of genus- g Riemann surfaces. This confirms a recent conjecture that the physics near the conifold singularity is governed by c = 1 string theory at the self-dual point.

C = 1 string as the topological theory of the conifold

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.

Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces

We extend the discussion of mirror symmetry, Picard-Fuchs equations, instanton corrected Yukawa couplings and the topological one-loop partition function to the case of complete intersections with higher dimensional moduli spaces. We will develop a new method of obtaining the instanton corrected Yukawa couplings through a study of the solutions of the Picard-Fuchs equations. This leads to closed formulas for the prepotential for the Kähler moduli fields induced from the ambient space for all complete intersections in nonsingular weighted projective spaces. As examples we treat part of the moduli space of the phenomenologically interesting three-generation models which are found in this class. We also apply our method to solve the simplest model in which a topology change was observed and discuss examples of complete intersections in singular ambient spaces.

N = 4 topological strings

We show how to make a topological string theory starting from an N = 4 superconformal theory. The critical dimension for this theory is c ̂ = 2 (c = 6) . It is shown that superstrings (in both the RNS and GS formulations) and critical N = 2 strings are special cases of this topological theory. Applications for this new topological theory include: (1) Proving the vanishing to all orders of all scattering amplitudes for the self-dual N = 2 string with flat background, with the exception of the three-point function and the closed-string partition function; (2) Showing that the topological partition function of the N = 2 string on the K3 background may be interpreted as computing the superpotential in harmonic superspace generated upon compactification of type II superstrings from 10 to 6 dimensions; and (3) Providing a new prescription for calculating s uperstring amplitudes which appears to be free of total-derivative ambiguities.