Using twistors, the * Gibbons-Hawking ansatz* is generalized to investigate 4n-dimensional hyperkähler metrics admitting an action of the n-torus T

There are in fact many explicit examples known of metrics on non-compact manifolds with SU(n) or Sp(2n) holonomy. The other holonomy groups automatically yielding Ricci-flat metrics are the special holonomy groups G_{2} in dimension 7 and Spin(7) in dimension 8. Until fairly recently only three explicit examples of complete metrics (in dimension 7) with * G_{2}-holonomy* and one explicit example (in dimension 8) with Spin(7)-holonomy were known. The G

The G_{2}-holonomy examples are all examples in which a Lie group G acts with low codimension orbits. This is a general feature of explicit examples of Einstein metrics. The simplest case of such a situation would be when there is a single orbit of a group action, in which case the metric manifold is homogeneous. For metrics on homogeneous manifolds, the Einstein condition may be expressed purely algebraically. Moreover, all homogeneous Ricci-flat manifolds are flat, and so no interesting metrics occur. Then what about cohomogeneity one with respect to G, i.e., the orbits of G are codimension one in general? Here, the Einstein condition reduces to a system of non-linear ordinary differential equations in one variable, namely the parameter on the orbit space. In the Ricci-flat case, * Cheeger-Gromoll theorem* implies that the manifold has at most one end. In the non-compact case, the orbit space is R

Brecht’s circular circuitry is **here**.

Allow me to make cross-sectional (both historically and geographically) references. I start with Mussolini, who talked of what use fascism could be put to by stating that capitalism throws itself into the protection of the state when it is in crisis, and he illustrated this point by referring to the Great Depression as a failure of laissez-faire capitalism and thus creating an opportunity for fascist state to provide an alternative to this failure. This in a way points to the fact that fascism springs to life economically in the event of capitalism’s deterioration. To highlight this point of fascism springing to life as a reaction to capitalism’s failure, let me take recourse to Samir Amin, who calls the fascist choice for managing a capitalist society in crisis as a categorial rejection of democracy, despite having reached that stage democratically. The masses are subjected to values of submission to a unity of socio-economic, political and/or religious ideological discourses. This is one reason why I call fascism not as a derivative category of capitalism in the sense of former being the historic phase of the latter, but rather as a coterminous tendency waiting in dormancy for capitalism to deteriorate, so that fascism could then detonate. But, are fascism and capitalism related in a multiple of ways is as good as how socialism is related with fascism, albeit only differently categorically.

It is imperative for me to add by way of what I perceive as financial capitalism and bureaucracy and where exactly art gets sandwiched in between the two, for more than anything else, I would firmly believe in Brecht as continuing the artistic practices of Marxian sociology and political-economy.

The financial capitalism combined with the impersonal bureaucracy has inverted the traditional schematic forcing us to live in a totalitarian system of financial governance divorced from democratic polity. It’s not even fascism in the older sense of the term, by being a collusion of state and corporate power, since the political is bankrupt and has become a mediatainment system of control and buffer against the fact of Plutocracies. The state will remain only as long as the police systems are needed to fend off people claiming rights to their rights. Politicians are dramaturgists and media personalities rather than workers in law. If one were to just study the literature and paintings of the last 3-4 decades, it is fathomable where it is all going. Arts still continue to speak what we do not want to hear. Most of our academics are idiots clinging on to the ideological culture of the left that has put on its blinkers and has only one enemy, which is the right (whatever the hell that is). Instead of moving outside their straightjackets and embracing the world of the present, they still seem to be ensconced in 19th century utopianism with the only addition to their arsenal being the dramatic affects of mass media. Remember Thomas Pynchon of Gravity’s Rainbow fame (I prefer calling him the illegitimate cousin of James Joyce for his craftiness and smoothly sailing contrite plots: there goes off my first of paroxysms!!), who likened the system of techno-politics as an extension of our inhuman core, at best autonomous, intelligent and ever willing to exist outside the control of politics altogether. This befits the operational closure and echoing time and time again that technology isn’t an alien thing, but rather a manifestation of our inhuman core, a mutation of our shared fragments sieved together in ungodly ways. This is alien technologies in gratitude.

We have never been natural, and purportedly so by building defence systems against the natural both intrinsically and extrinsically. Take for example, Civilisation, the most artificial construct of all humans had busied themselves building and now busying themselves upholding. what is it? A Human Security System staving off entropy of existence through the self-perpetuation of a cultural complex of temporal immortalisation, if nothing less and vulnerable to editions by scores of pundits claiming to a larger schemata often overlooked by parochiality. Haven’t we become accustomed to hibernating in an artificial time now exposed by inhabiting the infosphere, creating dividualities by reckoning to data we intake, partake and outtake. Isn’t analysing the part/whole dividuality really scoring our worthiness? I know the answer is yes, but merely refusing to jump off the tongue. Democracies have made us indolent with extremities ever so flirting with electronic knowledge waiting to be turned to digital ash when confronted with the existential threat to our *locus standi*.

But, we always think of a secret cabal conspiring to dehumanise us. But we also forget the impersonality of the dataverse, the infosphere, the carnival we simply cannot avoid being a part of. Our mistaken beliefs lie in reductionism, and this is a serious detriment to causes created *ex nihilo*, for a fight is inevitably diluted if we pay insignificance to the global meshwork of complex systems of economics and control, for these far outstrip our ability to pin down to a critical apparatus. This apparatus needs to be different from ones based on criticism, for the latter is prone to sciolist tendencies. Maybe, one needs to admit allegiance to perils of our position and go along in a Socratic irony before turning in against the admittance at opportune times. Right deserves tackling through the Socratic irony, lest taking offences become platitudinous. Let us not forget that the modern state is nothing but a PR firm to keep the children asleep and unthinking and believing in the dramaturgy of the political as real. And this is where Brecht comes right back in, for he considered creation of bureaucracies as affronting not just fascist states, but even communist ones. The above aside, or digression is just a reality check on how much complex capitalism has become and with it, its derivatives of fascism as these are too intertwined within bureaucratic spaces. Even when Brecht was writing in his heydays, he took a deviation from his culinary-as-ever epic theatre to found a new form of what he called theatre as learning to play that resembled his political seminars modeled on the rejection of the concept of bureaucratic elitism in partisan politics where the theorists and functionaries issued directives and controlled activities on behalf of the masses to the point of submission of the latter to the former. This point is highlighted not just for fascist states, but equally well for socialist/communist regimes reiterating the fact that fascism is potent enough to develop in societies other than capitalistic ones.

Moving on to the point when mentions of democracy as bourgeois democracy is done in the same breath as regards equality only for those who are holders of capital are turning platitudinous. Well, structurally yes, this is what it seems like, but reality goes a bit deeper and thereafter fissures itself into looking at if capital indeed is what it is perceived as in general, or is there more to it than meets the eye. I quip this to confront two theorists of equality with one another: Piketty and Sally Goerner. Piketty misses a great opportunity to tie the “r > g” idea (after tax returns on capital r > growth rate of economy g) to the “limits to growth”. With a careful look at history, there are several quite important choice points along the path from the initial hope it won’t work out that way… to the inevitable distressing end he describes, and sees, and regrets. It’s what seduces us into so foolishly believing we can maintain “g > r”, despite the very clear and hard evidence of that faiIing all the time… that sometimes it doesn’t. The real “central contradiction of capitalism” then, is that it promises “g > r”, and then we inevitably find it is only temporary. Growth is actually nature’s universal start-up process, used to initially build every life, including the lives of every business, and the lives of every society. Nature begins building things with growth. She’s then also happy to destroy them with more of the same, those lives that began with healthy growth that make the fateful choice of continuing to devote their resources to driving their internal and external strains to the breaking point, trying to make g > r perpetual. It can’t be. So the secret to the puzzle seems to be: Once you’ve taken growth from “g > r” to spoiling its promise in its “r > g” you’ve missed the real opportunity it presented. Sally Goerner writes about how systems need to find new ways to grow through a process of rising intricacy that literally reorganizes the system into a higher level of complexity. Systems that fail to do that collapse. So smart growth is possible (a cell divides into multiple cells that then form an organ of higher complexity and greater intricacy through working cooperatively). Such smart growth is regenerative in that it manifests new potential. How different that feels than conventional scaling up of a business, often at the expense of intricacy (in order to achieve so called economies of scale). Leaps of complexity do satisfy growing demands for productivity, but only temporarily, as continually rising demands of productivity inevitably require ever bigger leaps of complexity. Reorganizing the system by adopting ever higher levels of intricacy eventually makes things ever more unmanageable, naturally becoming organizationally unstable, to collapse for that reason. So seeking the rise in productivity in exchange for a rising risk of disorderly collapse is like jumping out of the fry pan right into the fire! As a path to system longevity, then, it is tempting but risky, indeed appearing to be regenerative temporarily, until the same impossible challenge of keeping up with ever increasing demands for new productivity drives to abandon the next level of complexity too! The more intricacy (tight, small-scale weave) grows horizontally, the more unmanageable it becomes. That’s why all sorts of systems develop what we would call hierarchical structures. Here, however, hierarchal structures serve primarily as connective tissue that helps coordinate, facilitate and communicate across scales. One of the reasons human societies are falling apart is because many of our hierarchical structures no longer serve this connective tissue role, but rather fuel processes of draining and self-destruction by creating sinks where refuse could be regenerated. Capitalism, in its present financial form is precisely this sink, whereas capitalism wedded to fascism as an historical alliance doesn’t fit the purpose and thus proving once more that the collateral damage would be lent out to fascist states if that were to be the case, which would indeed materialize that way.

That democracy is bourgeois democracy is an idea associated with Swedish political theorist Goran Therborn, who as recent as the 2016 US elections proved his point by questioning the whole edifice of inclusive-exclusive aspects of democracy, when he said,

Even if capitalist markets do have an inclusive aspect, open to exchange with anyone…as long as it is profitable, capitalism as a whole is predominantly and inherently a system of social exclusion, dividing people by property and excluding the non-profitable. a system of this kind is, of course, incapable of allowing the capabilities of all humankind to be realized. and currently the the system looks well fortified, even though new critical currents are hitting against it.

Democracy did take on a positive meaning, and ironically enough, it was through rise of nation-states, consolidation of popular sovereignty championed by the west that it met its two most vociferous challenges in the form of communism and fascism, of which the latter was a reactionary response to the discontents of capitalist modernity. Its radically lay in racism and populism. A degree of deference toward the privileged and propertied, rather than radical opposition as in populism, went along with elite concessions affecting the welfare, social security, and improvement of the working masses. This was countered by, even in the programs of moderate and conservative parties by using state power to curtail the most malign effects of unfettered market dynamics. It was only in the works of Hayek that such interventions were beginning to represent the road to serfdom thus paving way to modern-day right-wing economies, of which state had absolutely no role to play as regards markets fundamentals and dynamics. The counter to bourgeois democracy was rooted in social democratic movements and is still is, one that is based on negotiation, compromise, give and take a a grudgingly given respect for the others (whether ideologically or individually). The point again is just to reiterate that fascism, in my opinion is not to be seen as a nakedest form of capitalism, but is generally seen to be floundering on the shoals of an economic slowdown or crisis of stagflation.

On ideal categories, I am not a Weberian at heart. I am a bit ambiguous or even ambivalent to the role of social science as a discipline that could draft a resolution to ideal types and interactions between those generating efficacies of real life. Though, it does form one aspect of it. My ontologies would lie in classificatory and constructive forms from more logical grounds that leave ample room for deviations and order-disorder dichotomies. Complexity is basically an offspring of entropy.

And here is where my student-days of philosophical pessimism surface, or were they ever dead, as the real way out is a dark path through the world we too long pretended did not exist.

]]>Consider the surface S ⊆ (C^{∗})^{2} defined by the equation z_{1} + z_{2} + 1 = 0. Define the map log : (C^{∗})^{2} → R^{2} by log(z_{1}, z_{2}) = (log|z_{1}|, log|z_{2}|). Then log(S) can be seen as follows. Consider the image of S under the absolute value map.

The line segment r_{1} + r_{2} = 1 with r_{1}, r_{2} ≥ 0 is the image of {(−a, a−1)|0 < a < 1} ⊆ S; the ray r_{2} = r_{1} + 1 with r_{1} ≥ 0 is the image of {(−a, a−1)|a < 0} ⊆ S; and the ray r_{1} = r_{2} + 1 is the image of {(−a, a−1)|a > 1} ⊆ S. The map S → |S| is one-to-one on the boundary of |S| and two-to-one in the interior, with (z_{1}, z_{2}) and (z̄_{1}, z̄_{2}) mapping to the same point in |S|. Taking the logarithm of this picture, we obtain the amoeba of S, log(S) as depicted below.

Now consider S = S × {0} ⊆ Y = (C^{∗})^{2} × R = T^{2} × R^{3}. We can now obtain a six-dimensional space X, with a map π : X → Y, an S^{1}-bundle over Y\S degenerating over S, so that π^{−1}(S) →^{≅} S. We then have a T^{3}-fibration on X, f : X → R^{3}, by composing π with the map (log, id) : (C^{∗})^{2} × R → R^{3} = B. Clearly the discriminant locus of f is log(S) × {0}. If b is in the interior of log(S) × {0}, then f^{−1}(b) is obtained topologically by contracting two circles {p_{1}} × S^{1} and {p_{2}} × S^{1} on T^{3} = T^{2} × S^{1} to points. These are the familiar conical singularities seen in the special Lagrangian situation.

If b ∈ ∂(log(S) × {0}), then f^{−1}(b) has a slightly more complicated singularity, but only one. Let us examine how the “generic” singular fiber fits in here. In particular, for b in the interior of log(S) × {0}, locally this discriminant locus splits B into two regions, and these regions represent two different possible smoothings of f^{−1}(b).

Assume now that f : X → B is a special Lagrangian fibration with topology and discriminant locus ∆ being an amoeba. Let b ∈ Int(∆), and set M = f^{−1}(b). Set M^{o} = M\{x_{1}, x_{2}}, where x_{1}, x_{2} are the two conical singularities of M. Suppose that the tangent cones to these two conical singularities, C_{1} and C_{2}, are both cones of the form M^{0}. Then the links of these cones, Σ_{1} and Σ_{2}, are T^{2}’s, and one expects that topologically these can be described as follows. Note that M^{o} ≅ (T^{2}\{y_{1}, y_{2}}) × S^{1} where y_{1}, y_{2} are two points in T^{2}. We assume that the link Σ_{i} takes the form γ_{i} × S^{1}, where γ_{i} is a simple loop around y_{i}. If these assumptions hold, then to see how M can be smoothed, we consider the restriction maps in cohomology

H^{1}(M^{o}, R) → H^{1}(Σ_{1}, R) ⊕ H^{1}(Σ_{2}, R)

The image of this map is two-dimensional. Indeed, if we write a basis e^{i}_{1}, e^{i}_{2} of H^{1}(Σ_{i}, R) where e^{i}_{1} is Poincaré dual to [γ_{i}] × pt and e^{i}_{2} is Poincaré dual to pt × S^{1}, it is not difficult to see the image of the restriction map is spanned by {(e^{1}_{1}, e^{2}_{1})} and {(e^{1}_{2}, −e^{2}_{2})}. Now this model of a topological fibration is not special Lagrangian, so in particular we don’t know exactly how the tangent cones to M at x_{1} and x_{2} are sitting inside C^{3}, and thus can’t be compared directly with an asymptotically conical smoothing. So to make a plausibility argument, choose new bases f^{i}_{1}, f^{i}_{2} of H^{1}(Σ_{i}, R) so that if M(a,0,0), M(0,a,0) and M(0,0,a) are the three possible smoothings of the two singular tangent cones at the singular points x_{1}, x_{2} of M. Then Y(M_{i}^{(a,0,0)}) = πaf^{i}_{1}, Y(M_{i}^{(0,a,0)}) = πaf^{i}_{2}, and Y(M_{i}(0,0,a)) = −πa(f^{i}_{1} + f^{i}_{2}).

Suppose that in this new basis, the image of the restriction map is spanned by the pairs (f^{1}_{1}, rf^{2}_{2}) and (rf^{1}_{2}, f^{2}_{1}) for r > 0, r ≠ 1. Then, there are two possible ways of smoothing M, either by gluing in M_{1}^{(a,0,0)} and M_{2}^{(0,ra,0)} at the singular points x_{1} and x_{2} respectively, or by gluing in M_{1}^{(0,ra,0)} and M_{2}^{(a,0,0)} at x_{1} and x_{2} respectively. This could correspond to deforming M to a fiber over a point on one side of the discriminant locus of f or the other side. This at least gives a plausibility argument for the existence of a special Lagrangian fibration of the topological type given by f. To date, no such fibrations have been constructed, however.

On giving a special Lagrangian fibration with codimension one discriminant and singular fibers with cone over T^{2} singularities, one is just forced to confront a codimension one discriminant locus in special Lagrangian fibrations. This leads inevitably to the conclusion that a “strong form” of the * Strominger-Yau-Zaslow conjecture* cannot hold. In particular, one is forced to conclude that if f : X → B and f’ : X’ → B are dual special Lagrangian fibrations, then their discriminant loci cannot coincide. Thus one cannot hope for a fiberwise definition of the dualizing process, and one needs to refine the concept of dualizing fibrations. Let us see why the discriminant locus must change under dualizing. The key lies in the behaviour of the positive and negative vertices, where in the positive case the critical locus of the local model of the fibration is a union of three holomorphic curves, while in the negative case the critical locus is a pair of pants. In a “generic” special Lagrangian fibration, we expect the critical locus to remain roughly the same, but its image in the base B will be fattened out. In the negative case, this image will be an amoeba. In the case of the positive vertex, the critical locus, at least locally, consists of a union of three holomorphic curves, so that we expect the discriminant locus to be the union of three different amoebas. The figure below shows the new discriminant locus for these two cases.

Now, under dualizing, positive and negative vertices are interchanged. Thus the discriminant locus must change. This is all quite speculative, of course, and underlying this is the assumption that the discriminant loci are just fattenings of the graphs. However, it is clear that a new notion of dualizing is necessary to cover this eventuality.

1. When Brecht talks of acceding to the capitulation of Capitalism, in that, being a historic phase and new and old at the same time, this nakedest manifestation of Capitalism is attributed to relationality, which are driven by functionalist propositions and are non-linear, reversible schemas existing independently of the specific contents that are inserted as variables. This may sound a bit philosophical, but is the driving force behind Brecht’s understanding of Capitalism and is perfectly corroborated in his famous dictum, “Reality as such has slipped into the domain of the functional.” This dictum underlines what is new and what is old at the same time.

2. Sometime in the 30s, Brecht’s writings corroborated the linkages between Capitalism and Fascism, when the victories of European fascism prompted consideration of the relationship between collective violence and regressive social configurations. At its heart, his corpus during the times was a defining moment of finance capital, an elaborate systemic treatment of economic transactions within the literary narrative with fascistic overtones. It is here the capitalist is consummate par excellence motivated by the rational calculus (Ayn Rand rings the bells!!!). Eschewing the narrative desire of the traditional dramatic novel, Brecht compels the readers without any recourse to emotional intensity and catharsis, and capturing the attention via phlegmatic and sublimated pleasures of logical analysis, riddle solving, remainder less, and bookkeeping. This coming together of the financial capital with the rise in European Fascism, despite leading to barbaric times in due course, brought forth the progeny of corporation merging with the state incorporating social functions into integrated networks of production and consumption. What Brecht reflects as barbaric is incidentally penned in these tumultuous ear, where capital evolves from Fordist norms into Corporations and in the process atrophy human dimensions. This fact is extrapolated in contemporary times when capital has been financialized to the extent of artificial intelligences, HFTs and algorithmic decision making, just to sound a parallel to **Nature 2.0**.

But, before digressing a bit too far, where is Brecht lost in the history of class consciousness here? With capital evolving exponentially, even if there is no or little class consciousness in the proletariat, there will come a realization that exploitation is widespread. This is the fecund ground when nationalist and fascist rhetoric seeds into a full-grown tree, inciting xenophobias infused with radicalization (this happened historically in Italy and in Germany, and is getting replicated on micro-to-macro scales contemporarily). But, what Brecht has failed to come to terms with is the whole logic of fascists against the capitalist. Fascists struggle with the capitalist question within their own circles (a far-fetched parallel drawn here as regards India is the right ideologue’s opposition to FDI, for instance). Historically speaking and during times when Bertotl was actively writing, there were more working class members of the Italian fascists than anyone else with anti-capitalist numbers. In Nazi Germany, there were close to 30 per cent within stormtroopers as minimal identifies and sympathizers with communism. The rest looked up to fascism as a stronger alternative to socialism/communism in its militancy. The intellectual and for moral (might be a strikethrough term here, but in any case…) tonic was provided for by the bourgeois liberals who opposed fascism for their capitalist bent. All in all, Brecht could have been prescient to say the most, but was too ensconced, to say the least, in Marxist paradigms to analyze this suturing of ideological interests. That fascism ejected itself of a complete domineering to Capitalism, at least historically, is evident from the trajectory of a revolutionary syndicalist, Edmondo Rossoni, who was extremely critical of internationalism, and spearheaded Italian fascist unions far outnumbering Italian fascist membership. Failure to recognize this fractious relationship between Fascism and Capitalism jettisons the credibility of Brechtian piece linked.

3. Althusser once remarked that Brecht’s work displays two distinct forms of temporality that fail to achieve any mutual integration, which have no relation with one another, despite coexisting and interconnecting, never meet one another. The above linked essay is a prime example of Althusser’s remark. What Brecht achieves is demonstrating incongruities in temporalities of capital and the human (of Capitalism and Barbarianism/Fascism respectively), but is inadequate to take such incongruities to fit into the jigsaw puzzle of the size of Capitalism, not just in his active days, but even to very question of his being prescient for contemporary times, as was mentioned in point 2 in this response. Brecht’s reconstructing of the genealogy of Capitalism in tandem with Fascism parses out the link in commoditized linear history (A fallacy even with Marxian notion of history as history of class consciousness, in my opinion), ending up trapped in tautological circles, since the human mind is short of comprehending the paradoxical fact of Capitalism always seemingly good at presupposing itself.

It is for these reasons, why I opine that Brecht has a circular circuitry.

]]>All of this starts with the dictum, “There are **no** men at work”.

The notion of blockchain is a decentralized polity. Blockchain is immutable, for once written on to the block, it is practically un-erasable. And most importantly, it is collateralized, in that, even if there is a lack thereof of physical assets, the digital ownership could be traded as a collateral. So, once you have a blockchain, you create a stack that could be database controlled using a Virtual Machine, think of it as some sort of digital twin. So, what exactly are the benefits of this decentralized digital polity? One crucial is getting rid of intermediaries (unless, one considers escrow accounts as an invisible intermediary!, which seldom fulfills the definitional criteria). So, in short, digital twinning helps further social scalability by getting intermediaries o to an invisible mode. Now, when blockchains are juxtaposed with algorithmically run machines (AI is just one branch of it), one gets the benefits of social scalability with analytics, the ever-increasing ocean of raw data hermeneutically sealed into information for utilitarian purposes. The advantages of decentralized polity and social scalability compiles for a true democratic experience in an open-sourced modeling, where netizens (since we still are mired in the controversy of net neutrality) experience participatory democracy.

How would these combine with exigencies of scarce nature or resources? It is here that such hackathons combine the ingenuity of blockchain with AI in a process generally referred to as “mining”. This launch from the nature as we know is Nature 2.0. To repeat, decentralized polity and social scalability creates a self-sustaining ecosystem in a sense of Anti-Fragility (yes, Taleb’s anti-fragile is a feedback into this) with autonomously created machine learning systems that are largely correctional in nature on one hand and improving learning capacities from the environment on the other. These two hands coordinate giving rise to resource manipulation in lending a synthetic definition of materialities taken straight from physics textbooks and scared-to-apprehend materialities as thermodynamic quotients. And this is where AI steams up in a grand globalized alliance of machines embodying agencies always looking for cognitive enhancements to fulfill teleonomic life derived from the above stated thermodynamic quotient of randomness and disorder into gratifying sensibilities of self-sustenance. Synthetic biologists (of the Craig Venter and CRISPR-like lines) call this genetic programming, whereas singularitarians term it as evolution, a break away from simulated evolution that defined initial days of AI. The synthetic life is capable of decision making, the more it is subjected to the whims and fancies of surrounding environment via the process of machine learning leading to autonomous materialities with cognitive capabilities. These are parthenogenetic machines with unencumbered networking capacities. Such is the advent of self-ownership, and taking it to mean to nature as we have hitherto known is a cathectic fallacy in ethics. Taking to mean it differently in a sense of establishing a symbiotic relationship between biology and machines to yield bio machines with characteristics of biomachinations, replication (reproduction, CC and CV to be thrown open for editing via genetic programming) and self-actualization is what blockchain in composite with AI and Synthetic Biology is Nature 2.0.

Yes, there are downsides to traditional mannerisms of thought, man playing god with nature and so on and so on…these are ethical constraints and thus political in undertones, but with conservative theoretics and thus unable to come to terms with the politics of resource abundance that the machinic promulgates…

]]>On a seemingly daily basis, we see attacks against the intellectual culture of the academic humanities, which, since the 1960s, have opened up spaces for leftists to develop critical theories of all kinds. Attacks from supposedly liberal professors and centrist op-ed columnists, from well-funded conservative think tanks and white supremacists on college campus tours. All rail against the evils of feminism, post-modernism, and something called “neo-Marxism” with outsized agitation.

For students and professors, the onslaughts are exhausting, and not only because they have very real, often dangerous, consequences, but because they all attack the same straw men (or “straw people”) and refuse to engage with academic thought on its own terms. Rarely, in the exasperating proliferation of cranky, cherry-picked anti-academia op-eds do we encounter people actually reading and grappling with the ideas of their supposed ideological nemeses….

Visit

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i have been thinking over the last few weeks about the nature of hope and its role in my life or rather essential/existential life. the more i reflect on the nature of hope, the more so, the concept seems to be erasing itself from my existence. but, ultimately, what counts is the liquid nature of reflection based on passion. passion, per se, is the emotion par excellence and is indispensable to move towards either the positive or the negative axis of the concept of the question. it is like the arrest between the man born posthumously or the man who has lived without being born. in both cases, the strangeness is displaced or rather deliberately misplaced and placed marginally for the hope to take the center stage. hoping is clinging on to the position of hypothetical, placing oneself in an attempt, a never-say-die one at that, to keep the invasion of the strange-ness talked about a sentence before this. the real is the strange-ness that is brutally kept at a distant, for us to enjoin the virtual realm. but aren’t we all victims of the real at some point in time and space? we fail terminologically to ascribe real to it and at the same time masking the virtual with the virtually real/really virtual. this is the space we live in and could be called the schizosphere. we are an agent who act not on the world stage, but in our own schizoid lives. the radiation of the result herein is two-fold. one is the transcendental i.e. a passionate longing for the escape from the predicament and other is the immanental, a silent erasure of the acceptance of the fact. the latter is buried and made dormant as the repressive tendencies that we call hopelessness and that does strike all of us, however hard we call our transcendentalism a success. this just calls for the non-fixity of our hope in time, a process that is always happening, a process that knows no ends. in short, a procedural aporia. now, isn’t it clear, that procedural aporia would always entail a fight of hope on hopelessness, in order to gain this static role it has always been bent on acheiving. QED, isn’t this journey substitutability by hope, which would imply that hoping on hope is a ‘hope into hopelessness’ as slight fracturing of the subject. the old subject is cut by the new one, but the wounds to disappear as fast they are created, a suturing takes place to take us back from this impediment to the un-cut stage….

]]>Geometric Langlands Duality can be formulated as follows: Let C be a Riemann surface (compact, without boundary), G be a compact reductive Lie group, G_{C} be its complexification, and M_{flat}(G, C) be the moduli space of stable flat G_{C}-connections on C. The Langlands dual of G is another compact reductive Lie group ^{L}G defined by the condition that its weight and coweight lattices are exchanged relative to G. Let Bun(^{L}G, C) be the moduli stack of holomorphic ^{L}G-bundles on C. One of the statements of Geometric Langlands Duality is that the derived category of coherent sheaves on M_{flat}(G, C) is equivalent to the derived category of D-modules over Bun(^{L}G, C).

M_{flat}(G, C) is mirror to another moduli space which, roughly speaking, can be described as the cotangent bundle to Bun(^{L}G, C). The category of A-branes on T ^{∗} Bun(^{L}G, C) (with the canonical symplectic form) is equivalent to the category of B-branes on a noncommutative deformation of T ^{∗} Bun(^{L}G, C). The latter is the same as the category of (analytic) D-modules on Bun(^{L}G, C).

So, what exactly is, the relationship between A-branes and noncommutative B-branes. This relationship arises whenever the target space X is the total space of the cotangent bundle to a complex manifold Y. It is understood that the symplectic form ω is proportional to the canonical symplectic form on T ^{∗} Y. With the B-field vanishing, and Y as a complex, we regard ω as the real part of a holomorphic symplectic form Ω. If q^{i} are holomorphic coordinates on Y, and p_{i} are dual coordinates on the fibers of T ^{∗} Y, Ω can be written as

Ω = 1/ħdp_{i} ∧ dq^{i} = dΘ

Since ω (as well as Ω) is exact, the closed A-model of X is rather trivial: there are no nontrivial instantons, and the quantum cohomology ring is isomorphic to the classical one.

We would like to understand the category of A-branes on X = T ^{∗} Y. The key observation is that ∃ a natural coisotropic A-brane on X well-defined up to tensoring with a flat line bundle on X. Its curvature 2-form is exact and given by

F = Im Ω

If we denote by I the natural almost complex structure on X coming from the complex structure on Y , we have F = ωI, and therefore the endomorphism ω^{−1}F = I squares to −1. Therefore any unitary connection on a trivial line bundle over X whose curvature is F defines a coisotropic A-brane.

Now, what about the endomorphisms of the canonical coisotropic A-brane, i.e., the algebra of BRST-closed open string vertex operators? This is easy if Y is an affine space. If one covers Y with charts each of which is an open subset of C^{n}, and then argues that the computation can be performed locally on each chart and the results “glued together”, one gets closer to the fact that the algebra in question is the cohomology of a certain sheaf of algebras, whose local structure is the same as for Y = C^{n}. In general, the path integral defining the correlators of vertex operators does not have any locality properties in the target space. Each term in perturbation theory depends only on the infinitesimal neighbourhood of a point. This shows that the algebra of open-string vertex operators, regarded as a formal power series in ħ, is the cohomology of a sheaf of algebras, which is locally isomorphic to a similar sheaf for X = C^{n} × C^{n}.

Let us apply these observations to the canonical coisotropic A-brane on X = T ^{∗} Y. Locally, we can identify Y with a region in C^{n} by means of holomorphic coordinate functions q_{1}, . . . , q_{n}. Up to * BRST-exact terms*, the action of the A-model on a disc Σ takes the form

S = 1/ħ ∫_{∂Σ} φ ^{∗} (p_{i}dq^{i})

where φ is a map from Σ to X. This action is identical to the action of a particle on Y with zero Hamiltonian, except that q^{i} are holomorphic coordinates on Y rather than ordinary coordinates. The BRST-invariant open-string vertex operators can be taken to be holomorphic functions of p, q. Therefore quantization is locally straightforward and gives a noncommutative deformation of the algebra of holomorphic functions on T ^{∗} Y corresponding to a holomorphic Poisson bivector

P = ħ∂/∂p_{i} ∧ ∂/∂q^{i}

One can write an explicit formula for the deformed product:

(f ⋆ g)(p, q) = exp(ħ/2(∂^{2}/∂p_{i}∂q̃^{i} − ∂^{2}/∂q^{i∂p̃i })) f(p, q) g (p̃, q̃)|_{p̃ = p, q̃ = q}

This product is known as the Moyal-Wigner product, which is a formal power series in ħ that may have zero radius of convergence. To rectify the situation, one can restrict to functions which are polynomial in the fiber coordinates p_{i}. Such locally-defined functions on T ^{∗} Y can be thought of as symbols of differential operators; the Moyal-Wigner product in this case reduces to the product of symbols and is a polynomial in ħ. Thus locally the sheaf of open-string vertex operators is modelled on the sheaf of holomorphic differential operators on Y (provided we restrict to operators polynomial in p_{i}).

Locally, there is no difference between the sheaf of holomorphic differential operators D(Y ) and the sheaf of holomorphic differential operatorsD(Y, L) on a holomorphic line bundle L over Y. Thus the sheaf of open-string vertex operators could be any of the sheaves D(Y, L). Moreover, the classical problem is symmetric under p_{i} → −p_{i} combined with the orientation reversal of Σ; if we require that quantization preserve this symmetry, then the algebra of open-string vertex operators must be isomorphic to its opposite algebra. It is known that the opposite of the sheaf D(Y, L) is the sheaf D(Y, L^{−1} ⊗ K_{Y}), so symmetry under p_{i} → −p_{i} requires L to be a square root of the canonical line bundle K_{Y}. It does not matter which square root one takes, since they all differ by flat line bundles on Y, and tensoring L by a flat line bundle does not affect the sheaf D(Y, L). The conclusion is that the sheaf of open-string vertex operators for the canonical coisotropic A-brane α on X = T ^{∗} Y is isomorphic to the sheaf of noncommutative algebras D(Y, K^{1/2}). One can use this fact to associate Y to any A-brane β on X a twisted D-module, i.e., a sheaf of modules over D(Y, K^{1/2}). Consider the A-model with target X on a strip Σ = I × R, where I is a unit interval, and impose boundary conditions corresponding to branes α and β on the two boundaries of Σ. Upon quantization of this model, one gets a sheaf on vector spaces on Y which is a module over the sheaf of open-string vertex operators inserted at the α boundary. A simple example is to take β to be the zero section of T ^{∗} Y with a trivial line bundle. Then the corresponding sheaf is simply the sheaf of sections of K_{Y}^{1/2}, with a tautological action of D(Y, K_{Y}^{1/2}).

All relativistic quantum theories contain “antiparticles,” and allow the process of particle-antiparticle annihilation. This inspires a physical version of the Grothendieck construction of K-theory. Physics uses topological K-theory of manifolds, whose motivation is to organize vector bundles over a space into an algebraic invariant, that turns out to be useful. Algebraic K-theory started from K_{i} defined for i, with relations to classical constructions in algebra and number theory, followed by Quillen’s homotopy-theoretic definition ∀ i. The connections to algebra and number theory often persist for larger values of i, but in ways that are subtle and conjectural, such as special values of zeta- and L-functions.

One could also use the conserved charges of a configuration which can be measured at asymptotic infinity. By definition, these are left invariant by any physical process. Furthermore, they satisfy quantization conditions, of which the prototype is the Dirac condition on allowed electric and magnetic charges in Maxwell theory.

There is an elementary construction which, given a physical theory T, produces an abelian group of conserved charges K(T). Rather than considering the microscopic dynamics of the theory, all that is needed to be known is a set S of “particles” described by T, and a set of “bound state formation/decay processes” by which the particles combine or split to form other particles. These are called “binding processes.” Two sets of particles are “physically equivalent” if some sequence of binding processes convert the one to the other. We then define the group K(T) as the abelian group Z^{S} of formal linear combinations of particles, quotiented by this equivalence relation.

Suppose T contains the particles S = {A,B,C}.

If these are completely stable, we could clearly define three integral conserved charges, their individual numbers, so K(T) ≅ Z^{3}.

Introducing a binding process

A + B C —– (1)

with the bidirectional arrow to remind us that the process can go in either direction. Clearly K(T) ≅ Z^{2} in this case.

One might criticize this proposal on the grounds that we have assumed that configurations with a negative number of particles can exist. However, in all physical theories which satisfy the constraints of special relativity, charged particles in physical theories come with “antiparticles,” with the same mass but opposite charge. A particle and antiparticle can annihilate (combine) into a set of zero charge particles. While first discovered as a prediction of the Dirac equation, this follows from general axioms of quantum field theory, which also hold in string theory.

Thus, there are binding processes

B + B̄ Z_{1} + Z_{2} + · · · .

where B̄ is the antiparticle to a particle B, and Z_{i} are zero charge particles, which must appear by energy conservation. To define the K-theory, we identify any such set of zero charge particles with the identity, so that

B + B̄ 0

Thus the antiparticles provide the negative elements of K(T).

Granting the existence of antiparticles, this construction of K-theory can be more simply rephrased as the Grothendieck construction. We can define K(T) as the group of pairs (E, F) ∈ (Z^{S}, Z^{S}), subject to the relations (E, F) ≅ (E+B, F +B) ≅ (E+L, F +R) ≅ (E+R, F +L), where (L, R) are the left and right hand side of a binding process (1).

Thinking of these as particles, each brane B must have an antibrane, which we denote by B̄. If B wraps a submanifold L, one expects that B̄ is a brane which wraps a submanifold L of opposite orientation. A potential problem is that it is not *a priori* obvious that the orientation of L actually matters physically, especially in degenerate cases such as L a point.

Now, let us take X as a Calabi-Yau threefold for definiteness. A physical A-brane, which are branes of the A-model topological string and thereby a TQFT shadow of the D-branes of the superstring, is specified by a pair (L, E) of a special Lagrangian submanifold L with a flat bundle E. The obvious question could be: When are (L_{1}, E_{1}) and (L_{2}, E_{2}) related by a binding process? A simple heuristic answer to this question is given by the Feynman path integral. Two configurations are connected, if they are connected by a continuous path through the configuration space; any such path (or a small deformation of it) will appear in the functional integral with some non-zero weight. Thus, the question is essentially topological. Ignoring the flat bundles for a moment, this tells us that the K-theory group for A-branes is H_{3}(Y, Z), and the class of a brane is simply (rank E)·[L] ∈ H_{3}(Y, Z). This is also clear if the moduli space of flat connections on L is connected.

But suppose it is not, say π_{1}(L) is torsion. In this case, we need deeper physical arguments to decide whether the K-theory of these D-branes is H_{3}(Y, Z), or some larger group. But a natural conjecture is that it will be K_{1}(Y), which classifies bundles on odd-dimensional submanifolds. Two branes which differ only in the choice of flat connection are in fact connected in string theory, consistent with the K-group being H_{3}(Y, Z). For Y a simply connected Calabi-Yau threefold, K_{1}(Y) ≅ H_{3}(Y, Z), so the general conjecture is borne out in this case

There is a natural bilinear form on H_{3}(Y, Z) given by the oriented intersection number

I(L_{1}, L_{2}) = #([L_{1}] ∩ [L_{2}]) —– (2)

It has symmetry (−1)^{n}. In particular, it is symplectic for n = 3. Furthermore, by Poincaré duality, it is unimodular, at least in our topological definition of K-theory.

D-branes, which are extended objects defined by mixed * Dirichlet-Neumann boundary conditions* in string theory, break half of the supersymmetries of the type II superstring and carry a complete set of electric and magnetic Ramond-Ramond charges. The product of the electric and magnetic charges is a single Dirac unit, and that the quantum of charge takes the value required by string duality. Saying that a D-brane has RR-charge means that it is a source for an “RR potential,” a generalized (p + 1)-form gauge potential in ten-dimensional space-time, which can be verified from its world-volume action that contains a minimal coupling term,

∫C^{(p + 1)} —–(3)

where C^{(p + 1)} denotes the gauge potential, and the integral is taken over the (p+1)-dimensional world-volume of the brane. For p = 0, C^{(1)} is a one-form or “vector” potential (as in Maxwell theory), and thus the D0-brane is an electrically charged particle with respect to this 10d Maxwell theory. Upon further compactification, by which, the ten dimensions are R^{4} × X, and a Dp-brane which wraps a p-dimensional cycle L; in other words its world-volume is R × L where R is a time-like world-line in R^{4}. Using the Poincaré dual class ω_{L} ∈ H^{2n−p}(X, R) to L in X, to rewrite (3) as an integral

∫_{R × X} C^{(p + 1) }∧ ω_{L} —– (4)

We can then do the integral over X to turn this into the integral of a one-form over a world-line in R^{4}, which is the right form for the minimal electric coupling of a particle in four dimensions. Thus, such a wrapped brane carries a particular electric charge which can be detected at asymptotic infinity. Summarizing the RR-charge more formally,

∫_{L}C = ∫_{X}C^{ }∧ ω_{L} —– (5)

where C ∈ H∗(X, R). In other words, it is a class in H_{p}(X, R).

In particular, an A-brane (for n = 3) carries a conserved charge in H_{3}(X, R). Of course, this is weaker than [L] ∈ H_{3}(X, Z). To see this physically, we would need to see that some of these “electric” charges are actually “magnetic” charges, and study the * Dirac-Schwinger-Zwanziger quantization condition* between these charges. This amounts to showing that the angular momentum J of the electromagnetic field satisfies the quantization condition J = ħn/2 for n ∈ Z. Using an expression from electromagnetism, J⃗ = E⃗ × B⃗ , this is precisely the condition that (2) must take an integer value. Thus the physical and mathematical consistency conditions agree. Similar considerations apply for coisotropic A-branes. If X is a genuine Calabi-Yau 3-fold (i.e., with strict SU(3) holonomy), then a coisotropic A-brane which is not a special Lagrangian must be five-dimensional, and the corresponding submanifold L is rationally homologically trivial, since H

If X is a torus, or a K3 surface, the situation is more complicated. In that case, even rationally the charge of a coisotropic A-brane need not lie in the middle-dimensional cohomology of X. Instead, it takes its value in a certain subspace of ⊕_{p} H^{p}(X, Q), where the summation is over even or odd p depending on whether the complex dimension of X is even or odd. At the semiclassical level, the subspace is determined by the condition

(L − Λ)α = 0, α ∈ ⊕_{p} H^{p}(X, Q)

Let there be a morphism f : X → Y between varieties. Then all the information about f is encoded in the graph Γ_{f} ⊂ X × Y of f, which (as a set) is defined as

Γ_{f} = {(x, f(x)) : x ∈ X} ⊂ X × Y —– (1)

Now consider the natural projections p_{X}, p_{Y} from X × Y to the factors X, Y. Restricted to the subvariety Γ_{f}, p_{X} is an isomorphism (since f is a morphism). The fibres of p_{Y} restricted to Γ_{f} are just the fibres of f; so for example f is proper iff p_{Y} | Γ_{f} is.

If H(−) is any reasonable covariant homology theory (say singular homology in the complex topology for X, Y compact), then we have a natural push forward map

f_{∗} : H(X) → H(Y)

This map can be expressed in terms of the graph Γ_{f} and the projection maps as

f_{∗}(α) = p_{Y∗} (p_{X}^{∗}(α) ∪ [Γ_{f}]) —– (2)

where [Γ_{f}] ∈ H (X × Y) is the fundamental class of the subvariety [Γ_{f}]. Generalizing this construction gives us the notion of a “multi-valued function” or correspondence from X to Y, simply defined to be a general subvariety Γ ⊂ X × Y, replacing the assumption that p_{X} be an isomorphism with some weaker assumption, such as p_{X} |Γ_{f}, p_{Y} | Γ_{f} finite or proper. The right hand side of (2) defines a generalized pushforward map

Γ_{∗} : H(X) → H(Y)

A subvariety Γ ⊂ X × Y can be represented by its structure sheaf O_{Γ} on X × Y. Associated to the projection maps p_{X}, p_{Y}, we also have pullback and pushforward operations on sheaves. The cup product on homology turns out to have an analogue too, namely tensor product. So, appropriately interpreted, (2) makes sense as an operation from the derived category of X to that of Y.

A derived correspondence between a pair of smooth varieties X, Y is an object F ∈ D^{b}(X × Y) with support which is proper over both factors. A derived correspondence defines a functor Φ_{F} by

Φ_{F} : D^{b}(X) → D^{b}(Y)

(−) ↦ Rp_{Y∗}(Lp_{X}^{∗}(−) ⊗^{L} F)

where (−) could refer to both objects and morphisms in D^{b}(X). F is sometimes called the kernel of the functor Φ_{F}.

The functor Φ_{F} is exact, as it is defined as a composite of exact functors. Since the projection p_{X} is flat, the derived pullback Lp_{X}^{∗} is the same as ordinary pullback p_{X}^{∗}. Given derived correspondences E ∈ D^{b}(X × Y), F ∈ D^{b}(Y × Z), we obtain functors Φ_{E }: D^{b}(X) → D^{b}(Y), Φ_{F }: D^{b}(Y) → D^{b}(Z), which can then be composed to get a functor

Φ_{F} ◦ Φ_{E }: D^{b}(X) → D^{b}(Z)

which is a two-sided identity with respect to composition of kernels.